Abstract
We study the Muskat problem for one fluid in an arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that, for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem.
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1 Introduction
In this paper we study traveling wave solutions to the one-phase Muskat problem, which concerns the dynamics of the free boundary of a viscous fluid in homogeneously permeable porous media. The n-dimensional (\(n\geqq 2\)) wet region \(\Omega _{\zeta (\cdot ,t)}\) lies above the flat bed of depth \(b>0\) and below the free boundary that is the graph of an unknown time-dependent function \(\zeta \), i.e.
where the cross-section \(\Gamma \) is either \(\mathbb {R}^{n-1}\) or \(\mathbb {T}^{n-1}:= \mathbb {R}^{n-1} / (2\pi \mathbb {Z}^{n-1})\). Here, for any point \(x\in \Gamma \times \mathbb {R}\) we split its horizontal and vertical coordinates as \(x=(x', x_n)\). We denote the free-boundary and the flat bed respectively by \(\Sigma _{\zeta (\cdot ,t)}\) and \(\Sigma _{-b}\); that is,
We posit that when the cross-section is \(\mathbb {R}^{n-1}\), the free boundary \(\zeta (x', t)\) decays as \(|x'|\rightarrow \infty \).
The fluid is acted upon in the bulk by a uniform gravitational field \(-e_n\) pointing downward, where \(e_n\) is the upward pointing unit vector in the vertical direction, and a generic body force \(\tilde{\mathfrak {f}}(\cdot , t): \Omega _{\zeta (\cdot , t)}\rightarrow \mathbb {R}^n\). Then the fluid motion in the porous medium is modeled by the Darcy law
where, for the sake of simplicity, we have normalized the dynamic viscosity, the fluid density, and the permeability of the medium to unity. Here, w and P respectively denote the fluid velocity and pressure. On the surface, the fluid is acted upon by a constant pressure \(P_0\) from the dry region above \(\Omega _{\zeta (\cdot ,t)}\) and an externally applied pressure \(\phi (\cdot , t): \Sigma _{\zeta (\cdot , t)}\rightarrow \mathbb {R}\). This leads to the boundary condition
The no-penetration boundary condition is assumed on the flat bed:
Finally, the free boundary evolves according to the kinematic boundary condition
We shall refer to the following system as the (one-phase) Muskat problem
In the absence of the body force and the external pressure, i.e. \(\tilde{\mathfrak {f}} =0\) and \(\phi =0\), (1.7) is called the free Muskat problem.
The free Muskat problem can be recast as a nonlocal equation for the free boundary function \(\eta \) (see (2.17)). It was proved in [12] that the problem is locally-in-time well-posed for large data \(\eta _0\in H^s(\Gamma )\) for any \(1+\frac{n-1}{2}<s\in \mathbb {R}\), which is the lowest Sobolev index guaranteeing that \(\eta _0\in W^{1, \infty }(\Gamma )\). We also refer to [3] for local well-posedness for the case of non-graph free boundary. The free Muskat problem admits the following trivial stationary solutions
In fact, under mild regularity and decay assumptions, (1.8) are the only stationary solutions. They have been proved to be stable in various norms [4, 5, 11, 13]. To the best of our knowledge, (1.8) are the only solutions that are known to be stable.
In this paper we are interested in the construction of nontrivial special solutions and the stability of them. In view of the translation invariance of (1.7) in the horizontal directions, it is natural to consider traveling wave solutions. These are solutions that propagate along a fixed direction, which without loss of generality we may assume is the \(x_1\)-direction, with constant velocity \(\gamma \). To this end, we assume that
and make the traveling wave ansatz
This determines the unknown domain \(\Omega _\eta = \{x \in \Gamma \times \mathbb {R}\;\vert \;-b< x_n < \eta (x')\}\) as well as the free boundary \(\Sigma _\eta = \{x \in \Gamma \times \mathbb {R}\;\vert \;x_n = \eta (x')\}\) as before. We then define the traveling wave unknowns \(v: \Omega _\eta \rightarrow \mathbb {R}^n\) and \(q: \Omega _\eta \rightarrow \mathbb {R}\) via
In the latter we have subtracted off the hydrostatic pressure, as is often convenient. The new equations for \((v,q, \eta )\) read
where
We pause to remark that the only solutions to the free version of (1.12) (\( \mathfrak {f}=0\) and \(\phi =0\)) are the trivial solutions as given in (1.8). Indeed, assuming that \((v, q, \eta )\) is a decaying solution, then using Green’s theorem and the boundary conditions for v and q, we obtain
It follows that \(v=0\) and hence \(q=c\), a constant. Consequently \(\eta (x')=q(x', \eta (x'))=c\). When \(\Gamma = \mathbb {R}^{n-1}\) this implies that \(\eta =q=0\) since \(\eta \) decays. Thus \((v, q, \eta )=(0, 0, 0)\) is the trivial solution when \(\Gamma = \mathbb {R}^{n-1}\). In the periodic case, \(\Gamma = \mathbb {T}^{n-1}\), we obtain the trivial solutions \((v, q, \eta )=(0, c, c)\), for \(c\in \mathbb {R}\) (one can uniquely determine c by fixing a mass of the fluid). This is not a surprise since for free Muskat problem, the energy dissipates, so it cannot sustain the permanent structure of traveling waves. It is therefore necessary to have some sort of external energy in order for traveling wave solutions to exist. In the context of (1.12), this is provided by the external bulk force \( \mathfrak {f} \) and the external pressure \(\phi \).
Our first main result states that for suitably small \(\mathfrak {f}\) and \(\phi \), there exists a locally unique traveling wave solution to (1.12) in Sobolev-type spaces. Note that in the following statement we employ a reformulation of the problem (1.12) as well as some nonstandard function spaces; these will be explained after the theorem statement:
Theorem 1.1
(Proved in Sect. 4.2) Let \(\frac{n}{2} - 1 < s \in \mathbb {N}\) and consider the open set
with \(\delta >0\) as constructed in Theorem 4.3. Define the open set \(\mathfrak {C}\subseteq \mathbb {R}\) to be \(\mathbb {R}\) if \(\Gamma = \mathbb {T}^{n-1}\) and \(\mathbb {R}\backslash \{0\}\) if \(\Gamma = \mathbb {R}^{n-1}\). Then there exist open sets
such that the following hold:
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(1)
\(\mathfrak {C} \times \{0\} \times \{0\} \times \{0\} \times \{0\} \subseteq \mathcal {D}^s\) and \((0,0,0) \in \mathcal {S}^s\).
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(2)
For each \((\gamma , \varphi _0, \varphi _1, \mathfrak {f}_0, \mathfrak {f}_1) \in \mathcal {D}^s\) there exists a locally unique \((u,p,\eta ) \in \mathcal {S}^s\) classically solving
$$\begin{aligned} {\left\{ \begin{array}{ll} u + \nabla _{\mathcal {A}} p + \nabla _{\mathcal {A}} \mathfrak {P}\eta = \mathfrak {J}\mathcal {M}^T[ \mathfrak {f}_0 + \mathfrak {f}_1\circ \mathfrak {F}_\eta ] &{} \text {in }\Omega :=\Gamma \times (-b, 0) \\ {{\,\textrm{div}\,}}{u} = 0 &{}\text {in } \Omega \\ -\gamma \partial _1 \eta = u _n &{}\text {on } \Sigma :=\Gamma \times \{0\} \\ p = \varphi _0 + \varphi _1 \circ \mathfrak {F}_\eta &{}\text {on } \Sigma \\ u_n =0 &{} \text {on } \Sigma _{-b}. \end{array}\right. } \end{aligned}$$(1.17) -
(3)
The map \(\mathcal {D}^s \ni (\gamma , \varphi _0, \varphi _1, \mathfrak {f}_0, \mathfrak {f}_1) \mapsto (u,p,\eta ) \in \mathcal {S}^s\) is \(C^1\) and locally Lipschitz.
Some remarks are in order.
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(1)
(1.17) is a reformulation of (1.12) in the fixed domain \(\Omega \) and with \(\mathfrak {f}(x)=\mathfrak {f}_0(x')+\mathfrak {f}_1(x)\) and \(\varphi (x)=\varphi _0(x')+\varphi _1(x)\). See Sect. 2.1 for the derivation of (1.17) and for the precise meaning of \(\mathcal {A}\), \(\mathfrak {J}\), \(\mathcal {M}\) and \(\mathfrak {F}_\eta \). The preceding forms of \(\mathfrak {f}\) and \(\varphi \) are imposed since we assume less regularity for \(\mathfrak {f}\) and \(\varphi \) when they are independent of the vertical variable \(x_n\). Note also that the integer constraint for the regularity parameter s comes from the need to verify that the maps \((\mathfrak {f}_1,\eta ) \mapsto \mathfrak {f}_1 \circ \mathfrak {F}_\eta \) and \((\varphi _1,\eta ) \mapsto \varphi _1 \circ \mathfrak {F}_\eta \) are \(C^1\). If these forcing terms are ignored, then we may relax this requirement for s: see Theorems 6.1 and 6.3.
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(2)
The space \({_n}H^{s+1}(\Omega ;\mathbb {R}^n)\) is defined in Definition 3.13.
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(3)
When the cross-section is \(\mathbb {T}^{n-1}\), the boundary function \(\eta \) is constructed in \(\mathring{H}^s(\mathbb {T}^{n-1})\), the usual Sobolev space of zero-mean functions. On the other hand, when the cross-section is \(\mathbb {R}^{n-1}\), then \(\eta \) belongs to the anisotropic Sobolev space \(\mathcal {H}^{s+\frac{3}{2}}(\mathbb {R}^{n-1})\), as defined in Definition A.1. At high frequencies this space provides standard \(H^{s+3/2}\) Sobolev control, but at low frequencies it only controls
$$\begin{aligned} \int _{B(0,1)} \frac{\xi _1^2+|\xi |^4}{|\xi |^2}|\widehat{\eta }(\xi )|^2\textrm{d}\xi . \end{aligned}$$(1.18)The modulus \(\frac{\xi _1^2+|\xi |^4}{|\xi |^2}\) naturally arises from the linearized operator \(\partial _1-|D|\tanh (b|D|)\) and the following structure of the nonlinearity at low frequencies: \(\mathcal {N}=|D|\widetilde{\mathcal {N}}\). The anisotropic Sobolev space \(\mathcal {H}^s(\mathbb {R}^d)\), which satisfies the inclusions \(H^s(\mathbb {R}^d) \subset \mathcal {H}^s(\mathbb {R}^d) \subseteq H^s(\mathbb {R}^d) + C^\infty _0(\mathbb {R}^d)\), was introduced in [9] for the construction of traveling wave solutions to the free boundary Navier–Stokes equations and plays a key role in our construction here. We recall the definition and basic properties of \(\mathcal {H}^s\) in Appendix A.1.
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(4)
Theorem 1.1 asserts the uniqueness of traveling wave solutions in the small but does not exclude the possibility of nonuniqueness in the large.
Our proof of Theorem 1.1 is based on the implicit function theorem, applied in a neighborhood of the trivial solutions obtained with \(\gamma \in \mathfrak {C}\), \(\mathfrak {f}_0 = \mathfrak {f}_1 = 0\), \(\varphi _0 = \varphi _1 =0\), \(u=0\), \(p=0\), and \(\eta =0\). In order for this strategy to work, we need a good understanding of the solvability of the linear problem
which is obtained by linearizing the flattened reformulation of (1.12) given in (1.17) around the trivial solutions. More precisely, we need to identify appropriate function spaces \(\mathbb {E}\) and \(\mathbb {F}\) for which the linear map \(\mathbb {E} \ni (u,p,\eta ) \mapsto (F,G,H,K) \in \mathbb {F}\) induced by (1.19) is an isomorphism. When \(\Gamma = \mathbb {T}^{n-1}\) the function spaces we employ are standard \(L^2-\)Sobolev spaces, but when \(\Gamma = \mathbb {R}^{n-1}\) even identifying appropriate spaces turns out to be quite delicate for a couple reasons. First, in \(L^2-\)Sobolev spaces on the infinite domain \(\Omega = \mathbb {R}^{n-1} \times (-b,0)\) there are some subtle compatibility conditions that the data tuple (F, G, H, K) need to satisfy, and these need to be encoded in \(\mathbb {F}\). Second, as mentioned in the above remarks, even when the data satisfy the appropriate compatibility conditions, the free surface function \(\eta \) necessarily lives in the strange anisotropic Sobolev spaces given in Definition A.1, which behave like standard \(L^2-\)based Sobolev spaces at large frequencies but have unusual anisotropic behavior at low frequencies (for instance, these spaces are not closed under composition with rotations). Similar issues arose in the second author’s recent work on the construction of traveling wave solutions to the incompressible Navier–Stokes system [7, 9, 14], and fortunately, we were able to adapt some of the techniques used in those works to handle the Muskat construction of Theorem 1.1.
In identifying the appropriate function spaces, we also uncover the method for showing that (1.19) induces an isomorphism. We first take the divergence of the first equation and eliminate u to arrive at a problem for p and \(\eta \) only, (3.1). To solve this problem we initially ignore the \(\eta \) terms and view the resulting problem as an overdetermined problem for p, (3.13). This overdetermined problem is only solvable for data satisfying certain compatibility conditions, reminiscent of those from the closed range theorem, which we identify in Sect. 3.2. These turn out to be the key to solving (3.1), as they lead us to a pseudodifferential equation for \(\eta \) that can be solved independently of p:
Here \(\psi \) is a specific function determined linearly by the data in (3.1) (see (3.59) for the precise definition). It is this equation that forces \(\eta \) into the anisotropic Sobolev spaces, but in turn the spaces allow us to construct \(\eta \) and verify that it is a reasonably nice function. Note that when \(\Gamma = \mathbb {R}^{n-1}\) we require \(\gamma \ne 0\) precisely because this term is responsible for ensuring that \(\eta \) is a nice function; in the case \(\gamma =0\) we lose the ability to verify this. With \(\eta \) in hand, we can then solve for p and show that (3.1) induces an isomorphism (see Theorem 3.12). Then in Theorem 3.17 we show that we can return to (1.19) and uncover an isomorphism. Finally, in Sect. 4, we verify that our function spaces are nice enough to be used in an implicit function theorem argument and then employ the IFT to prove Theorem 1.1.
It is natural to investigate the stability of the traveling wave solutions constructed in Theorem 1.1, and we next turn to this topic. We expect that the stability analysis depends on the type and form of external forces. In our second main result, we prove that under the sole effect of the external pressure (i.e. \(\mathfrak {f}_0=\mathfrak {f}_1=0\)), the small periodic traveling wave solutions constructed in Theorem 1.1 are asymptotically stable. For simplicity we state the result for \(\varphi (x)=\varphi _0(x')\).
Theorem 1.2
(Proved in Sect. 6.2 ) Let \(\gamma \in \mathbb {R}\) and \(1+\frac{n-1}{2}<s\in \mathbb {R}\). There exists a small positive constant \(\varepsilon _*=\varepsilon _*(s, b, n)\) such that if \(\Vert \nabla \varphi _0\Vert _{H^{s-\frac{1}{2}}(\mathbb {T}^{n-1})}<\varepsilon _*\), then the unique steady solution \(\eta _*\in \mathring{H}^s(\mathbb {T}^{n-1})\) of (2.17) with \(\varphi (x)=\varphi _0(x')\) is asymptotically stable in \(\mathring{H}^s(\mathbb {T}^{n-1})\). More precisely, there exist positive constants \(\nu \) and \(\delta \), both depending only on (s, b, n), such that if \(\eta _0 \in \mathring{H}^s(\mathbb {T}^{n-1})\) satisfies \(\Vert \eta _0-\eta _*\Vert _{\mathring{H}^s}<\delta \), then the dynamic problem (2.17) with initial data \(\eta _0\) has a unique solution \(\eta \in \eta _*+B_{Y^s([0, T])}(0, \nu )\) for all \(T>0\), where
moreover, we have the estimates
and
where \(c_0=c_0(s, b, d)\).
To be best of our knowledge, Theorem 1.2 provides the first class of nontrivial stable solutions to the one-phase Muskat problem with graph free boundary.
