Abstract
We show the local in time well-posedness of the Prandtl equations for data with Gevrey 2 regularity in x and Sobolev regularity in y. The main novelty of our result is that we do not make any assumption on the structure of the initial data: no monotonicity or hypothesis on the critical points. Moreover, our general result is optimal in terms of regularity, in view of the ill-posedness result of Gérard-Varet and Dormy (J Am Math Soc 23(2):591–609, 2010).
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Introduction
We are interested in the 2D Prandtl equations
set in the domain \(\Omega = {\mathbb {T}}\times \mathbb {R}_+\), completed with boundary conditions
These equations are a degenerate Navier-Stokes model, introduced by Prandtl in 1904 to describe the boundary layer, which is the region of high velocity gradients that forms near solid boundaries in incompressible flows at high Reynolds number. They can be derived from the Navier-Stokes equation under the formal asymptotics
where \(\nu \) is the inverse Reynolds number, and \((u^\nu , v^\nu )\) is the Navier-Stokes solution. This asymptotics is supposed to apply to the flow in the boundary layer region: the typical scale \(\sqrt{\nu }\) of the boundary layer in this model is inspired by the heat part of the Navier-Stokes equation. Away from the boundary, one rather expects an inviscid asymptotics of the type
where \((u^E, v^E)\) is the solution of the Euler equation. In order to match the two asymptotic expansions, one must impose the condition
which yields the boundary condition for \(y \rightarrow \infty \) in (2). The other two boundary conditions at \(y = 0\) express the usual no-slip condition at the boundary. We refer to [6] for a more detailed derivation. Let us stress that the pressure in the Prandtl model is independent of y: its value is given by the pressure in the Euler flow at \(z=0\). This explains the right-hand side of (1), which depends only on t, x, and is coherent with the third boundary condition in (2).
The Prandtl system (1)–(2) is very classical, as it appears in most textbooks on fluid dynamics. Still, it is well-known from physicists that its range of applications is narrow, due to underlying instabilities. Among those instabilities, one can mention the phenomenon of separation, which is related to the development of a reverse flow in the boundary layer [4, 8]. Another example is the so-called Tollmien-Schlichting wave, that is typical of viscous flows at high but finite Reynolds number [5, 15]. Of course, such instability mechanisms create difficulties at the PDE level, making the mathematical analysis of boundary layer theory an interesting topic. The two main problems that one needs to address are the well-posedness of the reduced model (1), and the validity of the asymptotics (3). We shall focus on the former in the present paper. About the validity of boundary layer expansions in the unsteady setting, there are many possible references, among which [11, 14, 16, 26, 31, 32]. About the steady setting, see the recent works [10, 17, 19].
To analyse the well-posedness of the Prandtl model is uneasy, even at the level of local in time smooth solutions. The key difference with Navier-Stokes is that there is no time evolution for the vertical velocity, which is recovered only through the divergence-free condition. Hence, the term \(v \partial _y u\) can be seen as a first order nonlinear operator in x. Moreover, this operator is not skew-symmetric in \(H^s\). As the diffusion in (1) is only transverse, this prevents the derivation of standard Sobolev estimates. The first rigorous study of the Prandtl equations goes back to Oleinik [29], who tackled the case of data \(U\vert _{t=0}\) that are monotonic in y. She established local well-posedness of the system using the so-called Crocco transform, a tricky change of variables and unknowns. Let us stress that such monotonicity assumption excludes the phenomenon of reverse flow and therefore prevents boundary layer separation. More recently, the local well-posedness result of Oleinik was revisited using the standard Eulerian form of the equations, see [1, 21, 27] for the local theory in Sobolev spaces.
The analysis of non-monotonic data is much more recent, and has experienced some strong impetus over the last years. Surprisingly, it was shown in [9] that the Prandtl system is ill-posed in the Sobolev setting (cf. [13, 18, 24] for improvements). Specifically, paper [9] centers on the linearization of (1)–(2) around shear flows, given by \((U,V) = (U_s(y),0)\). The linearized system reads
In the case where \(U_s\) has one non-degenerate critical point, one can show that (4) has unstable solutions of the form \(u(t,x,y) = \mathrm {e}^{\mathrm {i}k x} \mathrm {e}^{\sigma _k t} u_k(y)\) for k arbitrarily large and \(\mathfrak {R}\sigma _k \sim \lambda \sqrt{k}\). Such high frequency instability forbids the construction of Sobolev solutions. To obtain positive results, one must start from initial data \(u_\mathrm {in}\) that are strongly localized in Fourier, typically for which \(|\hat{u}_0(k,y)| \lesssim \mathrm {e}^{-\delta |k|^\gamma }\) for some positive \(\delta > 0\), \(\gamma \le 1\). Such localization condition corresponds to Gevrey regularity in x (Gevrey class \(1/\gamma \)). The first result in this direction is due to Sammartino and Caflisch [30], who established existence of local in time solutions in the analytic setting (\(\gamma = 1\)). See also the nice paper [22]. Note that the requirement for analyticity is natural in view of standard estimates. For instance, at the level of the linearized equations (4), one gets directly by testing against u that
so that \(\Vert \hat{u}(t,k,\cdot )\Vert _{L^2_y} \le \mathrm {e}^{C|k|t} \Vert \hat{u}_0(k, \cdot )\Vert _{L^2_y}\). Hence, if \(\Vert \hat{u} _0(k,\cdot )\Vert _{L^2_y} \lesssim \mathrm {e}^{-\delta |k| }\), a uniform control will be provided as long as \(t \le \delta /C\).
