Abstract
The general problem of shock formation in three space dimensions was solved by D. Christodoulou in [2]. In this work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but in addition has singular initial data because of the behavior of the solution at the blowup surface. The present work delivers the solution to this problem in the case of spherical symmetry for a barotropic fluid. A complete description of the singularities associated to the development of shocks in terms of smooth functions is given.
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Introduction
Overview
The Euler equations are a set of nonlinear hyperbolic partial differential equations. Physically they represent the conservation of energy, momentum and mass. It is well known that, given smooth initial data, solutions of equations of this type can blow up in finite time. In the case of the Euler equations the gradients of the solution become infinite. The mechanism of the blowup is called formation of a shock and has first been studied in one space dimension by Riemann in 1858 [6]. The general problem of shock formation in three space dimensions for a fluid with an arbitrary equation of state was solved by Christodoulou in the monograph [2]. In this work also a complete description of the maximal development of the initial data is provided. This description properly sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem and is stated in the epilogue of [2]. It belongs to the category of free boundary problems but possesses the additional difficulty of having singular data due to the behavior of the solution at the blowup surface. The present work gives the solution to this problem in the physically important case of spherical symmetry for a fluid with barotropic equation of state. The result is a step in understanding the development of shocks in fluids. It provides the basis on which the continuation, interaction and breakdown of shocks in spherical symmetry can be studied. Furthermore, the mathematical tools invented to deal with the problem will be of importance in studying solutions to nonlinear hyperbolic equations beyond shock formation.
Shock Development
The general problem of shock formation in a relativistic fluid has been studied in the monograph [2] by Christodoulou. This work is in the framework of special relativity. The theorems in this monograph give a detailed picture of shock formation in 3-dimensional fluids. In particular a detailed description is given of the geometry of the boundary of the maximal development of the initial data and of the behavior of the solution at this boundary. The notion of maximal development in this context is not that relative to the background Minkowski metric \(\eta _{\mu \nu }\), but rather the one relative to the acoustical metric \(g_{\mu \nu }\). This is a Lorentzian metric, the null cones of which are the sound cones. In the monograph it is shown that the boundary of the maximal development in the ’acoustical’ sense (relative to g) consists of a regular part and a singular part. Each component of the regular part \(\underline{C}\) is an incoming characteristic (relative to g) hypersurface which has a singular past boundary. The singular part of the boundary is the locus of points where the density of foliations by outgoing characteristic (relative to g) hypersurfaces blows up. It is the union \(\partial _-\mathcal {B}\cup \mathcal {B}\), where each component of \(\partial _-\mathcal {B}\) is a smooth embedded surface in Minkowski spacetime, the tangent plane to which at each point is contained in the exterior of the sound cone at that point. On the other hand, each component of \(\mathcal {B}\) is a smooth embedded hypersurface in Minkowski spacetime, the tangent hyperplane to which at each point is contained in the exterior of the sound cone at that point, with the exception of a single generator of the sound cone, which lies on the hyperplane itself. The past boundary of a component of \(\mathcal {B}\) is the corresponding component of \(\partial _-\mathcal {B}\). The latter is at the same time the past boundary of a component of \(\underline{C}\). This is the surface where a shock begins to form. The maximal development in the case of spherical symmetry is shown in figure 1. In spherical symmetry a component of \(\partial _-\mathcal {B}\) corresponds to a sphere and therefore to a point in the t-r-plane, the cusp point, which we denote by O.
The maximal development in a neighborhood of a blowup point in spherical symmetry. \(\underline{C}\) denotes the regular part of the boundary of the maximal development. \(\underline{C}\) is incoming characteristic and originates at the cusp point O. The cusp point O corresponds to \(\partial _-\mathcal {B}\). \(\mathcal {B}\) denotes the singular part of the boundary of the maximal development. The family of outgoing characteristic curves is drawn as straight lines for simplification
Now the maximal development in the acoustical sense, or ’maximal classical solution’, is the physical solution of the problem up to \(\underline{C}\cup \partial _-\mathcal {B}\), but not up to \(\mathcal {B}\). In the last part of the monograph the problem of the physical continuation of the solution is set up as the shock development problem. This is a free boundary problem associated to each component of \(\partial _-\mathcal {B}\). In this problem one is required to construct a hypersurface of discontinuity \(\mathcal {K}\), the shock, lying in the past of the corresponding component of \(\mathcal {B}\) but having the same past boundary as the latter, namely the given component of \(\partial _-\mathcal {B}\), the tangent hyperplanes to \(\mathcal {K}\) and \(\mathcal {B}\) coinciding along \(\partial _-\mathcal {B}\). Moreover, one is required to construct a solution of the differential conservation laws in the domain in Minkowski spacetime bounded in the past by \(\underline{C}\cup \mathcal {K}\), agreeing with the maximal classical solution on \(\underline{C}\cup \partial _-\mathcal {B}\), while having jumps across \(\mathcal {K}\) relative to the data induced on \(\mathcal {K}\) by the maximal classical solution. For reasons which will be made clear below we call this solution state behind while the solution in the maximal development we call state ahead. The jumps across \(\mathcal {K}\) have to satisfy the jump conditions which follow from the integral form of the conservation laws (the relativistic form of the Rankine-Hugoniot jump conditions). Finally, \(\mathcal {K}\) is required to be spacelike relative to the acoustical metric g induced by the maximal classical solution, which holds in the past of \(\mathcal {K}\), and timelike relative to the new solution, which holds in the future of \(\mathcal {K}\) (the last condition is equivalent to the condition that the jump in entropy is positive). The maximal classical solution thus provides the boundary conditions on \(\underline{C}\cup \partial _-\mathcal {B}\), as well as a barrier at \(\mathcal {B}\). The situation in spherical symmetry is shown in figure 2.
The state behind in dark shade and the state ahead in light shade, separated by \(\underline{C}\) and the shock \(\mathcal {K}\) where outgoing characteristics meet. The family of outgoing characteristic curves is drawn as straight lines for simplification
In the present work the shock development problem is solved in the case of spherical symmetry and under the assumption that the fluid is described by a barotropic equation of state. The presence of spherical symmetry represents an important physical case, also from the point of view of applications, and reduces the problem to one on the t-r-plane, where t denotes Minkowski time and r denotes the radial coordinate. The assumption of a barotropic equation of state is appropriate for liquids and also for a radiation gas. For a radiation gas we have \(p=(1/3)\rho \) where p is the pressure and \(\rho \) is the energy density in the rest frame of the fluid. This model applies in particular to the early, radiation dominated, phase of the history of the universe. The fluid being barotropic the energy-momentum conservation law decouples from the particle conservation law. The system of partial differential equations reduces to an inhomogeneous system with two unknowns. One of the key concepts used to deal with the system of equations are the Riemann Invariants \(\alpha \), \(\beta \) of the principal part of the system of equations. The equations are reformulated in terms of characteristic coordinates (u, v). These coordinates are defined by the outgoing and incoming null rays with respect to the acoustical metric g, u being constant along outgoing null rays and v being constant along incoming null rays. In addition the coordinates are set up such that the shock \(\mathcal {K}\) is given by \(u=v\). The system of equations for the time and radial coordinates (t, r) in terms of (u, v) is the Hodograph system. The Hodograph system together with the system of equations for the Riemann Invariants is a non-linear four by four system. This system is then solved using a double iteration consisting of an inner and an outer iteration. In the outer iteration the position of the free boundary in the t-r-plane is iterated, providing, through the jump conditions, in each step the boundary conditions for a fixed boundary problem. The equations being non-linear, this fixed boundary problem is then solved using again an iteration, the inner iteration. The solution of the fixed boundary problem then allows to set the position of the free boundary in the t-r-plane for the next iterate. This is accomplished as follows. The solution of the fixed boundary problem provides the values of r and t in terms of the characteristic coordinates along the shock \(\mathcal {K}\), i.e. r(v, v), t(v, v). In the formation problem the solution in the maximal development (denoted by \((\cdot )^*\)) is given in terms of the acoustical coordinates (t, w), where w is a function which is constant along outgoing characteristic hypersurfaces. Now r(v, v) is set equal to the radial coordinate given by the acoustical coordinates, i.e. \(r^*(t,w)\), when \(t=t(v,v)\) is substituted, i.e. \(r(v,v)=r^*(t(v,v),w)\). This equation is called the identification equation since it identifies the radial coordinates of events in spacetime, with respect to the solution of the fixed boundary problem and with respect to the solution in the maximal development, along the shock \(\mathcal {K}\). It plays a very important role, and the study of it is at the heart of the solution to the problem. The identification equation has to be solved for w in terms of v in order to be able to apply the jump conditions and in order to compute the boundary data for the next iterate in the outer iteration. This is not possible offhandedly. Only after correctly guessing the asymptotic form of the solution as we approach the sphere \(\partial _-\mathcal {B}\), can the identification equation be reduced to an equation which is solvable for w in terms of v. The iteration then yields the local existence of a continuously differentiable solution to the shock development problem. Also uniqueness of this solution is proven. Finally it is proven that the solution is, away from the shock \(\mathcal {K}\), smooth.
The problem is solved in the framework of special relativity. Nevertheless, no special care is needed to extract information on the non-relativistic limit. This is due to the fact that the non-relativistic limit is a regular limit, obtained by letting the speed of light in conventional units tend to infinity, while keeping the sound speed fixed.
Relation to Other Work
We first remark that the methods of the present work trivially apply to the case of one-dimensional isentropic flow. The equations in this case can be written in the form of a non-linear hyperbolic conservation law
There exists a quite complete theory for studying equations of this form. In the following we are going to put the present work in relation to that theory.
The fundamental building block of that theory is the solution of the Riemann problem [4] which is the problem of studying the above equation for data which is piecewise constant with a single jump at the origin. The solution of this problem is self similar and consists of several constant states connecting the piecewise solution on both sides of the jump. Approximate solutions for more general data are then constructed by patching together several solutions of Riemann problems. This is done using the Glimm scheme [3] whose deterministic version has been established by Liu [5]. In that approach, the initial data is approximated by a piecewise constant function and the algorithm produces a sequence of approximate solutions whose convergence relies on a compactness argument based on uniform bounds on the total variation. Therefore, these methods establish the existence of solutions in the space of functions of bounded variation.
We should keep in mind that the objective is to determine solutions which arise by evolving given smooth initial data according to the physical laws, not only existence, uniqueness and continuous dependence on the data but also the description of the main qualitative features of the solution, chief among which is the precise description of the singularities which arise. The aim being of course to derive results that can be compared with experiment.
From given smooth initial data singularities develop naturally. But from a solution in the space of functions of bounded variation no regularity information can be extracted, not even on what is the set of points at which the solution fails to be continuous.
This is in contrast to the present work where we obtain complete knowledge of the solution in terms of smooth functions. That is, we obtain a complete resolution of the singularity. In comparison with the theory of elliptic problems, what we develop is the analogue of a complete regularity theory rather than only the analogue of an existence theory in the class of functions admissible in the variational problem.
Furthermore, the approach using the space of functions of bounded variation is unsuitable to address the physical three dimensional problem, being in principle confined to one spatial dimension.
Overview of the Article
We give an overview of the content of each section. We also state the places where important theorems can be found.
Section 2
In 2.1 we present the model of a perfect fluid in special relativity. Based on the conservation laws for energy-momentum and particle number (see (6) and (7)) we derive the equations of motion (the Eulerian system given by (6), (10) and (11)). We analyze the characteristics of this system and introduce the sound speed \(\eta \) as well as the acoustical metric \(g_{\mu \nu }\) which describes the sound cone (see (23) together with (24)). In 2.2 we derive the jump conditions across a hypersurface of discontinuity, (see (40) and (41)). They follow from the integral form of the conservation laws which we derive on the basis of the conservation laws for energy-momentum and particle number. In 2.3 we define which hypersurfaces of discontinuity are shocks. Then we present the determinism condition and the entropy condition. These are both conditions on the solution of the equations of motion across a shock. The determinism condition is the condition that the shock is supersonic relative to the state ahead and subsonic relative to the state behind the shock. The determinism condition is illustrated in figure 4. The entropy condition is the condition that the jump in entropy per particle is positive across a shock. In 2.3 we also show that for suitably small jumps, the determinism condition and the entropy condition are equivalent. In 2.4 we restrict our analysis to the case where the fluid is barotropic, i.e. where \(p=f(\rho )\) where p is the pressure and \(\rho \) is the mass-energy density. Due to this assumption we get a decoupling of one of the equations of motion (see the system (107), (108), (109) in contrast to the system (36), (37), (38)). Furthermore, we then restrict to the irrotational case which implies that one of the remaining equations is identically satisfied (the one dealing with vorticity), thus reducing the equations of motion to a single equation for the potential \(\phi \) (see (116)). This equation takes the form of a nonlinear wave equation when rewriting it using the acoustical metric (see (119)).
Most of the material in this section can be found in the first chapter and the epilogue of [2] and in the first section of [1].
Section 3
We restrict to the spherically symmetric case. In 3.1 we look at the radial null vectors \(L_\pm \) (null with respect to the acoustical metric), introduce the normalization \(L^t=1\) and rewrite the nonlinear wave equation using these null vectors (see (128)). In 3.2 we keep only the principal part of the nonlinear wave equation and derive the Riemann invariants \(\alpha \), \(\beta \) corresponding to it. These Riemann invariants correspond to the Riemann invariants of the solution under the assumption of plain symmetry, i.e. the purely one dimensional problem. In 3.3 we rewrite the nonlinear wave equation for spherically symmetric solutions using \(\alpha \), \(\beta \) (see (154)). Then we introduce double null coordinates (with respect to the acoustical metric). The equations of motion then become the equations for the derivatives of the Riemann invariants together with the system satisfied by the space-time coordinates t, r (the Hodograph system). We refer to this set of equations as the characteristic system. In 3.4 we rewrite the jump conditions as an equation for the shock speed V and an equation in terms of jumps (see (170), (171)). Then we derive the relation \(\left[ \beta \right] =\left[ \alpha \right] ^3G(\left[ \alpha \right] )\), where \(\left[ f\right] =f_+-f_-\) and G is a smooth function of its arguments with G(0) a constant. In 3.5 we describe the boundary of the maximal development of the initial data. Here we make use of the result [2] where one can find a detailed description of this boundary in chapter 15. While in [2] no assumption on the symmetry of the problem was made, here we restrict the result to spherical symmetry. We then derive the behavior of the Riemann invariants \(\alpha \), \(\beta \) at the cusp point, which is the first point of blowup. This behavior is given in (266), (267). We then describe the incoming characteristic originating at the cusp point by giving t in terms of the acoustical coordinate w along this characteristic. In 3.6 we state the shock development problem in all its details and outline the strategy for its solution. See in particular (325).
Section 4
As described in the outline of the strategy in the end of section 3, in each step of the iteration, which is used to solve the shock development problem, we need to solve the characteristic system with given initial data for \(\alpha \) and t, boundary data for \(\beta \) and a given shock speed V which enters the free boundary condition (see (301)). The characteristic system being nonlinear we use again an iteration. In 4.1 we set up the iteration scheme. We also prove (see proposition 4.1) a preliminary result concerning the equation for t. This second order partial differential equation follows once r has been eliminated from the Hodograph system (see equation (336)). In 4.1 we also establish the inductive step of the iteration. This is the content of lemma 4.1. In the second subsection we show convergence. This result is the content of lemma 4.2. The section concludes with the existence result for the fixed boundary problem given by proposition 4.2.
Section 5
As described in the outline of the strategy in the end of section 3 we solve the shock development problem using an iteration. In 5.1 we specify the form of the boundary functions to be iterated and the corresponding function spaces. Then we establish the inductive step of the iteration. This is the content of proposition 5.1. In subsection 5.2 we show convergence. We first show three lemmas, each of them corresponding to a particular step in the induction process. Lemma 5.1 gives estimates for the solution to the fixed boundary problem. Lemma 5.2 gives estimates for the solution of the identification equation (see equation (298) in the description of the strategy). Lemma 5.3 gives estimates for the quantities related by the jump conditions. These three lemmas are then used to close the convergence argument in the proof of proposition 5.2. The above then leads to the existence of a continuously differentiable solution to the free boundary problem. This is the content of theorem 5.1.
Section 6
The proof of uniqueness of the solution is done in two steps. In 6.1 we first prove that any solution of the characteristic system which satisfies the smoothness conditions from the existence theorem (theorem 5.1) possesses the same leading order behavior as the solution given by the existence theorem. This is the content of proposition 6.1. In 6.2 we prove uniqueness of the solution of the shock development problem, assuming the solution has the given leading order behavior. The result is the content of theorem 6.1. In 6.3 we prove that \(L_+\alpha \) and \(L_+\beta \) are continuous across the incoming characteristic originating at the cusp point.
Section 7
Up to this point we established a solution to the shock development problem in the class of continuously differentiable functions. Now we prove that this solution is smooth (see theorem 7.1). This is accomplished by induction with respect to the order of differentiation. The inductive hypothesis is stated in 7.1 and the base case of the induction is shown in 7.2. In 7.3 we show the inductive step. In 7.4 we show that the derivatives of the Riemann invariants (and therefore the derivatives of the physical quantities) with respect to \(L_+\) of order greater than the first blow up as we approach the incoming characteristic originating at the cusp point from the state behind the shock.
Relativistic Fluids
Relativistic Perfect Fluids
The motion of a perfect fluid in special relativity is described by a future-directed unit time-like vector field u and two positive functions n and s, the number of particles per unit volume (in the local rest frame of the fluid) and the entropy per particle, respectively. Let us denote the Minkowski metric by \(\eta \). The conditions on the velocity u are then
The mechanical properties of the fluid are specified once we give the equation of state, which expresses the mass-energy density \(\rho \) as a function of n and s
Let \(e=\rho /n\) be the energy per particle. According to the first law of thermodynamics we have
where p is the pressure, \(v=1/n\) the volume per particle and \(\theta \) the temperature. We have
The functions \(\rho \), p, \(\theta \) are assumed to be positive. The equations of motion for a perfect fluid are given by the particle conservation law and the energy-momentum conservation law, i.e.
where T and I are the energy-momentum-stress tensor and the particle current, respectively, given by
The component of (7) along u is the energy equation
Using (6) in (9) together with (5) we deduce
i.e. modulo the particle conservation law, the energy equation is equivalent to the entropy being constant along the flow lines. Nevertheless the equivalence of the energy and entropy conservation only holds for \(C^1\) solutions. Let \(\Pi ^\mu _\nu \,{:=}\,\delta ^\mu _\nu +u^\mu u_\nu \) denote the projection onto the local simultaneous space of the fluid. The projection of (7) is the momentum conservation law
The symbol \(\sigma _\xi \) of the Eulerian system (6), (10), (11) at a given covector \(\xi \) is the linear operator on the space of variations \((\dot{n},\dot{s},\dot{u})\) whose components are
We note that
The characteristic subset of \(T_x^*M\), that is the set of covectors \(\xi \) such that the null space of \(\sigma _\xi \) is nontrivial, consists of the hyperplane \(P_x^*\):
and the cone \(C_x^*\):
where \(\eta \) is the sound speed
We assume that the equation of state satisfies the basic requirement
The characteristic subset of \(T_xM\) corresponding to \(P_x^*\), i.e., the set of vectors \(\dot{x}\in T_xM\) of the form
is simply the vector u(x), while the characteristic subset of \(T_xM\) corresponding to \(C_x^*\), i.e., the set of vectors \(\dot{x}\in T_xM\) of the form
is the sound cone \(C_x\):
We define the acoustical metric \(g_{\mu \nu }\) by
\(C_x\) is then given by
We assume that the equation of state satisfies the basic requirement
which is equivalent to the condition that the sound cone is contained within the light cone. For \(\xi \in P_x^*\) the null space of \(\sigma _\xi \) consists of the variations satisfying
(the isobaric vorticity waves). For \(\xi \in C_x^*\) the null space of \(\sigma _\xi \) consists of the variations satisfying
(the adiabatic sound waves). We note that the inverse acoustical metric is given by
We define the one form \(\beta \) by
where h is the enthalpy per particle given by
We deduce
where for the first equality we used the definition of the Lie derivative together with the first of (2) while for the second equality we used (11). By (30) in conjunction with (4) the expression in the last parenthesis is equal to \(-\theta \nabla _\nu s\). Therefore, by (10), the last term vanishes and we have
We define the vorticity two form by
Let us denote by \(i_X\) contraction from the left by X. From (29) we deduce
Since for any exterior differential form \(\vartheta \) it holds that \(\mathcal {L}_X\vartheta =i_Xd\vartheta +di_X\vartheta \), we obtain from (32)
We conclude that the equations of motion (6), (7) are equivalent to the system
In fact (37) follows from (38).
Jump Conditions
It is well known that the solution of the equations (6), (7), in general, develop discontinuities. Let \(\mathcal {K}\) be a hypersurface of discontinuity, i.e. a \(C^1\) hypersurface \(\mathcal {K}\) with a neighborhood \(\mathcal {U}\) such that \(T^{\mu \nu }\) and \(I^\mu \) are continuous in the closure of each connected component of the complement of \(\mathcal {K}\) in \(\mathcal {U}\) but are not continuous across \(\mathcal {K}\). Let \(N_\mu \) be a covector at \(x\in \mathcal {K}\), the null space of which is the tangent space of \(\mathcal {K}\) at x
Then, denoting by \(\left[ \cdot \right] \) the jump across \(\mathcal {K}\) at x, we have the jump conditions
These follow from the integral form of the conservation laws (6), (7). Consider the 3-form \(I_{\alpha \beta \gamma }^*\) dual to \(I^\mu \), that is,
where \(\varepsilon _{\mu \alpha \beta \gamma }\) is the volume 4-form of the Minkowski metric \(\eta \). In terms of \(I^*\) equation (6) becomes
Also, given any vector field X, we can define the vector field
By virtue of (7), P satisfies
where
In terms of the 3-form \(P^*\) dual to P
equation (45) reads
Consider now an arbitrary point \(x\in \mathcal {K}\) and let \(\mathcal {U}\) be a neighborhood of x in Minkowski spacetime. We denote \(\mathcal {W}=\mathcal {K}\cap \mathcal {U}\). Let Y be a vector field without critical points in some larger neighborhood \(\mathcal {U}_0\supset \mathcal {U}\) and transversal to \(\mathcal {K}\). Let \(\mathcal {L}_\delta (y)\) denote the segment of the integral curve of Y through \(y\in \mathcal {W}\) corresponding to the parameter interval \((-\delta ,\delta )\)
where \(F_s\) is the flow generated by Y. We then define the neighborhood \(\mathcal {V}_\delta \) of x in Minkowski spacetime by
Integrating equations (43), (48) in \(\mathcal {V}_\delta \) and applying Stokes’ theorem we obtain
Now the boundary of \(\mathcal {V}_\delta \) consists of the hypersurfaces
together with the lateral hypersurface
Since this lateral component and \(\mathcal {V}_\delta \) are bounded in measure by a constant multiple of \(\delta \), we take the limit \(\delta \rightarrow 0\) in (51), (52) to obtain
That these are valid for any neighborhood \(\mathcal {W}\) of x in \(\mathcal {K}\) implies that the corresponding 3-forms induced on \(\mathcal {K}\) from the two sides coincide at x, or, equivalently, that
The first of these equations coincides with (41), while the second, for four vector fields X constituting at x a basis for \(T_xM\), implies (40).
Determinism and Entropy Condition
By virtue of (25) only time-like hypersurfaces of discontinuity can arise. Since \(T_x\mathcal {K}\) is time-like, the normal vector \(N^\mu =(\eta ^{-1})^{\mu \nu }N_\nu \) is space-like and we can normalize it to have unit magnitude
We must still determine the orientation of \(\mathcal {K}\). Let \(N^\mu \) point from one side of \(T_x\mathcal {K}\), which we label \(+\) and which we say is behind \(T_x\mathcal {K}\), to the other side of \(T_x\mathcal {K}\), which we label \(-\) and which we say is ahead of \(T_x\mathcal {K}\). Then for any quantity q we have \(\left[ q\right] =q_+-q_-\). If we define
then the jump condition (41) reads
where the quantity f is called particle flux. If \(f\ne 0\), the discontinuity is called a shock. In this case we choose the orientation of N such that \(f{>}0\), that is, the fluid particles cross the hypersurface of discontinuity \(\mathcal {K}\) from the state ahead to the state behind (see figure 3). If \(f=0\) the discontinuity is called a contact discontinuity and in this case the orientation of \(N^\mu \) is merely conventional.