Inspired by the proof of stability of the trivial solution for the Muskat problem in [11], we obtain the stability of small periodic traveling wave solutions by linearizing the Dirichlet–Neumann operator about the flat surface,
and establish good boundedness and contraction estimates for the remainder \(R(\eta )\). More precisely, the results obtained in Sect. 5 imply the estimates
and
where \(\eta _\delta =\eta _1-\eta _2\) and \(d=n-1\). The estimates in Sect. 5 for the Dirichlet–Neumann operator are obtained for the free boundary belonging to the anisotropic Sobolev spaces \(\mathcal {H}^s(\Gamma )\), \(\Gamma \in \{\mathbb {R}^d, \mathbb {T}^d\}\), and are of independent interest.
Now, fix a traveling wave solution \(\eta _*\) with data \(\varphi _0\). The perturbation \(f=\eta -\eta _*\) then satisfies
Assuming that \(f_0\) has zero mean in \(H^s(\mathbb {T}^d)\), then f(t) has zero mean for \(t>0\). When performing the \(H^s\) energy estimate, the dissipation term m(D) yields a gain of \(\frac{1}{2}\) derivative:
On the other hand, by virtue of (1.25) and (1.26), we can control the nonlinear terms in (1.27) in \(H^{s-\frac{1}{2}}(\mathbb {T}^d)\) by
where the coefficient \(\alpha \) is small when \(\varphi _0\) and \(\eta _*\) are small. Therefore, if \(\Vert f(t)\Vert _{H^s}\) is small globally, then it decays exponentially. On the other hand, the global existence and smallness of \(\Vert f(t)\Vert _{H^s}\) are proved by appealing to the estimates (1.25) and (1.26) again for the mild-solution formulation of (1.27).
2 Problem Reformulations
In this section we present two reformulations we will use in proving our two main theorems. The first one is a reformulation for the general traveling wave system (1.12) in a flattened domain. When the generic body force \(\mathfrak {f}\) is absent, we present a reformulation for the dynamic problem (1.7) using the Dirichlet–Neumann operator.
2.1 Flattening the Traveling Wave System
Consider the flat domain \(\Omega := \Omega _0 = \Gamma \times (-b,0)\) and write \(\Sigma = \Sigma _0 = \Gamma \times \{0\}\). We define the Poisson extension operator \(\mathfrak {P}\) as in Appendix 7. Assuming that \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\) (see Definition A.1 for the precise definition of this anisotropic Sobolev space), we define the flattening map \(\mathfrak {F}_\eta : \bar{\Omega } \rightarrow \bar{\Omega }_\eta \) via
Note that \(\mathfrak {F}_\eta \vert _{\Sigma _{-b}} = I\) and \(\mathfrak {F}_\eta (\Sigma ) = \Sigma _{\eta }\). We compute
We define the functions \(\mathfrak {J},\mathfrak {K}: \Omega \rightarrow (0,\infty )\) via
It will be useful to introduce the matrix field \(\mathcal {M}: \Omega \rightarrow \mathbb {R}^{n \times n}\) via
Our interest in the field \(\mathcal {M}\) comes from a trio of useful identities it satisfies. The first is Piola identity,
The second and third are a pair of identities on \(\Sigma \) and \(\Sigma _{-b}\):
To see the utility of the Piola identity note that \(v: \Omega _\eta \rightarrow \mathbb {R}^n\) satisfies \({{\,\textrm{div}\,}}{v}=0\) if and only if \(\hat{v} = v \circ \mathfrak {F}_\eta : \Omega \rightarrow \mathbb {R}^n\) satisfies \(\mathfrak {J}\mathcal {M}_{ij} \partial _j \hat{v}_i =0\) (the summation convention is used here), but
so a further equivalent condition is that \(u = \mathfrak {J}\mathcal {M}^T \hat{v}: \Omega \rightarrow \mathbb {R}^n\) satisfies \({{\,\textrm{div}\,}}{u}=0\).
In light of the previous calculations, we use \(\mathfrak {F}_\eta \) and \(\mathcal {M}\) to rephrase the traveling wave Muskat system (1.12) in the fixed domain \(\Omega \) by defining \(u = \mathfrak {J}\mathcal {M}^T v \circ \mathfrak {F}_\eta \) and \(p = -\mathfrak {P}\eta + q \circ \mathfrak {F}_\eta \). The new system reads
where \(\mathcal {A}: \Omega \rightarrow \mathbb {R}^{n \times n}_{sym}\) is defined by
and we write
2.2 Dirichlet–Neumann Reformulation
We consider the dynamic Muskat problem (1.7) with \(\widetilde{\mathfrak {f}}=0\) and \(\phi (x,t) = \varphi (x-\gamma t e_1)\). In the moving frame \(x\mapsto x-\gamma t e_1\), we make the change of variables
to obtain the system
This problem can be recast on the free boundary by means of the Dirichlet–Neumann operator (2.14) defined as follows. Let \(\psi \) be the solution of
The Dirichlet–Neumann operator associated to \(\Omega \) is denoted by \(G(\eta )\) and
By taking the divergence of the first equation in (2.12), we deduce that q satisfies
It follows from the third equation in (2.12) that
where \(G(\eta )\) denotes the Dirichlet–Neumann operator for \(\Omega _\eta \). Therefore, \(\eta \) obeys the equation
3 Linear Analysis for the Traveling Wave System
In this section we study the linearization of (2.8) around the trivial solution, which is the system (1.19), where (F, G, H, K) are given data. Note that for the purposes of studying the linearization of (2.8) we could reduce to the case \(G=0\) and \(H=0\); we have retained these terms here for the sake of generality.
We can eliminate u to get an equivalent formulation of the problem. Indeed, we take the divergence of the first equation and then use the first equation to remove u from the boundary conditions. This results in the problem
We will study this form of the problem and eventually show that it is equivalent to (1.19).
Remark 3.1
Throughout what follows we will often abuse notation by identifying
in order to allow us to handle linear combinations of functions defined on \(\Sigma \), \(\Sigma _{-b}\), and \(\Gamma \) in a simple way. In reality we actually identify these through the natural isometric isomorphism, but this is obvious and the corresponding notation is too cumbersome to introduce.
3.1 The upper-Dirichlet–lower-Neumann Isomorphism
Consider the problem
for given \((f,k,l) \in H^s(\Omega ) \times H^{s+3/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b})\). Associated to this PDE is the bounded linear map
given by
Theorem 3.2
The map \(T_0\) is an isomorphism for every \(0 \leqq s \in \mathbb {R}\).
Proof
To see that \(T_0\) is injective we suppose that \(T_0 p =0\), multiply the resulting equation \(-\Delta p =0\) by p and integrate by parts. Using the boundary conditions contained in the identity \(T_0 p =0\), we deduce that \(\int _{\Omega } \left| \nabla p\right| ^2 = 0,\) and so \(p=0\) in \(\Omega \) since \(p=0\) on \(\Sigma \). Thus \(p=0\), and injectivity is proved.
It remains to prove that \(T_0\) is surjective, and this ultimately boils down to the weak solvability and elliptic regularity associated to the problem (3.3), which we will briefly sketch. We initially define the space \({^0}H^1(\Omega ) = \{f \in H^1(\Omega ) \;\vert \;f=0 \text { on } \Sigma \},\) which we can equip with the inner-product \(\left( f,g \right) _{{^0}H^1} = \int _\Omega \nabla f \cdot \nabla g\). This in indeed an inner-product and generates the usual \(H^1\) topology thanks to a Poincaré-type inequality provided by the vanishing on \(\Sigma \). Then by Riesz representation, for any \(\mathcal {F} \in ( {^0}H^1(\Omega ))^*\), there exists a unique \(p \in {^0}H^1(\Omega )\) such that
Next we consider \(s \in \mathbb {N}\) and data \(f \in H^{s}(\Omega )\) and \(l \in H^{s+1/2}(\Sigma _{-b})\). According to standard trace theory and the above, we can then find a unique \(p \in {^0}H^1(\Omega )\) such that
Standard interior elliptic regularity shows that \(p \in H^{s+2}_{\text {loc}}(\Omega )\) and \(-\Delta p =f\) in \(\Omega \). Using horizontal difference quotients, we may deduce in turn that
We then recover control of the vertical derivatives by using the identity \(-\partial _n^2 p = \Delta ' p + f\) together with a simple iteration argument; this yields the inclusion \(p \in H^{s+2}(\Omega )\) with the estimate
Returning to the weak formulation and integrating by parts, we find that
and hence that \(-\partial _n p = l\) on \(\Sigma _{-b}\). Thus, \(p \in H^{s+2}(\Omega ) \cap {^0}H^1(\Omega )\) satisfies \(T_0 p = (f,0,l)\).
For each \(s \in \mathbb {N}\) this analysis defines a bounded linear map \(S_0: H^{s}(\Omega ) \times H^{s+1/2}(\Sigma _{-b}) \rightarrow H^{s+2}(\Omega ) \cap {^0}H^1(\Omega )\) via \(S_0(f,l) = p\). Employing the usual Sobolev interpolation theory (see, for instance, [2, 15]), we deduce that \(S_0\) extends to a map between the same spaces but for all \(0 \leqq s \in \mathbb {R}\).
Now suppose that \(f \in H^s(\Omega )\), \(k \in H^{s+3/2}(\Sigma )\), and \(l \in H^{s+1/2}(\Sigma _{-b})\) for some \(0\leqq s \in \mathbb {R}\). By trace theory, we can pick \(K \in H^{s+2}(\Omega )\) such that \(P = k\) on \(\Sigma _b\) H:\(K=k\) on \(\Sigma \). Using the above, we then find \(P = S_0(f+\Delta K,l+\partial _n K) \in H^{s+2}(\Omega )\cap {^0}H^1(\Omega )\), which satisfies \(T_0 P = (f + \Delta K, 0, l + \partial _n K)\). Then \(p:= P+K \in H^{s+2}(\Omega )\) satisfies \(T_0 p = (f,k,l)\), and we conclude that \(T_0\) is surjective. \(\square \)
Later in our analysis we will need to consider the following bounded linear operator:
Definition 3.3
We define the bounded linear map \(\Xi : H^{s+3/2}(\Sigma ) \rightarrow H^{s+2}(\Omega )\) via \(\Xi k = T_0^{-1}(0,k,0)\).
The next result records a crucial property of \(\Xi \).
Theorem 3.4
The map \(\Xi \) from Definition 3.3 satisfies \(\widehat{\Xi k} (\xi ,x_n) = \hat{k}(\xi ) Q(\xi ,x_n)\) for \(\xi \in \hat{\Gamma }\), where \(Q: \mathbb {R}^{n-1} \times (-b,0) \rightarrow \mathbb {R}\) is defined by
Note that in the case \(\Gamma = \mathbb {T}^{n-1}\) we have that the dual group is \(\hat{\Gamma } = \mathbb {Z}^{n-1} \subset \mathbb {R}^{n-1}\) and Q is given by restriction to \(\hat{\Gamma }\).
Proof
Write \(p = T_0^{-1}(0,k,0) \in H^{s+2}(\Omega )\), and let \(\hat{p}\) denote its horizontal Fourier transform. Then \(\hat{p}\) satisfies the ordinary differential boundary value problem
From this it’s an elementary exercise to verify that \(\hat{p}(\xi ,x_n) = \hat{k}(\xi ) Q(\xi ,x_n)\), and the result follows. \(\square \)
3.2 The Over-Determined Problem: Compatibility Conditions
Consider the over-determined problem
for given \((f,h_+,h_-,k) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b}) \times H^{s+3/2}(\Sigma )\).
Associated to (3.13) are a pair of compatibility conditions. The first actually is associated to a sub-system of (3.13).
Proposition 3.5
Suppose that \((f,h_+,h_-) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b})\) and \(p \in H^{s+2}(\Omega )\) satisfy
Then the following hold:
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(1)
If \(\Gamma = \mathbb {R}^{n-1}\), then
$$\begin{aligned} \int _{-b}^0 f(\cdot ,x_n) \textrm{d}x_n - (h_+ - h_-) \in \dot{H}^{-2}(\Sigma ) \cap \dot{H}^{-1}(\Sigma ) \end{aligned}$$(3.15)and we have the bounds
$$\begin{aligned}{} & {} \left[ \int _{-b}^0 f(\cdot ,x_n) \textrm{d}x_n - (h_+ - h_-) \right] _{\dot{H}^{-2}} \lesssim \left\| p\right\| _{L^2}\nonumber \\{} & {} \qquad \text { and } \left[ \int _{-b}^0 f(\cdot ,x_n) \textrm{d}x_n - (h_+ - h_-) \right] _{\dot{H}^{-1}} \lesssim \left\| \nabla ' p\right\| _{L^2}. \end{aligned}$$(3.16) -
(2)
If \(\Gamma = \mathbb {T}^{n-1}\), then
$$\begin{aligned} \int _{-b}^0 \hat{f}(0,x_n) \textrm{d}x_n - (\hat{h}_+(0) - \hat{h}_-(0)) =0. \end{aligned}$$(3.17)
Proof
We will only record the proof for \(\Gamma = \mathbb {R}^{n-1}\), as the other case follows from similar but simpler analysis. We have that
We then compute
and
Combining these, we see that
and the \(\dot{H}^{-2}\) inclusion and estimate then follow from an application of Cauchy–Schwarz, Fubini–Tonelli, and Parseval:
The \(\dot{H}^{-1}\) inclusion and estimate follow similarly. \(\square \)
Next we identify the formal adjoint of the over-determined problem as an under-determined problem, given here in homogeneous form:
We can augment this problem with an extra Dirichlet condition at the upper boundary in order to introduce the upper-Dirichlet–lower-Neumann problem (3.3). Indeed, we can parameterize solutions to (3.23) by letting \(q = \Xi \psi \) for some \(k \in H^{s+3/2}(\Sigma )\), where \(\Xi \) is as in Definition 3.3. With this in mind we borrow an idea from the closed range theorem to deduce a second compatibility condition.
Proposition 3.6
Suppose that \((f,h_+,h_-,k) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b}) \times H^{s+3/2}(\Sigma )\) and \(p \in H^{s+2}(\Omega )\) satisfy (3.13). Then the data \((f,h_+,h_-,k)\) satisfy both of the following equivalent conditions:
-
(1)
For every \(\psi \in H^{s+3/2}(\Sigma )\), if we let \(q = \Xi \psi \in H^{s+2}(\Omega )\) for \(\Xi \) as in Definition 3.3, then
$$\begin{aligned} \int _{\Omega } f q - \int _{\Sigma } k \partial _n q + h_+ \psi + \int _{\Sigma _{-b}} h_- q = 0. \end{aligned}$$(3.24) -
(2)
For a.e. \(\xi \in \hat{\Gamma }\) we have that
$$\begin{aligned} 0{} & {} = \int _{-b}^0 \hat{f}(\xi , x_n) \frac{\cosh ( \left| \xi \right| (x_n+b))}{\cosh ( \left| \xi \right| b)} \textrm{d}x_n - \hat{k}(\xi ) \left| \xi \right| \tanh ( \left| \xi \right| b) - \hat{h}_+(\xi )\nonumber \\{} & {} \quad + \hat{h}_-(\xi ) {{\,\textrm{sech}\,}}( \left| \xi \right| b). \end{aligned}$$(3.25)
Proof
Let \(\psi \in H^{s+3/2}(\Sigma )\) and write \(q = \Xi \psi \in H^{s+2}(\Omega )\). Multiplying the first equation in (3.13) by q and integrating by parts, we find that
Rearranging yields (3.24). It remains to prove that (3.25) is equivalent to this.
Viewing k, \(h_+\), and \(h_-\) as functions on \(\hat{\Gamma }\) in the natural way, we may rearrange (3.26) and apply Fubini–Tonelli to see that
From this, Parseval’s theorem, and Theorem 3.4, we then find that
for all \(\psi \in H^{s+3/2}(\Sigma )\). This implies the identity
for a.e. \(\xi \in \hat{\Gamma }\), and (3.25) then follows by employing the formula for \(Q(\xi ,x_n)\) from Theorem 3.4. The fact that (3.25) implies (3.24) is readily seen by multiplying (3.25) by \(\overline{\hat{\psi }}\) and then working backward through the above argument with Parseval and Fubini–Tonelli. \(\square \)
Next we show that data obeying the conditions identified in this result must also obey an estimate in \(\dot{H}^{-2}\) as in Proposition 3.5.