To relax the analyticity condition is much harder. In the special case where \(u_\mathrm {in}\) has for each value of x a single non-degenerate critical point in y, the first author and N. Masmoudi proved the local well-posedness of system (1)–(2) for data that are in Gevrey class 7 / 4 with respect to x [12]. Well-posedness was extended to Gevrey class 2 in article [23], for data that are small perturbations of a shear flow with a single non-degenerate critical point. Note that this exponent (corresponding to \(\gamma = 1/2\)) is optimal in view of the instability mechanism of [9].
All the recent results mentioned above rely heavily on the structure of the initial data: monotonicity for the Sobolev setting, single non-degenerate critical points for the Gevrey setting. It is therefore natural to ask about the optimal regularity under which local well-posedness of the Prandtl equations holds, without additional structural assumption. This is the problem that we solve in the present paper: we establish the short-time well-posedness of the Prandtl equations for general data with Gevrey 2 regularity in x and Sobolev regularity in y. We recall once more that such regularity framework is the best possible. Indeed, from the results of [9], high frequency modes k in x may experience exponential growth with rate \(\sqrt{k}\). This means that to hope for short time stability, the amplitude of these modes should be \(O(\mathrm {e}^{-C\sqrt{k}})\), which is the Fourier translation of a Gevrey 2 requirement.
Result
Let \(\gamma \ge 1\), \(\tau > 0\), \(r \in \mathbb {R}\). For functions \(f = f(x)\) of one variable, we define the Gevrey norm
and for functions \(f = f(x,y)\) of two variables, the norm
where \(\Vert \cdot \Vert _j\), \(j \ge 0\), denotes a family of weighted \(L^2\) norms. Namely,
where \(\rho _j\), \(j \ge 0\), is the family of weights given by
for fixed constants \(\alpha \ge 0\) and \(m \ge 0\) chosen later (m large enough and \(\alpha \) matching the constraints found from the estimates). The need for this family of weights will be clarified later. Let us note that locally in y, this family of norms is comparable to more classical families such as
For instance, for functions f which are zero for \(|y|\ge M\), one has
The only difference is when y goes to infinity, where the family of weights \(\rho _j\) puts less constraints on the decay of the derivatives compared to a fixed weight \(\rho _0\) for derivatives of any order.
With these spaces, we can now state our main result.
Theorem 1
There exists m and \(\alpha \) such that: for all \(0< \tau _1 < \tau _0\), \(r \in \mathbb {R}\), for all \(T_0 > 0\), for all \(U^E\) satisfying
for all \(U^P_\mathrm {in}\) satisfying
and under usual compatibility conditions (see the last remark below), there exists \(0 < T \le T_0\) and a unique solution \(U^P\) of (1)–(2) over (0, T) with initial data \(U^P_\mathrm {in}\) that satisfies
Remark
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The main novelty of the theorem is that we reach the optimal Gevrey regularity although no structural assumption is made on the data: no monotonicity, or hypothesis on the number and order of the critical points is needed. Only Gevrey regularity of the data and natural compatiblity conditions are required.
-
Our method of proof, explained below, is inspired by the hyperbolic part of the Prandtl equations. It is based on both a tricky change of unknown and appropriate choice of test function. This method would also allow to recover the Sobolev well-posedness of the hyperbolic version of the Prandtl system by means of energy methods. As far as we know, the well-posedness of this inviscid Prandtl model had been only established in \(C^k\) spaces using the method of characteristics: see [20] for more. This part will be detailed elsewhere. In the case of the usual Prandtl equations studied here, our methodology has to be slightly modified to handle in an optimal way the diffusion term. Still, commutators are responsible for the loss of Sobolev regularity: only Gevrey 2 smoothness in x can be established.
-
There is a loss on the Gevrey radius \(\tau \) of the solutions through time, going from \(\tau _0\) to \(\tau _1\). This loss, which appears technical in the paper, is actually unavoidable. This is due to the instabilities described in [9]: exponential growth of perturbations at rate \(\sqrt{k}\) causes a decay of the Gevrey radius linearly with time.
-
Besides the regularity requirements mentioned in Theorem 1, the initial data must satisfy compatibility conditions. It is typical of parabolic problems in domains with boundaries, cf. [28, Chapter 3] for a general discussion. Here, the value of \(U^P_\mathrm {in}\) and of some of its derivatives at \(y=0\) cannot be arbitrary: they must be related to \(U^E\) accordingly to the equation and to the amount of regularity asked for u (with respect to the y-variable). Let us note that locally near \(y=0\), most of our estimates only involve \(U^P-U^E\) in \(L^2_t H^2_y\) (not mentioning the Gevrey regularity in x). Such estimates could be carried with the single compatibility condition \(U^P_\mathrm {in}\vert _{y=0}=0\). Still, the low norm \(\Vert (U^P, V^P) \Vert _{low}\) introduced in (19) involves more y-derivatives: its control through Lemma 15 implies therefore a few more compatibility conditions. For the sake of brevity, we do not provide their explicit expressions, and refer to [33, Proposition 2.3] for a detailed discussion on a variation of the Prandtl equations.
Outline of the strategy As mentioned earlier, our analysis of the Prandtl equations relies on the identification of new controlled quantities because the usual unknown u and kinetic energy do not give enough information. To help to identify the relevant quantities, it is a good idea to start from the study of the linearized system (4). After Fourier transform in x and Laplace transform in time, we are left with the ODE
where \(\Psi \) corresponds to the Fourier-Laplace transform of the stream function. At high frequencies k, a natural idea (although not legitimate in the end) is to neglect the diffusion term. We are then left with the first order ODE
We note that the standard estimate (based on taking \(\partial _y \overline{\Psi }\) as a test function) yields a control of the type
where the last line comes from the Hardy inequality (as soon as \(|U'_s(y)| = \mathcal {O}(y^{-1})\) at infinity). Such bound ensures the solvability of the resolvent equation (10) only for \(\lambda \sim k\). This in turn yields a semigroup bound of the type \(\mathrm {e}^{C kt}\), only compatible with stability in the analytic setting.