Four fluid flow lines crossing the shock \(\mathcal {K}\) from the state ahead to the state behind
In terms of v, the volume per particle, we have
The jump condition (40) reads
Substituting (61) into (62) the latter reduces to
where we used (30). According to (63) the vectors \(u_+\), \(u_-\) and N all lie in the same timelike plane. Taking the \(\eta \)-inner product of (63) with N we obtain
Substituting from (61) this becomes
On the other, taking the \(\eta \)-inner product of each side of (63) with itself we obtain
Equations (65) and (66) together imply whenever \(f\ne 0\), as is the case for a shock, the following relation
This is the relativistic Hugoniot relation, first derived by A. Taub [7]. We note that in the case of a contact discontinuity (\(f=0\)) (63) reduces to \(p_+=p_-\).
The only shock discontinuities which arise naturally are those which are supersonic relative to the state ahead and subsonic relative to the state behind. We call this the determinism condition. The condition that \(\mathcal {K}\) is supersonic relative to the state ahead means that, for each \(x\in \mathcal {K}\), \(N_\mu \) is a time-like covector relative to \(g^{-1}_-\), i.e.
while the condition that \(\mathcal {K}\) is subsonic relative to the state behind means that, for each \(x\in \mathcal {K}\), \(N_\mu \) is a space-like covector relative to \(g_+^{-1}\), i.e.
In view of (28), conditions (68) and (69) are
Substituting from (61), these become
We conclude that the determinism condition reduces to
The determinism condition is illustrated in figure 4.
Illustration of the determinism condition: Sound cones at a point on \(\mathcal {K}\). Dotted: Backward sound cone with respect to the state ahead. Dashed: Forward sound cone with respect to the state behind. Thin solid line: fluid flow line
We now look at the entropy condition which is
In the following we will show the equivalence of the entropy condition to the determinism condition. Since (recall (4), (30))
the expansion of \(\left[ h\right] =h_+-h_-\) in powers of \(\left[ p\right] \) and \(\left[ s\right] \) is
Hence
Also \(\left[ v\right] =v_+-v_-\) is expanded as
Hence
Comparing (76) and (78) with the Hugoniot relation (67) we conclude
Consider next the condition (72). Defining the quantity
the condition (72) is seen to be equivalent to
Hence, in view of (30) and \(v=1/n\), q is given by
We then obtain
In view of the fact that by (79) \(\left[ s\right] =\mathcal {O}(\left[ p\right] ^3)\), we obtain
Therefore, the condition (81) is equivalent for suitably small \(\left[ p\right] \) to
provided that the quantity in the curly bracket is non-zero. This together with (79) is equivalent to (73). We have therefore established, for suitably small \(\left[ p\right] \), the equivalence of the determinism condition (72) to the entropy condition (73).
Remark 1
We will impose the determinism condition in the shock development problem and we will see that this condition is necessary for the solution to be uniquely determined by the data (see the formulation of the shock development problem together with the description of the boundary of the maximal development below).
Remark 2
By (86), the sign of the coefficient of \(\left[ p\right] \) in (86) is the same as the sign of \(\left[ p\right] \). Let now \(\Sigma \) be defined by
The coefficient of \(\left[ p\right] \) in (86) can be related to \((d\Sigma /dh)_-\) if the state ahead of the shock is isentropic, as will be the case under consideration. From (30) we have
where we use the subscript s to indicate isentropy. Hence
By (30) we have
which implies
Substituting (91) and its derivative with respect to p at constant s in (89) we obtain
Therefore, if the state ahead is isentropic, the quantity in the curly bracket in (86) is
We conclude that the jump in pressure \(\left[ p\right] \) behind the shock is \({>}0\) or \(<0\) according as to whether \((d\Sigma /dh)_-\) is \(<0\) or \({>}0\), at least for suitably small \(\left[ p\right] \).
Barotropic Fluids
In the barotropic case \(p=f(\rho )\) is an increasing function of \(\rho \). Therefore,
is a function of \(\rho \), which implies that for a barotropic perfect fluid, \(\rho \), and hence also p, is a function of the product \(\sigma \,{:=}\,nm\), where m is a function of s alone. In fact
and it satisfies
The positivity of \(\theta \) implies that m is a strictly increasing function. Therefore, we can eliminate s in favor of m. (37) becomes
We define
where
Comparing (98) with (29) we see that
Defining now
we obtain
From the second of (5) we have
Therefore
which implies through (38)
From the particle conservation (36) and the adiabatic condition (97) we deduce
which, through (96), is equivalent to the energy equation (9). Therefore, imposing the energy equation (106) as well as the adiabatic condition (97) the conservation of particle number follows. We conclude that in the barotropic case the system of equations reduces to the system
The unknowns are u, m and \(\sigma \). Equation (108) is decoupled from the other two. We may thus ignore it and consider only the system consisting of (107), (109).
The irrotational barotropic case is characterized by the existence of a function \(\phi \) such that
which implies
Therefore, (109) is identically satisfied. By (98),
whenever X is a future-directed timelike vector. Therefore \(\phi \) is a time function. By (98), (110),
From (99)
which implies that \(\sigma \) can be expressed as a smooth, strictly increasing function of H, i.e. \(\sigma =\sigma (H)\). Defining
equation (107) becomes
where H is given by (113). Taking into account that (see (28), (98), (114))
where
(116) becomes
We note that
We also note that in terms of H and F, the acoustical metric is given by
Since (107) and therefore in the irrotational case (119) is equivalent to the energy equation (9), in the barotropic case we only need to consider the energy-momentum jump conditions (40).
Setting the Scene
Nonlinear Wave Equation in Spherical Symmetry
We choose spherical coordinates \((t,r,\vartheta ,\varphi )\). Then \(\eta =\text {diag}(-1,1,r^2,r^2\sin ^2\vartheta )\) and spherically symmetric solutions \(\phi =\phi (t,r)\) of equation (119) satisfy
The radial null vectors \(L_\pm \) with respect to the acoustical metric satisfy
Using the normalization condition \(L^t=1\), we obtain
where \(\eta \) is the sound speed (see (18)) and v is the fluid spatial velocity given by
where we recall \(\psi _\mu =\partial _\mu \phi \). Using the null vectors the inverse acoustical metric can be written as
From (117) we have
From (120) in conjunction with (113) we see that the assumption \(\eta ^2<1\) is expressed by the condition that \(F{>}0\). The nonlinear wave equation can be written as
Riemann Invariants of the Principal Part
Keeping only the principal part of (128) we are left with
The Riemann invariants are defined to be the functions \(\alpha (\psi _t,\psi _r)\), \(\beta (\psi _t,\psi _r)\) such that
where \(l_{\pm \mu }\) are the basis 1-forms dual to the basis vector fields \(L_{\pm }^\mu \). From (130) we deduce
for some functions \(\xi \), \(\lambda \). Using (129) we obtain
which shows that (129) is equivalent to the system
We now proceed to determine \(\alpha \) and \(\beta \). The basis 1-forms dual to the basis vector fields \(L_\pm ^\mu \) satisfy
Therefore
Defining the operators
(130) becomes
We introduce the functions \(\tilde{h}\), \(\zeta \) as coordinates in the positive open cone in the \(\psi _t\)-\(\psi _r\) plane by
(For \(\tilde{h}\) see (99), (113)). Note that by (125),
Defining then the operators
(138) becomes
Let us now define the thermodynamic potential \(\tilde{\rho }\) by
\(\tilde{\rho }\) is defined up to an additional constant. We may fix \(\tilde{\rho }\) by setting it equal to zero in the surrounding constant state. Since (142) takes in terms of \(\tilde{\rho }\) the form
the solutions of (143) are
up to composition on the left with an arbitrary increasing function. Using
we see that our expressions \(\alpha \), \(\beta \) agree with (5.16) of [7] where \(\phi \) is in the role of \(\tilde{\rho }\) and u is in the role of v. \(\alpha \) and \(\beta \) are the relativistic version of the Riemann invariants introduced in [6].
Characteristic System
In analogy to the equivalence of (129) and (134) it follows from (131) that (128) is equivalent to the system
We now proceed to determine \(\xi \), \(\lambda \) for the choice (146) of Riemann invariants of the principal part. From \(\tilde{h}^2=\psi _t^2-\psi _r^2\) we have
From (147) we obtain
Using (149), (150) together with (144) we obtain
Using (131) in the case \(\mu =t\) and recalling (124) we deduce from (151), (152)
The system of equations (148) becomes
where
Now we introduce characteristic coordinates u, v such that \(u=\text {const.}\) represents the outgoing and \(v=\text {const.}\) the incoming characteristic curves. Furthermore, as the characteristic speeds we set \(c_\pm \,{:=}\,L_\pm ^r\). It follows that the space-time coordinates t, r satisfy
The system (154) becomesFootnote 1
Defining
the characteristic system (156), (159) becomes
We note that (162) is the Hodograph system.
Remark 3
The characteristic system is invariant under the conformal map
where f and g are increasing functions.
Remark 4
In view of (113), (120), (125), (155) we can express \(\tilde{A}\), \(\tilde{B}\) in terms of r, the sound speed \(\eta \) and the spatial fluid velocity v as
Jump Conditions
Let N be the unit vector normal to \(\mathcal {K}\)
where \(V=V(t,r)\) is the shock speed. We define
and reformulate (40) as
In components these are the two jump conditions
which are equivalent to
Since
we obtain from (8)
Using (see (96), (99), (113), (115), (125))
the components of the energy-momentum-stress tensor become
Let
(139) become
Using (see (144))
we get
Now (see (118), (120), (144), (146))
And similarly
From (114), (144), (146) we have
And similarly
From (125) together with (179) we obtain
Using (179), (180), (181), (182), (183), (184) to compute the partial derivatives of the components of T, given by (175), we arrive at
Let us denote \(c_\pm \,{:=}\,L_\pm ^r\) (see (124)). From (185) we have
while from (186) we have
Let us define
We note that with \(\tilde{\Sigma }\) defined by
we have
where \(\Sigma \) is given by (87). So
and
We have the following proposition:
Proposition 3.1
where the coefficients on the right are evaluated at \((\alpha _-,\beta _-)\).
Proof
We prove the Proposition by showing the following statements:
-
(i)
J is symmetric under the interchange of \(\alpha \) and \(\beta \).
-
(ii)
$$\begin{aligned} J(\alpha _-,\alpha _-,\beta _-,\beta _-)&=\frac{\partial J}{\partial \alpha _+}(\alpha _-,\alpha _-,\beta _-,\beta _-)\nonumber \\&=\frac{\partial ^2 J}{\partial \alpha _+^2}(\alpha _-,\alpha _-,\beta _-,\beta _-)\nonumber \\&=\frac{\partial ^3J}{\partial \alpha _+^3}(\alpha _-,\alpha _-,\beta _-,\beta _-)\nonumber \\&=0. \end{aligned}$$(195)
-
(iii)
$$\begin{aligned} \frac{\partial ^2 J}{\partial \alpha _+ \partial \beta _+}(\alpha _-,\alpha _-,\beta _-,\beta _-)=(G\psi _t^2(1-v^2))^2(\alpha _-,\beta _-). \end{aligned}$$(196)
-
(iv)
$$\begin{aligned} \frac{\partial ^4J}{\partial \alpha _+^4}(\alpha _-,\alpha _-,\beta _-,\beta _-)=\left( \frac{(G\psi _t^2)^2}{8\eta ^2}(1-v^2)^2\mu ^2\right) (\alpha _-,\beta _-). \end{aligned}$$(197)
To check (i) we note that \(\rho \) and p, like all thermodynamic variables, are functions of \(\tilde{\rho }=\frac{1}{2}(\alpha +\beta )\), therefore symmetric under the interchange of \(\alpha \) and \(\beta \). On the other hand \(v=-\psi _r/\psi _t=-\tanh \zeta \) and \(\zeta =\frac{1}{2}(\beta -\alpha )\), therefore v is antisymmetric under the interchange of \(\alpha \) and \(\beta \). It follows from (171) (173) that J is symmetric under the interchange of \(\alpha \) and \(\beta \).
Since J is quadratic in the differences of components of T, the first two of (ii) are satisfied. Let
(Where the notation \(\partial T_+^{\mu \nu }/\partial \alpha _+=(\partial T^{\mu \nu }/\partial \alpha )(\alpha _+,\beta _+)\) is used). Now, (185) implies
Therefore
which implies
From (200) we get
which, in conjunction with (199), implies
Now we turn to (iii). We have
Now we turn to (iv). From (202) we obtain
From
we deduce
where
To proceed we need expressions for the second derivatives of the components of T. Using
together with (180), the first of (184) and the first of (179), it follows from (185) by a straightforward computation
with \(\mu \) given by (189). Therefore,
This concludes the proof of the proposition. \(\square \)
We will use the following proposition.
Proposition 3.2
Any smooth function f(x, y) can be written as
where
and \(\overline{g}(x,y)\) is the mean value of g in the rectangle \(R(x,y)\,{:=}\,\{(x^{\prime },y^{\prime })\in \mathbb {R}^2:0\le x^{\prime }\le x,0\le y^{\prime }\le y\}\).
Proof
Integrating (214) on the rectangle R(x, y) yields the result. \(\square \)
We now consider \(J(\alpha _+,\alpha _-,\beta _+,\beta _-)\) as a function of \(\left[ \alpha \right] \), \(\left[ \beta \right] \) with given \(\alpha _-\), \(\beta _-\). We denote this function again by J. Using propositions 3.1, 3.2 we can write
where the coefficients are evaluated at \((\alpha _-,\beta _-)\). Here M, L and N are smooth functions of their arguments and \(M(0,0)=L(0)=N(0)=1\).
Proposition 3.3
Let f(x, y) be a smooth function on \(\mathbb {R}^2\) of the form
with \(m(0,0)=1\), where m, l, n are smooth functions. For small enough x, the equation
has a unique solution for y, given by
where g(x) is a smooth function and \(g(0)=-l(0)\).
Proof
Setting \(y=x^3z\), (217) becomes
where
Since \(m(0,0)=1\), the pair \((x_0,z_0)\,{:=}\,(0,-l(0))\) satisfies (219). Now, since
we can apply the implicit function theorem to deduce that there exists a smooth function g(x) such that for small enough \(x-x_0\) we have \(z=g(x)\) with \(g(0)=z_0\). It follows that for small enough x, \(f(x,y)=0\) has a solution
Applying this proposition to \(J\left( \left[ \alpha \right] ,\left[ \beta \right] \right) =0\) and taking into account (215) it follows that there is a smooth function \(G\left( \left[ \alpha \right] \right) \) such that
We recall that above we considered \(J(\alpha _+,\alpha _-,\beta _+,\beta _-)\) as a function of \(\left[ \alpha \right] \), \(\left[ \beta \right] \) with given \(\alpha _-\), \(\beta _-\). In the following we will make use of (223) in the form (with a different function G)
Boundary of the Maximal Development
Let initial data be given on a spacelike hypersurface which coincides with the initial data of a constant state outside a bounded domain. According to [2] the boundary of the domain of the maximal solution consists of a regular part \(\underline{C}\) and a singular part \(\partial _-\mathcal {B}\cup \mathcal {B}\). Each component of \(\partial _-\mathcal {B}\) is a smooth, space-like (w.r.t. the acoustical metric), 2-dimensional submanifold, while the corresponding component of \(\mathcal {B}\) is a smooth embedded 3-dimensional submanifold ruled by curves of vanishing arc length (w.r.t. the acoustical metric), having past end points on the component of \(\partial _-\mathcal {B}\). The corresponding component of \(\underline{C}\) is the incoming null (w.r.t. the acoustical metric) hypersurface associated to the component of \(\partial _-\mathcal {B}\). It is ruled by incoming null geodesics of the acoustical metric with past end points on the component of \(\partial _-\mathcal {B}\). The result of [2] holds for a general equation of state.
In the following we will restrict ourselves to the barotropic case. We also assume the initial data to be spherically symmetric. Therefore, also the solution is spherically symmetric and it suffices to study the problem in the t-r-plane, where t, r are part of the standard spherical coordinates \((t,r,\vartheta ,\varphi )\).
In the t-r-plane the boundary of the maximal development corresponds to a curve consisting of a regular part \(\underline{C}\) and a singular part \(\partial _-\mathcal {B}\cup \mathcal {B}\). Each component of \(\mathcal {B}\) corresponds to a smooth curve of vanishing arc length with respect to the induced acoustical metric, having as its past end point the point corresponding to \(\partial _-\mathcal {B}\). The corresponding component of \(\underline{C}\) corresponds in the t-r-plane to an incoming null geodesic with respect to the induced acoustical metric with past end point being the point corresponding to \(\partial _-\mathcal {B}\). We denote this point by O. See figure 5 on the right.
Left: Part of the maximal development in acoustical coordinates. Right: Part of the maximal development as a subset of spacetime. \(\underline{C}\) denotes the incoming characteristic originating at the cusp point O while \(\mathcal {B}\) denotes the singular part of the boundary of the maximal development. Both figures show the maximal development just in a neighborhood of a cusp point
In the following we will use (t, w) as the acoustical coordinates (in contrast to [2], where (t, u) are playing the corresponding roles). We recall that the level sets of w are the outgoing characteristic hypersurfaces with respect to the acoustical metric. The solution in the maximal development is a smooth solution with respect to the acoustical coordinates. In terms of these coordinates the solution also extends smoothly to the boundary. We recall the function \(\mu \) which plays a central role in [2], given by
(See (2.13) of [2]). \(\mu \) vanishes on the singular part of the boundary. On the other hand, \(\mu \) is positive on the regular part \(\underline{C}\) and the solution extends smoothly to this part also in the (t, r) coordinates.
We now show that
We use the vector field T, given in acoustical coordinates by (cf. (2.31) of [2])
We have
Therefore
Now we use the function \(\kappa \) as defined by (2.24) of [2]
So
The minus sign appears due to the initial condition
where k is a positive constant (see page 39 of [2]). We now recall the function \(\alpha \) (see (2.41) of [2])
and the relation (see (2.48) of [2])
Since (see (117))
we obtain
Together with (234) we arrive at
which, in conjunction with (231), implies (226).
Remark 5
The acoustical metric \(h_{\mu \nu }\) as introduced in [2] coincides with \(g_{\mu \nu }\), but quantities such as \(\phi \), \(\beta \), F used in [2] do not coincide with the quantities denoted in the same way which were introduced in the present work. Making a distinction by putting a tilde on the quantities from [2] we have for example
where \(\tilde{\beta }\) is introduced in (1.44) of [2], while \(\beta \) is the one form defined in (29). For m see (95). Therefore, despite the fact that the wave equations of the present work and [2] have the same form, the physical meaning of the wave function is different. Nevertheless, functions such as \(\alpha \), \(\kappa \), \(\mu \) and relations thereof such as (234) are related only to the Lorentzian geometry given by the acoustical metric and can therefore be used in the present context as well.
In the following we restrict ourselves to one component of \(\partial _-\mathcal {B}\cup \mathcal {B}\) and the corresponding component of \(\underline{C}\) with past end point \(\partial _-\mathcal {B}\) which we denote by O. Now, the function \(\mu \) vanishes on \(O\cup \mathcal {B}\). From (226) together with \(\eta {>}0\), it follows that \(\partial r/\partial w\) vanishes on \(O\cup \mathcal {B}\). In particular
where the index 0 denotes evaluation at the cusp point O. Let the singular part of the boundary of the maximal solution be given by \(t=t_*(w)\) and let us set \(w_0=0\), \(t_0=0\), i.e. the cusp point O is the origin of the acoustical coordinate system. In spherical symmetry, the results at the end of Chapter 15 of [2] translate into
Using (226) and (240) we obtain
where we defined
From Chapter 15 of [2] we have
It follows that \(\kappa {>}0\). Since \(\mu (t_*(w),w)=0\), we obtain
which, using (241), implies
Taking the partial derivative of (226) with respect to w and evaluating at the cusp point yields
where we used (240). Taking the second partial derivative of (226) with respect to w and evaluating at the cusp point we obtain
where we defined
From Chapter 15 of [2] we have
which implies that \(\lambda {>}0\). From (241), (242), (243), (249) we deduce that \(\mathcal {B}\), i.e. the singular part of the boundary of the maximal development, is given in a neighborhood of the cusp point by
In the following we will also make use of the definition
We summarize the behavior of the radial coordinate at the cusp point.
We made the definitions
The boundary of the domain of the maximal solution close to a cusp point is shown in Figure 5.
Behavior of \(\alpha \) and \(\beta \) at the Cusp Point
In [2] the null vector fields L, \(\underline{L}\) are used. In the t-r-plane they are given in terms of acoustical coordinates by (see page 933 of [2])
Therefore,
Now, since the solution is smooth with respect to the acoustical coordinates (t, w) and the Riemann invariants are given smooth functions of \(\psi _\mu \) we have
Remark 6
We note that in (256), (257) (and also in (262) and in the second line of (265) below) \(\alpha \) refers to the quantity given by (233) as in [2]. However in (258) and everywhere else in the present work \(\alpha \) denotes the Riemann invariant defined by the first of (146). Also in (260), (261), (265) below \(\kappa \) denotes, as in [2], the inverse spatial density of the outgoing characteristic hypersurfaces, defined in (230) while everywhere else in the present work it denotes the quantity defined in (244).
Now we look at the partial derivative of \(\alpha \) and \(\beta \) with respect to w. From the second of (131)
Let us now use the vector field \(\hat{T}\), collinear and in the same sense as T and of unit magnitude with respect to the acoustical metric (see (2.57) of [2])
Using now \(X^\mu Y\psi _\mu =Y^\mu X\psi _\mu \) (recall that \(\psi _\mu =\partial _\mu \phi \)), we deduce from (259)
where f is a smooth function of (t, w). Therefore, in conjunction with (234),
which implies
Now,
where we used the first of (131). Using again the vector field \(\hat{T}\) we get
where we used (234). Substituting this in \(L_-\alpha \) we see that \(\left( L_-\alpha \right) _0\) blows up.
We note that from (261) together with (247) and the relation (234) we have
while from (265) we have
Incoming Characteristic Originating at the Cusp Point
Let in acoustical coordinates \(\underline{C}\) be given by \(w=\underline{w}(t)\). Setting \(\underline{r}(t)=r(t,\underline{w}(t))\), we obtain
Since
we have
Therefore, the inverse function \(\underline{t}(w)\) satisfies
Using (254) we deduce
Taking the derivative of (271) with respect to w we obtain from (254), in view of (272), that
Taking a second derivative of (271) and using (254), (255) yields
Taking the third derivative of (271) and evaluating the result at the cusp point yields, in conjunction with the above results
where we used the definition (253) and we defined
We conclude from (272), (273), (274) and (275) that
The function \(\alpha \) along \(\underline{C}\) is given by
Taking into account (272), (273) we obtain
Defining
we have
For the function \(\beta \) along \(\underline{C}\) given by
we find, taking into account (266),
Now, since
applying d / dw to this, evaluating at \(w=0\) and using (244), (266), (272) and (279) we obtain
Shock Development Problem
The notion of maximal development of the initial data is reasonable from the mathematical point of view and also the correct notion from the physical point of view up to \(\underline{C}\cup \partial _-\mathcal {B}\). However, it is not the correct notion from the physical point of view up to \(\mathcal {B}\). Let us consider a given component of \(\mathcal {B}\) which we again denote by \(\mathcal {B}\). Its past end point we denote by O (this corresponds to \(\partial _-\mathcal {B}\)). We also consider the corresponding component of \(\underline{C}\), i.e. the incoming null curve originating at O which we again denote by \(\underline{C}\) (see figure 5 on the right).