Proposition 3.7
If \((f,h_+,h_-,k) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b}) \times H^{s+3/2}(\Sigma )\) satisfy either (and thus both) of the conditions in Proposition 3.6. Then the following hold.
-
(1)
If \(\Gamma = \mathbb {R}^{n-1}\), then
$$\begin{aligned} \int _{-b}^0 f(\cdot ,x_n) \textrm{d}x_n - (h_+ - h_-) \in \dot{H}^{-2}(\mathbb {R}^{n-1}) \end{aligned}$$(3.30)and
$$\begin{aligned} \left[ \int _{-b}^0 f(\cdot ,x_n) \textrm{d}x_n - (h_+ - h_-) \right] _{\dot{H}^{-2}} \lesssim \left\| f\right\| _{L^2} + \left\| h_+\right\| _{L^2} + \left\| h_-\right\| _{L^2} + \left\| k\right\| _{L^2}. \end{aligned}$$(3.31) -
(2)
If \(\Gamma = \mathbb {T}^{n-1}\), then
$$\begin{aligned} \int _{-b}^0 \hat{f}(0,x_n) \textrm{d}x_n - (\hat{h}_+(0) - \hat{h}_-(0)) =0. \end{aligned}$$(3.32)
Proof
We will only record the proof when \(\Gamma = \mathbb {R}^{n-1}\) as the other case is simpler. The condition (3.25) implies that
Upon making routine Taylor expansions and applying Cauchy–Schwarz and Parseval, we see that
This yields the low frequency control of the \(\dot{H}^{-2}\) seminorm, but the high frequency control comes directly from Cauchy–Schwarz and Parseval:
Thus, the inclusion (3.30) holds, and upon summing (3.34) and (3.35) we deduce the estimate (3.31). \(\square \)
3.3 A Pair of Useful Function Spaces
We now introduce a couple function spaces that will be useful in our study of the over-determined problem (3.13).
Definition 3.8
For \(0 \leqq s \in \mathbb {R}\) we define the following spaces:
-
(1)
For \(\Gamma = \mathbb {R}^{n-1}\) and \(0 < t \in \mathbb {R}\) we define the space
$$\begin{aligned} Y^s_{t}= & {} \{(f,h_+,h_-,k) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma ) \times H^{s+3/2}(\Sigma _{-b}) \;\vert \;\nonumber \\{} & {} \int _{-b}^0 f(\cdot ,x_n) \textrm{d}x_n - (h_+ - h_-) \in \dot{H}^{-t}(\Sigma )\} \end{aligned}$$(3.36)and endow this space with the square-norm
$$\begin{aligned}{} & {} \left\| (f,h_+,h_-,k)\right\| _{Y^s_t}^2 = \left\| f\right\| _{H^s}^2 + \left\| h_+\right\| _{H^{s+1/2}}^2 + \left\| h_-\right\| _{H^{s+1/2}}^2 + \left\| k\right\| _{H^{s+3/2}}^2\nonumber \\{} & {} \quad + \left[ \int _{-b}^0 f(\cdot ,x_n) \textrm{d}x_n - (h_+ - h_-)\right] _{\dot{H}^{-t}}^2 \end{aligned}$$(3.37)and its associated inner-product.
-
(2)
For \(\Gamma = \mathbb {T}^{n-1}\) and \(0 < t \in \mathbb {R}\) we define the space
$$\begin{aligned} Y^s_{t}= & {} \{(f,h_+,h_-,k) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma ) \times H^{s+3/2}(\Sigma _{-b}) \;\vert \;\nonumber \\{} & {} \int _{-b}^0 \hat{f}(0,x_n) \textrm{d}x_n - (\hat{h}_+(0) - \hat{h}_-(0)) =0\} \end{aligned}$$(3.38)and endow this space with the square-norm
$$\begin{aligned} \left\| (f,h_+,h_-,k)\right\| _{Y^s_t}^2 = \left\| f\right\| _{H^s}^2 + \left\| h_+\right\| _{H^{s+1/2}}^2 + \left\| h_-\right\| _{H^{s+1/2}}^2 + \left\| k\right\| _{H^{s+3/2}}^2 \nonumber \\ \end{aligned}$$(3.39)and its associated inner-product.
-
(3)
We define the space
$$\begin{aligned} Z^s= & {} \{(f,h_+,h_-,k) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b}) \times H^{s+3/2}(\Sigma ) \;\vert \;\nonumber \\{} & {} \int _{\Omega } f q - \int _{\Sigma } k \partial _n q + h_+ \psi + \int _{\Sigma _{-b}} h_- q = 0 \text { for every } \psi \in H^{s+3/2}(\Sigma ),\nonumber \\{} & {} \text { where } q = \Xi \psi \}. \end{aligned}$$(3.40)Here we recall that \(\Xi : H^{s+3/2}(\Sigma ) \rightarrow H^{s+2}(\Omega )\) is defined in Definition 3.3. We endow \(Z^s\) with the square norm
$$\begin{aligned} \left\| (f,h_+,h_-,k)\right\| _{Z^s}^2 = \left\| f\right\| _{H^s}^2 + \left\| h_+\right\| _{H^{s+1/2}}^2 + \left\| h_-\right\| _{H^{s+1/2}}^2 + \left\| k\right\| _{H^{s+3/2}}^2 \nonumber \\ \end{aligned}$$(3.41)and its associated inner-product.
The next result establishes some key properties of these spaces.
Proposition 3.9
Let \(0 \leqq s \in \mathbb {R}\), \(0 < t \in \mathbb {R}\), and let \(Y^s_t\) and \(Z^s\) be as in Definition 3.8. Then the following hold.
-
(1)
\(Y^s_t\) and \(Z^s\) are Hilbert spaces.
-
(2)
If \(t < r \in \mathbb {R}\) then we have the continuous inclusion \(Y^s_r \hookrightarrow Y^s_t\).
-
(3)
We have the continuous inclusion \(Z^s \hookrightarrow Y^s_2\).
Proof
The completeness of \(Y^s_t\) is routine to verify, and since \(\Xi \) is a bounded linear map it is easy to see that \(Z^s\) is a closed subspace of \(H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b}) \times H^{s+3/2}(\Sigma )\) and thus complete. This proves the first item. The second item is trivial when \(\Gamma = \mathbb {T}^{n-1}\), and when \(\Gamma = \mathbb {R}^{n-1}\) it follows from the fact that
when \(t < r\) and \(\psi \) is measurable. The continuous inclusion \(Z^s \hookrightarrow Y^s_2\) follows from Proposition 3.7. \(\square \)
3.4 The Over-Determined Problem: Isomorphism
We now aim to establish an isomorphism associated to the over-determined problem (3.13).
Theorem 3.10
The bounded linear map \(T_1: H^{s+2}(\Omega ) \rightarrow Z^s\) associated to (3.13), which is given by
is well-defined and is an isomorphism for every \(0 \leqq s \in \mathbb {R}\).
Proof
The map \(T_1\) is obviously a bounded linear map into \(H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b}) \times H^{s+3/2}(\Sigma )\), but the range of \(T_1\) lies in \(Z^s\) by virtue of Proposition 3.6. Thus, \(T_1: H^{s+2}(\Omega )\rightarrow Z^s\) is a well-defined and bounded linear map. If \(T_1 p =0\), then in particular \(T_0 p =0\), where \(T_0\) is the isomorphism from Theorem 3.2, and so \(p=0\). This means that \(T_1\) is injective.
Now let \((f,h_+,h_-,k) \in Z^s\). Then Theorem 3.2 allows us to set \(p = T_0^{-1}(f,k,h_-) \in H^{s+2}(\Omega )\). Set \(H_+ = -\partial _n p \vert _{\Sigma } \in H^{s+1/2}(\Sigma )\). Then p solves the over-determined problem
and so Proposition 3.6 tells us that
for every \(\psi \in H^{s+3/2}(\Sigma )\), where \(q = \Xi \psi \in H^{s+2}(\Omega )\) for \(\Xi \) defined by Definition 3.3. On the other hand, the compatibility condition on \((f,h_+,h_-,k)\) built into the definition of \(Z^s\) requires that
for all such \(\psi \) and q. Equating these then shows that
from which we conclude that \(h_+ = H_+\). Hence p solves (3.13), or equivalently \(T_1 p = (f,h_+,h_-,k)\). Thus \(T_1\) is surjective and so defines an isomorphism. \(\square \)
3.5 The Isomorphism for the Pressure-Free Surface System
Next we aim to show that the PDE system
induces an isomorphism between appropriate Banach spaces. As a first step, in the next lemma we establish that the linear mapping associated to our PDE system actually takes values in \(Y^s_1\) and is bounded.
Lemma 3.11
Let \(0 \leqq s \in \mathbb {R}\). If \((p,\eta ) \in H^{s+2}(\Omega ) \times \mathcal {H}^{s+3/2}(\Sigma )\) then we have the inclusion
and
Proof
Write the tuple in (3.49) as \((f,h_+,h_-,k)\). From Theorems A.2, A.4, A.7 and standard trace theory we see that
and so in particular \((f,h_+,h_-,k) \in H^s(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+1/2}(\Sigma _{-b}) \times H^{s+3/2}(\Sigma )\).
Suppose now that \(\Gamma = \mathbb {R}^{n-1}\). Proposition 3.5 implies that
We know from Theorem A.2 that \(\left[ \partial _1 \eta \right] _{\dot{H}^{-1}} \lesssim \left\| \eta \right\| _{\mathcal {H}^{s+3/2}}\), and we know from Proposition A.8 that
so we deduce that
Thus \((f,h_+,h_-,k) \in Y^s_1\) and the estimate (3.50) holds when \(\Gamma = \mathbb {R}^{n-1}\).
Now consider the case \(\Gamma =\mathbb {T}^{n-1}\). In this case Proposition 3.5 shows that
but Proposition A.8 shows \(\widehat{\partial _n \mathfrak {P}\eta \vert _{\Sigma }}(0) = \widehat{\partial _1 \eta }(0) = \widehat{\partial _n \mathfrak {P}\eta \vert _{\Sigma _{-b}}}(0)=0,\) so
Thus, \((f,h_+,h_-,k) \in Y^s_1\) and the estimate (3.50) holds when \(\Gamma = \mathbb {T}^{n-1}\). \(\square \)
We can now state our isomorphism theorem associated to (3.48).
Theorem 3.12
If \(\Gamma = \mathbb {R}^{n-1}\), then assume that \(\gamma \ne 0\). Then the bounded linear map \(T_2: H^{s+2}(\Omega ) \times \mathcal {H}^{s+3/2}(\Sigma ) \rightarrow Y^s_1\) associated to (3.48), which is defined by
is an isomorphism for every \(0 \leqq s \in \mathbb {R}\).
Proof
First note that Lemma 3.11 tells us that \(T_2\) is a well-defined bounded linear map. If \(T_2(p,\eta ) =0\), then
and so \(p + \mathfrak {P}\eta =C\) for some constant \(C \in \mathbb {R}\). However, on \(\Sigma \) we have that \(p=0\) and \(\mathfrak {P}\eta =\eta \), so \(\eta =C\). In turn this requires that \(\eta =0\) (since \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\)) and \(p =0\), and so \(T_2\) is injective.
Now let \((f,h_+,h_-,k) \in Y^s_1\). Define the function \(\psi : \hat{\Gamma } \rightarrow \mathbb {C}\) via
Note that we may rewrite
When \(\Gamma = \mathbb {R}^{n-1}\), we readily deduce from this and standard Taylor expansion that
Similarly, when \(\Gamma = \mathbb {T}^{n-1}\), we must have that \(\psi (0) =0\). On the other hand, in both cases we can bound
Combining these bounds shows that
with the understanding that the first integral is replaced with 0 when \(\Gamma = \mathbb {T}^{n-1}\).
Next note that for \(\xi \in \hat{\Gamma }\) we have that
and in particular the quantity on the left side vanishes if and only if \(\xi =0\). Consequently, we can define the measurable function \(\hat{\eta }: \hat{\Gamma } \rightarrow \mathbb {C}\) via the identity
for \(\xi \ne 0\) and \(\hat{\eta }(0)=0\). It may be easily checked that since the data are real-valued we have that \(\overline{\psi (\xi )} = \psi (-\xi )\). The multiplier on the left side of (3.65) satisfies the same identity, and so we conclude that \(\overline{\hat{\eta }(\xi )} = \hat{\eta }(-\xi )\), which means that \(\eta \) is also real-valued. Synthesizing (3.63) and (3.64), we see from (3.65) that \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\) and
again with the understanding that the integrals over B(0, 1) are replaced by 0 when \(\Gamma = \mathbb {T}^{n-1}\), and recalling that \(\mathcal {H}^{s+3/2}(\mathbb {T}^{n-1}) = H^{s+3/2}(\mathbb {T}^{n-1})\).
We now know that \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\), so we can use Theorem A.7 to see that \(\mathfrak {P}\eta \in \mathbb {P}^{s+2}(\Omega )\), as defined in Definition A.3. In particular, this, Theorem A.4, and standard trace theory show that \(\partial _n \mathfrak {P}\eta \vert _{\Sigma } \in H^{s+3/2}(\Sigma )\) and \(\partial _n \mathfrak {P}\eta \vert _{\Sigma _{-b}} \in H^{s+3/2}(\Sigma _{-b})\). Moreover, a simple computation shows that
for \(\xi \in \hat{\Gamma }\). From these and the properties of \(\mathcal {H}^{s+3/2}(\Sigma )\) given in Theorem A.2, we readily deduce that we have the inclusion
We claim that, in fact, this modified tuple belongs to the space \(Z^s\). To show this it suffices to check that the modified tuple satisfies the compatibility condition of Proposition 3.6. Using the identities (3.67), we compute
Thus, the identity
is equivalent to the identity (3.65), which is satisfied by the construction of \(\eta \). Thus, for the modified tuple we have the inclusion
as claimed.
In light of (3.71) and Theorem 3.10 we may then define
which satisfies
Thus, \(T_2(p,\eta ) = (f,h_+,h_-,k)\), and we conclude that \(T_2\) is surjective and hence an isomorphism. \(\square \)
3.6 The Isomorphism for the Velocity-Pressure-Free Surface System
Finally, we aim to show that the PDE system (1.19) induces an isomorphism between appropriate Hilbert spaces. First we must identify the domain and codomain by introducing two definitions. The first defines a closed subspace of \(H^s(\Omega ;\mathbb {R}^n)\).
Definition 3.13
For \(1/2 < s \in \mathbb {R}\) we define the space
Standard trace theory shows that this is a closed subspace of \(H^s(\Omega ;\mathbb {R}^n)\) and thus a Hilbert space.
The second definition introduces a container space for the data in the problem (1.19).
Definition 3.14
Let \(0 \leqq s \in \mathbb {R}\). For \(\Gamma = \mathbb {R}^{n-1}\) we define the space
and endow it with the square norm
and the associated inner-product. On the other hand, for \(\Gamma = \mathbb {T}^{n-1}\) we define the space
and endow it with the square norm
and the associated inner-product. It’s easy to see that in both cases \(V^s\) is a Hilbert space.
Remark 3.15
Note that
and so
When \(\Gamma = \mathbb {R}^{n-1}\) this provides the estimate
and this means that the term appearing in the \(\dot{H}^{-1}\) seminorm in the definition of the \(V^s\) norm can be replaced with
to produce an equivalent norm. Similarly, when \(\Gamma = \mathbb {T}^{n-1}\) these calculations show that a data tuple \((F,G,H,K) \in H^{s+1}(\Omega ;\mathbb {R}^n) \times H^{s}(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+3/2}(\Sigma )\) belongs to \(V^s\) if and only if
Our next lemma shows that the linear map associated to (1.19) takes values in \(V^s\) and is bounded.