To reach stability in lower regularity, an important point is to notice that the homogeneous equation has \(\Psi _s = (\lambda + i k U_s)\) as a special solution. With the integrating factor method in mind, it is then natural to set \(\Psi = (\lambda + i k U_s) \psi \). The first order equation (10) becomes
which is much better than the original formulation. Indeed, we can test the equation against \(\phi = \frac{1}{\lambda + i k U_s} \partial _y \overline{\psi }\) to obtain a control of \(\partial _y \psi \) in terms of \(u_\mathrm {in}\), and from there a control of \(\Psi \) for any \(\lambda > 0\).
Back to the full resolvent equation (9) we find for the same unknown \(\psi \)
Testing again against \(\phi = \frac{1}{\lambda + i k U_s} \partial _y \overline{\psi }\), the LHS allows the control
In the commutator at the RHS, the worst error term is \(3\mathrm {i}k \partial _y U_s \partial _y^2 \psi \), which is bounded as
We see that under the constraint \(\mathfrak {R}\lambda \sim k^{2/3}\), the estimate can be closed, and this can be shown to imply short time stability for data with Gevrey regularity 3 / 2. This estimate around a shear flow is detailed as Lemma 4.1 in [3].
In order to reach the optimal Gevrey exponent 2, we need to get rid of the commutator term containing \(\partial _y^2 \psi \), which comes with a worse control than \(\partial _y \psi \). To do so, we change a bit our new unknown \(\psi \): we now define \(\psi \) through the relation
including the diffusion term. Hence, (9) becomes
Testing this time against the solution \(\phi \) of \((\lambda +\mathrm {i}k U_s - \partial _y^2) \phi = \partial _y \overline{\psi }\) (again with the diffusion term), the LHS yields the same control, but the error term is now
From the definition of \(\phi \) it can be shown that \(\Vert \phi \Vert \lesssim \lambda ^{-1} \Vert \partial _y \psi \Vert \) so that the error can be bounded by
The estimate can now be closed for \(\mathfrak {R}\lambda \sim k^{1/2}\) yielding Gevrey regularity 2.
Obviously, such approach is no longer applicable as such to the nonlinear system (1)–(2): we not only lose the linearity of the equations, but the coefficients are no longer of shear flow type. They notably depend on t and x, which forbids an easy use of Fourier or Laplace transform. Rather than turning to the characterization of Gevrey spaces in the Fourier variable k, we consider norms based on the x-variable, see (5) and (6). Roughly, the idea is to work with time dependent norms, that is with the quantities
By differentiating j-times the first line of the Prandtl system, we can derive an equation on
that can be written as
Roughly, inspired by the shear flow case, the idea will be to introduce as a new unknown the solution \(\psi _j = \int _0^y H_j\) of
which is reminiscent of the Fourier relation (11). The test function \(\phi _j\) should then solve the reverse equation
and be solved backward in time. Performing the same estimate as in the shear flow case, we expect to find an inequality of the type
By exploiting a relation of the form \(\Vert \partial _x \partial _y \psi _j\Vert \sim j^\gamma \Vert \partial _y \psi _{j+1}\Vert \) (that needs to be shown!) and using that \(\gamma \le 2\), we will then be able to sum over j and establish for large enough \(\beta \) a control of \(\sum _j \Vert \partial _y \psi _j\Vert ^2\) in terms of \(\sum _j \Vert F_j \Vert ^2\).
In fact, in implementing this strategy, several refinements are necessary, and the relations satisfied by \(\psi _j = \int _0^y H_j\) or \(\phi _j\) need to be slightly modified. Particularly problematic is the term \(V^P \partial _y\) because \(V^P \sim -\partial _x U^E y\) increases linearly with y: this prevents from closing an energy estimate with a fixed weigth \(\rho = \rho (y)\). This difficulty appears in various places in the literature on the Prandtl equations. This is for instance the reason why article [12] is limited to the special case \(U^E = 0\) and decaying initial data. One can also mention [22], where this difficulty is overcome by a clever change of variables, which is reminiscent of the method of characteristics and allows to remove the bad part of the convection term from the momentum equation. Energy estimates can then be established in these new coordinates \(x',y'\), and yield some local well-posedness result, with solutions that are analytic in \(x'\) and \(L^2\) in \(y'\). The disadvantage of this approach is that the regularity of the solution in the original variables x and y is no longer clear at positive times. Here, we stick to the eulerian variables, but overcome the difficulty by introducing the family of weights \(\rho _j\), \(j \ge 0\). These weights allow to trade a power of y against a derivative in x, which is appropriate to the commutator terms. Moreover, they put very little conditions on the derivatives of the solution, so that they provide a very general framework for well-posedness. Note that the specific expression of \(\rho _j\) is important: it could not be for instance replaced by the more natural guess \((1+y)^{2(m-j)}\), as commutators with the diffusion term would not be under control. Note also that the strategy used in [27], where Sobolev well-posedness is established under monotonicity assumptions by increasing the weight with the number of y-derivatives, does not extend to the Gevrey framework in variable x.