The shock development problem is the following:
Find a timelike curve \(\mathcal {K}\) in the t-r plane, lying in the past of \(\mathcal {B}\) and originating at O, together with a solution of the equations of motion in the domain in Minkowski spacetime bounded in the past by \(\mathcal {K}\) and \(\underline{C}\), such that the data induced by this solution on \(\underline{C}\) coincides with the data induced by the prior maximal solution, while across \(\mathcal {K}\) the new solution displays jumps relative to the prior maximal solution, jumps which satisfy the jump conditions. The past of \(\mathcal {K}\), where the prior maximal solution holds, is called the state ahead, and the future of \(\mathcal {K}\), where the new solution holds, is called the state behind (see 2.3). \(\mathcal {K}\) is to be space-like relative to the acoustical metric induced by the maximal solution and time-like relative to the new solution which holds in the future of \(\mathcal {K}\). The requirement in the last sentence is the determinism condition.
Let \(T_\varepsilon \) be the subset bounded by \(\underline{C}\), \(\mathcal {K}\) and the outgoing characteristic originating at the point on \(\underline{C}\) with acoustical coordinate \(w=\varepsilon {>}0\). In \(T_\varepsilon \) we use characteristic coordinates. We first shift the origin of the (t, w) coordinate plane so that the cusp point O has coordinates (0, 0). We then assign to a point in \(T_\varepsilon \) the coordinates (u, v) if it lies on the outgoing characteristic which intersects \(\underline{C}\) at the point \(w=u\) and on the incoming characteristic which intersects \(\mathcal {K}\) at the point where the outgoing characteristic through the point \(w=v\) on \(\underline{C}\) intersects \(\mathcal {K}\). It follows that (see figure 6)
The domain \(T_\varepsilon \)
Remark 7
We note that to set up the characteristic coordinates in this way we have to a priori assume that the solution is smooth in these coordinates. This is shown to be true below.
In the following we will denote \(\alpha \), \(\beta \) and r corresponding to the solution in the maximal development by \(\alpha ^*\), \(\beta ^*\) and \(r^*\) to distinguish them from \(\alpha \), \(\beta \), r which we use in referring to the solution in \(T_\varepsilon \). The quantities corresponding to the prior maximal solution are expressed in (t, w) coordinates. The solution in \(T_\varepsilon \) has to satisfy the characteristic system (see (161), (162))
together with initial data (for \(\underline{t}\) see (277))
and
The system consisting of the second of (288) and the second of (289) together with (292), (293) constitutes, at \(v=0\), a system of ordinary differential equations for \(\beta \) and r. Hence the above conditions on \(\beta \) and r at O imply that the data for \(\beta \) and r along \(\underline{C}\) coincide with the data induced by the prior maximal solution.
Let
Condition (171) is
where
the right hand sides given by the solution in \(T_\varepsilon \) and
the right hand sides given by the solution in the maximal development, where z(v) is the solution of the identification equation
identifying the radial coordinate of points on \(\mathcal {K}\) coming from the solution in the maximal development and from the solution in \(T_\varepsilon \). Condition (170) is
f(v) and g(v) have to satisfy
We restate the free boundary problem as follows.
For small enough \(\varepsilon \) find in \(T_\varepsilon \) a solution of (288), (289) which attains along \(\underline{C}\) the given data and along \(\mathcal {K}\) satisfies (295), (300), where V(v) is given by (299) and z(v) is given by (298).
We solve the problem using an iteration whose strategy is the following. We start with approximate solutions \(z_m(v)\), \(\beta _{+,m}(v)\), \(V_m(v)\). Then we solve the characteristic system (288), (289) with \((\alpha _{m+1},\beta _{m+1},t_{m+1},r_{m+1})\) in the role of \((\alpha ,\beta ,t,r)\) with initial data \(t_{m+1}(u,0)=h(u)\), \(\alpha _{m+1}(u,0)=\alpha _i(u)\) (on \(\underline{C}\)), boundary data \(\beta _{m+1}(v,v)=\beta _{+,m}(v)\) (on \(\mathcal {K}\)) and \(r(0,0)=r_0\) together with the requirement that
where
We then substitute \(f_{m+1}(v)\), \(g_{m+1}(v)\) for f(v), g(v), respectively, in the identification equation (298) and solve for z in terms of v. The solution we define to be \(z_{m+1}(v)\). Using now \(z_{m+1}(v)\), \(f_{m+1}(v)\) we obtain through (297) \(\alpha _{-,m+1}(v)\), \(\beta _{-,m+1}(v)\). We then use these together with \(\alpha _{+,m+1}(v)\) to solve (295) for \(\beta _{+}(v)\) which we define to be \(\beta _{+,m+1}(v)\). Note that \(\beta _{m+1}(v,v)=\beta _{+,m}(v)\) but \(\alpha _{m+1}(v,v)=\alpha _{+,m+1}(v)\). We then define \(V_{m+1}(v)\) by (299) where the jumps on the right hand side correspond to \(\alpha _{\pm ,m+1}(v)\), \(\beta _{\pm ,m+1}(v)\). We summarize the strategy as follows
In the following we shall call the triplet \((z_m,\beta _{+,m},V_m)\), which are functions on the boundary \(\mathcal {K}\), boundary functions corresponding to the m’th iterate.
We make the following crucial observation. Let
Since (see (240))
it is not possible to use the standard implicit function theorem to directly solve the identification equation (298) for z in our iteration scheme. We will use leading order expansions of g(v), f(v), z(v) in the identification equation to arrive, through a cancellation, at a reduced identification equation which can then be solved for the remainder function of z(v).
Solution of the Fixed Boundary Problem
Setup of Iteration Scheme and Inductive Step
The goal is to find a solution of the system of equations
together with initial data \(\alpha (u,0)=\alpha _i(u)\) (on \(\underline{C}\)), boundary data \(\beta (v,v)=\beta _+(v)\) (on \(\mathcal {K}\)), initial data \(t(u,0)=h(u)\) (on \(\underline{C}\)) and \(r(0,0)=r_0\), together with the requirement that for a given function V(v) the equation
is to be satisfied, where
and the requirement that t is a time function, i.e.
Since we set \(t_0=0\) we obtain from (309) \(f(0)=g(0)=0\). For the initial data of t along \(\underline{C}\) we assume (see (277))
For the initial data of \(\alpha \) along \(\underline{C}\) we assume (see (282))
Furthermore, we assume \(\beta _+(v)\in C^1[0,\varepsilon ]\), \(V(v)\in C^0[0,\varepsilon ]\) and
where \(y\in C^1[0,\varepsilon ]\) is a given function with
We define
The solution is to be found in a domain \(T_\varepsilon \) for small enough \(\varepsilon \), where
Taking the derivative of the first of (307) with respect to u and of the second of (307) with respect to v and subtracting yields
Defining
equation (317) becomes
Using (307) equation (308) becomes
where we use the definitions
where
Let us recall that in acoustical coordinates (t, w), the boundary of the singular part of the maximal development \(\mathcal {B}\) is given by \(t_*(w)\) (see (252)). We have
Since we are looking for a solution in which the shock \(\mathcal {K}\) lies in the past of \(\mathcal {B}\) we should have
where \(f(v)=t(v,v)\) describes the shock curve \(\mathcal {K}\). By this assumption together with (311) we get
From the first of (313) together with the second of (306) in conjunction with (311) we obtain
Taking the derivative of the second of (306) with respect to v we find
where we also used the first of (306) together with the first of (307). Using the first of (306) we obtain
Using this in (317) we get
Using this for the first term in (327) we get
Along \(\underline{C}\) (329), (330) build a system of the form
Together with the initial conditions given by (325), (326), we arrive at
Hence, we expect
Therefore, we base our iteration scheme with this expectation in mind.
We construct a solution of the fixed boundary problem as the limit of a sequence of functions \(((\alpha _n,\beta _n,t_n,r_n);n=0,1,2,\ldots )\). Given \((\alpha _n,\beta _n)\) we find \((\alpha _{n+1},\beta _{n+1})\) in the following way. We set
where
Let \(t_{n}\) be the solution of the linear equation
together with the initial data on \(\underline{C}\) (cf. (311)) and the boundary condition on \(\mathcal {K}\)
We then find \(r_{n}\) by integrating (307), i.e.
We then define \(\alpha _{n+1}\) and \(\beta _{n+1}\) by
where
We have thus found \((\alpha _{n+1} ,\beta _{n+1})\).
To initiate the sequence we set
\(t_0(u,v)\) is given by the solution of (336) (with 0 in the role of n) with \(\mu _0\), \(\nu _0\) given by (334) (with 0 in the role of n). \(r_0(u,v)\) is then given by (338).
The way we set things up we see that to each pair \((\alpha _n,\beta _n)\) there corresponds a unique pair \((t_n,r_n)\) given by (336), (338). It therefore suffices to show that the iteration mapping maps the respective spaces to itself (by induction) and the convergence only for the sequence \(((\alpha _n,\beta _n);n=0,1,2,\ldots )\). Let us denote by \((\alpha ,\beta )\) the limit of \(((\alpha _n,\beta _n);n=0,1,2,\ldots )\). The convergence of \((\alpha _n,\beta _n)\) to \((\alpha ,\beta )\) will imply the convergence of \((t_n,r_n)\) to (t, r), where t is the solution of (336) with the coefficients \(\mu \), \(\nu \) given by \(c_\pm (\alpha ,\beta )\) and r is given by (338) such that when t, r are substituted into the right hand sides of (352) below, the left hand sides of (352) are \(\alpha \), \(\beta \) respectively. Therefore this will imply the existence to a solution of the fixed boundary problem.
The first of (313) is equivalent to
We now define
where the constant C is the constant from (342). Therefore,
Let now
Let \(\delta {>}0\) and let \(R_{\delta }\) be the rectangle given by (see figure 7)
The rectangle \(R_\delta \)
We define
where we defined
Let
We now choose a constant \(N_0\) such that
Defining
we have
Let X be the closed subspace of \(C^1(T_\varepsilon ,\mathbb {R}^2)\) consisting of those functions \(F=(F_1,F_2)\) which satisfy
-
(i)
$$\begin{aligned} F_1(u,0)=0,\qquad F_2(v,v)=0, \end{aligned}$$(353)
-
(ii)
$$\begin{aligned} \Vert F\Vert _X\,{:=}\,\max \left\{ \sup _{T_\varepsilon }\left| \frac{1}{u}\frac{\partial F_1}{\partial u}\right| ,\sup _{T_\varepsilon }\left| \frac{1}{v}\frac{\partial F_1}{\partial v}\right| ,\sup _{T_\varepsilon }\left| \frac{1}{u}\frac{\partial F_2}{\partial u}\right| ,\sup _{T_\varepsilon }\left| \frac{1}{v}\frac{\partial F_2}{\partial v}\right| \right\} \le N_0. \end{aligned}$$(354)
As a preliminary result concerning the linear equation (336) we have the following proposition.
Proposition 4.1
Let
Then, provided we choose \(\varepsilon \) small enough, depending on Y, the solution of the equation
with initial condition
and boundary condition
is in \(C^1(T_\varepsilon )\) and satisfies
for \((u,v)\in T_\varepsilon \), where C(Y) are non-negative, non-decreasing, continuous functions of Y. Furthermore, let \(f(v)\,{:=}\,t(v,v)\), then
for \(v\in [0,\varepsilon ]\).
Proof
Integrating (359) with respect to v from \(v=0\) yields
while integrating (359) with respect to u from \(u=v\) yields
where we used the definitions
From the first of (366) together with (356) we obtain
Evaluating (364) at \(u=v\) yields
Defining (see (355) for the definition of \(\tau (u,v)\))
we have
where we used
which implies, using (357), (370),
Using the definitions
equation (373) becomes
Since (cf. (311))
we can write
where
Integrating (376) from \(v=0\) yields
Substituting this back into (376) gives
Using now (378) we find
where N(v) is linear in \(\hat{B}(v)\), while M(v) is independent of \(\hat{B}(v)\), i.e.
with
and
where we split \(\hat{B}(v)\) into a part depending on I(v) and a part independent of I(v), i.e.
In view of (357) we have
In the arguments to follow \(q{>}0\) will denote a number which we can make as small as we wish by choosing \(\varepsilon \) suitably small. From \(y(0)=-1\) we have
Now, since we can make \(1-\tfrac{1}{2}vY\) as close to 1 as we wish by choosing \(\varepsilon \) suitably small depending on Y (in the following we will not state this dependence explicitly anymore) and since (cf. (391))
we obtain
for \(p{>}1\) but as close to 1 as we wish by choosing \(\varepsilon \) suitably small. Hence
where \(q{>}0\) as small as we wish by restricting \(\varepsilon \) (and therefore \(\varepsilon Y\)) suitably (cf. (391)). Therefore,
We now look at \(\hat{B}_{0}\). From (311) it follows that the asymptotic form of the second bracket on the second line of (388) is \(\mathcal {O}(v^3)\). Taking into account (367), (390), as well as (395), we obtain
(374) together with (394) yields
From (374) we obtain
where for the first integral we use (cf. (394))
and we again choose \(\varepsilon \) sufficiently small. \(q'\), \(q''\) are like q, positive and as small as we wish by restricting \(\varepsilon \) suitably. From the two bounds in (397) and (398) it follows that the contribution of the \(\mathcal {O}(v)\) term in \(\hat{B}_{0}\) (cf. (396)) to \(N_{0}\) (cf. (387) with \(i=0\)) has the asymptotic form \(\mathcal {O}(v^2)\).
We now look at the contribution of the first term in (396) to \(N_{0}\) (cf. (387)). This contribution is
Since the function \((1-e^{x})/x\) is bounded for \(x\in [-1,1]\), it follows from (398) that the first term in the second curly bracket in (400) is bounded by
Now, since
where we used (399), we obtain
where we again used (399). Choosing \(\varepsilon \) sufficiently small such that \(Y^2\varepsilon \le 1\), it follows that \(C(Y+1)Yv^3\le Cv^2\), i.e. the first term in the second curly bracket of (400) is bounded in absolute value by \(Cv^2\).
Now we look at the second term in the second curly bracket in (400). We will use
Those are consequences of (397) and (399). Using these and (398), we have
where in the last step we again use the assumption that \(Y^2\varepsilon \le 1\). We conclude that the second term in the second curly bracket of (400) is bounded in absolute value by \(Cv^2\). Therefore, the second curly bracket in (400) is bounded in absolute value by \(Cv^2\).
We rewrite the first curly bracket in (400) as
For the curly brackets we use the estimate
where we used (391). We deduce that the second line in (406) is bounded in absolute value by \(Cv^2\) where we again make use of the assumption that \(Y^2\varepsilon \le 1\). We conclude that
Integrating by parts yields
We now turn to \(N_{1}\). For this we have to estimate \(\hat{B}_{1}\). In view of (389) we have
Now we look at I(v). From (367), (355) we have
These imply
which yields
Now,
where to go from the second to the third line we changed the order of integration and in the fourth line we used the definition
Using (413) and (414) in (387) with \(i=1\) we obtain
We postpone the estimation of T until the estimates for M are completed.
We now look at \(M_{1}\) (cf. (385)). We rewrite \(M_{1}\) as
Since
we can rewrite the term on the first line as
We now rewrite
where for the first term we used integration by parts while for the second term we used (407) together with the assumption that \(\varepsilon Y^2\le 1\). This implies that the term in the first line of (417) is equal to
Using the fact that the function \((1-e^{-x}-x)/x^2\) is bounded for \(x\in [-1,1]\), implies that the term on the second line in (417) is bounded by
Now, since
where we used (404), we obtain
where we used the assumption \(\varepsilon Y^2\le 1\). This together with (421) implies
We now look at \(M_{2}\) (cf. (386)). We rewrite \(M_{2}\) as
For the first term we have
where we used (407) together with the assumption \(\varepsilon Y^2\le 1\). Taking into account that the function \((1-e^{-x})/x\) is bounded for \(x\in [-1,1]\) and integrating by parts we find that the second term in (426) can be estimated in absolute value by
Using the second of (404) we get
where we again used the assumption \(\varepsilon Y^2\le 1\). From (427), (429) we obtain
We shall now turn to the estimate of T(u, v) and prove that \(T(u,v)\le Cv\). To accomplish this we will use the ode (382) but we will use less delicate estimates for \(M_{i}\) and \(N_{i}\) than derived above. We derive these rough estimates from the delicate estimates (409), (425), (430) by using the two estimates (cf. (391))
Using the resulting (rough) estimates together with (384), (416) in (382) we arrive at
Using the asymptotic form of \(\mu (u,v)\) as given by (355) it follows from the second of(366) that \(|L(u,v)|\le C\varepsilon \). We deduce from (365) together with the asymptotic form of \(\nu \) given by (356), very roughly,
From (364) we deduce, very roughly,
Substituting in the integral in (433) the bound (434) yields
Now,
which implies
where for the first inequality we used
which follows from (357), and for the second inequality in (437) we used (432). Substituting (435) and (437) into (433) yields
Since T(u, v) is non-decreasing in its first argument (cf. (415)) we have
Taking the supremum over all \(u'\in [v,u]\) we find
Defining
equation (441) becomes
Since \(\Sigma _u(0)=0\), we get
Therefore,
Plugging this estimate into the estimate for \(N_{1}\), i.e. into (416), we obtain
Using now the delicate estimates for \(M_{0}\), \(N_{0}\), \(M_{1}\), \(M_{2}\) as given by (384), (409), (425), (430) in the ode (382), we arrive at
This is (363).
Using now
in (447) we obtain
Since
we obtain from (438),
Together with (449) we conclude
Recalling the definition (415) and using (445) we get
Substituting this into (434) yields
We now revisit (365), in particular the term
From (355) we derive
Using now (356) and (454) we have
We now look at the first term in (365) and rewrite it as
The second term can be estimated using (452), while for the first term we use (355) to estimate
Now,
which, together with (459), implies
Using the estimates (452), (461) in (458) and the resulting estimate together with (457) in (365) yields
This is the first of (362).
We now revisit (364), in particular the term
We rewrite it as
Using (355), the first term on the right of (464) possesses the following asymptotic form
By (355) and (462), the second integral in (464) can be bounded in absolute value by \(C(Y)uv^2\). Together with (311) we obtain from (364),
This is the second of (362). This completes the proof of the proposition. \(\square \)
Lemma 4.1
For \(\varepsilon \) sufficiently small depending, on \(N_0\), Y, the sequence \(((\alpha '_n,\beta '_n);n=0,1,2,\ldots )\) is contained in X.
Proof
Therefore
We now show the inductive step. In the following the generic constants will depend on \(N_0\) but we shall not specify this dependence. It suffices for us that this constants are non-negative non-decreasing continuous functions of \(N_0\). The inductive hypothesis is
We start with
where we used \(\alpha _n'(u,0)=0\).
where we used \(\beta '_n(v,v)=0\). Choosing \(\varepsilon \) small enough we deduce
Therefore, the functions \(c_{\pm ,n}\) and derivatives thereof are bounded, i.e.
(also for higher order derivatives). From (351) we have
We now derive bounds on \(\mu _n\), \(\nu _n\). From the first of (318) we have
For the second term we have
where we used the first of (475) together with the inductive hypothesis (469). Defining the functions
the first term in (476) becomes
For \(f_{1,2}\) we note that \(f_{1,2}\in C^1[0,\varepsilon ]\) with \(f_{1,2}(0,0)=0\), which implies
For \(f_3\) we have
where we used the first of (474). The inductive hypothesis together with (312) yields
From (480), (482) together with
where we recalled \((\partial c_+/\partial \alpha )_0\dot{\alpha }_0=\kappa \) (see (286)), we obtain
Together with (477) we conclude
We now look at
For the first term we use the second of (474) which together with the inductive hypothesis (469) implies
For the second term in (486) we use the second of (475) together with the inductive hypothesis (469) and the property of the boundary data given by the first of (313). We get
Therefore
We now look at \(\gamma _n(v)\). Since
we have
Now,
From \(\alpha '(u,0)=0\) together with the inductive hypothesis (469) we obtain
Hence
From (342) we have
From
together with the inductive hypothesis we have
Expanding now \((\partial c_+/\partial \alpha )(\alpha ,\beta )\), \((\partial c_+/\partial \beta )(\alpha ,\beta )\) and making use of (494), (495) we obtain
Using these together with (342), (497) in (491) and recalling \(\left( \partial c_+/\partial \alpha \right) _0\dot{\alpha }_0=\kappa \) we find
Hence
Therefore, using the second of (313),
From the fact that \(\bar{c}_{-n}(v)=c_-(\alpha _{+n}(v),\beta _+(v))\) is in \(C^1[0,\varepsilon ]\) we obtain, in conjunction with the second of (313),
Now, using (502) and (503) in the third of (321) we find
where
Remark 8
There is no index missing on \(\beta _+\) in (490). This is because \(\beta _+\) originates from the boundary data which is not subjected to the iteration while \(\alpha _{+n}\) is part of the solution of the fixed boundary problem, therefore subjected to the iteration.
In view of (311), (485), (489), (504), (505) we are in the position to apply proposition 4.1 with \((\mu _n,\tau _n,\nu _n,\gamma _n,\rho _n,t_n)\) in the role of \((\mu ,\tau ,\nu ,\gamma ,\rho ,t)\). From proposition 4.1 we have (recalling the constant \(C_0\) given in (349))
provided that \(\varepsilon \) is sufficiently small. Using (506) in (338) together with (473) we get
provided that \(\varepsilon \) is sufficiently small. It follows from (472), (507) and the definitions (346), (347)
Taking the derivative of the first of (352) with respect to v and of the second of (352) with respect to u we obtain from (506) together with (508) that
Now we establish bounds for \(\partial \alpha _{n+1}'/\partial u\), \(\partial \beta _{n+1}'/\partial v\). (472), (507) imply
From the first of (352),
where for the third term we used the second of (307). From (336), (355), (356), (506) we obtain
Now we bound (511). For the first three terms we use the second of (506), (510) and the inductive hypothesis (469) in conjunction with (351) (and the fact that we have bounds on the derivative of the initial data \(\alpha _i\)). For the fourth term in (511) we use (512). We conclude
From the second of (351),
We can bound the integral in the same way as we did for (511) (and using the first of (313)). For the term \(B_n(v,v)\) we use
where we used the second of (509). Therefore,
From (509), (513), (516) we have
Replacing u by \(u'\), taking the supremum over \(u'\in [v,u]\) and choosing \(\varepsilon \) sufficiently small we find
This completes the inductive step and therefore the proof of the lemma. \(\square \)
Convergence
Lemma 4.2
For \(\varepsilon \) sufficiently small depending on \(N_0\), Y, the sequence \(((\alpha _n',\beta _n');n=0,1,2,\ldots )\) converges in X.
Proof
We use the notation
From the first of (352) we have
Therefore
From the first of (351) we have
which implies
Similarly we get
For an estimate of \(|\Delta _nr(u,v)|\) we use (338), which gives
We first look at the terms \(\Delta _nc_\pm \).
where for the second inequality we use (523), (524). To find an estimate for the differences of partial derivatives of t we subtract (319) with \(t_{n-1}\) in the role of t from (319) with \(t_n\) in the role of t (analogously for the roles of \(\mu \) and \(\nu \)). We arrive at
where
The initial conditions for (527) are \(\Delta _nt(u,0)=0\). Furthermore we have the boundary condition
Integrating (527) with respect to v from \(v=0\) yields
where we used the first of (366). Evaluating (530) at \(u=v\) gives
where we used the definitions
For the definition of \(\tau _n(v)\) see (355).