Lemma 3.16
Let \(0 \leqq s \in \mathbb {R}\). Suppose \((u,p,\eta ) \in {_n}H^{s+1}(\Omega ;\mathbb {R}^n) \times H^{s+2}(\Omega ) \times \mathcal {H}^{s+3/2}(\Sigma )\) and set
Then \((F,G,H,K) \in V^s\), \((G - {{\,\textrm{div}\,}}{F}, H - F_n \vert _{\Sigma }, - F_n\vert _{\Sigma _{-b}}, K) \in Y^s_1\), and we have the bound
Proof
We may readily bound
On the other hand, if we define \(f = G- {{\,\textrm{div}\,}}{F} \in H^{s}(\Omega )\), \(h_+ = H - F_n \vert _{\Sigma } \in H^{s+1/2}(\Sigma )\), \(h_- = - F_n \vert _{\Sigma _{-b}} \in H^{s+1/2}(\Sigma _{-b})\), and \(k=K \in H^{s+3/2}(\Sigma )\), then we see that
and so Lemma 3.11 implies that \(\left\| (f,h_+,h_-,k) \right\| _{Y^s_1} \lesssim \left\| p\right\| _{H^{s+2}} + \left\| \eta \right\| _{\mathcal {H}^{s+3/2}}\). When \(\Gamma = \mathbb {R}^{n-1}\), the \(\dot{H}^{-1}\) control provided by the \(Y^s_1\) norm is exactly the \(\dot{H}^{-1}\) control in the \(V^s\) norm, and the stated estimate follows by summing our two bounds. Similarly, when \(\Gamma = \mathbb {T}^{n-1}\), the vanishing zero mode condition required for inclusion in \(Y^s_1\) corresponds with the vanishing condition needed for inclusion in \(V^s\). \(\square \)
Finally, we can state the isomorphism theorem for the map associated to (1.19).
Theorem 3.17
Assume that \(0 \leqq s \in \mathbb {R}\), and if \(\Gamma = \mathbb {R}^{n-1}\), then assume that \(\gamma \ne 0\). Then the bounded linear map \(T_3: {_n}H^{s+1}(\Omega ;\mathbb {R}^n) \times H^{s+2}(\Omega ) \times \mathcal {H}^{s+3/2}(\Sigma ) \rightarrow V^s\) associated to (1.19), which is defined by
is an isomorphism.
Proof
Lemma 3.16 tells us that \(T_3\) is well-defined and bounded. If \(T_3(u,p,\eta ) =0\), then in particular \(u+ \nabla p + \nabla \mathfrak {P}\eta =0\), and in turn this means that \(T_2(p,\eta ) = 0\). Theorem 3.12 then implies that \(p=0\) and \(\eta =0\), and so \(u=0\) as well. Thus, \(T_3\) is injective.
Now let \((F,G,H,K) \in V^s\). Lemma 3.16 shows that
and so we may use Theorem 3.12 to define \((p,\eta ) = T_2^{-1}(f,h_+,h_-,k) \in H^{s+2}(\Omega ) \times \mathcal {H}^{s+3/2}(\Sigma )\). In other words, \((p,\eta )\) satisfy
and upon setting \(u = F - \nabla p -\nabla \mathfrak {P}\eta \in {_n}H^{s+1}(\Omega ;\mathbb {R}^n)\) (where we have used Theorems A.2, A.4, A.7 to handle the \(\nabla \mathfrak {P}\eta \) term) we deduce that \(T_3(u,p,\eta ) = (F,G,H,K)\). Thus, \(T_3\) is surjective and so is an isomorphism. \(\square \)
4 Nonlinear Analysis for the Traveling Wave System
Now we aim to invoke the implicit function theorem to solve (2.8).
4.1 The Nonlinear Mapping
To employ the implicit function theorem we first check that a number of basic nonlinear maps are well-defined.
Proposition 4.1
Let \(s > n/2 -1\). Then there exists \(\delta _0 >0\) such that the following hold:
-
(1)
If \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\) and \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _0\), then
$$\begin{aligned} \left\| b^{-1} \mathfrak {P}\eta + \tilde{b} \partial _n \mathfrak {P}\eta \right\| _{C^0_b} \leqq \frac{1}{2}, \end{aligned}$$(4.1)where \(\tilde{b}(x) = 1+x_n/b\). In particular, for such \(\eta \) the functions \(\mathfrak {K}\) and \(\mathcal {A}\), defined in terms of \(\eta \) via (2.3) and (2.9), are well-defined.
-
(2)
If \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\) and \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _0\), then the flattening map \(\mathfrak {F}_\eta : \Omega \rightarrow \Omega _\eta \) given by (2.1) is a \(C^{1 + \lfloor s-n/2 +1 \rfloor }\) orientation-preserving diffeomorphism.
-
(3)
For \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\) such that \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _0\), the functions \(\mathfrak {J}\) and \(\mathfrak {K}\), given in terms of \(\eta \) as in (2.3), define \(H^{s+1}(\Omega )\) multipliers. Moreover, the maps
$$\begin{aligned} \begin{aligned} B_{\mathcal {H}^{s+3/2}(\Sigma )}(0,\delta _0)&\ni \eta \mapsto \mathfrak {J}\in \mathcal {L}(H^{s+1}(\Omega )) \text { and } \\ B_{\mathcal {H}^{s+3/2}(\Sigma )}(0,\delta _0)&\ni \eta \mapsto \mathfrak {K}\in \mathcal {L}(H^{s+1}(\Omega )) \end{aligned} \end{aligned}$$(4.2)are smooth.
-
(4)
For \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\) such that \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _0\), the functions \(\mathcal {M}\) and \(\mathcal {A}\), given in terms of \(\eta \) as in (2.4) and (2.9), define \(H^{s+1}(\Omega ;\mathbb {R}^n)\) multipliers. Moreover, the maps
$$\begin{aligned} \begin{aligned} B_{\mathcal {H}^{s+3/2}(\Sigma )}(0,\delta _0)&\ni \eta \mapsto \mathcal {M}\in \mathcal {L}(H^{s+1}(\Omega ;\mathbb {R}^n)) \text { and }\\ B_{\mathcal {H}^{s+3/2}(\Sigma )}(0,\delta _0)&\ni \eta \mapsto \mathcal {A}\in \mathcal {L}(H^{s+1}(\Omega ;\mathbb {R}^n)) \end{aligned} \end{aligned}$$(4.3)are smooth.
Proof
We will only provide the proof for the case \(\Gamma = \mathbb {R}^{n-1}\), as the case \(\Gamma = \mathbb {T}^{n-1}\) is similar but simpler. Since \(s+1 > n/2\), the existence of a \(\delta _1 >0\) such that \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _1\) implies the bound (4.1) follows readily from the results in Theorems A.4 and A.7, which show that \(\mathfrak {P}\eta \in \mathbb {P}^{s+2}(\Omega ) \hookrightarrow C^1_b(\Omega )\). With this bound in hand, we may appeal to (2.3) to write
In turn, (2.4) allows us to write
and we conclude that \(\mathfrak {K}\) and \(\mathcal {B}\), and hence \(\mathfrak {J}= \mathfrak {K}^{-1}\), \(\mathcal {M}=I + \mathcal {B}\), and \(\mathcal {A}= \mathfrak {J}(I+\mathcal {B})^T (I+ \mathcal {B})\) are well-defined when \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _1\).
Next we use Theorem A.5 (again noting that \(s+1> n/2\)) to see that the power series
converges and defines an analytic function for \(\left\| z\right\| _{\mathbb {P}^{s+1}} < \delta _2\), for some \(\delta _2 >0\). Again employing Theorems A.4 and A.7, we may choose \(\delta _3 >0\) such that \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _3\) implies that \(\left\| b^{-1} \mathfrak {P}\eta + \tilde{b}\partial _n \mathfrak {P}\eta \right\| _{\mathbb {P}^{s+1}} < \delta _2\).
Set \(\delta _0 = \min \{\delta _1,\delta _2,\delta _3\}\). Then for \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _0\) we have that \(\mathfrak {K}\), \(\mathfrak {J}\), \(\mathcal {M}\), and \(\mathcal {A}\) are well-defined pointwise, and the formulas (4.4) and (4.5) then show that the maps given in (4.2) and (4.3) are smooth.
Finally, suppose \(\left\| \eta \right\| _{\mathcal {H}^{s+3/2}} < \delta _0\). Then according to Theorems A.7 and A.4, the map \(\mathfrak {F}_\eta \) is \(C^{1 + \lfloor s-n/2+1 \rfloor }\). Moreover, the bound (4.1) implies that for each \(x' \in \mathbb {R}^{n-1}\), the map \((-b,0) \ni x_n \mapsto e_n \cdot \mathfrak {F}_\eta (x',x_n) \in (-b, \eta (x'))\) is increasing since its derivative is greater than 1/2 everywhere. From this and the fact that \(\mathfrak {F}_\eta (x)' = x'\) we conclude that \(\mathfrak {F}_\eta \) is a bijection from \(\Omega \) to \(\Omega _\eta \). On the other hand, the bound (4.1) also shows that \(\det \nabla \mathfrak {F}_\eta (x) \geqq 1/2\) for \(x \in \Omega \), so \(\mathfrak {F}_\eta \) is a \(C^{1 + \lfloor s-n/2 +1\rfloor }\) diffeomorphism by virtue of the inverse function theorem. \(\square \)
We next introduce some useful notation.
Definition 4.2
Let \(0 \leqq s \in \mathbb {R}\) and V be a finite dimensional real inner-product space. We define the bounded linear map \(L_\Omega : H^s(\Gamma ;V) \rightarrow H^s(\Omega ;V)\) via \(L_\Omega f(x) = f(x')\).
The next theorem verifies that the nonlinear maps associated to the problem 2.8 are well-defined and \(C^1\), which is essential for our subsequent use of the implicit function theorem.
Theorem 4.3
Let \(n/2-1 < s \in \mathbb {N}\) and for \(\delta >0\) define the set
There exists a constant \(\delta >0\) such that if \(\gamma \in \mathbb {R}\), \(\varphi _0 \in H^{s+3/2}(\Sigma )\), \(\varphi _1 \in H^{s+3}(\Gamma \times \mathbb {R})\), \(\mathfrak {f}_0 \in H^{s+1}(\Omega ;\mathbb {R}^n)\), \(\mathfrak {f}_1 \in H^{s+2}(\Gamma \times \mathbb {R};\mathbb {R}^n)\), and \((u,p,\eta ) \in U^s_\delta \), and we define \(F: \Omega \rightarrow \mathbb {R}^n\), \(G: \Omega \rightarrow \mathbb {R}\), and \(H,K: \Sigma \rightarrow \mathbb {R}\) via
where \(\mathfrak {F}_\eta \), \(\mathfrak {J}\), \(\mathcal {M}\), and \(\mathcal {A}\) are determined by \(\eta \) via (2.1), (2.3), (2.4), and (2.9), then \((F,G,H,K) \in V^s\), where \(V^s\) is as in Definition 3.14. Moreover, the map
defined by \(\Psi (\gamma , \varphi _0, \varphi _1, \mathfrak {f}_0, \mathfrak {f}_1, u,p,\eta ) = (F,G,H,K)\) is \(C^1\).
Proof
Again, we will only write the proof for the case \(\Gamma = \mathbb {R}^{n-1}\), as the case \(\Gamma = \mathbb {T}^{n-1}\) is similar but simpler.
Let \(\delta >0\) be the smaller of \(\delta _0>0\) from Proposition 4.1 and \(\delta _*>0\) from Corollary A.12. Proposition 4.1, Theorems A.4 and A.7, and the first item of Corollary A.12, applied with \(r = \sigma = s+1\), show that the map
is well-defined and \(C^1\). Next, we note that the maps
are smooth since the former is linear and the latter is a sum of the linear trace map and a quadratic map. Finally, Proposition 4.1 and the second item of Corollary A.12, applied with \(r = \sigma +1 = s+2\), show that the map \((\varphi _0,\varphi _1,p,\eta ) \mapsto p -\varphi _0 - \varphi _1 \circ \mathfrak {F}_\eta \vert _{\Sigma }\) is \(C^1\). These combine to show initially that \(\Psi \) is well-defined and \(C^1\) as a map into \(H^{s+1}(\Omega ;\mathbb {R}^n) \times H^{s}(\Omega ) \times H^{s+1/2}(\Sigma ) \times H^{s+3/2}(\Sigma )\).
On the other hand, the map
is well-defined (thanks to Theorem A.2) and quadratic, and thus smooth. Combining the above observations with Remark 3.15, we conclude that \(\Psi \) is actually a \(C^1\) mapping into the space \(V^s\). \(\square \)
4.2 Invoking the Implicit Function Theorem: Proof of the Main Existence Theorem
Finally, we are ready to invoke the implicit function theorem to prove the existence of solutions to the system (2.8).
Proof of Theorem 1.1
For the sake of brevity, write \(X^s = \mathbb {R}\times H^{s+3/2}(\Sigma ) \times H^{s+3}(\Gamma \times \mathbb {R}) \times H^{s+1}(\Omega ;\mathbb {R}^n) \times H^{s+2}(\Gamma \times \mathbb {R};\mathbb {R}^n)\) and \(W^s ={_n}H^{s+1}(\Omega ;\mathbb {R}^n) \times H^{s+2}(\Omega ) \times \mathcal {H}^{s+3/2}(\Sigma )\) and note that \(U^s_\delta \subseteq W^s\) is an open subset. Let \(\Psi : X^s \times U^s_\delta \rightarrow V^s\) be the \(C^1\) map given in Theorem 4.3 and note that a given tuple \((\gamma , \varphi _0, \varphi _1, \mathfrak {f}_0, \mathfrak {f}_1, u,p,\eta ) \in X^s \times U^s_\delta \) satisfies (1.17) if and only if \(\Psi (\gamma , \varphi _0, \varphi _1, \mathfrak {f}_0, \mathfrak {f}_1, u,p,\eta ) = (0,0,0,0) \in V^s\).
Given the product structure \(\Psi : X^s \times U^s_\delta \rightarrow V^s\), we can construct the partial derivatives of \(\Psi \) with respect to each factor:
It is then a simple matter to check that, for \(\gamma \in \mathfrak {C}\),
where \(T_3: W^s \rightarrow V^s\) is the isomorphism constructed in Theorem 3.17. For any \(\gamma _*\in \mathfrak {C}\) we may then employ the implicit function theorem to find open sets \(\mathcal {D}^s(\gamma _*) \subseteq X^s\) and \(\mathcal {S}^s(\gamma _*) \subseteq U^s_\delta \) and a \(C^1\) and Lipschitz function \(\Xi _{\gamma _*}: \mathcal {D}^s(\gamma _*) \rightarrow \mathcal {S}^s(\gamma _*)\) such that \((\gamma _*,0,0,0,0) \in \mathcal {D}^s(\gamma _*)\), \((0,0,0) \in \mathcal {S}^s(\gamma _*)\), and
for all \((\gamma , \varphi _0,\varphi _1, \mathfrak {f}_0, \mathfrak {f}_1) \in \mathcal {D}^s(\gamma _*)\). The implicit function theorem also guarantees that \(\Xi _{\gamma _*}\) parameterizes the unique such solution within \(\mathcal {S}^s(\gamma _*)\).
We now define the open sets
By construction, we have the inclusions listed in the first item. Using the above analysis, we can define the map \(\Xi : \mathcal {D}^s \rightarrow \mathcal {S}^s\) via \(\Xi (\gamma , \varphi _0,\varphi _1, \mathfrak {f}_0, \mathfrak {f}_1) = \Xi _{\gamma _*} (\gamma , \varphi _0,\varphi _1, \mathfrak {f}_0, \mathfrak {f}_1)\) whenever \((\gamma , \varphi _0,\varphi _1, \mathfrak {f}_0, \mathfrak {f}_1) \in \mathcal {D}^s(\gamma _*)\). This is well-defined, \(C^1\), and locally Lipschitz by the above consequences of the implicit function theorem. The second and third items then follow by setting \((u,p,\eta ) = \Xi (\gamma , \varphi _0,\varphi _1, \mathfrak {f}_0, \mathfrak {f}_1)\). \(\quad \square \)
5 Analysis of the Dirichlet–Neumann Operator
This section is devoted to the analysis of the Dirichlet–Neumann operator \(G(\eta )\) when \(\eta \) is small in \(\mathcal {H}^s\). Since we will primarily work with the horizontal coordinates, it is more convenient to denote a point in \(\Omega _\eta \) by (x, y), where \(x\in \mathbb {R}^d\), \(d=n-1\geqq 1\), and \(y\in \mathbb {R}\). Then, we recall that
where \(\psi \) solves the problem
We straighten the domain \(\Omega _\eta =\{(x, y)\in \mathbb {R}^d\times \mathbb {R}: -b<y<\eta (x)\}\) using the mapping
Note that \(\mathfrak {F}_\eta \) is the mapping (2.1) written our new notation. Since
if
then \(\mathfrak {F}_\eta \) is a Lipschitz diffeomorphism. A direct calculation shows that if \(g:\Omega _\eta \rightarrow \mathbb {R}\) then \(\widetilde{g}(x, z):=(g\circ \mathfrak {F}_\eta )(x, z)=g(x, \varrho (x, z))\) satisfies
where
\(I_d\) being the \(d\times d\) identity matrix. \(\mathcal {A}\) is the matrix (2.9) written in our new notation. Since \(\psi \) is harmonic in \(\Omega _\eta \), \(v=\psi \circ \mathfrak {F}_\eta \) satisfies \( {{\,\textrm{div}\,}}_{x, z}(\mathcal {A}\nabla _{x, z}v)=0\) in \(\Omega _\eta \). We write \(\mathcal {A}\) as a perturbation of the identity matrix
Consequently, v satisfies
where
Setting
we decompose \(\Delta _{x, z}=(\partial _z+\mathcal {D}(z))(\partial _z-\mathcal {D}(z))\). Then, (5.9) is equivalent to the following system of forward and backward parabolic equations
Since \(\mathcal {D}(-b)=0\) and \(\partial _zv(x, -b)=\partial _z\varrho (x, -b)\partial _y\phi (x, -b)=0\), we have \(Q_a[v](x, -b)\) and \(w(x, -b)=0\).