The plan of the paper is as follows. In the next section, we first collect several properties of the weight \(\rho _j\). We then write the equations satisfied by the x-derivatives of the Prandtl solution in a form analogue to (12). This means that we put most of the nonlinear terms at the right-hand side, and consider those equations as linear. We finish the section by introducing the adapted quantities \(H_j\) and \(\phi _j\). The main section is Sect. 4: a priori Gevrey estimates for the linear equations are performed, that provide a control of the \(u_j\)’s in terms of the nonlinear terms \(F_j's\). Note that such estimates are obtained under a condition of the form \(\beta > C (1+ \Vert (U^P, V^P) \Vert _{low})^2\), where \(\Vert (U^P, V^P)\Vert _{low}\) is a low regularity norm of the solution. The treatment of the nonlinearity \(F_j\) is then handled in Sect. 5. The last step in the derivation of a priori estimates is to recover the control of the low regularity norm \(\Vert (U^P,V^P)\Vert _{low}\), see Sect. 6. Finally, issues regarding the construction and uniqueness of solutions are discussed in Sect. 7.
Preliminaries
The explicit form of the weights \(\rho _j\) is only needed in the Sect. 5. In the other parts, we just need a sufficient control of the logarithmic derivative (Lemma 2), a bound for antiderivatives (Lemma 3) and relate \(\rho _j\) to \(\rho _{j+1}\) (Lemma 4).
Lemma 2
Let \(m \ge 0\) and \(\alpha \ge 0\). There exists a constant \(C_l\) such that for all \(y \in \mathbb {R}^+\), \(j \in \mathbb {N}\)
and
Proof
Given the explicit form of \(\rho _j\), we can compute the logarithmic derivative of \(\rho \) directly as
From this expression the result follows directly. \(\square \)
Lemma 3
For \(m > \frac{1}{2}\) introduce the constant
Then for all \(\alpha \ge 0\), \(j \in \mathbb {N}\) and all \(f = f(y)\),
More generally, for \(0 \le n \le j\) with \(n < m - \frac{1}{2}\) one has
Eventually, for all \(A = A(x,y)\) and \(B = B(x,y)\), the following inequality holds:
Proof
Note that \(\rho _j/\rho _n\) for \(j \ge n\) is non-increasing. Hence
where we used the Cauchy-Schwarz inequality in the second inequality.
As \(\alpha \ge 0\) we find directly that
whose integral over \(y \in \mathbb {R}^+\) gives \(C_{m-n}^2\). This proves the first and second bounds. The remaining estimate with A and B follows directly. \(\square \)
The weights are decaying so that \(\rho _j \le \rho _k\) for \(j \ge k\). As \(\alpha \ge 0\), we have for \(j \in \mathbb {N}\) that \((1+y)^2 \rho _{j+1} \le (j{+}1)^{2\alpha } \rho _j\) and \(\rho _{j+1} \ge \frac{\rho _j}{(1+y)^2}\). This shows:
Lemma 4
Let \(\alpha \ge 0\). For \(j\in \mathbb {N}\), \(A = A(x,y)\) and \(B = B(x,y)\) it holds that for
and
Let us insist again that most parts of the proof would work with constant weight \(\rho \) instead of \(\rho _j\). The dependency on j will be only needed to treat the commutator terms coming from \(V^P \partial _y U^P\). The difficulty is that \(V^P\) grows like y as soon as \(U^E\) is non-constant. Here the crucial property that we will use is that we can control \(\Vert (1+y) A \Vert _{j+1}\) by \((j{+}1)^{\alpha } \Vert A \Vert _j\).
The Prandtl equations are given for \((U^P,V^P)\) with inhomogeneous boundary conditions at \(y \rightarrow \infty \). In order to work with homogeneous boundary conditions at zero and infinity, we introduce
and set \(u = U^P - U^e\), \(v = -\int _0^y \partial _x u = V^P - V^e\). Then,
where
In the new variables (u, v) the boundary conditions are
The condition at \(y\rightarrow \infty \) will be encoded in the functional space of u.
To prove Theorem 1, the point is to obtain good estimates for Gevrey norms of u of type (6) for time-dependent radius \(\tau = \tau (t)\). More precisely, we give ourselves parameters \(m,\alpha ,\gamma ,r\), to be fixed later, as well as the time-dependent radius \(\tau (t) = \tau _0 \mathrm {e}^{-\beta t}\), with \(\beta > 0\) to be fixed later. Then, for any function \(f = f(t,x)\) or \(f=f(t,x,y)\) and \(j \in \mathbb {N}\) we set
Taking j derivatives in x of (13) and multiplying by \(M_j\) yields
where \(F_j\) collects all terms with less than j derivatives in x as well as the weighted derivative of the forcing \(f^e\). It is given by
We now introduce our crucial auxiliary functions \(H_j(t,x,y)\) defined by
For the existence of \(H_j\), one can consider (17) as a convection-diffusion equation for \(A_j = \int _0^y H_j\, \mathrm {d}z\), with boundary conditions \(A_j|_{y=0} = \partial _y A_j|_{y\rightarrow \infty } = 0\), which has a solution by the classical theory of parabolic PDEs. The PDE (17) itself then implies that \(\partial _y^2 A_j|_{y=0}=0\) so that taking \(H_j = \partial _y A_j\) gives the required solution.
We further introduce the corresponding test functions \(\phi _j\) by
Note here that the operator acting on \(\phi _j\) is the formal adjoint operator of the operator acting on \(\int _0^y H_j \, \mathrm {d}z\) in (17), with respect to the \(L^2(\rho _j)\) scalar product, denoted \(\left\langle \,,\right\rangle _j\). This is a backward heat equation solved backward in time for \(t \in [0,T]\).