We now estimate the differences \(\Delta _n\mu \), \(\Delta _n\nu \). From the first equality in (476) we obtain
The first term can be bounded using the estimates (526). For the second term we use
which implies
For the first and third difference we use
For the difference of derivatives of \(c_{n,\pm }\) we use
where we used (523), (524). The same result holds with \(\beta \) in the role of \(\alpha \). Using (537), (538) in (536) and the resulting estimate together with (535) in (534), we obtain
From (486) we get
The first term can be bounded using the estimate (526) and taking into account that
where we used the inductive hypothesis (469). For the second term in (540) we use the expression for \(\partial \Delta _nc_-/\partial v\) analogous to (536) and take into account the expressions analogous to (537). We arrive at
Defining (see (354))
Recalling (506) we deduce from (544) that
Therefore,
Since
the boundary condition (529) gives
where we use the definition
Using (505) we obtain
Substituting (531) we get
where \(A_n(v)\) is given by (374) and
Integrating (551) from \(v=0\) to v yields
Substituting this back into (551) gives
We decompose \(\Delta _nB\) according to
where \(\overset{1}{\Delta }_nB\) contains the terms of \(\Delta _nB\) which are linear in \(\Delta _nI\). The right hand side of (554) being linear in \(\Delta _nB\), we decompose analogous to the decomposition (555), i.e.
\(\overset{1}{R}_n\) being linear in \(\Delta _nI\). Recalling the third of (321) together with the second of (313) we deduce
where we used (523) (notice that in the third of (321) we have \(\gamma _n\) in the role of \(\gamma \) and \(\bar{c}_{\pm ,n}(v)=c_\pm (\alpha _{+n}(v),\beta _+(v))\) in the role of \(\bar{c}_\pm (v)\) but \(\beta _+(v)\) as well as V(v) are given by the boundary data). From the first of (506) we then get
From (397) we have
Therefore, through (554),
We now estimate \(\Delta _nI\). From (367) we get
while from (355) we have
Using (562) and (563) in (532) we obtain
Therefore,
which implies
where
Since, from (565)
we conclude, together with (566), that the part of the right hand side of (554) which is linear in \(\Delta _nI\) (which we denote by \(\overset{1}{R}_n\)) satisfies the estimate
Integrating (527) with respect to u from \(u=v\) yields
The second of these implies, through the second of (366), that
Using the first of (572) as well as (573) and (545) we obtain
Now, from (530) in conjunction with (545) and the second of (572) we get
Using this for the integral of the right hand side of (574) we estimate
Substituting this into (574) we obtain
Using this estimate in (577) we find
Taking the supremum over u in [v, u] we deduce
Setting
(581) becomes
Integrating yields
Using this in (581), we find
The above imply, through (580),
Using this in (575) we conclude
Using these estimates together with (506) and (526) in (525) we deduce
It then follows from (521) together with (523), (524), (586), (588) that
From the second of (352) we deduce
Therefore (cf. (521)),
Using now (523), (524), (587) (588) we arrive at
Now, from the first of (352) we deduce
For the second term on the right of (593) we use
where for the last inequality we used (523), (524), (586), (588). The fourth term on the right of (593) we treat analogous to the second. For the fifth term we use (587) and for the sixth we use (526). The seventh and the last term in (593) can be bounded in the same way as the second and the fourth. This leaves us with the eighth term. From (527) in conjunction with (545), (586), (587) we get
Putting things together we deduce from (593)
Using the second of (352) we see that for the difference
we get an analogous equation as we got in (593) with the exception of the additional term
For the terms analogous to the ones showing up in (593) we use the analogous estimates while for the term (598) we use the first equality in (590) with \(u=v\) and the estimate (592). Therefore,
Using (589), (592), (596), (599), it follows
It follows that for \(\varepsilon \) small enough we have convergence of the sequence in the space X. This concludes the proof of the lemma. \(\square \)
The two lemmas above show that the sequence \((\alpha _n',\beta _n')\) converges to \((\alpha ',\beta ')\in X\) uniformly in \(T_\varepsilon \). Therefore we also have uniform convergence of \((\alpha _n,\beta _n)\) to \((\alpha ,\beta )\in C^1(T_\varepsilon )\) (see (351)). Now, (586), (587) show the convergence of the derivatives of \(t_n\). Therefore, the pair of integral equations (364), (365) are satisfied in the limit. We denote by t the limit of \((t_n)\). It then follows that the mixed derivative \(\partial ^2t/\partial u\partial v\) satisfies (359). In view of the Hodograph system (307) the partial derivatives of \(r_n\) converge and the limit satisfies the Hodograph system. Let us denote by r the limit of \((r_n)\). We have thus found a solution of the fixed boundary problem. Since every member of the sequence \((t_n)\) satisfies the expressions for the asymptotic form as given in the statement of proposition 4.1 these expressions (i.e. (362), (363)) also hold for the limit t. We have therefore proven the following proposition.
Proposition 4.2
Let h(u) and \(\alpha _i(u)\) be given by
Furthermore, let \(\beta _{+}(v)\in C^1[0,\varepsilon ]\), \(V(v)\in C^0[0,\varepsilon ]\) satisfy
where y is a given function such that
Let
Let \(r_0{>}0\) and let \(N_0\) be given as in (350). Then there exists a solution \((\alpha , \beta ,t,r)\in C^1(T_\varepsilon )\) of the characteristic system (306), (307) such that \(\alpha (u,0)=\alpha _i(u)\), \(\beta (v,v)=\beta _{+}(v)\), \(t(u,0)=h(u)\), \(r(0,0)=r_0\) and
provided \(\varepsilon \) is sufficiently small depending on \(N_0\), Y.
Construction
Inductive Step
We recall briefly the strategy of the iteration. We start with the boundary functions corresponding to the m’th iterate \((z_m,\beta _{+,m},V_m)\). We then solve the corresponding fixed boundary problem using the result from the previous chapter. The solution of the fixed boundary problem provides us with the functions \(\alpha _{+,m+1}\), \(f_{m+1}\), \(g_{m+1}\). Using \(f_{m+1}\), \(g_{m+1}\) we solve the identification equation, the solution of which we denote by \(z_{m+1}\). Using then \(\alpha _{+,m+1}\), \(z_{m+1}\) in the jump conditions we obtain \(\beta _{+,m+1}\), \(V_{m+1}\). We have thus obtained the boundary functions corresponding to the \((m+1)\)’th iterate \((z_{m+1},\beta _{+,m+1},V_{m+1})\). This concludes the iteration.
The input for the construction problem are the following assumptions for the boundary functions \(z_m\), \(\beta _{+,m}\), \(V_m\):
with \(y_m,\hat{\beta }_{+,m}\in C^1[0,\varepsilon ]\), \(\hat{V}_{m}\in C^0[0,\varepsilon ]\). \(\lambda \), \(\kappa \) and \((\partial \beta ^*/\partial t)_0\) are given by the solution in the maximal development. We recall (see (244), (250))
We choose closed balls in the function spaces as follows
where Y, \(\delta _1\), \(\delta _2\) are to be chosen appropriately.
We initiate the sequence by
Proposition 5.1
Choosing the constants Y, \(\delta _1\), \(\delta _2\) appropriately, the sequence \(((y_m,\hat{\beta }_{+,m},\hat{V}_m);m=0,1,2,\ldots )\) is contained in \(B_Y\times B_{\delta _1}\times B_{\delta _2}\), provided we choose \(\varepsilon \) suitably small.
Proof
We see that
The inductive hypothesis is
Therefore,
In the arguments to follow \(q{>}0\) will denote a number which we can make as small as we wish by choosing \(\varepsilon \) suitably small. From \(y_m(0)=-1\) we obtain
In the following we use the notation \(g(v)=\mathcal {O}_d(v^n)\) to denote
where the constant C is a non-negative, non-decreasing, continuous function of d.
provided that \(\varepsilon \) is sufficiently small. This can be seen as follows. The statements (623) are equivalent, respectively, to
From the inductive hypothesis (619) we have
which implies
for a fixed numerical constant C if we choose \(\varepsilon \) sufficiently small. Using this in (see (611))
we obtain
if we choose \(\varepsilon \) sufficiently small. This is equivalent to the first of (624). The second of (624) follows directly from the inductive hypothesis.
Recalling the definition of \(N_0\) in the fixed boundary problem (see (342),...,(350)), we note that since the constant in the first of (624) is a fixed numerical constant, also \(N_0\) is a fixed numerical constant.
We now apply proposition 4.2 with \((y_m,\beta _{+,m},V_m)\) in the role of \((y,\beta _+,V)\). The resulting solution we denote by \((\alpha _{m+1},\beta _{m+1},t_{m+1},r_{m+1})\). We also denote
From the solution of the fixed boundary problem we have
Defining the function \(\hat{f}_{m+1}\) by
we deduce
where
and we used (632). Integrating by parts we obtain
This implies
Hence
Defining the function \(\delta _m(v)\) by
we obtain
Remark 9
The appearance of the indices m and \(m+1\) originates from the basic strategy (see (303)) in which the solution of the fixed boundary problem carrying the index \(m+1\) satisfies
This confusion did not appear up to now since we dropped the index m from the outer iteration during the solution process of the fixed boundary problem. An analogous situation appears when evaluating the function \(\beta _{m+1}\) on the boundary \(u=v\). There we have
since \(\beta _{+,m}\) is the boundary value for the fixed boundary problem whose solution carries the index \(m+1\). The function \(\beta _{+,m+1}\) is determined later on by making use of the jump conditions. The appearance of the indices m and \(m+1\) in (632) are explained in the same way.
We define the function \(\phi (v)\) by
We split up the function \(\delta _{m+1}(v)\) according to
where the functions \(\delta _0(v)\) and \(\delta _1(v)\) are given by \(\delta _0(0)=0\), \(\delta _1(0)=0\) and
(We make use of \(V_m(v)=c_{+0}+\frac{\kappa }{2}(1+y_m(v))v+\mathcal {O}_{\delta _2}(v^2)\)). Defining the functions \(\hat{\delta }_i(v),i=0,1\) by \(\delta _i(v)=v^3\hat{\delta }_i(v)\) we get
where we integrated by parts.
From \(V_m(v)-c_{+0}=\frac{\kappa }{2}(1+y_m(v))v+\mathcal {O}_{\delta _2}(v^2)\) we have (see (621))
Together with (see the right hand side of (632))
we obtain
Therefore, for \(\varepsilon \) sufficiently small,
Using this in
we deduce
Using (646), (652) we arrive at
Hence
In view of (632), (640), (647) we have
Therefore, in conjunction with \(\delta _{m+1}(0)=g_{m+1}(0)-c_{+0}f_{m+1}(0)=0\), we have
which implies
Now we look at the identification equation, i.e. at
Here the function on the right hand side is the solution \(r^*(t,w)\) given in the maximal development (recall that we set \(t_0=w_0=0\)), while the left hand side is given by the solution of the fixed boundary problem (The identification equation is an equation for \(y_{m+1}\) as a function of v given the functions \(g_{m+1}(v)\), \(f_{m+1}(v)\)). We have
In the following discussion of the identification equation we will omit the index \(m+1\). We define the function
Thus
Remark 10
The function h(t, w) is introduced in order to represent the terms \(\mathcal {O}(v^k)\) for \(k\ge 5\) in the expansion of \(r^*(t,w)\), when \(v^2\hat{f}(v)\) for t and vy for w are being substituted. We use
Let now
The identification equation becomes
Using now
and expressing (t, w) in terms of v and y according to \(t=f(v)=v^2\hat{f}(v)\), \(w=vy\) and making use of (659) the function F(v, y) becomes
We note that
where H is a smooth function of its two arguments.
Defining the function \(\hat{F}\) by the relation
(666) is equivalent to
where the remainder R is given by
The identification equation is now equivalent to
At \(v=0\) this becomes (we recall that \(\hat{\delta }(0)=0\) and note that \(\hat{f}(0)=\lambda /6\kappa ^2\) (see (632), (633)))
The only physical solution is \(y=-1\). We set \(v_0\,{:=}\,0\), \(y_0\,{:=}\,-1\). We now see that
Therefore we are able to solve the identification equation for y as a function of v for v small enough. We have
Differentiating (671) implicitly yields
We have
We first derive bounds for the remainder R and its derivatives of first order. From (638) we have, for \(\varepsilon \) small enough,
Therefore,
We now examine (675). Using (676) we find for the denominator
Now, for small enough \(\varepsilon \), we have
which, together with the second of (679) implies
Now we look at the numerator of (675). Making use of (677) we get
From (638) together with (621) we have, for small enough \(\varepsilon \),
Together with the first and the third of (679) we obtain
which implies, through (637), (653),
Substituting (687) for the numerator in (675), using the estimate (683) together with (673) for the denominator in (675) and putting back the indices m and \(m+1\) we arrive at
where
Taking the absolute value and then taking the supremum over \(v\in [0,\varepsilon ]\) yields
Choosing then \(\varepsilon \) suitably small such that
where C is the constant appearing in the denominator of (690), we obtain
Therefore, choosing now \(Y=2C'(\delta _2)\),
Remark 11
Y depends on \(\delta _2\).
In the following we will establish closure for the iterations of the functions \(\hat{\beta }_m\), \(\hat{V}_m\). The balls for the respective iterations have been chosen according to (615), (616). Since there are no more indices m to appear, only indices \(m+1\), we will omit in the following the index \(m+1\). We have
From (638) it follows (using \(\hat{f}(0)=\lambda /6\kappa ^2\))
This implies, through (633) and the first line of (634),
From \(z_{m+1}(v)=vy_{m+1}(v)\) we obtain
Therefore,
Now we look at the asymptotic form of \(\beta _-(v)\). We have
where the function on the right is \(\beta ^*(t,w)\) from the state ahead (i.e. given by the solution in the maximal development) and we recall that \(t_0=w_0=0\). We have
Expanding \((\partial \beta ^*/\partial t)(t,w)\) to first order and substituting \(t=f(v)\), \(w=z(v)\) we obtain
while expanding \((\partial \beta ^*/\partial w)(t,w)\) to second order, substituting \(t=f(v)\), \(w=z(v)\) and using (see (266))
yields
Therefore,
Hence,
Now we find the asymptotic form of \(\alpha _-(v)\). From
we have
Expanding now \((\partial \alpha ^*/\partial t)(t,w)\) and \((\partial \alpha ^*/\partial w)(t,w)\) to first order and substituting \(t=f(v)\), \(w=z(v)\) we obtain
Therefore,
Thus,
where the last equality holds provided we choose \(\varepsilon \) suitably small.
We now deal with the asymptotic form of \(\alpha _+(v)\). From
we obtain
where we made use of \(\alpha (v,0)=\alpha _i(v)\). This implies
The solution of the fixed boundary problem satisfies (see proposition 4.2)
Here the constants depend on Y and \(\delta _2\). Therefore (cf. (694)),
The first implies
From
we deduce, together with (717) and (cf. (485), (489))
that
where the second equality holds provided we choose \(\varepsilon \) small enough. Therefore
Using the first of (717) we find
where we denote by \((\ldots )\) the bracket in the last line of (715). We conclude from (718), (722), (723) that
which implies
where the second equality holds provided we choose \(\varepsilon \) small enough.
We now turn to the jumps \(\left[ \alpha (v)\right] \), \(\left[ \beta (v)\right] \). The first line of (725) together with the first line of (712) yields (note that \(\tilde{A}_0=\left( \partial \alpha ^*/\partial t\right) _0\))
where the second equality holds provided we choose \(\varepsilon \) sufficiently small. Using now (224), i.e.
we get from (726)
where
Therefore, in view of (706), we get
From \(\beta _+(v)=\beta _0+v^2\hat{\beta }_+(v)\) we have
Taking the derivative of (727) we obtain
From (698), (711), (724) we have
Using this together with (726) in (732) we obtain
Using now (705) we find
Therefore, substituting (730), (735) into (731), we conclude (putting back the index \(m+1\))
We now find an expression for the asymptotic form of \(V_{m+1}(v)\). We again omit the index \(m+1\) for now. We have (see (170))
We rewrite the numerator as
We expand
and
Similar expressions hold for the denominator of (737). Using \(\left[ \beta \right] =\left[ \alpha \right] ^3G(\alpha _+,\alpha _-,\beta _-)\) (see (224)) it follows
where for the second equality we used \(\partial T^{tt}/\partial \alpha \ne 0\), which follows from the first of (185). We now use the first of (187), i.e.
which implies
We look at \(c_+(\alpha _+,\beta _-)\). We have
Using \(\left( d\beta _-/dv\right) _0=0\) we obtain
where we also used \(\left( \partial c_+/\partial \alpha \right) _0=\kappa /\dot{\alpha }_0\). We also have
Using now (745) and (746) together with the asymptotic forms of \(d\alpha _+/dv\), \(d\beta _-/dv\) given by (724), (735) respectively, we obtain
Therefore,
where we introduced
Therefore, provided we choose \(\varepsilon \) small enough,
Using (726), (745), (750) in (743) we arrive at
Using now \(\hat{V}(v)=(1/v^2)(V(v)-c_{+0}-(\kappa /2)(1+y(v))v)\) we deduce, putting back the index \(m+1\),
i.e.
for a fixed numerical constant C. Choosing now the size of the ball for the iteration of the function \(\hat{V}_m\), i.e. \(\delta _2\), to be equal to the numerical constant C appearing in (753) we see that \(\hat{V}_{m+1}\in B_{\delta _2}\). From (736) we then have
for a fixed numerical constant C. Choosing now the size of the ball for the iteration of the function \(\hat{\beta }_{+,m}\), i.e. \(\delta _1\), to be equal to the numerical constant appearing in (754) we see that \(\hat{\beta }_{+,m+1}\in B_{\delta _1}\). From (694) we get
for a fixed numerical constant C. In view of (693) this shows that \(y_{m+1}\in B_Y\). This concludes the proof of the proposition. \(\square \)
Convergence
We define
For differences between successive members of the iteration we will use the notation \(\Delta _m f\,{:=}\,f_m-f_{m-1}\). For any sequence of functions \(h_m\in C^1\) with identical values \(h_m(0)=:h_0\) for any m we have
which implies
Estimates for the Solution of the Fixed Boundary Problem
Lemma 5.1
Let \((t_{m+1},r_{m+1},\alpha _{m+1},\beta _{m+1})\) be the solution of the fixed boundary problem as given by proposition 4.2, corresponding to the boundary functions \((z_m,\beta _{+,m},V_m)\). Then the following estimates hold
Proof
The difference \(\Delta _{m+1}t\) satisfies
where
In addition we have the boundary condition
where
and the initial condition
Before we study equation (762) we estimate the difference \(\Delta _{m+1}(1/\gamma )\) and the term \(\Xi _{m+1}\). We have
where
Remark 12
The mixture of indices in (765), (768) arises because the index increases with the solution of the fixed boundary problem as described in the basic strategy. \(\alpha _+\) and \(c_\pm \) carry the index of the solution of the fixed boundary problem while \(\beta _+\) and \(V_m\) carry the index of the data which goes into the fixed boundary problem.
We consider first the denominator of (767). From
together with \(\left( \partial c_+/\partial \alpha \right) _0=\kappa /\dot{\alpha }_0\) and the asymptotic forms of \(d\alpha _{+,m}/dv\), \(d\beta _{+,m}/dv\) given by (724), (735) respectively, we find
Using this together with
(recall that \(V_m(v)=c_{+0}+\frac{\kappa }{2}(1+y_m(v))v+v^2\hat{V}(v)\), \(y_m(v)=-1+\mathcal {O}(v)\)) we get
We turn to the numerator of (767). We rewrite it as
We have
Now we study the difference \(\Delta _{m+1}\bar{c}_\pm \). Since \(\bar{c}_{\pm ,m}(v)=c_\pm (\alpha _{+,m}(v),\beta _{+,m-1}(v))\) we need to estimate the difference \(\Delta _{m+1}\alpha _+\). From (see the first of (306))
we obtain
From this we deduce that for \(\varepsilon \) small enough
For the difference \(|\Delta _{m+1}r|\) we integrate (307) (cf. also (338)) to get
which implies
Using this in (777) and choosing \(\varepsilon \) sufficiently small we find
To estimate the difference \(\Delta _{m+1}\beta \) we use (see the second of (306))
which implies
Using (779) and choosing \(\varepsilon \) small enough, this in turn implies
Using this in (780) we obtain
provided that we choose \(\varepsilon \) suitably small. For future reference we use this in (783) which implies
From (784) we have
It now follows for the difference \(\Delta _{m+1}\bar{c}_\pm \)
Using this we get
where for (788) we used \(\bar{c}_{+,m}-V_{m-1}=\mathcal {O}(v)\) and for (789) we used \(V_{m-1}-\bar{c}_{-,m}=c_{+0}-c_{-0}+\mathcal {O}(v)\).
The remaining part of the numerator of (767) is
Using now (774), (788), (789), (790) together with (772) in (767) we arrive at
We turn to \(\Xi _{m+1}\). For an estimate of this we need estimates for the differences \(\Delta _{m+1}\mu \), \(\Delta _{m+1}\nu \). From the first of (318) we have
which implies
For the last term we use
For the partial derivatives of \(c_+\) with respect to \(\alpha \), \(\beta \) we have
For the partial derivative of \(\beta \) with respect to u we use the second of (306). I.e. we have
which implies
Using (795), (797) in (794) and the resulting estimate in (793) we obtain
For the term involving the partial derivative of \(\alpha \) with respect to u we use
which implies
We rewrite this as
where the \(I_i\) denote the nine terms appearing in (800) (the \((\ldots )\) bracket counting as two terms). The first term we will absorb on the left hand side. For \(I_2\) we use
which implies
For \(I_3\) we use
Using now (797) this implies
\(I_4\) can be treated in the same way as \(I_2\). For \(I_5\) we use (804) with r in the role of \(\beta \), which implies
For \(I_6\) we use
which implies
For \(I_7\) we use
which implies
For \(I_8\) we use
From this it follows that
which implies
For \(I_9\) we use
which implies
Using the estimates for the integrals of \(I_2\ldots I_9\) in (801), taking the supremum of the resulting estimate in \(T_u\) and absorbing the term involving \(I_1\) on the left hand side it follows that for small enough \(\varepsilon \) we have
Using this estimate in (798) it follows that for \(\varepsilon \) small enough we have
To get an estimate for \(\Delta _{m+1}\nu \) we apply a similar procedure. The estimate (793) is replaced by
Using now
it follows that in the role of the estimate (798) we have
Now for the estimate of \(\partial \Delta _{m+1}\beta /\partial v\) we use
This follows from the second of (306) integrated with respect to u from \(u=v\) up to u. Using this we get for the difference \(\partial \Delta _{m+1}\beta /\partial v\) terms which are analogous to the ones in (800) (A replaced by B, \(\partial /\partial u\) replaced by \(\partial /\partial v\)) but in addition to the integral we have the terms
For the second difference in (822) we use (796), (797). It follows
Therefore, for \(\varepsilon \) small enough, we have
From (817), (824) together with (779) we obtain
Substituting now (784), (785) we get
where we also used \(\sup _{[0,u]}|\Delta _{m+1}\beta _+|\le u\sup _{[0,u]}|d\Delta _{m+1}\beta /dv|\). Using the estimates (827), (828) in (763) we arrive at
where
We now look at equation (762). We are going to deal with this equation in a similar way as we dealt with the one appearing in the convergence proof of the fixed boundary problem. Integrating with respect to v yields (for \(K_{m+1}\) recall the first of (366))
We define
We recall that \(\tau _{m+1}(u,v)\) is given by
and that \(\tau _{m+1}(u,v)=\mathcal {O}(u)\). We obtain from (831) (recall that \(b(v)=(\partial t/\partial u)(v,v)\))
Defining
we obtain from (764) (recall \(a(v)=(\partial t/\partial v)(v,v)\))
which implies
Using the estimate (829) in (833) we find
Using (791) together with the second of (362) we obtain
We substitute (835) into (838) and arrive at
where (for \(\rho \) see (357))
Integrating (841) from \(v=0\) gives
Substituting back into (841) yields
Now we decompose \(\Delta _{m+1}B\) into
where \(\overset{1}{\Delta }_{m+1}B\) contains the terms of \(\Delta _{m+1}B\) which are linear in \(\Delta _{m+1}I\). Analogously we decompose
We estimate \(\overset{0}{\Delta }_{m+1}B\) using (839),(840). We obtain
This implies
where
We now estimate \(\overset{1}{\Delta }_{m+1}B\). For this we need an estimate for \(\Delta _{m+1}I\). From \(K_{m+1}(u,v)=\mathcal {O}(v^2)\) we have
Since we also have \(\tau _{m+1}(v)=\mathcal {O}(v)\) we obtain from (832)
which implies
Therefore,
where we use the definition
Using this definition also in (853) we obtain
From the estimates (854), (856) we find
This together with (849) gives
Integrating (762) with respect to u from \(u=v\) yields
Using now \(\nu _{m+1}(u,v)=\mathcal {O}(v)\) (cf. (489)) together with (829) we obtain
Now, from (831) we get
From this we obtain
Using this in (860) we find
From (837) we have
Using the estimates for \(\Lambda _{m+1}\) and \(d\Delta _{m+1}f/dv\) given by (840), (858) respectively, we obtain
Using this in (863) we get
where we use the definition
We also define
From (866) with \(u'\) in the role of u and taking the supremum over \(u'\in [v,u]\) we deduce
Defining now
(869) becomes
which implies
Therefore,
i.e.