By the chain rule and (5.13), the Dirichlet–Neumann operator can be written in terms of f and w as
where we have denoted
Using the identities
we can integrate (5.12) and (5.13) to obtain
for \(z\in [-b, 0]\). It then follows from (5.14) that
Clearly \(R(\eta )f\) is a derivative.
We denote \(I=[-b, 0]\) and
where the definition of the Chemin–Lerner spaces \(\widetilde{L}^p H^s\) is recalled in Definition A.15. In Appendix A.4, we establish estimates in Chemin–Lerner norms for the operators appearing in (5.17) and (5.18). The next proposition uncovers the low-frequency structure and provides the boundedness of the remainder \(R(\eta )\).
Proposition 5.1
Let \(\sigma \geqq \sigma _0> 1+\frac{d}{2}\). There exist a small positive constant \(c_1=c_1(\sigma , \sigma _0, b, d)\) such that if \(\Vert \eta \Vert _{\mathcal {H}^{\sigma _0}}<c_1\) then the following assertions hold.
(1) We have
where and \(C=C(\sigma , \sigma _0, b, d)\).
(2) For any continuous symbol \(\ell :\mathbb {R}\rightarrow \mathbb {R}\) satisfying
we have
where \(C=C(\ell , \sigma , \sigma _0, b, d)\). Moreover, for \(\eta ,~f: \mathbb {T}^d\rightarrow \mathbb {R}\) we have \(\widehat{R(\eta )f}(0)=0\) and
where \(C=C(\sigma , \sigma _0, b, d)\).
Proof
Applying the estimates (A.42) and (A.43) to (5.18) and (5.17), we obtain
and
where \(C=C(b)\). It follows that
On the other hand, equation (5.13) gives \(\partial _zv= \mathcal {D}(z)v+w+Q_a[v]\). Since \(\mathcal {D}(z)=|D|\tanh ((z+b)|D|)\) and \(0\leqq \tanh ((z+b)|\xi |)\leqq 1\) for \(z\in [-b, 0]\), we obtain \(\Vert \mathcal {D}(z)v\Vert _{U^{\sigma -1}}\leqq \Vert \nabla _xv\Vert _{U^{\sigma -1}}\). Combining this with (5.26) and (5.27), we deduce
For \(\sigma \geqq \sigma _0>1+\frac{d}{2}\) and \(\Vert \eta \Vert _{\mathcal {H}^{\sigma _0}}<c_1\) small enough, we can apply the product estimate (A.57) and the nonlinear estimate (A.68) to obtain
where \(\mathcal {F}:\mathbb {R}^+\rightarrow \mathbb {R}^+\) is nondecreasing and depends only on \((\sigma , \sigma _0, b, d)\).
A combination of (5.28), (5.29) and (5.30) yields
where \(C=C(\sigma , \sigma _0, b, d)\). Applying (5.31) with \(\sigma =\sigma _0\) we deduce that there exists \(c_0=c_0(\sigma _0, b, d)>0\) small enough such that if \(\Vert \eta \Vert _{\mathcal {H}^{\sigma _0}}<c_0\), then \(\Vert \nabla _{x,z}v\Vert _{U^{\sigma _0-1}}\leqq C(\sigma _0, b, d)\Vert \nabla _xf\Vert _{H^{\sigma _0-1}}\). Inserting this into (5.31) yields
Therefore, for some \(c_1=c_1(\sigma , \sigma _0, b, d)\leqq c_0\) small enough, we have
provided that \(\Vert \eta \Vert _{\mathcal {H}^{\sigma _0}}<c_1\). This concludes the proof of (5.22).
We turn to prove (5.24). Using the formula (5.20) and the estimate (A.44) we obtain
Noticing that
we deduce
Therefore, (5.24) follows from (5.36), (5.30) and (5.33). Finally, (5.25) can be proved analogously. \(\square \)
Next we establish the contraction estimate for \(R(\eta )\).
Proposition 5.2
Let \(\sigma \geqq \sigma _0>1+\frac{d}{2}\) and consider \(f\in H^\sigma \) and \(\eta _j\in H^\sigma \), \(j=1, 2\). Set \(\eta _\delta =\eta _1-\eta _2\). There exists a positive constant \(c_2=c_2(\sigma , \sigma _0, b, d)\leqq c_1\) such that if \(\Vert \eta _j\Vert _{\mathcal {H}^{\sigma _0}}<c_2\), \(j=1, 2\), then the following estimates hold:
(1) For any symbol \(\ell \) satisfying (5.23), we have
where \(C=C(\ell , \sigma , \sigma _0, b, d)\).
(2) We have
where \(C=C(\sigma , \sigma _0, b, d)\).
Proof
We shall only prove (5.37) since the proof of (5.38) is similar. Consider \(\eta _j,~f: \mathbb {R}^d\rightarrow \mathbb {R}\) such that \(\Vert \eta _j\Vert _{\mathcal {H}^{s_0}}<c_1\). We recall from (5.20) that
where \(\Vert \nabla _{x,z}v_j\Vert _{U^{\sigma -1}}\lesssim \Vert \nabla _xf\Vert _{H^{\sigma -1}}\) by virtue of (5.22).
We shall adopt the notation \(g_\delta =g_1-g_2\). Arguing as in (5.36), we obtain
Using the product estimate (A.57), the nonlinear estimate (A.68) and the bound
one can prove that
Combining this with the easy inequalities
we obtain
Assuming that \(\Vert \eta _j\Vert _{\mathcal {H}^{\sigma _0}}<c_1\), we can invoke the estimate (5.22) to have
Consequently,
On the other hand, the proof of (5.28) and (5.29) yields
Arguing as in the proof of (5.46) one can show that
It follows from (5.47) and (5.48) that
With \(\sigma =\sigma _0\), (5.49) implies that if \(\Vert \eta _j\Vert _{\mathcal {H}^{\sigma _0}}<\widetilde{c_1}\leqq c_1\) then
We then insert the preceding estimate into (5.49) to obtain that if \(\Vert \eta _1\Vert _{\mathcal {H}^{\sigma _0}}<c_2\leqq \widetilde{c_1}\) then
Finally, inserting (5.51) into (5.46) we arrive at (5.37). \(\square \)
6 Stability of Traveling Wave Solutions
In this section, we consider the Muskat problem without external bulk force (\(\mathfrak {f}=0\)). In order to simplify the presentation, we shall assume that the external pressure \(\varphi \) is independent of the vertical variable, i.e. \(\varphi (x, y)=\varphi _0(x)\), where we adopt the notation (x, y) for the horizontal and vertical components of a point in the fluid domain. More precisely, we study equation (2.17) for the free boundary \(\eta \):
The proofs of all the results in this section can be generalized to the more general case \(\varphi (x, y)=\varphi _0(x)+\varphi _1(x, y)\) with extra regularity assumption of \(\varphi _1\) as in Theorem 1.1.
6.1 Existence of Traveling Wave Solutions
The existence and uniqueness of steady solutions to (6.1) have been obtained in Theorem 1.1 by means of the implicit function theorem. In this subsection, we shall apply the results in Sect. 5 for the Dirichlet–Neumann operator to provide an alternative proof in this special case of the data, which serves to motivate and inform the strategy we will employ in studying the time-dependent problem (6.1).
Solutions to the steady equation
shall be constructed by a fixed point argument. To this end, we first use the expansion \(G(\eta )=m(D)+R(\eta )\) to equivalently rewrite (6.2) as
We note that the symbol \(\gamma i\xi _1-m(\xi )\) vanishes only at \(\xi =0\), and it follows from the definition (5.20) of \(R(\eta )\) that the right-hand side of (6.3) vanishes at zero frequency. Therefore, we may seek solutions that vanish at zero frequency by solving the fixed point problem
where we adopt the convention that \(\widehat{\mathcal {T}_{\varphi _0}(\eta )}(0)=0\).
Theorem 6.1
Let \(d\geqq 1\), \(s>1+\frac{d}{2}\) and \(\gamma \in \mathbb {R}{\setminus }\{0\}\). There exist small positive constants \(r_0\) and \(r_1\), both depending only on \((\gamma , s, b, d)\), such that for \(\Vert \nabla {\varphi _0}\Vert _{H^{s-1}(\mathbb {R}^d)}<r_1\), \(\mathcal {T}_{\varphi _0}\) is a contraction mapping on \(B_{\mathcal {H}^s(\mathbb {R}^d)}(0, r_0)\). Moreover, the mapping that maps \({\varphi _0}\) to the unique fixed point of \(\mathcal {T}_{\varphi _0}\) in \(B_{\mathcal {H}^s(\mathbb {R}^d)}(0, r_0)\) is Lipschitz continuous.
Proof
In the finite depth case, we have that \(m(D)=|D|\tanh (b|D)\) satisfies
Consequently, for low frequencies \(|\xi |<1\), we have
where \(\mathscr {F}\) denotes the Fourier transform. On the other hand, for high frequencies \(|\xi |\geqq 1\), we have
We note that the condition \(\gamma \ne 0\) was used in the low-frequency estimate (6.6) but not in the high-frequency estimate (6.7).
It follows from (6.5), (6.6) and (6.7) that
From (6.8) and the definition of \(\mathcal {T}_{\varphi _0}(\eta )\), we deduce
where we have used the fact that the norms \(\Vert m^\frac{1}{2}(D){\varphi _0}\Vert _{H^{s-\frac{1}{2}}}\) and \(\Vert \nabla {\varphi _0}\Vert _{H^{s-1}}\) are equivalent.
Suppose that \(\Vert \eta \Vert _{\mathcal {H}^s}<c_1\), where \(c_1\) is given in Proposition 5.1. Then the estimate (5.24) with \(\sigma =\sigma _0=s\) yields
where we have used the inequality \(\Vert \nabla \eta \Vert _{H^{s-1}}\leqq C\Vert \eta \Vert _{\mathcal {H}^s}\).
It follows from (6.9) and (6.10) that for \(\Vert \eta \Vert _{\mathcal {H}^s}<c_1\) we have
where \(C_1=C_1(\gamma , s, b, d)\). If \(\Vert \nabla {\varphi _0}\Vert _{H^{s-1}}<\frac{1}{2C_1(2C_1+1)}\), then \(\mathcal {T}_{\varphi _0}\) maps the ball \(B_{\mathcal {H}^s}(0, r_0)\subset \mathcal {H}^s\) to itself, where \(r_0=\min \{c_1, \frac{1}{2C_1+1}\}\). By virtue of Proposition 5.2, one can reduce the size of \(\Vert \nabla {\varphi _0}\Vert _{H^{s-1}}\) and the radius \(r_0\) so that \(\mathcal {T}_{\varphi _0}\) is a contraction on \(B_{\mathcal {H}^s}(0, r_0)\). By the Banach contraction mapping principle, \(\mathcal {T}_{\varphi _0}\) has a unique fixed point \(\eta \) in \( B_{\mathcal {H}^s}(0, r_0)\). The Lipschitz continuous dependence of \(\eta \) on \({\varphi _0}\) again follows from Proposition 5.2. \(\square \)
We now define the space of Sobolev functions with average zero on the torus.
Definition 6.2
For \(0 \leqq s \in \mathbb {R}\) we define the space \(\mathring{H}^s(\mathbb {T}^d)=\big \{f\in H^s(\mathbb {T}^d) \;\vert \;\int _{\mathbb {T}^d} f=0\big \}\).
We now record a variant of Theorem 6.1 for the torus case.
Theorem 6.3
Let \(d\geqq 1\), \(s>1+\frac{d}{2}\) and \(\gamma \in \mathbb {R}\). There exist small positive constants \(r_0\) and \(r_1\), both depending only on (s, b, d), such that for \(\Vert \nabla {\varphi _0}\Vert _{H^{s-1}(\mathbb {T}^d)}<r_1\), \(\mathcal {T}_{\varphi _0}\) is a contraction mapping on \(B_{\mathring{H}^s(\mathbb {T}^d)}(0, r_0)\). Moreover, the mapping that maps \({\varphi _0}\) to the unique fixed point of \(\mathcal {T}_{\varphi _0}\) in \(B_{\mathring{H}^s(\mathbb {T}^d)}(0, r_0)\) is Lipschitz continuous.
Proof
The proof mostly follows from obvious modifications to the proof of Theorem 6.1 with the following caveat. Since the zero mode of \(\mathcal {T}_{\varphi _0}\) vanishes in the periodic setting, there is no need for a low frequency estimate such as (6.6) and only the high frequency estimate (6.7) is needed. Consequently, \(\gamma \ne 0\) is not needed, and the constants then do not depend on \(\gamma \). \(\square \)
6.2 Stability of Periodic Traveling Wave Solutions
In this subsection, we prove that small periodic traveling wave solutions obtained in Theorem 6.3 are nonlinearly asymptotically stable. The remainder of this section is devoted to the proof of Theorem 1.2.
Suppose that, for fixed \((\gamma , {\varphi _0})\), \(\eta _*\) is a steady solution of (2.17). We perturb \(\eta _*\) by \(f_0\) and set \(f(x, t)=\eta (x, t)-\eta _*(x)\), where \(\eta \) is the solution of (2.17) with initial data \(\eta _0=\eta _*+f_0\). We have that
Using the expansion \(G(\eta )g=m(D)g+R(\eta )f\) we rewrite (6.12) as
The solution of (6.13) with initial data \(f_0\) will be sought as the fixed point of the mapping
where
To that end, we shall appeal to the following fixed point lemma:
Lemma 6.4
Let \((E, \Vert \cdot \Vert )\) be a Banach space and let \(\nu >0\). Denote by \(B_\nu \) the open ball of radius \(\nu \) centered at 0 in E. Assume that \(\mathcal {L}:B_\nu \rightarrow E\) and \(\mathcal {B}:B_\nu \times E\rightarrow E\) satisfy the following conditions.
-
For all \(x\in B_\nu \), \(\mathcal {B}(x, \cdot )\) is linear.
-
There exists a constant \(C_\mathcal {L}\in (0, 1)\) such that \(\Vert \mathcal {L}(x)\Vert \leqq C_\mathcal {L}\Vert x\Vert \) for all \(x\in B_\nu \).