Testing (18) against \(\phi _j\) in \(\Vert \cdot \Vert _j\) and integrating over [t, T] yields
Hence we find
By Lemma 2, under the condition \(\alpha \ge \frac{1}{2}\), we get the following control:
Lemma 5
Fix \(m\ge 0\) and \(\alpha \ge \frac{1}{2}\). Then there exist a constant \(\mathcal {C} = \mathcal {C}(m,\alpha )\) such that for all \(j \in \mathbb {N}\) it holds that
if
Note that for \(\alpha < \frac{1}{2}\), the term with \(\Vert \frac{\partial _y \rho _j}{\rho _j} \Vert _{\infty }^2\) could not have been absorbed. This a priori estimate also ensure the existence of \(\phi _j\) as solution of (18). A similar estimate holds for \(H_j\) which ensures the existence of \(H_j\) as solution of (17).
Linear Estimates
In this section we analyse the linearised equation (16) and obtain an estimate for the solution in terms of the \(F_j\) containing the forcing and lower-order terms. For this, we shall first analyse (16) for a fixed j. We will obtain a control of \(H_j\) in terms of the forcing \(F_j\) and an error term \(\partial _x H_j\), which will be shown to be approximately \((j{+}1)^{\gamma } H_{j+1}\). By summing over j, we will find the following control.
Lemma 6
Fix \(m > \frac{1}{2}, \frac{1}{2} \le \alpha \le \frac{1}{2} + \gamma , 1 \le \gamma \le 2, r \in \mathbb {R}\). Then there exists a constant \(\mathcal {C} = \mathcal {C}(m,\alpha ,\gamma ,r)\) such that for all \(\tau _1\), \(\beta \) and T such that
the \(H_j\)’s defined by (17) for solutions \(u_j\)’s of (16) satisfy
Here we use a low-order control of \(U^P\) and \(V^P\) in order to control the commutator error terms. From the required bounds, we define the low-order norm as
Although a main ingredient of our proof, the unknown \(H_j\) is less natural than the usual \(u_j\), notably for the future treatment of the nonlinearity, which involves \(u_j\) and \(\omega _j = \partial _y u_j\). This is why we shall we relate the control of \(H_j\) to \(u_j\) and show:
Proposition 7
Fix \(m > \frac{1}{2}, \frac{1}{2} \le \alpha \le \frac{1}{2} + \gamma , 1 \le \gamma \le 2, r \in \mathbb {R}\). Then there exist constants \(C = C(m,\alpha ,\gamma ,r)\) and \(\mathcal {C} = \mathcal {C}(m,\alpha ,\gamma ,r)\) such that for all \(\tau _1\), \(\beta \) and T such that
the solution u of (16) satisfies
For \(\gamma \ge 5/4\) this is
Estimate for \(H_j\)
We focus first on Lemma 6. The idea is to use the solution \(\phi _j\) of (18) as a test function in (16). Taking the weighted scalar product and integrating over [0, T], we find for the first term in (16):
Note that there is no boundary term as \(u_j\) and \(\phi _j\) vanish at the boundaries. Differentiating (17), we can replace \(u_j\) in the last integral and find
By the boundary values of \(H_j\) there are again no boundary terms from partial integration in \(y\). In the last expression, the first line contains the good controlled terms, the second line will cancel the leading contribution from the bad terms \(\partial _y U^P v_j + j \partial _{xy} U^P \partial _x^{-1} v_j\) (see below), while the last two lines collect the error terms.
Next, we compute the contribution from the terms with \(v_j\) using \(v_j = -\partial _x \int _0^y u_j \, \mathrm {d}z\):
and
In both cases the leading order term cancels. Hence collecting the terms we arrive at
where \(E_1,\dots ,E_5\) collect the lower-order error terms as
Here \(E_3\) and \(E_4\) contain the worst terms, as they involve x-derivatives of \(H_j\). They are responsible for the Gevrey regularity requirement.
Assume \(m \ge 0\), \(\alpha \ge \frac{1}{2}\) and \(\beta \) large enough so that Lemma 5 applies. We can then estimate the forcing terms as
and
Absorbing the terms with \(H_j\) we therefore find
We now estimate the error terms, where we repeatedly use Lemma 3. For \(E_1\) we find
For \(E_2\) we also use Lemma 2 and assume \(\alpha \ge \frac{1}{2}\)
In the term \(E_3\) we have terms with \(\partial _x H_j\), which we want to estimate in \(\Vert \cdot \Vert _{j+1}\) as they will be later controlled by \(H_{j+1}\). Using Lemma 4 we find
where we used the identity
Similarly, we find for \(E_4\) that
And finally for \(E_5\) we find
where we took again advantage of (20).
We collect the various factors in constants \(D_1,D_2,D_3\) defined as folllows:
and
and
Then
With Lemma 5 the \(\phi \) integral can be estimated as
and thus can be absorbed in the LHS.
Here \(\Vert (U^P,V^P)\Vert _{low}\) has been designed such that we can find numerical constants \(c_1,c_2,c_3\) such that
Combining all the estimates we arrive at the following lemma.
Lemma 8
Assume \(\alpha \ge \frac{1}{2}\) and \(m > \frac{1}{2}\). Then there exist a constant \(\mathcal {C} = \mathcal {C}(m,\alpha )\) such that for
and \(j \in \mathbb {N}\) the \(H_j\) defined by (17) for a solution \(u_j\) of (13) satisfy
Proof
Use the previous estimates. Note that the condition on \(\beta \) also implies that the hypothesis of Lemma 5 is satisfied by choosing \(\mathcal {C}\) large enough. \(\square \)
Relating \(\partial _x H_j\) with \(H_{j+1}\)
To conclude the proof of Lemma 6, that will be achieved by summation of the previous estimate over j, we need first to control \(\partial _x H_j\) by \(H_{j+1}\).