which in turn implies, through (861),
It follows that for \(\varepsilon \) small enough
We call these the rough estimates. In the following we will use these to get more precise estimates.
which implies, through the first inequality of (853),
Now, from (848), together with the rough estimates, we obtain
Therefore,
which, in conjunction with (844), implies
Using the rough estimates in (839) yields
Now we use (878), (882), (883) in (835) and arrive at
Using the rough estimates in (840) we get
Using now (883), (885) in (843) we get
Together with (879) we find
Now we derive the precise estimate for \(A_{m+1}(v)\). In view of (842) we need first an estimate for \(\rho _{m+1}(v)\). From (357) we have
From (394) we have
Therefore,
which, in conjunction with (882), implies
Using (887), (891) in (841) we arrive at
Let us make the following generic definition. By \(E_m(v)\) we mean a function of the kind
Using now
(892) becomes
Therefore,
Integrating by parts we obtain
Now we express the above in terms of \(\Delta _{m+1}\hat{f}(v)\) given by \(\Delta _{m+1}f(v)=v^2\Delta _{m+1}\hat{f}(v)\). Using \(|\Delta _my(v)|\le v\Vert \Delta _my \Vert _{X}\), we obtain from (896),
Since
We turn to \(\delta _{m+1}(v)=g_{m+1}(v)-c_{+0}f_{m+1}(v)\). From (see (640))
we have
For the first term in (902) we use (see (632) and \(V_m(v)-c_{+0}=\frac{\kappa }{2}(1+y_m(v))v+\mathcal {O}(v^2)\))
and get
For the second term in (902) we use
which implies, in conjunction with (898), (900),
Therefore,
Putting things together we arrive at
Integrating (and also integrating by parts) we obtain
Now we use \(\hat{\delta }_{m+1}\) given by \(\delta _{m+1}(v)=v^3\hat{\delta }_{m+1}(v)\). We obtain from (908), (909)
which implies
We have
For the first term we use as a starting point (819), i.e.
which implies
To deal with the difference \(\Delta _{m+1}r\) we use (779). For the differences \(\Delta _{m+1}\alpha \), \(\Delta _{m+1}\beta \) we use (784), (785) respectively. Using also the estimates (876), (877) together with the definition (867), we find
For the second term in (912) we use (816). For the differences in the first line of (816) we use the same estimates which we used for the first term in (912). For the differences in the second line of (816) we use again the estimates (876), (877). We obtain
For the differences of \(\mu \) and \(\nu \) we use (827), (828) together with (876), (877). We obtain
Therefore,
i.e. (see (867) and take into account that \(\beta _{+,m}(v)=\beta _0+v^2\hat{\beta }_{+.m}(v)\))
In view of (900), (911) and (919), the proof of the lemma is complete. \(\square \)
Estimates for the Identification Equation
Lemma 5.2
Let \(f_m(v)=v^2\hat{f}_m(v)\), \(y_m(v)\), \(g_m(v)\) satisfy the identification equation
Then the following relation holds
where
Proof
We look at (see (675))
where (we omit the argument of y)
The difference we have to estimate is given by
Now
For the second term we use (cf. (638))
For the fourth term in (927) we recall (see (670))
where H is a smooth function of its two arguments. We estimate the difference of the first term by
The difference of the second term in (929) we estimate using (recall \(y(v)=\mathcal {O}(1)\))
while for the difference of the third term in (929) we use
For the difference of the fourth term in (929) we use
Therefore,
Now we look at the fifth term in (927). We have
where by \(\partial _1H\) we denote the partial derivative of H with respect to its first argument. For the difference of the first term we use
while for the difference of the second term we use
For the differences of the last two terms in (935) we use
Therefore,
It follows that
We conclude that
Now we look at the second term in (926). We use
We need to estimate the difference \(\Delta _{m+1}\left( (\partial \hat{F}/\partial y)(v,y)\right) \). We have
For the first term we have
For the third term in (943) we use
For difference of the first term we use
while for the difference of the second term we use
For the difference of the third term in (945) we have
Therefore, we obtain for the third term in (943)
We conclude that
To deal with (926) we also need (see (673), (676))
Using now the asymptotic forms as given by (941), (950) and (951) in (926) we arrive at (921). \(\square \)
Estimates for the Jump Conditions
Lemma 5.3
Let \(\alpha _{+,m}(v)\), \(\beta _{+,m}(v)\), \(f_m(v)\), \(z_m(v)\), \(V_m(v)\) satisfy
where
where the right hand sides are given by the state ahead, i.e. by the solution in the maximal development, and
Then, the following estimates hold:
Proof
We start with
We have
We start with an estimate for the difference of
where \(\beta ^*(t,w)\) is given by the state ahead, i.e. by the solution in the maximal development. We have
Here the partial derivatives of \(\beta \) possess the same arguments as on the right hand side of (959). Taking into account the second of (703) we obtain
In view of (696), (698), (702), (704) we arrive at
Similarly we find
(The factor \(v^2\) does not appear since the conditions (703) do not hold for the partial derivatives of \(\alpha \) with respect to w).
We split the first term on the right hand side of (958) into \(I_1+I_2\), where
For \(I_1\) we use
The second term is estimated by (963). Taking into account \(\left[ \alpha (v)\right] =\mathcal {O}(v)\) (see (726)) we get
For \(I_2\) we use
Together with (962), (963) we obtain
The second term in (958) we split into \(I_3+I_4\), where
Reasoning in a similar way as to arrive at (966), (968) we find
Using now (966), (968), (970), (971) in (958), we obtain
which implies
From (972) together with (962) we conclude
We turn to the jump condition
We have
where
Let us denote the numerator of (976) by N. We have
Now we rewrite
where
Now, for a smooth function F(x, y) we have
Suppose now \(g_2(v)-g_1(v)=\mathcal {O}(v)\). It follows
Using this we obtain
Now we recall (for the first see (726))
Therefore,
Similarly we find
Now we look at the difference \(\Delta _{m+1}B\). We have
where
We rewrite
where
Now, appealing to the first line of (982), we have
Therefore,
We rewrite
where
Again, by the first line of (982), we have
Defining
where \(\alpha ^*(t,w)\), \(\beta ^*(t,w)\) are given by the state ahead, we have
We have
Using
where we also use \((\partial \alpha ^*/\partial w)_0=\dot{\alpha }_0\), \((\partial ^2\alpha ^*/\partial w^2)_0=\ddot{\alpha }_0\) (see (281)), we obtain
where we also use \(\tfrac{1}{2}(\Delta _{m+1}(z^2))=-v(\Delta _{m+1}z)+\mathcal {O}(v^2|\Delta _{m+1}z|)\), since \(z(v)=-v+\mathcal {O}(v^2)\) and \(f(v)=\mathcal {O}(v^2)\). This together with (999) implies
From (995) and (1005) we obtain
Making use of the estimate (973) we arrive at
where we also used (962), (974). The analogous expression holds for \(\Delta _{m+1}A\) but with \(T^{tr}\) in the role of \(T^{tt}\). Putting things together we arrive at the following expression for the numerator of (976)
Let us inspect the curly bracket in (1008). For this we use (see the first and second of (185))
Using (180), (210), the first of (179) and the first of (184) we obtain
Therefore, we obtain the following expression for the curly bracket in (1008)
Using the definition of \(\mu \) (see (189)) we see that (1012) is equal to
Therefore,
We turn to the investigation of the denominator of (976). Let
Rewriting
where
and arguing as we did for the derivation of the asymptotic forms of \(\overset{\,\,1}{A}_m\) and \(\overset{\,\,2}{A}_m\), we arrive at
Similarly with \(m+1\) in the role of m. Therefore, we obtain
where we used \(y_m=-1+\mathcal {O}(v)\). Using the first of (1009) we find
Now, from \(c_+=(v+\eta )/(1+v\eta )\) together with (184), (189), (210) we have
Using \(\kappa =\dot{\alpha }_0(\partial c_+/\partial \alpha )_0\) we obtain from (1014), (1021)
\(\square \)
Closing the Argument
Proposition 5.2
For \(\varepsilon \) small enough the sequence \((y_m,\hat{\beta }_{+,m},\hat{V}_m)\) converges in \(B_Y\times B_{\delta _1}\times B_{\delta _2}\).
Proof
We first note
We now use the first estimate from lemma 5.3 together with the estimates for \(|d\hat{f}/dv|\) and \(\Vert \Delta _{m+1}\alpha _+ \Vert _{X}\) from lemma 5.1. We obtain
For an estimate of \(\Vert \Delta _{m+1}y \Vert _{X}\) we look at the estimate of lemma 5.2. Using also the first two estimates of lemma 5.1 we obtain
provided we choose \(\varepsilon \) suitably small. Now we note
We now use the second estimate from lemma 5.3 together with (1026). We obtain
We rewrite the estimates (1025), (1026), (1028) as
It follows that for \(\varepsilon \) small enough the sequence \((y_m,\hat{\beta }_{+,m},\hat{V}_m)\) converges in \(B_Y\times B_{\delta _1}\times B_{\delta _2}\).
The two propositions above show that the sequence \((y_m,\hat{\beta }_{+,m},\hat{V}_m)\) converges uniformly in \([0,\varepsilon ]\) to \((y,\hat{\beta }_+,\hat{V})\in B_Y\times B_{\delta _1}\times B_{\delta _2}\).
We see that \((F_m)\) as given by (867) converges to 0 as \(m\rightarrow \infty \) uniformly in \([0,\varepsilon ]\). Therefore, in view of (876), (877) also \(\partial t_m/\partial v\), \(\partial t_m/\partial u\) converge uniformly in \(T_\varepsilon \). Let us denote the limit of \((t_m)\) by t. This, in view of (827), (828), implies the convergence of \((\mu _m,\nu _m)\) to \((\mu ,\nu )\) uniformly in \(T_\varepsilon \) and, in view of (829), also the convergence of \((\Xi _m)\) to 0. Therefore, the pair of integral equations (831), (859) are satisfied in the limit. It then follows that t satisfies (359).
In view of (784), (785) we have convergence of \((\alpha _m,\beta _m)\) to \((\alpha ,\beta )\) uniformly in \(T_\varepsilon \), which implies convergence of \((c_{\pm ,m})\) to \(c_\pm \) uniformly in \(T_\varepsilon \). In view of the Hodograph system the partial derivatives of \((r_m)\) converge uniformly in \(T_\varepsilon \) and the limit satisfies this system.
Now, the uniform convergence of \((\alpha _m,\beta _m)\), \((r_m)\) and the partial derivatives of \((t_m)\) imply the uniform convergence of \((A_m)\), \((B_m)\) to A, B. Therefore, the partial derivatives \(\partial \alpha _m/\partial v\), \(\partial \beta _m/\partial u\) converge to \(\partial \alpha /\partial v\), \(\partial \beta /\partial u\), uniformly in \(T_\varepsilon \) and it holds \(\partial \alpha /\partial v=A\), \(\partial \beta /\partial u=B\), i.e. also the other two equations of the characteristic system are satisfied in the limit. In view of (816), (823) also the partial derivatives \((\partial \alpha _m/\partial u)\), \((\partial \beta _m/\partial v)\) converge to \(\partial \alpha /\partial u\), \(\partial \beta /\partial v\) uniformly in \(T_\varepsilon \).
From (759), (760) we see that \((\hat{f}_m)\), \((\hat{\delta }_m)\) converge to \(\hat{f}\), \(\hat{\delta }\) uniformly in \(C^1[0,\varepsilon ]\). Therefore, \(z=vy\) satisfies the identification equation when \(f\,{:=}\,v^2\hat{f}\), \(g\,{:=}\,v^3\hat{\delta }+c_{+0}v^2\hat{f}\) are substituted. Also V, \(\beta _+\) are given by the jump conditions when \(\alpha _+\), f, z are substituted. We have thus found a solution to the free boundary problem.
We have that z(v) is given by \(z(v)=vy(v)\), where \(y\in C^1[0,\varepsilon ]\), \(y(0)=-1\) (see (699)). f(v) is given by \(f(v)=v^2\hat{f}(v)\), with \(\hat{f}\in C^1[0,\varepsilon ]\), \(\hat{f}(0)=\lambda /6\kappa ^2\) (see (697)). \(\beta _+(v)\) is given by \(\beta _+(v)=\beta _0+v^2\hat{\beta }_+(v)\) with \(\hat{\beta }_+\in C^1[0,\varepsilon ]\), \(\hat{\beta }_+(0)=\left( \partial \beta /\partial t\right) _0\lambda /6\kappa ^2\) (see (730)). \(\alpha _+(v)=\alpha _i(v)+v^2\hat{\alpha }_+(v)\) with \(\hat{\alpha }_+\in C^1[0,\varepsilon ]\), \(\hat{\alpha }_+(0)=\lambda \tilde{A}_0/6\kappa ^2\) (see (725)). From (639) together with \(\delta (v)=v^3\hat{\delta }(v)\), we have \(g(v)=v^2\hat{g}(v)\) with \(\hat{g}\in C^1[0,\varepsilon ]\), \(\hat{g}(0)=\lambda c_{+0}/6\kappa ^2\).
We recall (252) which is the singular boundary of the maximal development in acoustical coordinates (t, w):
Therefore, using \(w=z(v)=-v+\mathcal {O}(v^2)\), we have
Comparing this with (see (697))
we see that for \(\varepsilon \) sufficiently small the shock curve \(\mathcal {K}\) lies in the past of the singular boundary of the maximal development \(\mathcal {B}\).
From (771) we have
From (770) we have that the characteristic speed of the outgoing characteristics along \(\mathcal {K}\) in the state behind is
Now, let us denote by \(\underline{c}_+\) the characteristic speed of the outgoing characteristics along \(\mathcal {K}\) in the state ahead. We have
Now,
Therfore, using (286) and
we find
which implies
From (1033), (1034) and (1039) we obtain
These imply the determinism condition, provided that \(\varepsilon \) is sufficiently small.
We have therefore proven the following existence theorem.
Theorem 5.1
Let initial data for t and \(\alpha \) be given along \(\underline{C}\). Let \(r_0{>}0\). Let the solution in the state ahead be given by \(\alpha ^*(t,w)\), \(\beta ^*(t,w)\), \(r^*(t,w)\). Then, for \(\varepsilon \) small enough, there exists a continuously differentiable solution \((t,r,\alpha ,\beta )\) of the characteristic system in \(T_\varepsilon \) such that
-
(i)
along \(\underline{C}\) it attains the initial data and \(r(0,0)=r_0\).
-
(ii)
\(\alpha _+(v)\,{:=}\,\alpha (v,v)\), \(\beta _+(v)\,{:=}\,\beta (v,v)\), \(\alpha _-(v)\,{:=}\,\alpha ^*(f(v),z(v))\), \(\beta _-(v)\,{:=}\,\beta ^*(f(v),z(v))\) satisfy the jump conditions
$$\begin{aligned} -\left[ T^{tt}(v)\right] V(v)+\left[ T^{tr}(v)\right]&=0, \end{aligned}$$(1042)$$\begin{aligned} -\left[ T^{rt}(v)\right] V(v)+\left[ T^{rr}(v)\right]&=0, \end{aligned}$$(1043)where
$$\begin{aligned} T^{\mu \nu }_+(v)=T^{\mu \nu }(\alpha _+(v),\beta _+(v)),\qquad T^{\mu \nu }_-(v)=T^{\mu \nu }(\alpha _-(v),\beta _-(v)), \end{aligned}$$(1044)V(v) satisfies
$$\begin{aligned} \frac{df}{dv}(v)V(v)=\frac{dg}{dv}(v) \end{aligned}$$(1045)and z(v) satisfies the identification equation
$$\begin{aligned} g(v)+r_0=r^*(f(v),z(v)), \end{aligned}$$(1046)where
$$\begin{aligned} f(v)\,{:=}\,t(v,v),\qquad g(v)\,{:=}\,r(v,v)-r_0. \end{aligned}$$(1047)Furthermore, we have \(\hat{V}\in C^0[0,\varepsilon ]\), where \(\hat{V}(v)\) is given by
$$\begin{aligned} V(v)=c_{+0}+\frac{\kappa }{2}(1+y(v))v+v^2\hat{V}(v). \end{aligned}$$(1048) -
(iii)
We have
$$\begin{aligned} z(v)&=vy(v),&y&\in C^1[0,\varepsilon ],&y(0)&=-1, \end{aligned}$$(1049)$$\begin{aligned} f(v)&=v^2\hat{f}(v),&\hat{f}&\in C^1[0,\varepsilon ],&\hat{f}(0)&=\frac{\lambda }{6\kappa ^2}, \end{aligned}$$(1050)$$\begin{aligned} g(v)&=v^2\hat{g}(v),&\hat{g}&\in C^1[0,\varepsilon ],&\hat{g}(0)&=\frac{\lambda c_{+0}}{6\kappa ^2}, \end{aligned}$$(1051)$$\begin{aligned} \alpha _+(v)-\alpha _i(v)&=v^2\hat{\alpha }_+(v),&\hat{\alpha }_+&\in C^1[0,\varepsilon ],&\hat{\alpha }_+(0)&=\frac{\lambda \tilde{A}_0}{6\kappa ^2}, \end{aligned}$$(1052)$$\begin{aligned} \beta _+(v)-\beta _0&=v^2\hat{\beta }(v),&\hat{\beta }_+&\in C^1[0,\varepsilon ],&\hat{\beta }_+(0)&=\frac{\lambda }{6\kappa ^2}\left( \frac{\partial \beta }{\partial t}\right) _0. \end{aligned}$$(1053) -
(iv)
The curve \(\mathcal {K}\) given in rectangular coordinates by
$$\begin{aligned} v\mapsto (f(v),g(v)+r_0) \end{aligned}$$(1054)lies in the past of the singular boundary of the maximal development \(\mathcal {B}\) and satisfies the determinism condition, i.e. it is supersonic relative to the state ahead and subsonic relative to the state behind.
Uniqueness
Asymptotic Form
Proposition 6.1
Let \((t,r,\alpha ,\beta )\) be a continuously differentiable solution of the free boundary problem and let z(v) be the corresponding solution of the identification equation. Let z(v) and
be given by
with
Then it follows that
and
Proof
We first note that the characteristic system together with the solution being continuously differentiable implies
We recall the initial data for t:
Taking into account
we deduce
Now,
Substituting
together with (1066) yields, after dividing by v and taking the limit \(v\rightarrow 0\),
Since by (1060)
and the second of (161), namely
gives, by the first of (1067),
it follows
Now, \(\partial t/\partial v\) and \(\partial \beta /\partial v\) satisfy along \(\underline{C}\) a system of the form
which, together with (1067), (1075) implies
Hence
the indices on the Landau symbols representing the variable with respect to which the limit is taken, the limit being uniform in the other variable.
Making use of the equations for \(\alpha \) and \(\beta \) from the characteristic system (see (161)) we obtain
Therefore,
where for the first we used \((\partial c_+/\partial \alpha )_0=\kappa /\dot{\alpha }_0\) (see (286)). Hence
We turn to the integral in (1068). Making again use of the characteristic system we obtain
Hence
which, in view of (1064), implies
We have
where we made use of (1070). Together with (see (1085))
we find
We have
where for the second term in the second line we substituted (1087), setting \(v=v'\). Together with (see (1081))
we find
Now, rewriting the integrand in (1068) using (1064) and then substituting (1089) and (1092), we obtain
Substituting now (1066), (1069), (1070), (1083), (1093) into (1068), noting that
and that by (1071) the terms linear in v cancel, dividing by \(v^2\) and taking the limit as \(v\rightarrow 0\), we find
We now consider the identification equation
In view of (1071), (1096) the left hand side is
For the right hand side of (1097) we expand \(r^*(t,w)\) up to third order. Substituting \(t=f(v)=v^2\hat{f}(v)\), \(w=z(v)=vy(v)\) we obtain
Therefore, setting (1098) equal to (1099), dividing by \(v^3\) and taking the limit \(v\rightarrow 0\) we obtain
Defining now
(we recall that for a physical solution we need \(p{>}0\)), this becomes
We now turn to the jump conditions. Using (cf. (1094))
and the analogous expansion for dg / dv(v) in
we obtain
Hence, in view of (1096) and the first of (1101),
We recall the jump conditions:
where
and
where
the functions \(\alpha ^*(t,w)\), \(\beta ^*(t,w)\) in (1112) being given by the solution in the maximal development. Using the initial condition for \(\alpha \) as given by (312), we obtain from (1059), (1060)
It would actually suffice to assume \(\hat{\alpha },\hat{\beta }\in C^0\). But since \(\alpha _+\) and \(\beta _+\) correspond to the solution of the fixed boundary problem, the assumption for \(\hat{\alpha }\) and \(\hat{\beta }\) to be continuously differentiable is consistent.
Expanding now \(T^{\mu \nu }(\alpha ,\beta )\) up to second order and substituting (1113), (1114) we obtain
Expanding \(\alpha ^*(t,w)\), \(\beta ^*(t,w)\) up to second order and substituting \(t=v^2\hat{f}(v)\), \(w=vy(v)\) we obtain
where for the second one we made use of (266). Expanding now \(T^{\mu \nu }(\alpha ,\beta )\) up to second order and substituting (1116), (1117) we obtain
Using the definitions
this becomes
Therefore, from (1115), (1120),
We now substitute (1121) together with (1106) into (1107), (1108). Making use of (see (187))
dividing by \(v^2\) and taking the limit as \(v\rightarrow 0\) we arrive at
where we used the definition
Let us define
Using these definitions together with \(p=-y(0)\) in (1123), (1124) we obtain
We now solve the system of equations (1102), (1129), (1130). Solving (1102) for m yields
Substituting this in (1129), (1130) gives
Multiplying (1132) by \(c_{02}\) and (1133) by \(c_{01}\) and then subtracting the resulting equations from each other yields
where we used the definitions
If \(d_1\), \(d_2\) have opposite sign (recall that \(\kappa {>}0\)) then \(p=1\) is the only root of (1134) (recall the requirement \(p{>}0\) for a physical solution). We look at \(d_1/d_2\). From (1122) we deduce
Therefore
where we made use of \((\partial c_+/\partial \alpha )_0=\kappa /\dot{\alpha }_0\) (see (286)). Hence,
and we conclude that \(p=1\) is the only root of (1134). From (1131) we deduce \(m=0\). (1129) then yields \(l_+=0\). Therefore, from (1071), (1101) and the second of (1119) together with (1125) we obtain
From (1096) we obtain
We now turn to \(\hat{\alpha }_+(0)\). We recall
Therefore, (this is (715))
Using (1090), (1093) we obtain
which implies (using the second of (1139))
Therefore,
This concludes the proof of the proposition. \(\square \)
Uniqueness
Theorem 6.1
Let \((t',r',\alpha ',\beta ')\), \((t'',r'',\alpha '',\beta '')\) be two continuously differentiable solutions to the free boundary problem and let \(z'(v)\), \(z''(v)\) be the corresponding solutions of the identification equation. Let \(z'(v)\), \(z''(v)\) and
be given by
with
Let \(\hat{V}'(v)\), \(\hat{V}''(v)\) be given by
with
Then it follows that for \(\varepsilon \) sufficiently small, the two solutions coincide.