-
There exists an increasing function \(\mathcal {G}_\mathcal {B}\) such that \(\Vert \mathcal {B}(x, y)\Vert \leqq \mathcal {G}_\mathcal {B}(\Vert x\Vert )\Vert y\Vert \) for all \(x\in B_\nu \) and \(y\in E\). There exists \(r_*>0\) such that
$$\begin{aligned} C_\mathcal {L}+\mathcal {G}_\mathcal {B}(r_*)<\frac{1}{2}. \end{aligned}$$(6.17) -
There exists an increasing function \(\mathcal {F}_\mathcal {L}:\mathbb {R}_+\rightarrow \mathbb {R}_+\) such that
$$\begin{aligned} \forall x_1, x_2\in B_\nu ,~\Vert \mathcal {L}(x_1)-\mathcal {L}(x_2)\Vert \leqq \Vert x_1-x_2\Vert \mathcal {F}_\mathcal {L}(\Vert x_1\Vert +\Vert x_2\Vert ). \end{aligned}$$(6.18) -
There exists an increasing function \(\mathcal {F}_\mathcal {B}:\mathbb {R}_+\rightarrow \mathbb {R}_+\) such that
$$\begin{aligned}{} & {} \forall x_1, x_2\in B_\nu ,~\forall y\in E,~\Vert \mathcal {B}(x_1, y)-\mathcal {B}(x_2, y)\Vert \nonumber \\{} & {} \quad \leqq \Vert x_1-x_2\Vert \Vert y\Vert \mathcal {F}_\mathcal {B}(\Vert x_1\Vert +\Vert x_2\Vert ). \end{aligned}$$(6.19)
Assume, moreover, that
Then, for all \(x_0\in E\) with \(\Vert x_0\Vert <\frac{1}{2}\min \{\nu , r_*\}\), the mapping \(B_\nu \ni x\mapsto \mathcal {N}(x):=x_0+\mathcal {L}(x)+\mathcal {B}(x, x) \in B_\nu \) has a unique fixed point \(x_*\) in \(B_\nu \) with \(\Vert x_*\Vert \leqq 2\Vert x_0\Vert \).
Proof
Let \(x_0\in E\), \(\Vert x_0\Vert <\frac{1}{2}\min \{\nu , r_*\}\). The fixed point of \(\mathcal {N}\) will be obtained by the Picard iteration \(x_{n+1}=\mathcal {N}(x_n)\), \(n\geqq 1\). It can be shown using induction with the aid of (6.17) that \(\Vert x_n\Vert <2\Vert x_0\Vert \) for all \(n\geqq 0\), hence \((x_n)\subset B_\nu \). Using the assumptions on \(\mathcal {L}\) and \(\mathcal {B}\), we obtain
Combining (6.21) with (6.20) yields
It follows that \((x_n)\) is a Cauchy sequence, hence \(x_n\rightarrow x_*\in E\). In particular, we have \(\Vert x_*\Vert \leqq 2\Vert x_0\Vert <\nu \), and thus (6.21) implies that \(\mathcal {N}(x_n)\rightarrow \mathcal {N}(x_*)\). Passing to the limit in the scheme \(x_{n+1}=\mathcal {N}(x_n)\) yields \(x_*=\mathcal {N}(x_*)\). The uniqueness of \(x_*\) in \(B_\nu \) again follows from (6.21). \(\square \)
We now have all the tools needed to prove Theorem 1.2.
Proof of Theorem 1.2
We consider \(\eta _*\) and \({\varphi _0}\) in \(H^s(\mathbb {T}^d)\) with \(s>1+\frac{d}{2}\) and \(\Vert {\varphi _0}\Vert _{H^s}<\frac{c_2}{3}\) and \(\Vert \eta _*\Vert _{H^s}<\frac{c_2}{3},\) where \(c_2\) is the constant given in Proposition 5.2.
We note that if \(\int _{\mathbb {T}^d}u=0\) then \(\Delta _0 u=0\) (see (A.33)). Consequently, for \(1\leqq q_2\leqq q_1\leqq \infty \), we have
provided that u and \(g(\cdot , t)\) have zero mean. These estimates can be proved as in Proposition A.17 with the aid of the dyadic estimate
Set \(Y^\mu ([\alpha , \beta ])=\widetilde{L}^\infty ([\alpha , \beta ]; H^{\mu }(\mathbb {T}^d))\cap \widetilde{L}^2([\alpha , \beta ]; H^{\mu +\frac{1}{2}}(\mathbb {T}^d))\) and
Consider \(f_0\in \mathring{H}^s(\mathbb {T}^d)\) and let \(T>0\) be arbitrary. By Proposition 5.1 2), R always has zero mean, and so do \(\mathcal {L}\) and \(\mathcal {B}\). Therefore, using (6.23), (6.24) and the fact that
we obtain
We want to apply Lemma 6.4 with
Let \(B_\nu \) be the open ball in \(E_T\) with center 0 and radius \(\nu <\frac{2c_2}{3}\).
Let \(f\in B_\nu \). We have \(\Vert f\Vert _{L^\infty ([0, T]; H^s)}\leqq \Vert f\Vert _{\widetilde{L}^\infty ([0, T]; H^s)}\), hence \(\Vert f(t)\Vert _{H^s}<\nu <c_2/3\) a.e. \(t\in [0, T]\). Consequently, \(\Vert \eta _*+f(t)\Vert _{H^s}<c_2\) a.e. \(t\in [0, T]\) and thus we can apply the contraction estimate (5.37) with \(\eta _1=\eta _*\), \(\eta _2=\eta _*+f\), \(\sigma =s+\frac{1}{2}\) and \(\sigma _0=s\),
a.e. \(t\in [0, T]\), where \(C=C(s, b, d)\) and we have used the embedding \(H^\mu \subset \mathcal {H}^\mu \) for \(\mu \geqq 0\). Combining (6.29) and (6.32) yields
where \(C=C(s, b, d)\).
Next, for \(g\in B_\nu \), we can apply the remainder estimate (5.24) with \(\sigma =s+\frac{1}{2}\) and \(\sigma _0=s\) to have
a.e. \(t\in [0, T]\), where \(C=C(s, b, d)\). It follows from (6.30) and (6.34) that
where \(C=C(s, b, d)\). By a completely analogous argument, we obtain that
for \(f_1\), \(f_2\in B_\nu \), and
for \(g_1\), \(g_2\in B_\nu \). In both (6.36) and (6.37), \(C=C(s, b, d)\).
In view of (6.33), (6.35), (6.36) and (6.37), we find that the conditions in Lemma 6.4 are satisfied with
where \(C=C(s, b, d)\). According to Theorem 6.3, if \(\Vert \nabla {\varphi _0}\Vert _{H^{s-\frac{1}{2}}}\) is small enough then \(\Vert \eta _*\Vert _{H^{s+\frac{1}{2}}}\leqq C(s, b, d)\Vert \nabla {\varphi _0}\Vert _{H^{s-\frac{1}{2}}}\). Therefore, for sufficiently small \(\Vert \nabla {\varphi _0}\Vert _{H^{s-\frac{1}{2}}}\), we have \(C_\mathcal {L}<\frac{1}{4}\) and the conditions (6.17) and (6.20) hold for sufficiently small \(r_*\) and \(\nu \). Therefore, by virtue of Lemma 6.4 and with the aid of (6.28), if \(\Vert f_0\Vert _{H^s}<\frac{1}{2}\min \{\nu , r_*\}:=\delta \) then \(\mathcal {N}\) has a unique fixed point f in \(B_\nu \), and
Since the smallness of \({\varphi _0}\), \(\nu \) and \(r_*\) is independent of T, f is a global solution.
Next, we prove that f decay exponentially in \(H^s\). Since \(f\in Y^s([0, T])\) for all \(T>0\), using (6.32) and (6.34) we deduce that \(\partial _t f\in L^2([0, T]; H^{s-\frac{1}{2}})\) for all \(T>0\). Then applying Theorem 3.1 in [10] yields \(f\in C([0, T]; H^s)\) for all \(T>0\). Appealing to (6.32) and (6.34) again, we deduce that
where we have used the facts that \(f(\cdot , t)\) has zero mean to get that
It follows that
where
We choose \(\Vert \nabla _x {\varphi _0}\Vert _{H^{s-\frac{1}{2}}}\) small enough so that \(\mu <\frac{c_0}{3C}\), and assume that \(\Vert f_0\Vert _{H^s}<\min \{\delta , \frac{c_0}{3C}\}\). Using the continuity of \(t\mapsto \Vert f(t)\Vert _{H^s}\), we deduce from (6.41) that \(\Vert f(t)\Vert _{H^s}<\frac{c_0}{3C}\) for all \(t>0\), and thus
Therefore, we obtain the exponential decay
and the global dissipation bound
This completes the proof of Theorem 1.2. \(\square \)
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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Acknowledgements
H. Nguyen was supported by an NSF Grant (DMS #2205710). I. Tice was supported by an NSF Grant (DMS #2204912). We thank B. Pausader for discussions on the finite-depth Dirichlet–Neumann operator.
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Appendix A. Analytic Tools
Appendix A. Analytic Tools
This appendix collects a number of analysis tools used throughout the paper.
1.1 A.1 Specialized Scales of Anisotropic Sobolev Spaces
In this subsection we recall the properties of a scale of anisotropic Sobolev spaces introduced in [9].
Definition A.1
Let \(0 \leqq s \in \mathbb {R}\).
-
(1)
We define the anisotropic Sobolev-type space
$$\begin{aligned} \mathcal {H}^s(\mathbb {R}^d) = \{ f \in \mathscr {S}'(\mathbb {R}^d) \;\vert \;f = \bar{f}, \hat{f} \in L^1_{loc}(\mathbb {R}^d), \text { and } \left\| f\right\| _{\mathcal {H}^s} < \infty \}, \nonumber \\ \end{aligned}$$(A.1)where the square-norm is defined by
$$\begin{aligned} \left\| f\right\| _{\mathcal {H}^s}^2 = \int _{B(0,1)} \frac{\xi _1^2 + \left| \xi \right| ^4 }{\left| \xi \right| ^2} \left| \hat{f}(\xi )\right| ^2 \textrm{d}\xi + \int _{B(0,1)^c} \langle \xi \rangle ^{2s} \left| \hat{f}(\xi )\right| ^2 \textrm{d}\xi .\nonumber \\ \end{aligned}$$(A.2)We endow the space \(\mathcal {H}^s(\mathbb {R}^d)\) with the obvious associated inner-product. We write \(\mathcal {H}^s(\Sigma ) = \mathcal {H}^s(\mathbb {R}^{n-1})\) with the usual identification of \(\Sigma \simeq \mathbb {R}^{n-1}\).
-
(2)
We define
$$\begin{aligned} \mathcal {H}^s(\mathbb {T}^d) = \mathring{H}^s(\mathbb {T}^d) = \left\{ f \in H^s(\mathbb {T}^d) \;\vert \;\int _{\mathbb {T}^d} f = 0 \right\} \end{aligned}$$(A.3)with the usual norm.
-
(3)
We write \(\mathcal {H}^s(\Sigma ) = \mathcal {H}^s(\Gamma )\) via the natural identification of \(\Sigma = \Gamma \times \{0\}\) with \(\Gamma \in \{\mathbb {R}^{n-1},\mathbb {T}^{n-1}\}\).
The following result summarizes the fundamental properties of this space:
Theorem A.2
Let \(0 \leqq s \in \mathbb {R}\). Then the following hold.
-
(1)
\(\mathcal {H}^s(\mathbb {R}^d)\) is a Hilbert space, and the set of real-valued Schwartz functions is dense in \(\mathcal {H}^s(\mathbb {R}^d)\).
-
(2)
\(H^s(\mathbb {R}^d) \hookrightarrow \mathcal {H}^s(\mathbb {R}^d)\), and this inclusion is strict for \(d\geqq 2\). If \(d =1\), then \(H^s(\mathbb {R}) = \mathcal {H}^s(\mathbb {R})\).
-
(3)
If \(t \in \mathbb {R}\) and \(s < t\), then we have the continuous inclusion \(\mathcal {H}^t(\mathbb {R}^d) \hookrightarrow \mathcal {H}^s(\mathbb {R}^d)\).
-
(4)
For \(R \in \mathbb {R}_+\) and \(f \in \mathcal {H}^s(\mathbb {R}^d)\) define the low-frequency localization \(f_{l,R} = (\hat{f} {\chi }_{B(0,R)})^{\vee }\) and the high-frequency localization \(f_{h,R} = (\hat{f} {\chi }_{B(0,R)^c})^\vee \). Then \(f_{l,R} \in \bigcap _{t \geqq 0} \mathcal {H}^t(\mathbb {R}^d) \cap \bigcap _{k\in \mathbb {N}} C^k_0(\mathbb {R}^d)\), and \(f_{h,R} \in \mathcal {H}^s(\mathbb {R}^d) \cap H^s(\mathbb {R}^d)\). Moreover, we have the estimates
$$\begin{aligned} \left\| f_{l,R}\right\| _{\mathcal {H}^s} \leqq \left\| f\right\| _{\mathcal {H}^s} \text { and } \left\| f_{h,R}\right\| _{\mathcal {H}^s} \leqq \left\| f\right\| _{\mathcal {H}^s} \end{aligned}$$(A.4)as well as
$$\begin{aligned} \left\| f_{l,R}\right\| _{C^k_0} \lesssim _k \left\| f\right\| _{\mathcal {H}^s}, \left\| f_{l,R}\right\| _{\mathcal {H}^t} \lesssim \left\| f\right\| _{\mathcal {H}^s}, \text { and } \left\| f_{h,R}\right\| _{H^s} \lesssim \left\| f\right\| _{\mathcal {H}^s}. \nonumber \\ \end{aligned}$$(A.5) -
(5)
For each \(k \in \mathbb {N}\) we have the continuous inclusion \(\mathcal {H}^s(\mathbb {R}^d) \hookrightarrow C^k_0(\mathbb {R}^d) + H^s(\mathbb {R}^d)\).
-
(6)
If \(s > d/2\) then \(\hat{f}\in L^1(\mathbb {R}^d;\mathbb {C})\), and
$$\begin{aligned} \left\| \hat{f}\right\| _{L^1} \lesssim \left\| f\right\| _{\mathcal {H}^s}. \end{aligned}$$(A.6) -
(7)
If \(k \in \mathbb {N}\) and \(s >k+ d/2\), then we have the continuous inclusion \(\mathcal {H}^s(\mathbb {R}^d) \hookrightarrow C^k_0(\mathbb {R}^d)\).
-
(8)
If \(s \geqq 1\), then \(\left\| \nabla f\right\| _{H^{s-1}} \lesssim \left\| f\right\| _{\mathcal {H}^s}\) for each \(f \in \mathcal {H}^s(\mathbb {R}^d)\). In particular, we have that \(\nabla : \mathcal {H}^s(\mathbb {R}^d) \rightarrow H^{s-1}(\mathbb {R}^d;\mathbb {R}^d)\) is a bounded linear map, and this map is injective.
-
(9)
If \(s \geqq 1\), then \(\left[ \partial _1 f\right] _{\dot{H}^{-1}} \lesssim \left\| f\right\| _{\mathcal {H}^s}\) for \(f \in \mathcal {H}^s(\mathbb {R}^d)\). In particular, we have that \(\partial _1: \mathcal {H}^s(\mathbb {R}^d) \rightarrow H^{s-1}(\mathbb {R}^d)\cap \dot{H}^{-1}(\mathbb {R}^d)\) is a bounded linear map, and this map is injective.
Proof
These are proved in Proposition 5.2 and Theorems 5.5 and 5.6 of [9]. \(\square \)
Next we recall another space introduced in [9] that will be useful in working with the Poisson extension of functions in \(\mathcal {H}^s(\Sigma )\).
Definition A.3
Let \(0 \leqq s \in \mathbb {R}\) and \(n \geqq 2\).
-
(1)
When \(\Gamma = \mathbb {R}^{n-1}\) we define the space
$$\begin{aligned} \mathbb {P}^s(\Omega )= & {} H^s(\Omega ) + \mathcal {H}^s(\Sigma ) = \{f \in L^1_{\text {loc}}(\Omega ) \;\vert \;\text {there exist } g \in H^s(\Omega ) \text { and } h \in \mathcal {H}^s(\Sigma ) \nonumber \\{} & {} \text { such that } f(x) = g(x) + h(x') \text { for almost every }x \in \Omega \}. \end{aligned}$$(A.7)We endow \(\mathbb {P}^s(\Omega )\) with the norm
$$\begin{aligned} \left\| f\right\| _{\mathbb {P}^s} = \inf \{ \left\| g\right\| _{H^s} + \left\| h\right\| _{\mathcal {H}^s} \;\vert \;f = g +h \text { for } g \in H^s(\Omega ), h\in \mathcal {H}^s(\mathbb {R}^{n-1}) \}. \nonumber \\ \end{aligned}$$(A.8) -
(2)
When \(\Gamma = \mathbb {T}^{n-1}\) we define the space \(\mathbb {P}^s(\Omega ) = H^s(\Omega )\) and endow it with the usual \(H^s(\Omega )\) norm.