Lemma 9
Let \(m > \frac{1}{2}\) and \(\alpha \ge \frac{1}{2}\). Then there exist constants \(\mathcal {C} = \mathcal {C}(m,\alpha )\) and \(C = C(m,\alpha ,r)\) such that for all \(\tau _1\), \(\beta \) and T with
it holds that
Proof
From the definition of \(u_j\), it holds that \(\partial _x u_j(t) = \left( \frac{j+2}{j+1}\right) ^r \frac{(j+1)^\gamma }{\tau (t)} u_{j+1}(t)\). Hence we anticipate that
Therefore we estimate the difference
From equation (17) (used with indices j and \(j{+}1\)), we find that
We stress that \(\int _0^y \Delta _j\, \mathrm {d}z\) does not converge to zero at infinity, so that one can not perform \(L^2\) estimates on this quantity. However, we can notice by 3 that
The square integrable quantity \(\delta _j = \bigl (\frac{\rho _{j+1}}{\rho _0}\bigr )^{1/2} \int _0^y \Delta _j\, \mathrm {d}z\) satisfies the equation
As in (22), we obtain
We also get
By Lemma 2, we find
Using again Lemma 2 and the identity
and defining
we obtain
Eventually,
for some constant \(C = C(\alpha )\). The previous bounds combined with an energy estimate yield that for \(\mathcal {C}\) large enough (we remind that \(\alpha \ge \frac{1}{2}\)):
We can then take the x-derivative of equation (23) and proceed as above. For \(\mathcal {C}\) large enough, we get
We then use that
and
and the bound (26) to end up with
To estimate directly \(\Delta _j\), we differentiate the Eq. (21) with respect to y, which gives
We take the \(\left\langle \,,\right\rangle _{j+1}\) scalar product with \(\Delta _j\):
By the 1d Sobolev imbedding theorem, we find that for a constant \(C = C(m)\) it holds that
Combining these last two inequalities with (26), (27) and the inequality
and taking \(\mathcal {C}\) large enough, we obtain
Lemma 9 follows straightforwardly. \(\square \)
Combining Lemmas 8 and 9, we will now prove Lemma 6.
Proof of Lemma 6
We choose \(\mathcal {C}\) such that Lemmas 8 and 9 apply. We multiply the inequality in Lemma 8 by \(\beta (j{+}1)^{2\gamma -1}\) and sum over j to get
Taking \(\mathcal {C}\) large enough, we can then find by Lemma 9 a constant \(C = C(m,\alpha ,r)\) such that
We have used here that \(\alpha \le \gamma +\frac{1}{2}\). Hence, the last term at the right-hand side can be absorbed if
which can be ensured by a suitable large \(\mathcal {C}\) if \(\gamma \le 2\). \(\square \)
Control of \(u_j\) and \(\omega _j\)
We now relate the estimates on \(H_j\) to \(u_j\) and start with an estimate for the \(L^2\) norm.
Lemma 10
Let \(m > \frac{1}{2}\) and \(\alpha \ge \frac{1}{2}\). Then there exists a constant \(\mathcal {C} = \mathcal {C}(m,\alpha )\) such that for
and for any \(\epsilon _1, \epsilon _2, \epsilon _3, \epsilon _4 > 0\) it holds that
where \(u_j\) is satisfying (16), \(A_{j,\alpha }\) is defined in (25) and \(H_j\) is defined by (17).
Proof
Using the definition (17) of \(H_j\) we find
By the evolution equation (16) for \(u_j\), the first term can be written (from the partial integration in \(y\) there is no boundary term as \(u|_{y=0} = 0\))
The terms can now be bounded using Lemma 2:
Recalling that \(v_j = -\partial _x \int _0^y u_j \, \mathrm {d}z\) we find
For the forcing terms we find
The diffusion terms give
The integrand in the second integral in (29) can be estimated as
Collecting the terms we find by choosing \(\mathcal {C}\) large enough that
Splitting the products gives the claimed estimate. \(\square \)
The missing terms can be estimated by the evolution of \(u_j\) and \(\omega _j = \partial _y u_j\). For \(u_j\) we find:
Lemma 11
Let \(m > \frac{1}{2}\) and \(\alpha \ge \frac{1}{2}\). Then there exists a constant \(\mathcal {C} = \mathcal {C}(m,\alpha )\) such that for
the solution \(u_j\) of (16) satisfies
Proof
By (16) we find
where there is no boundary term from the partial integration in \(y\) as \(u_j\) vanishes at the boundary and we used in the inequality that \(\mathcal {C}\) can be chosen large enough. Integrating this over [0, T] gives the claimed result. \(\square \)
By differentiating (16) in y and find
This immediately yields the following control for \(\omega _j\).
Lemma 12
Let \(m > \frac{1}{2}\) and \(\alpha \ge \frac{1}{2}\). Then there exists a constant \(\mathcal {C} = \mathcal {C}(m,\alpha )\) such that for
the vorticity \(\omega _j = \partial _y u_j\) satisfies
Proof
Integrate (30) against \((1+y)^2\omega _j\) in \(\Vert \cdot \Vert _j\). This yields
where we find a boundary term from the diffusion and there is no boundary term from \(V^P\partial _y\) because \(V^P|_{y=0}=0\).
From (16) we find \(\partial _y \omega _j|_{y=0} = F_j|_{y=0}\). For \(\omega _j|_{y=0}\) write
to get
By choosing \(\mathcal {C}\) large enough and using that \(\alpha \ge \frac{1}{2}\), the result follows after integration over time. \(\square \)
Combining the results, we can conclude this section.