Proof
In the following, whenever there is no prime or no double prime on a function it is meant that the statement holds for both the primed as well as for the double primed function. Let us make the following definition for any function f
We see that the assumptions from proposition 6.1 are satisfied. Therefore, we are able to make use of the statement and the proof of proposition 6.1. From proposition 6.1 we have
Using the asymptotic forms (1158), (1159) in
we obtain, in the same way as we arrived at (705), (706), (711), (712),
Substituting (1158) in (1164) we obtain
The second of (1161) together with the second of (1166) imply
Looking at the proof of lemma 5.3 we see that the above asymptotic forms constitute all the necessary requirements for this proof to hold. Therefore, we have the following estimates
where the X norm is defined by (756).
We now define
Furthermore, from \(\hat{f}, \hat{g}\in C^1[0,\varepsilon ]\) together with (1063) we have
Therefore, we can write
with
We now show that \(d\hat{\delta }/dv\) extends continuously to 0. From the definition (1170) we obtain
We have
Therefore,
and we obtain
where
I.e.
It follows that
where we used the first of (1159). Taking into account \(\delta (0)=0\), this implies
Hence,
Thus \(\hat{\delta }\) extends to a \(C^1\) function on \([0,\varepsilon ]\), i.e. we have
Looking now at the proof of lemma 5.2 we see that the above asymptotic forms constitute all the necessary requirements for this proof to hold. Therefore, we have
Now we look at the partial derivatives of t. We recall (1064):
Integrating with respect to v from \(v=0\) yields
where we recall
and
We also recall from the proof of proposition 6.1 the expressions (1087), (1090):
From the second of (1062) we obtain
From (1091) we have
From (1088) we have
Using this together with (1192) in (1188) and recalling (1065) and the second of (1062), we obtain
We now improve the expressions for \(\mu \) and \(\nu \). From
we have
From
we have
Therefore,
which implies
In view of (1079) and the second of (1189) we have (recall that \(\left( \partial c_+/\partial \alpha \right) _0\dot{\alpha }_0=\kappa \))
From (1193) and (1199) we have
From (1197) we have
Therefore
which implies
In view of (1084) we have
Now we look at \(1/\gamma (v)\) given by (see (321))
where
Using the above asymptotic forms for \(\alpha _+\), \(\beta _+\) and V we obtain, in exactly the same way as we did in the part on the fixed boundary problem,
with
The above asymptotic forms now allow us to follow the proof of lemma 5.1. We thus obtain
We now combine (1168), (1169), (1186), (1213), (1214), (1215) in the same way as we did in the proof of proposition 5.2 and find, for \(\varepsilon \) sufficiently small,
These imply, for \(\varepsilon \) sufficiently small,
Substituting (1217) in (1218) and (1219) yields
for \(\varepsilon \) sufficiently small. Substituting (1220) in (1221) gives, for \(\varepsilon \) sufficiently small,
which gives
which gives
In view of (1213), (1214) and (1215), the vanishing of these differences implies that also the differences of f, \(\delta \) and \(\alpha _+\) vanish. Now we make use of estimates appearing in the proof of lemma 5.1. In all these estimates there appear no indices in the present context. F(u) given by (867) vanishes. Therefore, in view of (876), (877) the differences of the partial derivatives of t vanish. In view of (784), (785) the differences of \(\alpha \) and \(\beta \) vanish, therefore also the differences of \(\mu \) and \(\nu \) vanish. In view of (779) also the difference of r vanishes. In view of (816), (823) the differences of \(\partial \alpha /\partial u\) and \(\partial \beta /\partial v\) vanish. In view of the characteristic system the differences of \(\partial \alpha /\partial v\) and \(\partial \beta /\partial u\) vanish. In view of the Hodograph system also the differences of the partial derivatives of r vanish. Therefore, the two solutions (prime and double prime) coincide. This completes the uniqueness proof. \(\square \)
Continuity of \(L_+\alpha \) and \(L_+\beta \) Across the Incoming Characteristic Originating at the Cusp Point
In the present subsection we carry the argument of the above proof further. In particular we will first improve the estimates (1193), (1204) and (1207). Then on the basis of these improved estimates we will show the continuity of \(L_+\alpha \) and \(L_+\beta \) across \(\underline{C}\).
Proposition 6.2
\(L_+\alpha \) and \(L_+\beta \) are continuous across \(\underline{C}\).
Proof
Let us consider \(\partial t/\partial v\), \(\partial \beta /\partial v\) along \(\mathcal {K}\). We have, along \(\mathcal {K}\),
and, by proposition 6.1,
Evaluating (1196) at \(u=v\) we obtain
From (1226) and (1228) we obtain, through the first of (1225),
From (1227) and (1230) we obtain, through the second of (1225),
Let us then consider the system (329), (330) along any incoming characteristic. It is a system of the form (331):
This is a linear homogeneous system with a coefficient matrix
which is continuous on \(T_\varepsilon \). The initial data are on \(\mathcal {K}\) and given by (1229), (1231). Let the matrix m be the solution of
Then the solution of (1232) is
Since
it follows that
which improve (1193) and (1207) respectively. Also, (1239) implies, through (1199),
which improves (1204).
Since \(\alpha \) and t are by construction continuous across \(\underline{C}\) while \(\beta \) and r satisfy along \(\underline{C}\) the o.d.e. system
while
at the cusp point, it follows that r and \(\beta \) are continuous across \(\underline{C}\) as well. Then,
is also continuous across \(\underline{C}\).
Let us consider
We see that the matrix a is continuous across \(\underline{C}\). This implies that the matrix m is continuous across \(\underline{C}\). By (1235),
Hence \(L_+\beta \) is continuous across \(\underline{C}\) as well. \(\square \)
Higher Regularity
In the following we denote by \(\bar{P}_{m,n}\) a polynomial in v of degree m starting with an n’th order term. We denote by \(P_{m,n}(v)\) a sum of \(\bar{P}_{m,n}\) and a function of \(\mathcal {O}(v^{m+1})\). We then define \(P_m(v)\,{:=}\,P_{m,0}(v)\). We also denote by \(Q_{m,n}(u,v)\) a sum of a polynomial in u and v of degree m starting with an n’th order term and a function of \(\mathcal {O}(u^{m+1})\) (we recall that in the domain in question, i.e. in \(T_\varepsilon \), we have \(0\le v\le u\le \varepsilon \)). We then define \(Q_m\,{:=}\,Q_{m,0}\). We extend the meaning of \(P_{m,n}\) and \(Q_{m,n}\) to the case \(n=m+1\) by
Furthermore we will use the definitions
and
In the following we prove that the solution established in the existence theorem is smooth. We do this by induction, showing that all derivatives of y, \(\hat{f}\) are bounded and all derivatives of t, r, \(\alpha \), \(\beta \) are in \(C^1\).
Inductive Hypothesis
We make the following inductive hypotheses: We assume that we have bounds for \(Y_m\), \(F_m\) for \(m=1,\ldots , n-1\), i.e.
For the function \(\alpha _+(v)=\alpha (v,v)\) we assume
For the function t(u, v) we assume, for \(n\ge 3\),
With the indices p and m we indicate that we refer to pure and mixed derivatives.
In the case \(n=2\), (\(t_{m,n-1}\)) is not present and (\(t_{p,n-1}\)) is
For the functions \(\alpha \) and \(\beta \) we assume
and
For \(n=2\) the properties (\(\alpha _{m,n-1}\)), (\(\beta _{m,n-1}\)) are not present and (\(\alpha _{p,n-1}\)), (\(\beta _{p,n-1}\)) are
Base Case \(n=2\)
We show that the inductive hypothesis holds for \(n=2\). We are going to use estimates established during the existence proof. The index on functions in estimates in the existence proof was in order to label the iterates. These estimates also hold in the limit hence without any indices. Also the dependencies on \(\delta _2\) of the bounds are not present anymore since \(\delta _2\) has been chosen appropriately.
Since \(Y_1=Y\), where Y was defined in (315), we have from (755),
From (638) we have
while from (724) we have
Using now the first of (716) in
together with the fact that \(t(u,0)=u^3\hat{h}(u)\), where \(\hat{h}\) is a smooth function, we obtain (\(t_{p,1}\)). From (1198), (1202), (1204), (1207) we have
Therefore, \((\alpha _{0,1})\) and \((\beta _{0,1})\) hold.
We have
From
together with the first of (716) we obtain
Using this together with the fact that \(\alpha _i\) is a smooth function in (1259), we deduce that \((\alpha _{p,1})\) holds.
We have
From
together with the second of (716) we obtain
From the second of (1162) we have
Using this together with (1264) in (1262) we deduce that \((\beta _{p,1})\) holds. We conclude that the inductive hypothesis holds in the case \(n=2\).
Inductive Step
We now show the inductive step, i.e. we show that (\(Y_{n-1}\)), ..., (\(\beta _{0,n-1}\)) hold with n in the role of \(n-1\). Once this is proved, we have proven the following regularity theorem.
Theorem 7.1
The solution whose existence is the content of theorem 5.1 and whose uniqueness is the content of theorem 6.1 is actually smooth.
We remark that to see that also the function r(u, v), which is not present in the inductive hypothesis, is a smooth function, we appeal to the Hodograph system, i.e. to (162).
We write
where we recall the notation \(\left( \cdot \right) _0\) for evaluation at the cusp point, i.e. for functions of v evaluation at \(v=0\). Since \(z=vy\), we have
Setting \(i=n-1\) and using (1266), we deduce from the inductive hypothesis (\(Y_{n-1}\))
This implies, through integration,
Recalling \(f(v)=v^2\hat{f}(v)\) and making use of the assumption (\(F_{n-1}\)) we obtain, analogous to the way we arrived at (1269),
Making use of assumption (\(\alpha _{+,n-1}\)) we obtain, through integration,
We now look at the behavior of \(\alpha _-(v)\) and \(\beta _-(v)\). Recalling
where \(\alpha ^*(t,w)\), \(\beta ^*(t,w)\) denotes the solution in the state ahead and \(t(v,v)=f(v)\), \(w=z(v)\) are substituted, we have
Now, by (1269) and (1270) we have
and for higher order derivatives we have
Analogous, but now taking into account the fact that \(\left( \partial \beta ^*/\partial w\right) _0=0\), we have
and for higher order derivatives we have
Taking \(m-1\) derivatives of the first of (1273) and making use of (1269), (1270), (1274) and (1275) we obtain
Taking \(m-1\) derivatives of the second of (1273) and making use of (1269), (1270), (1276) and (1277) we obtain
Now we look at the behavior of \(\beta _+(v)\). For this we recall \(\left[ \beta \right] =\left[ \alpha \right] ^3G(\alpha _+,\alpha _-,\beta _-)\). We have
From (1271), (1278) and taking into account \(\alpha _-(0)=\alpha _+(0)\), we obtain
and
and similarly
In view of (1271), (1278), (1279) we have
Using (1281), (1283) and (1284) in (1280) we obtain
Therefore, from (1279) we obtain
We note that (1271), (1278), (1279) and (1286) constitute the behaviors of \(\alpha _\pm (v)\), \(\beta _\pm (v)\).
Estimate for \(d^{n-1}V/dv^{n-1}\)
We turn to estimating \(d^{n-1}V/dv^{n-1}\). We recall
which implies
We use the notation
and we recall (see (187), (188))
(1288) becomes
Defining now a(v) and b(v) by
i.e.
we can rewrite (1291) as
Now we define \(\tilde{a}\) by
With
(1296) becomes
The \((i-1)\)’th order derivative of this can be written as
Differentiating this we obtain
which gives us the following recursion formulas
Solving these gives
In the expression for the n’th derivative for y there will be involved the n’th derivative of f which in turn involves the \((n-1)\)’th derivative of V. Therefore, we have to estimate
To estimate \(\tilde{a}_{n-2}\) we have to estimate the \(n-2\) order derivative of \(\tilde{a}\). To estimate \(\tilde{b}_{n-2}\) we need to estimate the \(n-2\) order derivative of \(\tilde{b}\) and the \(n-3\) order derivative of \(\tilde{a}\). We consider first \(\tilde{b}_{n-2}\). We derive expressions for a, b to \(\mathcal {O}(v^2)\). We start with b. Since \(\left[ T^{tt}(0)\right] =0\), we have to estimate the \(n-2\) order derivative of the numerator of b to \(\mathcal {O}(v^3)\). Let us denote by \(T_i\) the i’th term in the curly bracket of (1295) and let
Then
We have
Let us look at
Using (1267) with \(n-1\) in the role of i and in the resulting expression (1266) with n and \(n-1\) in the role of k we obtain
Using now (1310) with \(n-1\) in the role of n and using (1266) with n in the role of k for the resulting integrands yields
We rewrite (1310) and (1311) as
where (1314) follows directly from (1269) and (1315) follows from \(z=vy\), \(y(0)=-1\).
Since
we have, in view of (1278), (1279),
Taking the derivative of (1316) we obtain
Taking \(n-3\) derivatives of this and making use of (1269), (1270), (1278), (1279), (1315) and
we obtain
Substituting (1313) we obtain
where (1322) is (1317) with \(m=n-k\), (1323) follows from (1318), (1319) and (1324) follows from \(\mathring{c}_+(0)=c_{+0}\). We note that (1312), ..., (1315) and (1321), ..., (1324) express the derivatives of z and \(\mathring{c}_+-c_{+0}\) to \(\mathcal {O}(v^3)\) for \(n\ge 5\). In the case \(n=4\) the derivatives of z and \(\mathring{c}_+-c_{+0}\) to \(\mathcal {O}(v^3)\) are given by the same expressions with the exceptions of (1314) and (1322) which have to be excluded. In the case \(n=3\) also (1315) and (1323) have to be excluded, i.e. in the case \(n=3\) the derivatives of z and \(\mathring{c}_+-c_{+0}\) to \(\mathcal {O}(v^3)\) are given by (1312), (1313), (1321), (1324). Finally, in the case \(n=2\) the derivative of z to \(\mathcal {O}(v^3)\) is given by (1312) and \(\mathring{c}_+-c_{+0}\) to \(\mathcal {O}(v^3)\) is given by (1321). The fact that the latter is true is seen from the fact that (1313) is also true for \(n=2\) and then using this in (1320) which in the case \(n=2\) is the Taylor expansion of \(\mathring{c}_+-c_{+0}\) to \(\mathcal {O}(v^3)\).
Integrating by parts we have
By the inductive hypothesis this is \(\mathcal {O}(v^m)\). In the case \(m=0\) we have
which is \(\mathcal {O}(1)\) by the inductive hypothesis. (1325) and (1326) imply
Now, in the case \(n\ge 5\) we have
All terms in the sum are products of terms of the form (1314) and (1322) therefore they are all a \(P_2\). For the second line in (1328) we make use of (1312), (1313), (1315) and (1321), (1323), (1324) together with (1327). Therefore, we obtain in the case \(n\ge 5\),
where we defined
In the case \(n=4\) the sum in (1328) is not present and (1314), (1322) are not needed. Therefore, (1329) is valid in the case \(n=4\) as well.
Let us look at the case \(n=3\). From (1313) and \(\left( dz/dv\right) _0=-1\) we have
Using this together with (1319) in (1318) yields
Using now (1312) in the case \(n=3\) together with (1324), (1331), (1332) we obtain
which is (1329) with 3 in the role of n. Therefore, (1329) is valid in the case \(n=3\) as well.
In the case \(n=2\) we have from (1312) together with \(\left( dz/dv\right) _0=-1\)
while from (1321) in the case \(n=2\) together with \(\left( \mathring{c}_+\right) _0=c_{+0}\), \(\left( d(\mathring{c}_+-c_{+0})/dv\right) _0=-\kappa \),
From (1334) and (1335) we obtain
which agrees with (1329) in the case \(n=2\). We therefore that (1329) is valid for \(n\ge 2\).
Integrating (1329) we obtain
We note that these are valid in the case \(n\ge 4\). In the case \(n=3\) we have to make use of (1333) instead of (1337) ((1338) stays valid), while in the case \(n=2\) we have to make use of (1336) alone.
Now we look at
From (1270) we have
We note that (1340), ..., (1343) express derivatives of f for \(n\ge 5\). In the case \(n=4\) the derivatives of f are expressed by the same expressions with the exceptions of (1342) which has to be excluded. In the case \(n=3\) the second and first derivative of f are given by (1340) and (1343) respectively. Finally in the the case \(n=2\) the first derivative of f is given by (1343).
We start with the case \(n\ge 5\). We have
All terms in the sum are products of terms of the form (1322) and (1342) therefore they are all a \(P_2\). For the second line in (1344) we make use of (1321), (1323), (1324) and (1340), (1341), (1343) together with (1327). We see that in the case \(n\ge 5\) all terms in (1344) are a \(P_2\) and therefore so is (1339). In the case \(n=4\) the sum in (1344) is not present, hence (1322) and (1342) are not needed. Therefore, also in the case \(n=4\) (1339) is a \(P_2\). In the case \(n=3\) we use (1332) together with (1324), (1340), (1343). We obtain that also in the case \(n=3\) (1339) is a \(P_2\). Finally in the case \(n=2\) we use (1335) and (1343). We conclude that, for \(n\ge 2\),
By integration we obtain from (1345)
Here (1346) is only valid for \(n\ge 4\) and (1347) is only valid for \(n\ge 3\).
Let us define
where for the second we used \(\left( \partial \alpha ^*/\partial w\right) _0=\dot{\alpha }_0\).
Now, in the case \(n\ge 4\) we have
All terms in the sum are products of terms of the form (1337) and (1351) therefore they are all a \(P_2\). For the first term in the second line of (1353) we use (1329) together with (1352) and for the second term in the second line of (1353) we use (1338). We find
In the case \(n=3\) we have
Making use of (1333), (1338), (1351), (1352) we see that (1354) is also valid in the case \(n=3\). In the case \(n=2\) we make use of (1336), (1352) and see that (1354) is valid in the case \(n=2\) as well but we have in particular
Now, in the case \(n\ge 4\) we have
All terms in the sum are products of terms of the form (1346) and (1351) therefore they are all at least a \(P_2\). For the first term in the last line of (1357) we use (1347) together with (1351). For the second term in the last line in (1357) we use (1348) together with (1351). We find in the case \(n\ge 4\),
In the case \(n=3\) the sum in (1357) is not present and the second line in (1357) is dealt with in the same way as in the case \(n\ge 4\). We see that (1358) is also valid in the case \(n=3\). In the case \(n=2\) we use again (1348) together with (1351). We see that (1358) is also valid in the case \(n=2\) and we have in particular
From (1354), (1358), in view of (1308) and the definitions (1349), (1350), we deduce, for \(n\ge 2\),
In the case \(n=2\) we have in particular, from (1356) (1359),
We turn to
where (see (1295), (1306), (1307))
where we defined
Since \(\bar{c}_+(v)=c_+(\alpha _+(v),\beta _+(v))\), in view of (1271), (1286) we have
From this together with (1271) we obtain
We now look at the case \(n=2\) in more detail. In the case \(n=2\) we use
together with
to deduce (in agreement with the first of (1365))
From (1271) we have
Therefore, in the case \(n=2\), we obtain
We turn to \(T_3+T_4\) (see (1295), (1306), (1307)).
Defining
(1372) becomes
From (1271), (1278), (1279), (1286) we have
Now, in the case \(n\ge 4\),
All terms in the sum are products of terms of the form of the second case of (1279) and the second of (1377). For the second line in (1378) we use the first case of (1279) and the first of (1377). We find that in the case \(n\ge 4\),
In the case \(n=3\) the sum in (1378) is not present. For the remaining terms we argue as in the case \(n\ge 4\) and find again (1379). In the case \(n=2\) only the product of the first case of (1279) and the first of (1377) is present and we find
in agreement with (1379). In view of (1271), (1286), we have
Using this together with (1285) we obtain, for \(n\ge 3\),
In the case \(n=2\) we have, using the second case of (1285) together with (1381),
From (1379), (1380), (1382), (1383) we obtain that for \(n\ge 2\),
and, in particular in the case \(n=2\),
In view of (1306), we deduce from (1360), (1366) and (1384)
In the case \(n=2\) we have in particular, from (1361), (1371) and (1385),
Now we look at
We first restrict ourselves to the case \(n\ge 3\). Defining
we have
In particular, from \(\left( \partial \alpha /\partial w\right) _0=\dot{\alpha }_0\) we have
Now, in the case \(n\ge 4\) we have
In view of (1269) and the first of (1394) we see that all terms in the sum are a \(P_2\). For the remaining term we use (1313) together with (1395). Therefore, in the case \(n\ge 4\),
In the case \(n=3\) we only have the last term in (1396). Arguing as in the case \(n\ge 4\) we see that (1397) is also valid in the case \(n=3\). From (1270) together with the second of (1394) we obtain, in the case \(n\ge 3\),
In view of (1271), (1286) we have
Therefore, in the case \(n\ge 3\),
From (1397), (1398), (1400) and (1401) we deduce, in the case \(n\ge 3\),
where
Now we look at the case \(n=2\). From (1389) and the first of (1399) we have
Therefore, together with (1370) and using also (1286) and the second of (1399), we obtain
From (1270) and the second of (1394) we have
For the derivative of z in the case \(n=2\) we use (1312). Since \(y(0)=-1\), (1312) in the case \(n=2\) is
Together with (1395) we obtain
Using (1405), (1406) and (1408) in (1393) we obtain
Since
we obtain from (1409),
where we made use of \(\left[ T^{tt}(0)\right] =0\). From this we see that (1402) is also valid for \(n\ge 2\). In addition we see that \(\left[ T^{tt}\right] \) is divisible by v.
We now integrate (1386) \(n-2\) times to obtain N. If we integrate \(n-2\) times a \(P_2\) we obtain a \(P_n\). In view of the definition (1330) we must calculate the l-fold iterated integral of
Let
for an integrable function \(f_0\) (see (1431) below). We define
We integrate l times the function \(v^kg_0(v)\) and denote the result by \(G_{k,l}\). We claim
We prove the claim in (1417) by induction. We start with \(l=1\). Integrating by parts and making use of (1415), (1416), we obtain
which is (1417) in the case \(l=1\). Let then (1417) hold for \(l=1, \ldots , l\). We have
Using the above to integrate (1417) we obtain
Since the first line agrees already with the right hand side with \(l+1\) in the role of l, except for the last term in the sum, the thing left to show is
We have
where
where the tilde denotes omission. \(P_l\) is a polynomial in k of degree l. The coefficient of \(k^l\) is
Since if \(n\in \{1,\ldots ,l\}\)
we have
Therefore,
i.e. the roots of \(P_l(k)\) are \(k=-1,\ldots ,-l\), which implies, together with the fact that the coefficient of \(k^l\) in \(P_l(k)\) is given by (1425),
Using this in (1423) yields (1422). This completes the proof of (1417). We remark that since (1417) was proved by induction on l, a positive integer, it also holds when k is a negative integer, as long as \(k+l<0\) and for \(k<0\) the first factor in the first term of (1417) is interpreted as
We now set
such that (see (1416))
We also set \(k=2-i\). Then \(G_{k,l}\) given by (1417) is the l-fold iterated integral of \(v^kg_0=v^{2-i}I_{n,i}\) (cf. (1413)) and
Therefore, the l-fold iterated integral of \(v^{2-i}I_{n,i}\) is
Using this in \(J_n\), given by (1330), we obtain, after a straightforward computation, that the l-fold iterated integral of \(J_n\) is
where we recall the notation \(\hat{I}_{n,m}=(1/v^m)I_{n,m}\). From (1386) this gives \(d^{n-l-2}N/dv^{n-l-2}\). Setting then \(j=l+2\) we obtain
In particular, setting \(j=n\) yields (see (1387))
Making use of (1434), we obtain, after a straightforward computation, that the l-fold iterated integral of \(K_n\) (see the definition (1403)) is
Together with (1402) this gives \(d^{n-l-2}\left[ T^{tt}\right] /dv^{n-l-2}\). Setting then \(j=l+2\) we obtain
In particular, setting \(j=n\) yields (see (1412))
Since \(\left[ T^{tt}\right] \) and N are divisible by v, we define
Furthermore, we note that, in view of (1387), (1412) and using (see (1327))
we have
We then have (see (1307))
Let now f be a \(C^k\) function of v such that \(f(0)=0\). We define
We have
Since
(1446) is
where \(A_k\) is the operator
which is homogeneous of degree zero relative to scaling.