The key properties of this space are recorded in the following.
Theorem A.4
Let \(0 \leqq s \in \mathbb {R}\) and \(n \geqq 2\). Then the following hold.
-
(1)
If \(t \in \mathbb {R}\) and \(s < t\), then we have the continuous inclusion \(\mathbb {P}^t(\Omega ) \subset \mathbb {P}^s(\Omega )\).
-
(2)
For each \(f \in \mathcal {H}^s(\Sigma )\) we have that \(\left\| f\right\| _{\mathbb {P}^s} \leqq \left\| f\right\| _{\mathcal {H}^s}\), and hence we have the continuous inclusion \(\mathcal {H}^s(\Sigma ) \subset \mathbb {P}^s(\Omega )\).
-
(3)
If \(k \in \mathbb {N}\) and \(s >k+ n/2\), then \(\left\| f\right\| _{C^k_b} \lesssim \left\| f\right\| _{\mathbb {P}^s}\) for all \(f \in \mathbb {P}^s(\Omega )\). Moreover, we have the continuous inclusion
$$\begin{aligned} \mathbb {P}^s(\Omega ) \subseteq \{f \in C^k_b(\Omega ) \;\vert \;\lim _{\left| x'\right| \rightarrow \infty } \partial ^\alpha f(x) = 0 \text { for } \left| \alpha \right| \leqq k\} \subset C^k_b(\Omega ). \nonumber \\ \end{aligned}$$(A.9) -
(4)
If \(s \geqq 1\), then \(\left\| \nabla f\right\| _{H^{s-1}} \lesssim \left\| f\right\| _{\mathbb {P}^s}\) for each \(f \in \mathbb {P}^s(\Omega )\). In particular, we have that \(\nabla : \mathbb {P}^s(\Omega ) \rightarrow H^{s-1}(\Omega ;\mathbb {R}^n)\) is a bounded linear map.
-
(5)
If \(s > 1/2\), then the trace map \({{\,\textrm{Tr}\,}}: H^s(\Omega ) \rightarrow H^{s-1/2}(\Sigma )\) extends to a bounded linear map \({{\,\textrm{Tr}\,}}: \mathbb {P}^s(\Omega ) \rightarrow \mathcal {H}^{s-1/2}(\Sigma )\). More precisely, the following hold.
-
(a)
If \(f \in C^0(\bar{\Omega }) \cap \mathbb {P}^s(\Omega )\), then \({{\,\textrm{Tr}\,}}f = f \vert _{\Sigma }\).
-
(b)
If \(\varphi \in C_c^1(\mathbb {R}^{n-1} \times (-b,0])\), then
$$\begin{aligned} \int _{\Sigma } {{\,\textrm{Tr}\,}}f \varphi = \int _{\Omega } \partial _n f \varphi + f \partial _n \varphi \text { for all } f \in \mathbb {P}^s(\Omega ). \end{aligned}$$(A.10) -
(c)
We have the bound \(\left\| {{\,\textrm{Tr}\,}}f\right\| _{\mathcal {H}^{s-1/2}} \lesssim \left\| f\right\| _{\mathbb {P}^s}\) for all \(f\in \mathbb {P}^s(\Omega )\).
-
(a)
Proof
In the case \(\Gamma = \mathbb {R}^{n-1}\) these are proved in Theorems 5.7, 5.9, and 5.11 of [9]. In the case \(\Gamma = \mathbb {T}^{n-1}\) they follow from standard properties of Sobolev spaces. \(\square \)
Next we record a crucial fact about the \(\mathbb {P}^s\) spaces: they give rise to standard \(H^s\) multipliers.
Theorem A.5
Let \(n \geqq 2\) and \(s > n/2\). Then for \(0 \leqq r \leqq s\)
In particular, for \(1 \leqq k \in \mathbb {N}\) the mapping
is a bounded \((k+1)-\)linear map.
Proof
When \(\Gamma = \mathbb {T}^{n-1}\) this follows from the standard properties of Sobolev spaces. Assume then that \(\Gamma = \mathbb {R}^{n-1}\). The bound (A.11) is proved for \(r=s\) in Theorem 5.14 in [9]. When \(r=0\), the bound (A.11) follows immediately from the third item of Theorem A.4. The general case \(0< r < s\) then follows from these endpoint bounds and interpolation (see, for instance, [2, 15]). \(\square \)
1.2 A.2 Poisson Extension
We now study the Poisson extension operator, first on standard Sobolev spaces.
Proposition A.6
Let \(-1/2 \leqq s \in \mathbb {R}\). For \(\eta \in H^s(\Sigma )\) define \(\mathfrak {P}\eta : \Omega \rightarrow \mathbb {R}\) via
Then \(\mathfrak {P}\eta \in H^{s+1/2}(\Omega )\) and \(\left\| \mathfrak {P}\eta \right\| _{H^{s+1/2}} \lesssim \left\| \eta \right\| _{H^{s}}\). In particular, \(\mathfrak {P}: H^{s}(\Sigma ) \rightarrow H^{s+1/2}(\Omega )\) defines a bounded linear map.
Proof
We’ll only present the proof in the case \(\Gamma = \mathbb {R}^{n-1}\), and the case \(\Gamma = \mathbb {T}^{n-1}\) is similar and simpler. First note that
Suppose initially that \(m \in \mathbb {N}\) and that \(\eta \in H^{m-1/2}(\Sigma )\). Using (A.14), we may readily bound
Thus, \(\mathfrak {P}: H^{m-1/2}(\Sigma ) \rightarrow H^m(\Omega )\) defines a bounded linear operator for every \(m \in \mathbb {N}\). Standard interpolation theory (see, for instance, [2, 15]) then shows that \(\mathfrak {P}: H^{t-1/2}(\Sigma ) \rightarrow H^t(\Omega )\) defines a bounded linear operator for every \(0 \leqq t \in \mathbb {R}\), which is the desired result upon setting \(t-1/2 = s\). \(\square \)
Next we consider the Poisson extension on the anisotropic spaces \(\mathcal {H}^s(\Sigma )\), which requires the use of the \(\mathbb {P}\) spaces from Definition A.3.
Theorem A.7
Let \(0 \leqq s \in \mathbb {R}\) and \(\Gamma = \mathbb {R}^{n-1}\). For \(\eta \in \mathcal {H}^s(\Sigma )\) define \(\mathfrak {P}\eta : \Omega \rightarrow \mathbb {R}\) via
Then the following hold:
-
(1)
\(\mathfrak {P}\eta - \eta _l \in H^{s+1/2}(\Omega )\) and \(\left\| \mathfrak {P}\eta - \eta _l\right\| _{H^{s+1/2}} \lesssim \left\| \eta \right\| _{\mathcal {H}^s}\), where \(\eta _l = \eta _{l,1} \in \mathcal {H}^{s+1/2}(\Sigma ) \cap \bigcap _{k \in \mathbb {N}} C^k_0(\Sigma )\) in the notation of Theorem A.2.
-
(2)
\(\mathfrak {P}\eta \in \mathbb {P}^{s+1/2}(\Omega )\) and \(\left\| \mathfrak {P}\eta \right\| _{\mathbb {P}^{s+1/2}} \lesssim \left\| \eta \right\| _{\mathcal {H}^{s}}\).
-
(3)
The induced map \(\mathfrak {P}: \mathcal {H}^{s}(\Sigma ) \rightarrow \mathbb {P}^{s+1/2}(\Omega )\) is bounded and linear.
Proof
We split \(\eta \) into its high and low frequency parts: \(\eta = \eta _h + \eta _l\), where \(\hat{\eta }_h = {\chi }_{B(0,1)^c} \hat{\eta }\) and \(\hat{\eta }_{l} = {\chi }_{B(0,1)} \hat{\eta }\). Then we know from Theorem A.2 that \(\eta _l,\eta _h \in \mathcal {H}^s(\Sigma )\) with \(\left\| \eta _{j}\right\| _{\mathcal {H}^s} \leqq \left\| \eta \right\| _{\mathcal {H}^s}\) for \(j \in \{l,h\}\). We also know that \(\eta _h \in H^s(\Sigma )\) with \(\left\| \eta _h\right\| _{H^s} \lesssim \left\| \eta \right\| _{\mathcal {H}^s}\). Consequently, Proposition A.6 shows that \(\mathfrak {P}\eta _h \in H^{s+1/2}(\Omega )\) with
Now consider the function \(\mathfrak {P}\eta _l - \eta _l: \Omega \rightarrow \mathbb {R}\), which satisfies
We calculate
For any \(m \in \mathbb {N}\) this allows us to bound
Thus, \(\mathfrak {P}\eta _l - \eta _l \in \bigcap _{m \in \mathbb {N}} H^m(\Omega )\), but in particular we can choose a fixed \(s+1/2 \leqq m \in \mathbb {N}\) to see that \(\mathfrak {P}\eta _l - \eta _l \in H^{s+1/2}(\Omega )\) with
Finally, note that \(\eta _l \in \bigcap _{t > 0} \mathcal {H}^t(\Sigma )\) and that
In particular, \(\eta _l \in \mathcal {H}^{s+1/2}(\Sigma )\) with \(\left\| \eta _l\right\| _{\mathcal {H}^{s+1/2}} \leqq \left\| \eta \right\| _{\mathcal {H}^s}\). We may thus combine (A.17), (A.21), and (A.22) to see that \(\mathfrak {P}\eta = [\mathfrak {P}\eta _h + (\mathfrak {P}\eta _l - \eta _l)] + \eta _l\) with
Hence, \(\mathfrak {P}\eta \in \mathbb {P}^{s+1/2}(\Omega )\) with \(\left\| \mathfrak {P}\eta \right\| _{\mathbb {P}^{s+1/2}} \lesssim \left\| \eta \right\| _{\mathcal {H}^s},\) which is the desired bound. \(\square \)
Finally, we record some results about the normal derivative of the Poisson extension.
Proposition A.8
Let \(0 \leqq s \in \mathbb {R}\) and \(\eta \in \mathcal {H}^{s+3/2}(\Sigma )\). Then the following hold.
-
(1)
If \(\Gamma = \mathbb {R}^{n-1}\), then \(\partial _n \mathfrak {P}\eta (\cdot ,0) - \partial _n \mathfrak {P}\eta (\cdot ,-b) \in \dot{H}^{-1}(\mathbb {R}^{n-1})\) and
$$\begin{aligned} \left[ \partial _n \mathfrak {P}\eta (\cdot ,0) - \partial _n \mathfrak {P}\eta (\cdot ,-b)\right] _{\dot{H}^{-1}} \leqq b \left\| \nabla \mathfrak {P}\eta \right\| _{L^2} \lesssim \left\| \eta \right\| _{\mathcal {H}^{s+3/2}}. \end{aligned}$$(A.24) -
(2)
If \(\Gamma = \mathbb {T}^{n-1}\), then \(\widehat{\partial _n \mathfrak {P}\eta \vert _{\Sigma }}(0) = \widehat{\partial _n \mathfrak {P}\eta \vert _{\Sigma _{-b}}}(0)=0\).
Proof
We’ll only prove the first item, as the second is simpler and similar. Theorem A.7 tells us that \(\mathfrak {P}\eta \in \mathbb {P}^{s+2}(\Omega )\), and so Theorem A.4 then implies that \(\partial _n \mathfrak {P}\eta \in H^{s+1}(\Omega )\). Note that \(\Delta \mathfrak {P}\eta =0\) in \(\Omega \). Using this and the absolute continuity of Sobolev functions on lines (see, for instance, Theorem 11.45 in [8]), we may then compute
Thus, Cauchy–Schwarz, Fubini–Tonelli, and Parseval imply that
The stated inequality follows from this and Theorems A.4 and A.7. \(\square \)
1.3 A.3 Composition
In this subsection we aim to study some composition operators. We begin by introducing some notation that allows us to extend the flattening maps to full space.
Definition A.9
Let \(\chi \in C^\infty _c(\mathbb {R})\) be such that \(0 \leqq \chi \leqq 1\), \(\chi =1\) on \([-2b,2b]\), and \({{\,\textrm{supp}\,}}(\chi ) \subset (-3b,3b)\). Given \(\eta \in \mathcal {H}^{\sigma +1/2}(\Sigma )\) define \(\mathfrak {E}_\eta : \Gamma \times \mathbb {R}\rightarrow \Gamma \times \mathbb {R}\) via
where \(E: L^2(\Omega ) \rightarrow L^2(\Gamma \times \mathbb {R})\) is a Stein extension operator, \(\mathfrak {P}\) is the Poisson extension as defined in Proposition A.7 and Theorem A.7, and when \(\Gamma = \mathbb {R}^{n-1}\) we take \(\eta _l = \eta _{l,1} \in \bigcap _{t \geqq 0} \mathcal {H}^{t}(\Sigma ) \cap \bigcap _{k \in \mathbb {N}} C^k_0(\Sigma )\) in the notation of Theorem A.2, while when \(\Gamma = \mathbb {T}^{n-1}\) we take \(\eta _l =0\). Note that Proposition A.6 and Theorem A.7 show that \(\mathfrak {P}\eta - \eta _l \in H^{\sigma +1}(\Omega )\), and since the Stein extension restricts to a bounded map \(E: H^{\sigma +1}(\Omega ) \rightarrow H^{\sigma +1}(\Gamma \times \mathbb {R})\) we have that \(E(\mathfrak {P}\eta -\eta _l) \in H^{\sigma +1}(\Gamma \times \mathbb {R})\).
Next we record some properties of these maps.
Proposition A.10
Let \(\sigma > n/2\), \(\eta \in \mathcal {H}^{\sigma +1/2}(\Sigma )\), and define \(\mathfrak {E}_\eta : \Gamma \times \mathbb {R}\rightarrow \Gamma \times \mathbb {R}\) as in Definition A.9. Then the following hold.
-
(1)
The map \(\mathfrak {E}_\eta \) is Lipschitz and \(C^1\), and \(\left\| \nabla \mathfrak {E}_\eta - I\right\| _{C^0_b} \lesssim \left\| \eta \right\| _{\mathcal {H}^{s+1/2}}\).
-
(2)
If V is a real finite dimensional inner-product space and \(0 \leqq r \leqq \sigma \), then
$$\begin{aligned} \sup _{1\leqq j,k \leqq n} \left\| \partial _j \mathfrak {E}_\eta \cdot e_k f \right\| _{H^r} \lesssim (1 + \left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}}) \left\| f\right\| _{H^r} \end{aligned}$$(A.28)and
$$\begin{aligned} \sup _{1\leqq j,k \leqq n} \left\| (\partial _j \mathfrak {E}_\eta \cdot e_k - \partial _j \mathfrak {E}_\zeta \cdot e_k ) f \right\| _{H^r} \lesssim \left\| \eta -\zeta \right\| _{\mathcal {H}^{\sigma +1/2}} \left\| f\right\| _{H^r} \end{aligned}$$(A.29)for every \(\eta ,\zeta \in \mathcal {H}^{\sigma +1/2}(\Sigma )\) and \(f \in H^r(\Gamma \times \mathbb {R};V)\).
-
(3)
There exists \(0< \delta _*<1\) such that if \(\left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}} < \delta _*\), then \(\mathfrak {E}_\eta \) is a bi-Lipschitz homeomorphism and a \(C^1\) diffeomorphism, and we have the estimate \(\left\| \nabla \mathfrak {E}_\eta - I\right\| _{C^0_b} < 1/2\).
Proof
First note that \(\sigma +1 >n/2 +1\), so Proposition A.6, Theorem A.7 and standard Sobolev embeddings show that \(E(\mathfrak {P}\eta - \eta _l) \in C^1_b(\Gamma \times \mathbb {R})\). On the other hand, \(\eta _l \in \bigcap _{t \geqq 0} \mathcal {H}^t(\Sigma )\), so Theorem A.2 shows that \(\eta _l \in C^1_b(\Sigma )\). These observations and their associated bounds then imply the first item. Next we write \(\mathfrak {E}_\eta = I + \omega e_n\) so that \(\nabla \mathfrak {E}_\eta = I + e_n \otimes \nabla \omega \). To prove the second item it suffices to show that \(\left\| \partial _j \omega f\right\| _{H^r} \lesssim \left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}} \left\| f\right\| _{H^r}\) for \(0 \leqq r \leqq \sigma \) and \(1 \leqq j \leqq n\). To establish this we observe that on the one hand, thanks to Theorem A.2, \(\chi \eta _l \in \bigcap _{k \in \mathbb {N}} C^k_0(\Gamma \times \mathbb {R})\), and on the other \(E(\mathfrak {P}\eta - \eta _l) \in H^{\sigma +1}(\Gamma \times \mathbb {R})\). Thus, \(\partial _j \omega \) consists of linear combinations of terms in \(\bigcap _{k \in \mathbb {N}} C^k_0(\Gamma \times \mathbb {R})\) and in \(H^{\sigma }(\mathbb {R}^n)\), and so the sufficient bound follows from standard Sobolev multiplier results (see, for instance, Lemma A.8 in [9]).