Proof of Proposition 7
Adding the control of Lemma 11 with a factor \(\epsilon _3 (j{+}1)^{-\gamma } / \beta \) and Lemma 12 with a factor \((j{+}1)^{1-2\gamma } / \beta ^2\) to the inequality of Lemma 10 yields
Using that \(\partial _x u_j(t) = \left( \frac{j+2}{j+1}\right) ^r \frac{(j+1)^\gamma }{\tau (t)} u_{j+1}(t)\), we can sum over j and choose \(\epsilon _1,\epsilon _2,\epsilon _3,\epsilon _4\) appropriately to arrive for \(m> \frac{1}{2}, \alpha \ge \frac{1}{2},\gamma \ge 1, \tau _1 > 0, r \in \mathbb {R}\) at the control
if
where C and \(\mathcal {C}\) are constant only depending on \(m,\alpha , \gamma ,r\) (and not \(\tau _1\)).
Controlling H by Lemma 6 then yields the result for a fixed time \(T\). Applying this estimate for all T in \([0,T^*]\) then shows the claimed estimate.
For \(\gamma \ge 5/4\) we find that \((j{+}1)^{2\gamma -4} \ge (j{+}1)^{1-2\gamma }\) so that
which proves the expression in this case. \(\square \)
Nonlinear Estimates
In order to close the estimate, we have to estimate \(F_j\).
Proposition 13
Fix the parameters \(m,\alpha ,\gamma ,r\) and an additional parameter R such that
Then there exists a constant \(C = C(m,\alpha ,\gamma ,r)\) such that for \(\beta \), \(\tau _1\) and T with \(\tau (T) \ge \tau _1\),
We restrict to the case of \(\gamma \ge 3/2\) because we need \(\alpha \ge \gamma -1\) in order to control the terms \(\partial _x^k u \partial _x^{l-k+1} u\) in \(F_j\). Combined with the earlier requirement that \(\alpha \ge 1/2\) this yields \(\gamma \ge 3/2\).
Proof
Write \(F_j = f_j^e + \sum _{i=1}^6 F_j^i\) with
As \(\gamma \ge 3/2\) and \(\alpha \le \gamma -1\), we have \(2\gamma - \frac{5}{2} \ge \alpha \), so that
so that it suffices to bound the right-hand side.
Analysis of \(F_j^1\). We write
For \(F_{j,low}^1\), we notice that for \(l \le \left\lfloor \frac{j+1}{2} \right\rfloor \) there exist a constant \(C = C(r)\) with
This shows
Note that for an absolute constant \(C_a\),
From the 1d Sobolev embedding and Lemma 3, we find that for \(n \le \min (m-1,l)\):
where \(C_A\) is an absolute constant, C is a constant depending on m, r. Note that we used here Lemma 4 to bound \(\Vert \omega _{l+1}\Vert _l\) by \(\Vert (1+y)\omega _{l+1}\Vert _{l+1}\).The factor with the \(\rho \) is explicit:
For \(l \le m-1\), we take \(n=l\) and find that
For \(l > m-1\), we take \(n=m-1\) and find that
for a constant \(C = C(m,\alpha )\) and using that \(l \le \left\lfloor \frac{j+1}{2} \right\rfloor \).
Hence we find for a constant \(C = C(m,\alpha ,r)\) that
using that \(1-\gamma +\alpha \le 0\) and \(2\gamma -2 \le \alpha (m-1)\).
The discrete Young’s convolution inequality implies for all \(t \in [0,T]\) that
As \(4\gamma - 1 - 2r < -1\), the first integral is finite. Hence we arrive at the required estimate
with a constant \(C=C(m,\alpha ,\gamma ,r)\).
For the treatment of \(F^1_{j,high}\) swap the roles of \(u_l\) and \(u_{j-l+1}\) so that
In the given range \(l=2,\dots ,j-\left\lfloor \frac{j+1}{2} \right\rfloor \) we find
so that it can be bounded as \(F_{j,low}^1\).
Analysis of \(F_j^2\). We write
and note that it vanishes unless \(j \ge 3\).
By \(v_{j-l+1} = -\partial _x \int _0^y u_{j-l+1}\, \mathrm {d}z\) we find for \(n \le \min (m-1,j-l+1)\) using the 1d Sobolev inequality and Lemma 3 that
for a constant \(\mathcal {C} = \mathcal {C}(m,r)\).
In the range \(l=2,\dots ,\left\lfloor \frac{j+1}{2} \right\rfloor \) for \(F_{j,low}^2\) we find that \(j-l+1 \ge \frac{j+1}{2}\) and as we can assume that \(j \ge 3\) we can always ensure that this is at least 2.
For \(\frac{j+1}{2} \le m-1\), we can take \(n=2\) and find a constant \(C = C(m,r)\) such that
and otherwise we can taken \(n=m-1\) and find the same control as for \(F_{j,low}^1\) as
and we can conclude as for \(F_{j,low}^1\).
For \(F_{j,high}^2\) we find
For \(n = \min (m-1,l+1)\) we find
For \(l+1<m-1\) we can find a constant \(C = C(m)\) such that
Using the stronger assumption \(2\gamma -1 \le \alpha (m-1)\), we can then conclude as in the treatment of \(F_{j,low}^1\) that
Hence we find
As \(3\gamma -2r < -1\), this gives the required estimate
with a constant \(C=C(m,\alpha ,\gamma ,r)\). As \(r-1 \le r+1-\gamma \) this is the required control.