In view of
we have
where
We see that \(a_{k,m}=0\) for \(m\le k\), unless \(m=0\). Therefore,
i.e. the null space of \(A_k\) is the space of polynomials of degree k with no constant term, a k-dimensional space. In view of the above, applying \(d^{n-2}/dv^{n-2}\) to \(\hat{N}\), \(\widehat{\left[ T^{tt}\right] }\), we obtain
Setting \(k=n-2\) and \(j=n-i\) in (1449) we obtain
Since the null space of \(A_{n-2}\) is the space of polynomials of degree \(n-2\) with no constant term, only the terms corresponding to the powers \(v^{n-1}\), \(v^n\) in \(P_{n,2}\) survive when we apply \(A_{n-2}\) to N. Therefore, substituting (1436) into (1456), we obtain
We rewrite the double sum as
where
Let us consider
For \(m\ge 2\) we have \(\tilde{a}_{n,m}=a_{n,m}\), but for \(m=0,1\) we have
We set \(k=j-m\) in (1460) to obtain
This vanishes except in the case \(n=m\) and we have \(\tilde{a}_{n,n}=(-1)^n\). I.e.
We conclude
Substituting (1465) in (1458) we conclude that the double sum in (1457) is
Therefore, from (1454),
For \(n=2\) we have from (1437),
Using the inductive hypothesis (\(Y_{n-1}\)) we have
Integrating this yields
where for the second we used (1437). We note that the first of (1470) is only valid for \(n\ge 4\).
Setting \(k=n-2\) and \(j=n-i\) in (1449) we obtain
Substituting (1439) yields
We rewrite the double sum as
where the coefficients \(a_{n,m}\) are defined by (1459) and given by (1465). We find that the double sum in (1472) is
Therefore, from (1455)
For \(n=2\) we have, in view of (1412),
Using the inductive hypothesis (\(Y_{n-1}\)) we have
Integrating this yields
where for the second we used the second of (1440). We note that the first of (1478) is only valid for \(n\ge 4\).
We now go back to (1307) and calculate \(d^{n-2}b/dv^{n-2}\) using (1467) and (1475). In the case \(n\ge 4\) we have
Each of the terms in the sum involves products of terms of the form given by the first of (1470) and the first of (1478), therefore they are all a \(P_1\). Making use of
where for the second equality we used (1475) and the second of (1478), together with the second of (1470), we see that the first term in the second line of (1479) is a \(P_1\). For the second term in the second line of (1479) we use (1467) and the second of (1478). In the case \(n=3\) the sum in (1479) is not present. With the second line in this equation we deal in the same way as in the case \(n\ge 4\). In the case \(n=2\) we have
and we make use of (1468), (1476). We conclude
Integrating this we obtain
where for the second we used (1443). The first is valid in the case \(n\ge 3\).
We recall \(\tilde{b}=vb\). We have
Since
we obtain from (1482),
Substituting in (1484) and using again (1482) yields \(d^{n-2}\tilde{b}/dv^{n-2}\) for \(n\ge 3\). In the case \(n=2\) we use directly (1482). We conclude
(see (1330) for the definition of \(J_n\)). Using the inductive hypothesis (\(Y_{n-1}\)) we have
Therefore, by integration,
The first one is valid in the case \(n\ge 4\). The second and the third are valid in the case \(n\ge 3\). We note that in the case \(n\ge 2\) we have
We turn to a given by (1292). We have
where
In view of (1393) we have
As in (1398) we have
Using the first of (1394) together with (1269) we obtain
Therefore,
Now, from (1409) we have
Recalling the second of (1443), we define
We then have
which, recalling (1297), implies
Setting now
we have
Integrating (1498) in view of (1500) we obtain
We now use (1454) with \(\hat{M}\), \(M'\) in the role of \(\hat{N}\), N respectively. Since the null space of \(A_{n-2}\) is the space of polynomials of degree \(n-2\) with no constant term, we obtain
By (1477) we have
Integrating this yields
the second in view of the fact that \(\widehat{\left[ T^{tt}\right] }'(0)=0\). Then, from (1454) with \(\widehat{\left[ T^{tt}\right] }'\), \(\widehat{\widehat{\left[ T^{tt}\right] }}\) in the role of N, \(\hat{N}\) respectively, we obtain
The estimates (1507), (1510) imply, through integration,
(1511), (1512) imply, through (1505),
Integrating this yields
In order to estimate \(d^{n-1}\tilde{u}/dv^{n-1}\) we have to estimate \(\tilde{a}_{n-2}\), \(\tilde{b}_{n-2}\) (see (1305)). From the first of (1304) we have
Using now (1513), (1514) we obtain
We turn to \(\tilde{b}_{n-2}\). We restrict ourselves first to the case \(n\ge 3\). From the second of (1304) we have
For the first term we use (1488). The double sum we rewrite as
Now, in view of the first of (1304), the factor \(d^l\tilde{a}_{j-l}/dv^l\) involves \(d^j\tilde{a}/dv^j\). Therefore, each of the terms in the inner sum in (1518) involves
Here \(0\le j\le n-3\). In the case \(j=n-3\) we use the third of (1490) together with (1514). We obtain
In the case \(j=n-4\) we use the second of (1490) together with (1514). We note that this case only shows up for \(n\ge 4\). We obtain
In the case \(0\le j\le n-5\) we use the first of (1490) together with (1514). We note that this case only shows up for \(n\ge 5\). We obtain
In the case \(n=2\) we use \(\tilde{b}_{n-2}=\tilde{b}_0\) and (1488) in the case \(n=2\). In view of the above we conclude
We turn to \(d^{n-1}u/dv^{n-1}\). Let us investigate first the case \(n=2\). To find an estimate for \(\tilde{u}\) we integrate (1299) and obtain
From (1516) and (1523) we have
Therefore,
and we obtain, through (1524),
Using now
we obtain
We note that \(\tilde{u}=\mathcal {O}(v^3)\). From this together with (1516) and (1523), in view of (1299), we obtain
Substituting this together with (1530) into
we find
Now we turn to the case \(n\ge 3\). From \(\tilde{u}=\mathcal {O}(v^3)\) together with (1516) and (1523) in (1305) we obtain
The l-fold iterated integral of \(J_n\) is given by (1435). From (1534) this gives \(d^{n-1-l}\tilde{u}/dv^{n-1-l}\). Setting then \(j=l+1\) we obtain
The different behavior of the polynomial part is explained by the Taylor expansion of \(\tilde{u}\) beginning with a cubic term. Now we use (1454) with \(\tilde{u}\), u in the role of N, \(\hat{N}\), respectively, and with \(n+1\) in the role of n (recall that \(v\hat{N}=N\)). I.e. we use
Setting \(k=n-1\) and \(j=n-i\) in (1449) we have
Recalling that the null space of \(A_{n-1}\) consists of all polynomials of degree \(n-1\) with no constant term, we obtain, substituting (1535),
We rewrite the double sum in (1538) as
where
Recalling \(\tilde{a}_{n,m}\) given by (1460), we see that for \(m\ge 2\) we have
On the other hand,
and
Then (1539) is
and we conclude from (1536),
Now, from (1533), (1545) we conclude, in view of \(u=V-c_{+0}\),
Estimate for \(d^{n-1}\rho /dv^{n-1}\)
We recall the function \(\rho \) given by
where
Using also \(u=V-c_{+0}\) and defining
we obtain
We have (recall \(\bar{c}_+(v)=c_+(\alpha _+(v),\beta _+(v))\))
Recalling
and (1533) we obtain
Therefore,
Together with (1545) we obtain
We now use (1415), (1416) and (1417) to compute the l-fold iterated integral of this. In the role of \(f_0\) we have \(d^ny/dv^n\) and \(vd^ny/dv^n\). We obtain
Setting then \(j=l+1\), we obtain, after a straightforward computation,
where we also used the fact that \(\tilde{\nu }(0)=0\).
We now apply (1536), (1537) with \(\nu \), \(\tilde{\nu }\) in the role of u, \(\tilde{u}\) respectively. Since the null space of \(A_{n-1}\) consists of all polynomials of degree \(n-1\) with no constant term, we obtain
We rewrite the double sum as
where
Comparing with the coefficients \(\tilde{a}_{n,m}\) given by (1460) we see that for \(m\ge 1\) we have \(a'_{n,m}=\tilde{a}_{n,m}\), but for \(m=0\) we have
From (1464) we obtain
Therefore, (1560) reduces to \(-\hat{I}_{n,0}/(n-1)!\) and we conclude from (1559),
Hence, from (1536) with \(\nu \), \(\tilde{\nu }\) in the role of u, \(\tilde{u}\) respectively,
Using the inductive hypothesis (\(Y_{n-1}\)) in (1545) and (1565) we obtain
From (1271), (1286) we obtain, in view of \(\bar{c}_-=c_-(\alpha _+,\beta _+)\),
Integrating (1566), (1567), we obtain for \(m\le n-2\)
By (1554), (1565), (1568) we obtain (see (1550))
Setting \(f_0=\frac{d^ny}{dv^n}\) in (1414) so that \(g_0=\hat{I}_{n,0}\) (see (1415)), the l-fold iterated integral of \(\hat{I}_{n,0}\) is given by (1417) with 0 in the role of k, i.e. it is
From this we obtain \(d^{n-1-l}\rho /dv^{n-1-l}\). Setting \(j=l+1\) and taking into account that \(\rho (0)=0\) we find
Let \(\hat{\rho }\,{:=}\,\rho /v\). We now apply (1536), (1537) with \(\hat{\rho }\), \(\rho \) in the role of u, \(\tilde{u}\) respectively. Since the null space of \(A_{n-1}\) consists of all polynomials of degree \(n-1\) with no constant term, we obtain
We rewrite the double sum as
where
Comparing with the coefficients \(\tilde{a}_{n,m}\) given by (1460) we see that \(c_{n,m}=-\tilde{a}_{n,m+1}\). Then, from (1464),
Hence (1573) is \(\hat{I}_{n,n-1}/(n-1)!\) and (1572) is
Then (1536) with \(\hat{\rho }\), \(\rho \) in the role of u, \(\tilde{u}\) respectively, yields
Now we find the l-fold iterated integral of (1577). Since
we use (1417) with \(f_0=v^{n-1}\frac{d^ny}{dv^n}\) which then implies \(g_0=I_{n,n-1}\) and \(k=-n\), so \(g_{k+m}=I_{n,m-1}\) (see (1415), (1416)). Since \(k<0\) we use (1430) for the first term in (1417). Therefore, the l-fold iterated integral of (1577) is
From the inductive hypothesis (\(Y_{n-1}\)) (see also (1327)) we get
Inductive Step for Derivatives of t, \(\alpha \), \(\beta \) Part One
In the following, using the assumptions (\(t_{p,n-1}\)), ..., (\(\beta _{0,n-1}\)), we prove
This will then establish (\(t_{p,n}\)), (\(t_{m,n}\)). We recall the equation for t(u, v) satisfied for \((u,v)\in T_\varepsilon \)
where
We first prove (1581), (1582) for \(n=2\). For (1581) in the case \(n=2\) we make use of what has already been established in the existence proof (see the proposition in the end of the chapter dealing with the solution of the fixed boundary problem), i.e. we have
Using the inductive hypotheses (\(\alpha _{0,n-1}\)), (\(\beta _{0,n-1}\)) with \(n=2\) together with (1586) in (1583) we obtain
which is (1582) in the case \(n=2\).
We now consider the case \(n\ge 3\). We have, for \(i+j= n-2\),
Let us first consider the case \(i,j\ge 1\). We note that this case only shows up for \(n\ge 4\). The case \(n=3\) will be contained in the proof of the cases \(i=0\) and \(j=0\). From (\(\alpha _{m,n-1}\)), (\(\beta _{m,n-1}\)), (\(\alpha _{p,n-1}\)), (\(\beta _{p,n-1}\)) together with (1586) we obtain
From (\(t_{m,n-1}\)) we have
All other derivatives of \(\alpha \), \(\beta \) and t appearing in (1588) are of the order less than \(n-1\) and are, by the inductive hypothesis, all \(Q_1\). Therefore,
This is (1582) in the case \(i,j\ge 1\).
We now study (1588) in the case \(i=0\). We note that in the case \(n=3\) only this case (or the other case, namely \(j=0\)) shows up and not the case \(i,j\ge 1\). We define
and rewrite (1588) in the case \(i=0\) as
which implies
where we recall
We have
The terms in the sum involve derivatives of t w.r.t. v of order at most \(n-2\) which, by the inductive hypotheses (\(t_{p,n-1}\)), are all \(Q_1\). The terms in the sum also involve derivatives of \(\mu \) w.r.t. v of order at most \(n-3\) which in turn, through (1584), involve derivatives of \(\alpha \), \(\beta \) of order at most \(n-2\). By the inductive hypothesis (\(\alpha _{p,n-1}\)), (\(\alpha _{m,n-1}\)), (\(\beta _{p,n-1}\)), (\(\beta _{m,n-1}\)), each of these terms is a \(Q_1\). Therefore, the sum is a \(Q_1\). The second term in (1596) involves mixed derivatives of \(\alpha \) and \(\beta \) of order \(n-1\) and pure derivatives of order \(n-2\) which, by the inductive hypothesis (\(\alpha _{p,n-1}\)), (\(\alpha _{m,n-1}\)), (\(\beta _{p,n-1}\)), (\(\beta _{m,n-1}\)), are a \(Q_1\). Therefore,
We also have
The first term on the right involves pure derivatives of \(\alpha \), \(\beta \) w.r.t. v of order at most \(n-1\). These derivatives are, by the inductive hypothesis (\(\beta _{0,n-1}\)), all \(Q_0\). Making use of the first of (1586) we see that the first term is a \(Q_1\). In view of the inductive hypothesis each of the terms in the sum is also a \(Q_1\). Therefore,
From (1592), (1597), (1599) we obtain
We recall
where \(\gamma \) is given by (1548). Differentiating \(n-2\) times we obtain
Defining now
we have
Also, (1602) reads
Using now the relation (1605) with \(n-1\) in the role of k, substituting (1606) and solving for \(a_{n-1}\) yields
By the third case of (1270) the first term in the curly bracket is a \(P_1\). From (\(t_{p,n-1}\)), (\(t_{m,n-1}\)) we have
From (1547) we have
From (1569) we have
We therefore find (recall that \(\rho (0)=0\))
This implies
From the above we deduce
From (1603) with \(n-1\) in the role of k we have
By (\(t_{m,n-1}\)) all the terms in the sum are a \(P_1\). Therefore,
which, together with (1600), through (1594), implies
This is the second of (1581). Using this in (1593) we obtain (1582) in the case \(i=0\).
We now study (1588) in the case \(j=0\). We define
and rewrite (1588) in the case \(j=0\) as
which implies
where we recall
and the initial condition \(t(u,0)=h(u)\). We assume h to be a smooth function. Analogous to the treatment of \(R_{v,n-1}\) (see (1596), ..., (1599)) we find
which, through (1619), implies
This is the first of (1581). Using this in (1618) we obtain (1582) in the case \(j=0\). We have thus shown (1581), (1582), i.e. we have established (\(t_{p,n}\)), (\(t_{m,n}\)).
Now we turn to show
This will then establish (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)) and (\(\beta _{m,n}\)).
Let us recall the system of equations for \(\alpha \), \(\beta \)
which implies
Here \(\alpha _i\) is given by the initial conditions on \(\underline{C}\) and we assume \(\alpha _i\) to be smooth. \(\beta _+\) is given by the jump condition and we recall \(\left[ \beta \right] =\left[ \alpha \right] ^3G(\alpha _+,\alpha _-,\beta _-)\).
From (1629) we obtain
Here we use the abbreviation \(\tilde{A}\) for \(\tilde{A}(\alpha ,\beta ,r)\). We split the sum into
From
and recalling the second of the Hodograph system
we see that in the second term of (1632) there are involved the partial derivatives of \(\alpha \), \(\beta \) and t w.r.t. u of order at most \(n-1\). Using now the assumption (\(\alpha _{0,n-1}\)) for the partial derivatives of \(\alpha \) and \(\beta \) together with (1622) we obtain
Together with the second of (1586) we find that the second term in (1632) is a \(Q_{1,1}\). From (1582) in the case \(j=0\), which was established above, we find that the third term in (1632) is a \(Q_1\). Each of the terms in the sum in (1632) involves derivatives of \(\tilde{A}\) of order less than \(n-1\) and mixed derivatives of t of order less than n. These terms are being taken care of by the assumptions (\(t_{p,n-1}\)), (\(t_{m,n-1}\)), (\(\alpha _{p,n-1}\)), (\(\alpha _{m,n-1}\)), (\(\beta _{p,n-1}\)), (\(\beta _{m,n-1}\)). Therefore, taking into account \(\int _0^vQ_1(u,v')dv'=Q_2\), we find
We note that in the case \(n=2\) the sum in (1632) is not present and instead of using (1582) we use (1587) to deal with the last term in (1632).
From (1628) we obtain
We split the sum into
The first ones of (\(t_{p,n-1}\)), (\(\alpha _{p,n-1}\)), (\(\beta _{p,n-1}\)) in conjunction with (1635) imply
Using the first of (\(t_{p,n-1}\)) we find that the sum in (1639) is a \(Q_2\). Now, \(\alpha \), \(\beta \) and r being continuously differentiable as established in the existence proof, we obtain from (1629), (1630) together with (1586) and (1286) that \(\alpha =Q_2\), \(\beta =Q_2\), which implies
From the second of this together with (1622) we obtain that also the second term in (1639) is a \(Q_2\). Therefore,
We note that in the case \(n=2\) the sum in (1639) is not present and for the second term we use the first of (1586).
From (1627) we obtain
We split the sum into
Each of the second ones of (\(t_{p,n-1}\)), (\(\alpha _{p,n-1}\)), (\(\beta _{p,n-1}\)) in conjunction with (1634) imply
Using the second of (\(t_{p,n-1}\)) we find that the sum in (1644) is a \(Q_1\). From the first of (1641) together with (1616) we obtain that also the second term in (1644) is a \(Q_1\). Therefore,
We note that in the case \(n=2\) the sum in (1644) is not present and for the second term we use the second of (1586).
For \(f=f(u,v)\) we define
We claim
We prove this claim by induction. It is satisfied for \(k=1\). Let it now hold for k. Then
Rewriting the second sum as
the sum of the two sums is
Making use of
(1651) becomes
Using this in (1649) we obtain
which is (1648) with \(k+1\) in the role of k. This proves the claim.
In view of (1630) we now set \(f(u,v)=-B(v,u)\) and \(a=u\) in (1647), where we recall
From (1648) we then obtain
For the first term we use (1286) in the case \(m=n-1\). We split the sum in the integral into
We see that in the second term there are involved the partial derivatives of \(\alpha \), \(\beta \) and r w.r.t. v of order at most \(n-1\). From the assumptions (\(\alpha _{0,n-1}\)), (\(\beta _{0,n-1}\)) together with (1616) we obtain
Together with the first of (1586) we find that the second term in (1657) is a \(Q_1\). From (1582) in the case \(i=0\), which was established above, we find that the third term in (1657) is a \(Q_1\). All terms in the sum in (1657) involve derivatives of \(\tilde{B}\) of order less than \(n-1\) and mixed derivatives of t of order less than n. These terms are being taken care of by the assumptions (\(t_{p,n-1}\)), ..., (\(\beta _{m,n-1}\)). Therefore, the second term in (1656) is a \(Q_1\).
We split the sum in the second line of (1656) into
The second term involves a derivative of t with respect to u of order \(n-1\) which, by (1622) is a \(Q_2\). All other terms appearing in (1659) involve mixed derivatives of t of order at most \(n-1\) which, by the assumption (\(t_{m,n-1}\)), are a \(Q_1\). Furthermore, these terms involve derivatives of t, \(\alpha \), \(\beta \) and r of order at most \(n-2\). By the assumptions (\(t_{p,n-1}\)), ..., (\(\beta _{0,n-1}\)) and the Hodograph system (1634), (1635), all of them are a \(Q_1\). Therefore, the third term in (1656) is a \(Q_1\). We conclude,
We note that in the case \(n=2\) the sum in (1657) is not present and instead of using (1582) we use (1587) to deal with the last term in (1657).
In view of the system (1627), (1628) and the Hodograph system (1634), (1635), a mixed derivative of \(\alpha \) or \(\beta \) of order n is given in terms of a mixed derivative of t of order at most n and derivatives of \(\alpha \), \(\beta \) and t of order at most \(n-1\). By the assumptions (\(t_{p,n-1}\)), ..., (\(\beta _{0,n-1}\)) together with (1581), (1582), (1637), (1642), (1646), (1660) we conclude that each of the mixed derivatives of \(\alpha \) and \(\beta \) of order n is a \(Q_1\), i.e.
From (1581), (1582), (1623), (1624), (1625), (1626) we conclude that (\(t_{p,n}\)), (\(t_{m,n}\)), (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)) hold.
In the following we prove
Putting n in the role of \(n-1\) in the equations (1619), (1631) and (1638) we have
From (1617) with n in the role of \(n-1\), together with (\(t_{p,n}\)), (\(t_{m,n}\)), (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)), we have
which implies
We split the sum in (1665) into
In view of (\(t_{p,n}\)), (\(t_{m,n}\)), (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)) and the Hodograph system, each of the terms in the sum is a \(Q_1\) and for the second term in (1669) we have
From (1588) with \(i=n-1\), \(j=0\) we obtain, using again the results for the partial derivatives of \(\alpha \), \(\beta \) and t,
Substituting now (1670), (1671) in (1669) and the resulting expression in (1665) we obtain
For (1666) we make use of (1668). We obtain
Defining
and taking the sum of the absolute values of (1668), (1672) and (1673) we obtain
which implies
which in turn implies
Therefore, using this in (1668), (1672) and (1673), we obtain
the first and the second of which are (1663). For the analogous expression for derivatives with respect to v see 7.3.8.
Estimate for \(d^n\hat{f}/dv^n\)
We recall the function A given by
where we recall
We set \(f(u,v)=-\nu (u,v)\) and \(a=0\) in (1647) which implies \(F(0,v)=K(v,v)\). We obtain from (1648)
The integrand involves partial derivatives of \(\alpha \), \(\beta \) of order at most n, where the pure derivatives with respect to v of order n do not show up. By the above results for the partial derivatives of \(\alpha \), \(\beta \) these are all a \(Q_0\) which implies that the first term is a \(P_1\). The second term involves derivatives of \(\alpha \), \(\beta \) of order at most \(n-1\). Again by the results for the partial derivatives of \(\alpha \), \(\beta \) these are all a \(P_1\). We conclude
which implies, by integration,
where for the second we took into account \(K(0,0)=0\).
We apply \(d^{n-1}/dv^{n-1}\) to (1679). Using (1577), (1580) and (1683) we obtain
Since the expression for \(d^{n-1}A/dv^{n-1}\) is formally identical to the one for \(d^{n-1}\hat{\rho }/dv^{n-1}\) given by the right hand side of (1577), the l-fold iterated integral of (1684) is given by the right hand side of (1579), i.e.
Let us recall (here \(f(v)=t(v,v)\))
and
as well as
and
We also recall
Differentiating (1686) \(n-1\) times we obtain
Setting \(a=0\) and
in (1647), and letting \(\tilde{F}(v)\,{:=}\,F(0,v)\), we have
and
where the right hand side is given by (1648). Since
the Taylor expansion of \(\tilde{F}\) begins with quadratic terms. Since
we have
where \(B_k\) is the linear kth order operator
This operator is homogeneous w.r.t. scaling. Hence \(B_k\) takes a polynomial to a polynomial of the same degree. Let G be a polynomial which begins with quadratic terms. Then \(M\,{:=}\,G/v^2\) is analytic, hence so is \(v^{-k-2}B_kG\). This follows from (1700) with G in the role of \(\tilde{F}\), M in the role of \(M_1\) and \(k+1\) in the role of n. It follows that the polynomial \(B_kG\) begins with terms of degree \(k+2\). We conclude that the null space of \(B_k\) consists of all polynomials of degree \(k+1\) which begin with quadratic terms, i.e.