To prove the third item we note that if \(\omega \) has Lipschitz constant less than unity, then \(\omega e_n\) is contractive on \(\mathbb {R}^n\), and so the Banach fixed point theorem implies that \(\mathfrak {E}_\eta \) is a bi-Lipschitz homeomorphism. To control the Lipschitz constant of \(\omega \) we use the supercritical Sobolev embeddings as above to verify that this constant is less than unity provided that \(\left\| \eta \right\| _{\mathcal {H}^{\sigma +1/2}} < \delta _*\) for some sufficiently small universal constant \(\delta _*\in (0,1)\). \(\square \)
The next result studies the smoothness properties of composition with the maps from Definition A.9.
Theorem A.11
Let \(n/2 < \sigma \in \mathbb {N}\), \(0< \delta _*<1\) be as in the third item of Proposition A.10, and V be a real finite dimensional inner-product space. Let \(r \in \mathbb {N}\) satisfy \(0 \leqq r \leqq \sigma +1\) and let \(k \in \{0,1\}\). Consider the map \(\Lambda : H^{r+k}(\Gamma \times \mathbb {R};V) \times B_{\mathcal {H}^{\sigma +1/2}(\Sigma )}(0,\delta _*) \rightarrow H^{r}(\Gamma \times \mathbb {R};V)\) given by \(\Lambda (f,\eta ) = f\circ \mathfrak {E}_\eta ,\) where \(\mathfrak {E}_\eta : \Gamma \times \mathbb {R}\rightarrow \Gamma \times \mathbb {R}\) is as defined in Definition A.9. Then \(\Lambda \) is well-defined and \(C^k\), and if \(k =1\) then \(D\Lambda (f,\eta )(g,\zeta ) = \chi \tilde{b} (\eta _l + E(\mathfrak {P}\eta - \eta _l) (\partial _n f \circ \mathfrak {E}_\eta )\zeta + g \circ \mathfrak {E}_\eta ,\) where \(\tilde{b}(x) = (1+x_n/b)\).
Proof
With Proposition A.10 established, the result follows from minor and evident modifications of the argument used to prove Theorem 1.1 in [6] (see also Theorem 5.20 in [9]). \(\square \)
Finally, as a byproduct of this theorem we obtain smoothness properties associated to composition with the flattening maps \(\mathfrak {F}_\eta \).
Corollary A.12
Let \(n/2 < \sigma \in \mathbb {N}\), \(0< \delta _*<1\) be as in the third item of Proposition A.10, and V be a real finite dimensional inner-product space. Let \(r \in \mathbb {N}\) satisfy \(0 \leqq r \leqq \sigma +1\). For \(\eta \in \mathcal {H}^{\sigma +1/2}(\Sigma )\) define \(\mathfrak {F}_\eta : \Omega \rightarrow \Omega _\eta \) via (2.1). Then the following hold.
-
(1)
The map \(\Lambda _\Omega : H^{r+1}(\Gamma \times \mathbb {R};V) \times B_{\mathcal {H}^{\sigma +1/2}(\Sigma )}(0,\delta _*) \rightarrow H^{r}(\Omega ;V)\) given by \(\Lambda (f,\eta ) = f\circ \mathfrak {F}_\eta \) is well-defined and \(C^1\) with \(D\Lambda _\Omega (f,\eta )(g,\zeta ) = \tilde{b} \mathfrak {P}\eta (\partial _n f \circ \mathfrak {F}_\eta )\zeta + g \circ \mathfrak {F}_\eta ,\) where \(\tilde{b}(x) = (1+x_n/b)\).
-
(2)
Assume \(r \geqq 1\). Then the map \(\mathfrak {S}_\Sigma : H^{r+1}(\Gamma \times \mathbb {R};V) \times B_{\mathcal {H}^{\sigma +1/2}(\Sigma )}(0,\delta _*) \rightarrow H^{r-1/2}(\Sigma ;V)\) given by \(\mathfrak {S}_\Sigma (f,\eta ) = f \circ \mathfrak {F}_\eta \vert _{\Sigma }\) is well-defined and \(C^1\) with \(D\mathfrak {S}_\Sigma (f,\eta )(g,\zeta ) = \eta (\partial _n f \circ \mathfrak {F}_\eta )\zeta \vert _{\Sigma } + g \circ \mathfrak {F}_\eta \vert _\Sigma \).
Proof
The first item follows from Theorem A.11 and the observation that \(\Lambda _\Omega (f,\eta ) = R_\Omega \Lambda (f,\eta )\), where \(R_\Omega : H^{r}(\Gamma \times \mathbb {R};V) \rightarrow H^{r}(\Omega ;V)\) is the bounded linear map given by restriction to \(\Omega \). This identity follows directly from the fact that, by construction, \(\mathfrak {E}_\eta = \mathfrak {F}_\eta \) in \(\Omega \). The second item follows by composing the first item with the bounded linear trace map. \(\square \)
1.4 A.4 Littlewood–Paley Analysis for the Anisotropic Sobolev Space \(\mathcal {H}^s\)
In this subsection we develop some Littlewood–Paley theory for the anisotropic spaces.
Definition A.13
Let \(\chi \in C^\infty (\mathbb {R}^d)\) be a radial function such that \(\chi (\xi )=1\) for \(|\xi |\leqq \frac{1}{2}\), \(\chi (\xi )=0\) for \(|\xi |\geqq 1\). Set
The Littlewood–Paley dyadic block \( \Delta _j\) is defined by the Fourier multiplier
The low-frequency cut-off operator \( S_j\) is defined by
The above Fourier multipliers can act on functions (distributions) defined on \(\mathbb {R}^d\) or \(\mathbb {T}^d\), and the Fourier transform is defined accordingly. In particular, for \(u:\mathbb {T}^d\rightarrow \mathbb {R}\) we have
Since \(\sum _{j=0}^\infty \varphi _j(\xi )=1\) for all \(\xi \in \mathbb {R}^d\), we have that \(\sum _{j=0}^\infty \Delta _j=\text {Id}\). Moreover, we have \({{\,\textrm{supp}\,}}\varphi _j\subset \{ 2^{j-2}<|\xi |<2^j\}\) for \(j\geqq 1\) and \(\chi \varphi _j=0\) for \(j\geqq 2\).
Bony’s decomposition for product of functions is
where
We note that \({{\,\textrm{supp}\,}}\widehat{S_{j-3}f\Delta _j g}\subset \{2^{j-3}<|\xi |<2^{j+1} \}\) for \(j\geqq 1\).
We recall the following result from [1].
Lemma A.14
([1, Lemma 2.2]) Let \(\mathcal {C}\) be an annulus in \(\mathbb {R}^d\), \(m\in \mathbb {R}\), and \(k=2[1+\frac{d}{2}]\), where [r] denotes the integer part of r. Let \(\sigma \) be a k-times differentiable function on \(\mathbb {R}^d\setminus \{0\}\) such that for all \(\alpha \in \mathbb {R}^d\) with \(|\alpha |\leqq k\), there exists a constant \(C_\alpha \) such that
There exists a constant C, depending only on the constants \(C_\alpha \), such that for any \(p\in [1, \infty ]\) and any constant \(\lambda >0\), we have, for any function \(u\in L^p(M^d)\), \(M\in \{\mathbb {R}, \mathbb {T}\}\), with Fourier transform supported in \(\lambda \mathcal {C}\),
Next we recall the definition of the Chemin–Lerner norm.
Definition A.15
Let M be either \(\mathbb {R}\) or \(\mathbb {T}\). For \(I\subset \mathbb {R}\) and \(s\in \mathbb {R}\), the Chemin–Lerner norm is defined by
When the low-frequency part is removed, we denote
It what follows, unless otherwise specified, when the set M is omitted in function space notation, it can be either \(\mathbb {R}\) or \(\mathbb {T}\). We recall another result from [1], this time about products.
Proposition A.16
([1, Corollary 2.54]) For \(I\subset \mathbb {R}\), \(q\in [1, \infty ]\) and \(s>0\), there exists \(C=C(d, s)\) such that
provided that the right-hand sides are finite.
Next we study the boundedness of some key operators in the Chemin–Lerner norm.
Proposition A.17
The following hold.
-
(1)
There exists an absolute constant C such that for all \(1\leqq p\leqq \infty \), \(\sigma \in \mathbb {R}\) and \(u\in H^\sigma (\mathbb {R}^d)\), we have
$$\begin{aligned} \left\| \frac{\cosh ((z+b)|D|)}{\cosh (b|D|)}u\right\| _{\widetilde{L}^p_z([-b, 0]; H^{\sigma +\frac{1}{p}})}\leqq \max \{2b^{\frac{1}{p}}, C\}\Vert u\Vert _{H^\sigma }. \end{aligned}$$(A.42) -
(2)
There exists an absolute constant C such that for all \(1\leqq q_2\leqq q_1\leqq \infty \), \(\sigma \in \mathbb {R}\) and \(f\in \widetilde{L}^{p_2}_z([-b, 0]; H_x^{\sigma -1+\frac{1}{p_2}}) \), we have
$$\begin{aligned}{} & {} \left\| \int _{-b}^z\frac{\cosh ((z'+b)|D|)}{\cosh ((z+b)|D|)}f(x, z')\textrm{d}z'\right\| _{\widetilde{L}^{q_1}_z\left( [-b, 0]; H^{\sigma +\frac{1}{q_1}}\right) }\nonumber \\{} & {} \quad \leqq \max \left\{ b^\frac{q_1+q_2'}{q_1q_2'}, C\right\} \Vert f\Vert _{\widetilde{L}^{q_2}_z\left( [-b, 0]; H^{\sigma -1+\frac{1}{q_2}}\right) }, \end{aligned}$$(A.43)where \(\frac{1}{q_2}+\frac{1}{q_2'}=1\). In addition, for any \(z\in [-b, 0]\), we have
$$\begin{aligned}{} & {} \left\| \int _{-b}^z\frac{\cosh ((z'+b)|D|)}{\cosh ((z+b)|D|)}f(x, z')\textrm{d}z'\right\| _{H^\sigma }\nonumber \\{} & {} \quad \leqq \max \{b^\frac{q_1+q_2'}{q_1q_2'}, C\}\Vert f\Vert _{\widetilde{L}^{q_2}_z\left( [-b, 0]; H^{\sigma -1+\frac{1}{p_2}}\right) }, \end{aligned}$$(A.44)
Proof
For all \(-b\leqq z_1\leqq z_2\), we have \(0\leqq z_2-z_1\leqq z_2+b\) and hence
for all \(c\geqq 0\).
To prove the first item we note that (A.45) implies
for all \(z\in [-b, 0]\). Consequently, for \(j\geqq 1\) and \(u\in \dot{H}^\sigma \), we have
since \(|\xi |\geqq 2^{j-2}\) on the support of \(\widehat{\Delta _j}u(\xi )\). It follows that
where C is an absolute constant. On the other hand, the low frequency part can be bounded as
Combining (A.48) and (A.49) yields
This completes the proof of the first item.
We now turn to the proof of the second item. To prove (A.43), we set
For \(z\in [-b, 0]\) and \(j\geqq 1\), using (A.45) we estimate
Applying Young’s inequality in z we deduce
where \(\frac{1}{q}=1+\frac{1}{q_1}-\frac{1}{q_2}\) and C is an absolute constant. On the other hand, it is readily seen that
and hence
A combination of (A.53) and (A.55) leads to (A.43).
Finally, the proof of (A.44) is similar to the case \(q_1=\infty \) of (A.43). \(\square \)
Next we consider some more product estimates.
Proposition A.18
Let \(s>0\), \( p\in [1, \infty ]\), and \(I\subset \mathbb {R}\). Then, there exists \(C=C(d, s)\) such that the estimate
holds provided that the right-hand side is finite. Consequently, for \(s>0\) and \(s_0>\frac{d}{2}\), there exists \(C=C(d, s, s_0)\) such that
Proof
We first note that for \(M=\mathbb {T}\), (A.57) is a consequence of (A.41) and the continuous embedding \(H^{s_0}(\mathbb {T}^d)\subset L^\infty (\mathbb {T}^d)\) for \(s_0>\frac{d}{2}\).
To prove (A.56) and (A.57) for \(M=\mathbb {R}\), we shall consider functions f(x, z) and g(x, z) defined on \(\mathbb {R}^d\times I\). For fixed \(z\in I\), we use Bony’s decomposition (A.34): \(fg=T_fg+T_gf+R(f, g)\), where \(T_fg=\sum _{j\geqq 3}S_{j-3}f\Delta _j g\). For \(j\geqq 3\) we have \({{\,\textrm{supp}\,}}\widehat{S_{j-3}f\Delta _j g}\subset \{2^{j-3}<|\xi |<2^{j+1}\}\) and hence \(\Delta _k (S_{j-3}f\Delta _j g)=0\) for all \(k\geqq 0\) satisfying \(|j-k|\geqq 3\). Thus, for \(k\geqq 0\) using Bernstein’s inequality we obtain
where \(C=C(d, s)\). Since \(f\in \widetilde{L}^p(I; H^s_\sharp )\), we have \(\Delta _j f\in L^2_x\) a.e. \(z\in I\) for \(j\geqq 3\). Consequently, the preceding estimate for \(T_fg\) also holds for \(T_gf\); that is,
It follows that
and similarly we have
As for the remainder \(R(f, g)=\sum _{j\geqq 0} \sum _{|\nu |\leqq 2} \Delta _j f\Delta _{j+\nu }g\), we note that \({{\,\textrm{supp}\,}} \widehat{ \Delta _j f\Delta _{j+\nu }g}\subset \{|\xi |<2^{j+3}\}\). Thus \(\Delta _k(\Delta _j f\Delta _{j+\nu }g)=0\) for \(k\geqq j+5\) and
where
and
It follows that
By Young’s inequality for series, we deduce
We thus obtain
Combining (A.60), (A.61) and (A.67) we obtain (A.56). Finally, (A.57) follows from (A.56) and (A.6). \(\square \)
Our next result records some estimates for nonlinear maps of the form \((f,g) \mapsto g(1+f)^{-1}\).
Proposition A.19
Let \(I\subset \mathbb {R}\), \(p\in [1, \infty ]\), \(s>0\), and \(s_0>\frac{d}{2}\). There exists a positive constant \(C=C(d, s, s_0)\) such that if \(\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}<\frac{1}{2C}\) then
Proof
By virtue of (A.6) we have \(\Vert f\Vert _{L^\infty (I; L^\infty )}\leqq C_1\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\), \(C_1=C_1 (d, s, s_0)\), and hence \(|f|\leqq \frac{1}{2}\) a.e. if \(\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\leqq \frac{1}{2C_1}\). Then the expansion
holds a.e on \(\mathbb {R}^d\). We claim that with \(C_2=\max \{C_1, C\}\), where C is given in (A.57), we have
for all \(j\geqq 1\). Indeed, the case \(j=1\) follows at once from (A.57). Assume that (A.70) holds for some \(j\geqq 1\). Applying (A.57) once again, we deduce
Combining this with the estimate
we obtain (A.70) for \(j+1\).
Finally, for \(\Vert f\Vert _{L^\infty (I; \mathcal {H}^{s_0})}\leqq \frac{1}{2C_2}\) we can sum (A.70) over \(j\geqq 1\) to obtain
This implies (A.68). \(\square \)
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Nguyen, H.Q., Tice, I. Traveling Wave Solutions to the One-Phase Muskat Problem: Existence and Stability. Arch Rational Mech Anal 248, 5 (2024). https://doi.org/10.1007/s00205-023-01951-z
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DOI: https://doi.org/10.1007/s00205-023-01951-z