This is a k-dimensional space.
Since \(f(v,v)=0\), we have
which implies
Substituting this into (1648) and making use of (note that this is only valid for \(l\le k-2\))
we obtain
To deal with the first term in (1706) we have to study \(\partial ^kf/\partial u^k\). From (1685) together with the inductive hypothesis (\(Y_{n-1}\)) (see also the second of (1327)) we obtain
Using this we deduce from (1695)
which implies
Setting now \(l=n-k\) in (1685), equation (1709) becomes
We now look at the second term in (1706). From (1695) we deduce, using (1707),
Therefore,
Substituting now (1685) with \(n-m+1\), \(n-m+2\), \(n-m+3\) in the role of l into the first, second and third term on the right hand side, respectively, and using the resulting expression together with (1710) in (1706), we obtain, after a straightforward computation,
Using this now in (1700) (see (1701) with \(n-1\) in the role of k for the operator \(B_{n-1}\)), we obtain
We rewrite the double sum in (1717) as
where
We consider
For \(l\ge 3\) we have \(\tilde{c}_{n,l}=c_{n,l}\). For \(l=2\) we have
while for \(l=1\) we have
To compute \(\tilde{c}_{n,l}\) we express it as
where
and
Hence
We obtain from (1721), (1722) and (1726)
Substituting in (1718) and the resulting expression in (1717) we obtain
To see that the polynomial part in (1728) has no term of order zero in the case \(n=2\) we consider
Since (1685) in the case \(n=2\), \(l=1\) (this is (1684) in the case \(n=2\) integrated once) is
we have from the inductive hypothesis (\(Y_{n-1}\))
Using this in (1729) we obtain
We turn to \(d^{n-1}M_2/dv^{n-1}\). Recalling (1689), (1690) we can write
Applying \(d^{n-1}/dv^{n-1}\) to the first term we obtain
For the first term we use (1684) while for the second and third we use (1685) with \(l=1\), \(l=2\) respectively. We arrive at
Applying \(d^{n-1}/dv^{n-1}\) to the second term in (1733) we obtain
From (1728) together with the inductive hypothesis (\(Y_{n-1}\)) we have
which, through integration and taking into account that the Taylor expansion of \(M_1\) starts with quadratic terms, implies
Using (1707) and (1738) in (1736) yields
which, together with (1735), implies
Noting that only in the case \(n=2\) we have the contribution \(dM_0/dv=2/3\) (see (1688)), we obtain from (1728), (1740)
We turn to \(d^{n-1}N/dv^{n-1}\). We look at (1691) and write
where
We first establish estimates for the derivatives of \(\hat{B}_0\). In view of (1692) this involves estimates for the derivatives of K(v, v) and \(\rho (v)\). The derivatives of K(v, v) are given by (1683). We recall that the second bracket in the second line of (1692) is a smooth function whose Taylor expansion begins with cubic terms. This implies
From (1610), through integration, we obtain
where we made use of \(\rho (0)=0\) (see the second of (1554)). Therefore,
Using this together with (1707) and making use of arguments analogous to the ones we used for derivatives of \(\hat{f}=f/v\) (see (1445)) and \(M_1=\tilde{F}/v^2\) (see (1700)) to take care of the terms involving prefactors of 1 / v and \(1/v^2\) respectively, we arrive at
We now turn to the principal term in (1742) in the case \(i=0\), which is \(N_0'\). We define
and rewrite
Since the Taylor expansion of \(G_0\) begins with quartic terms, we can apply (1700) with \(N_0'\), \(G_0\) in the roles of \(M_1\), \(\tilde{F}\), respectively, i.e.
where \(B_k\) is the operator (1701). We claim
For the proof of the claim in (1753) we suppress the index 0. For \(m=1,2,3\) it is true as can be seen by direct computation. For \(m=3\) we have
For \(m\ge 3\) we write
From (1754) we have
Differentiating (1755)
gives us the following recursion formulas
From the first of (1756) and (1758),
Substituting in (1759) gives
Therefore, since \(c_{1,1}=0\),
Substituting in (1760) gives
Therefore, since \(c_{0,2}=0\),
In view of (1761), (1763) and (1765) the claim (1753) is proven.
We substitute (1746) with m, \(m-1\), \(m-2\), \(m-3\) in the role of j into (1753). For the expressions involving derivatives of \(\rho \) we use (1571). We obtain, after a straightforward computation,
Now we go back to (1752). We must calculate \(B_{n-1}G_0\). From (1701),
Since the null space of \(B_{n-1}\) consists of all polynomials of degree n which begin with quadratic terms, \(B_{n-1}\) applied to the polynomial part of \(G_0\) gives a \(\bar{P}_{n+2,n+1}\). Then
The double sum is
where the coefficients \(c_{n,l}\) are given by (1719). Using (1727) we obtain, after a straightforward computation,
Substituting in (1752) and using the resulting expression together with (1749) in the \((n-1)\)’th order derivative of (1742) with \(i=0\), we arrive at
To see that the polynomial part in (1771) has no term of order zero in the case \(n=2\) we take the derivative of (1691)
From (1707), (1748) we have, in the case \(n=2\),
Using these in (1772) with \(i=0\) we obtain
We turn to derive an estimate for \(d^{n-1}N_1/dv^{n-1}\). In view of (1693) we need an estimate for \(d^{n-1}I/dv^{n-1}\), where we recall
where
Defining
we have
Making now use of (1647), (1648), we obtain
We see that we need estimates for mixed derivatives of f of order at most \(n-2\), for derivatives of f with respect to u of order at most \(n-1\) and for derivatives of f with respect to v of order at most \(n-2\). Therefore we need expressions for derivatives of \(\alpha \) and \(\beta \) with respect to u of order at most n, expressions for mixed derivatives of \(\alpha \) and \(\beta \) of order at most n and expressions for derivatives of \(\alpha \) and \(\beta \) with respect to v of order at most \(n-1\). We also need expressions for mixed derivatives of t of order at most n and expressions for derivatives of t with respect to v of order at most \(n-1\). We define
For the first term in (1779) we have
By the results for the partial derivatives of \(\alpha \), \(\beta \) and t established above we see that each of the terms in the sum is a \(Q_1\). By the same results we see that
Together with the second of (1586) we see that the second term in the second line in (1781) is a \(Q_{1,1}\). Therefore,
For the second term in (1779) we write
We first look at the case \(i=j=0\). From (1271), (1286) we have \(\alpha _+,\beta _+=P_2\). Therefore, \(\bar{c}_\pm =P_2\). Using this together with the first ones of (\(\alpha _{p,n-1}\)), (\(\beta _{p,n-1}\)) in (1584) we obtain
which, together with
implies
Therefore, together with (1682),
From (\(t_{p,n}\)), (\(t_{m,n}\)) we have
From (1788), (1789) we deduce that each of the terms in (1784) with \(i=j=0\) is a \(P_{2,1}\).
Now we look at the case \(i+j=n-2\). The only term in (1784) satisfying this condition is
The first factor involves derivatives of \(\alpha \) and \(\beta \) of order at most \(n-1\). Therefore, by (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)) the first factor in (1790) is a \(Q_1\). By (1270) and the first of (1586) we have for \(n\ge 3\)
Therefore, for \(n\ge 3\),
In the case \(n=2\) (1790) is \(H(\partial t/\partial v)\). Using the second of (1586) and (1788) we obtain that (1792) is also valid in the case \(n=2\). We note that for \(n=2\) this is the only non-vanishing term in (1784). For \(n=3\) the two cases \(i=j=0\) and \(i+j=n-2=1\) cover all the terms appearing in (1784).
Let now \(n\ge 4\). For the terms in (1784) with \(1\le i+j\le n-3\) we need estimates for
We set
Then
Setting \(p=n-k-2\) and \(j=q+l\) implies, through (1789),
This yields, through integration,
i.e.,
Hence,
We also need estimates for
These in turn require estimates for
We set
Then
Setting \(p=n-k-2\) and \(j=q+l\) and using (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)) implies
This yields, through integration,
i.e.,
Hence,
The same procedure applies to derivatives of \(\beta \) using \((\beta _{p,n})\), \((\beta _{m,n})\). Therefore,
From (1799) together with (1808) we obtain that each of the terms in (1784) with \(1\le i+j\le n-3\) is a \(P_2\). We conclude
which, together with (1783) implies
Making use of
we obtain
Therefore, by integrating (1810), we deduce
Now we apply (1700) with I in the role of \(\tilde{F}\) (see (1696) for the relation between \(M_1\) and \(\tilde{F}\)).
where the operator \(B_k\) is given in (1701). We find
which, through integration, implies
Now we apply \(d^{n-1}/dv^{n-1}\) to \(\hat{B}_1\). We rewrite
where
Using (1747) and (1683) we obtain
Now,
Therefore, from (1816) and (1819),
which, through integration, implies
We see that, in particular, (1748) with 1 in the role of 0 holds. Therefore, also (1749) with 1 in the role of 0 holds, i.e. we have
Now we define (this is (1750) with 1 in the role of 0)
and have
We then have, as in (1752) with 1 in the role of 0,
where (this is (1767) with 1 in the role of 0)
Now we use (1822) in (1753), setting successively \(j=m,m-1,m-2,m-3\). We obtain
Using this in (1827) and taking into account that the null space of \(B_{n-1}\) consists of all polynomials of degree n which begin with quadratic terms, we obtain
Substituting this in (1826) yields
which, together with (1823), yields
To see that the polynomial part has no term of order zero in the case \(n=2\) we use (1821), (1822) together with the first and the second of (1773) in (1772) with \(i=1\).
Combining the estimates (1771) and (1831) we obtain
We introduce
We have
Combining now (1741) with (1832) we obtain (see (1686))
We note that since
we have
We recall \(f=v^2\hat{f}\). Using (1700) we obtain
where (see (1701)) the linear n’th order operator \(M_n\) is given by
We recall that the null space of \(B_n\) (and hence the null space of \(M_n\)) is the space of all polynomials of degree \(n+1\) which begin with quadratic terms.
Now we use (1417) to compute, for \(0\le l\le n\), the l-fold iterated integral of \(v\hat{I}_{n,0}\), \(v\hat{I}_{n,1}\), \(v\hat{I}_{n,n+2}\) appearing in (1835). Setting \(f_0=\frac{d^ny}{dv^n}\) in (1415), so that \(g_0=I_{n,0}=\hat{I}_{n,0}\), and \(k=1\) in (1417), we find that the l-fold iterated integral of \(v\hat{I}_{n,0}\) is given by
Setting \(f_0=v\frac{d^ny}{dv^n}\) in (1415), so that \(g_0=I_{n,1}=v\hat{I}_{n,1}\), and \(k=0\) in (1417), we find that the l-fold iterated integral of \(v\hat{I}_{n,1}\) is given by
Setting \(f_0=v^{n+2}\frac{d^ny}{dv^n}\) in (1415), so that \(g_0=I_{n,n+2}=v^{n+2}\hat{I}_{n,n+2}\), and \(k=-n-1\), we find that the l-fold iterated integral of \(v\hat{I}_{n,n+2}\) is given by
We note that here we made use of (1417) in the case \(k<0\), which is valid since \(k+l\le -1\). Therefore we interpret the first factor in the first term of (1417) as in (1430).
Using now (1840), (1841) and (1842) for the l-fold iterated integral of (1835) in conjunction with (1839), yields
Here we made use of the fact that, when (1835) is iteratively integrated n times, the polynomial part of \(\Phi \) is of degree \(n+1\) and, in view of (1837) begins with cubic terms. Therefore, the polynomial part of \(\Phi \) is annihilated by \(B_n\) hence it is annihilated by \(M_n\). The first sum in (1843) is
since \(n\ge 2\).
In view of this the second sum in (1843) is
since \(n\ge 2\).
The third sum in (1843) is
Finally, we rewrite the double sum in (1843) as
where
We see that in (1847) the terms with \(m=n\) and \(m=n-1\) vanish. Therefore, we can restrict to the case \(1\le m\le n-2\). We have
since \(n-m\ge 1\). Also,
since \(n-m\ge 2\). We conclude that
Hence the double sum in (1843) vanishes.
We deduce from the above
Therefore, substituting in (1838), we conclude
Estimate for \(d^n\hat{\delta }/dv^n\)
We recall the function \(\delta \) given by
where
We also recall the function \(\hat{\delta }\) given by \(\delta (v)=v^3\hat{\delta }(v)\). Using
we deduce
where \(L_n\) is the n’th order differential operator
which is homogeneous w.r.t. scaling. Hence \(L_n\) takes a polynomial to a polynomial of the same degree. Let G be a polynomial which begins cubic terms. Then \(M\,{:=}\,G/v^3\) is analytic, hence so is \(v^{-k-3}L_nG\). This follows from (1857) with G in the role of \(\delta \) and M in the role of \(\hat{\delta }\). It follows that the polynomial \(L_nG\) begins with terms of degree \(k+3\). We conclude that the null space of \(L_n\) consists of all polynomials of degree \(n+2\) which begin with cubic terms, i.e.
This is a n-dimensional space.
We now estimate \(d^m\delta /dv^m\). Let us recall the splitting of \(\delta (v)\)
where the functions \(\delta _0\) and \(\delta _1\) are given by
where we recall the function \(\phi \)
We note that
where \(\Phi \) is given in (1833).
Now we apply \(d^{m-1}/dv^{m-1}\) to \(d\delta _0/dv\). We claim
where \(\bar{P}\) is a generic polynomial and we interpret \(d^ky/dv^k\) for \(k<0\) as the k-fold iterated integral of y. For \(m\ge 1\) this follows directly and we note that for \(m\ge 3\) the polynomial \(\bar{P}\) is the zero polynomial. For the case \(m=0\) we define
Then the right hand side in (1865) becomes
while from (1861) we have
We write
Substituting in (1870) we see that (1870) coincides with (1869) up to a polynomial. Therefore, (1865) holds for \(m=0\) also, hence it holds for \(m\ge 0\).
From
together with the expression for the l-fold iterated integral of \(\hat{I}_{n,0}\) given by (1570) we obtain \(d^{n-1-l}y/dv^{n-1-l}\). Setting \(m=n-1-l\) we obtain
Replacing m by \(m-1\), \(m-2\), \(m-3\) and substituting in (1865) we obtain
The polynomial part follows from \(\delta _0=\mathcal {O}(v^4)\), which in turn follows from \(d\hat{\delta }_0/dv=\mathcal {O}(1)\).
Now we estimate \(L_n\delta _0\), where the operator \(L_n\) is given in (1858). Recalling that the null space of \(L_n\) consists of all polynomials of degree \(n+2\) which begin with cubic terms, we see that \(L_n\) annihilates the polynomial part of \(\delta _0\). Therefore,
Since
for \(n\ge 1\), the first sum in (1875) vanishes.
We rewrite the double sum in (1875) as
where
We were able to include the term \(m=n+1\) for \(j=1\) trivially because it vanishes. We have
Therefore,
However, the terms with the coefficients \(a_{n,n}\) and \(a_{n,n+1}\) do not contribute in the sum in (1877) and we obtain that (1877) collapses to the term \(j=n+2\), i.e. (1877) is equal to
Therefore,
which, when substituted in (1857) with \(\delta _0\) in the role of \(\delta \), gives
We turn to \(d^n\hat{\delta }_1/dv^n\). We apply \(d^{n-1}/dv^{n-1}\) to \(d\delta _1/dv\). For \(n\ge 3\) we obtain,
From (1872) and (1873) with \(m=n-2\) we have, for \(n\ge 3\),
Using (1546) we obtain
which implies that the first line in (1884) is a \(P_{2,1}\).
From (1873) with \(m=n-3\) we have
Using
we obtain, through integration of (1546),
From (1886), (1888) and (1891) we obtain
i.e. the second line in (1884) is a \(P_2\).
From (1546) together with the fact that \(V(v)-c_{+0}=\mathcal {O}(v^2)\) we have
In view of (1864) we have from (1835), (1837),
Using now (1893) and (1894) we deduce that each term in the sum in (1884) is a \(P_2\). We conclude that for \(n\ge 3\), (1884) is a \(P_2\).
In the case \(n=2\) in place of (1884) we have
Using
in (1872) with \(n=2\) we obtain (recall that \(y(0)=-1\))
Together with (1546) and (1872), both with \(n=2\), we obtain
Using (1889), (1890), both with \(n=2\), in (1546) with \(n=2\), we deduce
This together with (1897) implies
From (1898), (1900) and (1901) we obtain
Together with the above conclusion for \(n\ge 3\) we conclude
From (1903) we have
In view also of (650) we have, in particular,
It follows that \(L_n\) annihilates the polynomial part of \(\delta _1\). From (1857) (see also (1858)) with \(\delta _1\) in the role of \(\delta \) we obtain
Combining with (1883) we conclude
Estimate for \(d^ny/dv^n\)
We recall
where
We recall that R is given by
where H is a smooth function of its arguments.
Setting \(f_0=v^{n+2}\frac{d^ny}{dv^n}\) in (1415) so that \(g_0=I_{n,n+2}\) and using (1417) with \(k=-n-3\) we obtain the l-fold iterated integral of \(\frac{1}{v}\hat{I}_{n,n+2}\). Using this in (1853), (1907) we obtain
From (1873) we have
Recalling \((\partial \hat{F}/\partial y)(0,-1)=\frac{\lambda }{3\kappa }\), we deduce from (1912), (1913) and (1914)
We apply \(d^{n-1}/dv^{n-1}\) to (1908). Using (1853), (1907), (1915) and (1916) we obtain
i.e.
Setting
we have
Integrating gives
Therefore,
which, when substituted in (1918), yields
i.e. \(Y_n\) is bounded, hence \((Y_n)\) holds and the inductive step for the derivatives of the function y is complete.
Bound for \(d^n\hat{f}/dv^n\) and \(d^n\alpha _+/dv^n=P_1\)
We recall (1853)
In view of the bound on \(Y_n\) we have
Therefore, \(F_n\) is bounded. Hence \((F_n)\) holds and the inductive step for the derivatives of the function \(\hat{f}\) is complete. We note that (1925) implies (recall \(f=v^2\hat{f}\))
We turn to \(d^n\alpha _+/dv^n\). From (1629) we obtain
The first term is taken care of by the assumption on the initial data. In the following we will make use of \((t_{p,m})\), \((t_{m,n})\), \((\alpha _{p,n})\), \((\alpha _{m,n})\), \((\beta _{p,n})\), \((\beta _{m,n})\) and (1678) without any further reference. Each of the terms in the sum of the second term is at least a \(Q_0\). Therefore, the second term is a \(P_{1,1}\). We split the third term according to
The first term involves mixed derivatives of t of order n and less and derivatives of \(\alpha \), \(\beta \) and t of order \(n-1\) and less which are all \(Q_1\). Therefore, the first term in (1928) is a \(P_1\). For the second term in (1928) we have
where for the first term we reason as above. To deal with the second term in (1929) we use (1603) with n in the role of k and (1607) with n in the role of \(n-1\) which is
For the first term in the bracket we use (1926). For the sum we note that each of the \(a_{n-l}\) involves derivatives of t of order \(n-1\) and less and is therefore a \(P_1\). From (1923) we have
Therefore, from (1569),
which, through (1609), implies, recalling that \(\rho (0)=0\),
Therefore,
Therefore, from (1603) with n in the role of k we find
Using this in (1929) we see that the second term in (1928) is a \(P_1\) and therefore the whole sum in (1928) is a \(P_1\) which in turn implies that the third term in (1927) is a \(P_1\). We conclude
i.e. \((\alpha _{+,n})\) holds.
Inductive Step for Derivatives of t, \(\alpha \), \(\beta \) Part Two
In the following we prove
Together with (1663) this will then establish (\(\alpha _{0,n-1}\)), (\(\beta _{0,n-1}\)) with n in the role of \(n-1\), i.e. it will establish (\(\alpha _{0,n}\)), (\(\beta _{0,n}\)). With n in the role of \(n-1\) in the equations (1594), (1643), (1656) we have
Analogous to (1667) we find
Therefore, taking into account (1935),
For (1939) we make use of (1942) together with (\(t_{p,n}\)), (\(t_{m,n}\)), (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)). We find
Now, (\(Y_n\)) implies that (1269) holds with n in the role of \(n-1\). By (1926) and (1936) also (1270) and (1271) hold with n in the role of \(n-1\). Therefore, we obtain, in the same way as we derived (1286), that (1286) holds with n in the role of \(n-1\). Therefore, the first term in (1940) is a \(P_1\).
We split the sum in the integral of (1940) into
In view of (\(t_{p,n}\)), (\(t_{m,n}\)), (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)) and the Hodograph system, each of the terms in the sum is a \(Q_1\) and for the second term in (1944) we have
From (1588) with \(i=0\), \(j=n-1\) we obtain, using again (\(t_{p,n}\)), (\(t_{m,n}\)), (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)),
We now look at the last term in (1940). We split the sum into
In view of (\(t_{p,n}\)), (\(t_{m,n}\)), (\(\alpha _{p,n}\)), (\(\alpha _{m,n}\)), (\(\beta _{p,n}\)), (\(\beta _{m,n}\)) and the Hodograph system, each of the terms in the sum is a \(Q_1\) and for the second term we use (1678) and conclude that this term is a \(Q_0\). Using now (1945), (1946) in (1944) and the resulting expression in (1940) we find
Defining
and taking the sum of the absolute values of (1942), (1943) and (1948), we obtain
which implies
which in turn implies
Therefore, using this in (1942), (1943), (1948) we obtain
the first and second of which are (1937). From (1663) and (1937) we conclude that (\(\alpha _{0,n}\)), (\(\beta _{0,n}\)) hold. This completes the proof of the inductive step for the derivatives of \(\alpha \), \(\beta \) and t. Therefore, the inductive step is complete.
Blowup on the Incoming Characteristic Originating at the Cusp Point
We recall the following asymptotic forms
As established above, \(\alpha \), \(\beta \) and t are smooth functions of u and v in the state behind the shock. Let us now consider an outgoing characteristic originating at a point on \(\underline{C}\) corresponding to the coordinates (u, 0). According to (1954), (1955) and (1956), along this outgoing characteristic the Taylor expansions in v of \(\alpha \), \(\beta \) as well as t do not contain linear terms but do contain odd powers beginning with the third. Therefore, \(\alpha \) and \(\beta \) hence also the \(\psi _\mu \) are smooth functions not of the parameter t but rather of the parameter
Therefore, the derivatives of the \(\psi _\mu \) with respect to \(L_+\) of order greater than the first blow up as we approach \(\underline{C}\) from the state behind the shock (recall (124) for \(L_+\)).
Notes
We use
$$\begin{aligned} \frac{\partial }{\partial v}=\frac{\partial t}{\partial v}\frac{\partial }{\partial t}+\frac{\partial r}{\partial v}\frac{\partial }{\partial r}=\frac{\partial t}{\partial v}\left( \frac{\partial }{\partial t}+\frac{\frac{\partial r}{\partial v}}{\frac{\partial t}{\partial v}}\frac{\partial }{\partial r}\right) =\frac{\partial t}{\partial v}\left( \frac{\partial }{\partial t}+c_+\frac{\partial }{\partial r}\right) =\frac{\partial t}{\partial v}L_+. \end{aligned}$$(157)Similarly
$$\begin{aligned} \frac{\partial }{\partial u}=\frac{\partial t}{\partial u}L_-. \end{aligned}$$(158)
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Acknowledgments
Work supported by ERC Advanced Grant 246574 Partial Differential Equations of Classical Physics.
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Christodoulou, D., Lisibach, A. Shock Development in Spherical Symmetry. Ann. PDE 2, 3 (2016). https://doi.org/10.1007/s40818-016-0009-1
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DOI: https://doi.org/10.1007/s40818-016-0009-1