Analysis of \(F_{j}^3\) and \(F_{j}^5\). We write
with the convention that
Using the definition of \(U^e\) and the 1d Sobolev embedding theorem we find
As \(l\ge 2\), this implies
For \(l=2,\dots ,\left\lfloor \frac{j+1}{2} \right\rfloor \) we find for a constant \(C = C(\gamma ,r)\)
so that as \(l \ge 2\)
Hence we find
As \(2\gamma -R < -1\) this gives the bound
For \(l=\left\lfloor \frac{j+1}{2} \right\rfloor +1,\dots ,j\) we find
so that
Hence we find
As \(r > \frac{1}{2}\) this gives the bound
which is the required bound as \(R \ge r+3\gamma -2\).
Analysis of \(F_j^4\). This term is creating trouble with the integrability in y as \(V^e \sim y\) and is the reason for most technical difficulties.
We write
As \(l \ge 1\) we find
where \(C=C(r)\) is constant. In the last line we used the 1d Sobolev inequality and that
For \(l=1,\dots ,\left\lfloor \frac{j+1}{2} \right\rfloor \) we find
so that
Hence we find
as \(4\gamma -2R < -1\). This gives the bound
which is the required bound as \(-1 \le 1-\gamma \).
For \(F_{j,high}^4\) we find
As \(-1+\gamma -r < - \frac{1}{2}\) this gives the bound
As \(R \ge r+3\gamma -2\) this is the required result.
Analysis of \(F_j^6\). We write
As \(\partial _y U^e\) is exponentially decaying, we find
For \(F_{j,low}^6\) we find (using that \(\left( {\begin{array}{c}j\\ l\end{array}}\right) \frac{M_j}{M_l M_{j-l+1}}\le C (l+1)^{-r}\) for \(l=2,\ldots , \left\lfloor \frac{j+1}{2} \right\rfloor \)):
As \(\gamma -R < - \frac{1}{2}\) this gives the control
For \(F_{j,high}^6\) we find
As \(\gamma -r < - \frac{1}{2}\) this gives the control
which is the required control as \(R \ge r+1-\gamma \). \(\square \)
As a direct consequence of Proposition 7 and Proposition 13, we can state the following corollary, where we use that \(F_j|_{y=0} = f^e_j|_{y=0}\) as \(u\) and \(v\) vanish at \(y=0\).
Corollary 14
Fix the parameters \(m, \alpha , \gamma , r, R\) as in (32) and \(\alpha \ge 1/2\). There exists \(\mathcal {C}\) and \(\mathbf {C}\) such that for all \(\beta , \tau _1, T\) with
we have
where
Control of the Low Norm and Final A Priori Estimate
Corollary 14, which shows an a priori bound on the Gevrey norm of u, was derived under a lower bound on \(\beta \) involving \(\Vert (U^P,V^P) \Vert _{low}\). The last step is to see how this low norm relates to \({\left| \left| \left| u \right| \right| \right| }\). A convenient approach is to establish an additional estimate on a weighted Sobolev norm, namely
where the summation variable is the multiindex \(\bar{\alpha } = (\bar{\alpha }_1,\bar{\alpha }_2)\). In this setting, we can state the following estimate.
Lemma 15
Let \(s \ge 3\) be an even integer, \(m\ge s+2\), \(\alpha \ge 0\), \(r \in \mathbb {R}\), \(\gamma \ge 1\), and define \({\left| \left| \left| u \right| \right| \right| }\) as in (35). Then, there exists C depending on \(s,m,\alpha ,\gamma ,r\) such that
where \(\omega ^P = \partial _y U^P\).
Proof
A similar estimate was established in [27, Proposition 5.6], so that we will only explain the main steps. The starting point is the advection-diffusion equation on the vorticity
One applies \(\partial ^\alpha \) to the equation, test it against \((1+y)^{2\alpha _2} \rho _0 \partial ^\alpha \omega ^P\), and sum over \(|\alpha | \le s\). Then,
Using the equation on \(U^P\), one can obtain recursively boundary conditions for the odd derivatives \(\partial _y^{2k+1} \omega ^P\), starting from the Neumann condition
More precisely, the boundary data \(\partial _y^{2k+1} \omega ^P\vert _{y=0}\) can be expressed in terms of the data \(U^E\) and of products of mixed derivatives \(\partial _x^{\rho _1} \partial _y^{\rho _2}\omega ^P\vert _{y=0}\) with \(\rho _2\le 2k-2\). We refer to [27, Lemma 5.9] for the expressions of these boundary conditions. This allows to establish the following bound, cf equations (5.20)-(5.22) in [27]:
The diffusion term does not raise any difficulty: we find
where C depends on s and m. Also, through standard estimates, we find
and
The other part of the commutator is slightly more delicate. First, one can show that
Note that the weight (that grows with the number of y-derivatives) allows to compensate for the linear growth in y of \(V^P\). The success of this trick comes from the fact that we are interested here in Sobolev estimates (contrary to the former Gevrey estimates). When \(\alpha _1 = s\), namely \(\alpha = (s,0)\), one can show similarly that
However, the term where the s derivatives with respect to x apply to \(V^P\) cannot be handled with usual manipulations. It is the well-known loss of x-derivative peculiar to the Prandtl equation: in particular, one cannot control \(\Vert (1+y)^{-1}\partial ^s_x V^P \Vert _{L^2(\rho _0)}\) by \(\Vert \omega ^P\Vert _{\mathcal {H}^s}\). This is where \({\left| \left| \left| u \right| \right| \right| }\) is involved. We find that
using that \(\Vert (1+y)^{-1} \partial _x^s V^P\Vert _{\infty } \le C \Vert \partial _x^{s+1}u^P\Vert _\infty \le C {\left| \left| \left| u \right| \right| \right| }\) as soon as \(m \ge s+2\). Putting together the previous estimates yields the result. \(\square \)
We conclude this section with
Proposition 16
Let us fix \(s=6\), \(m \ge s+2\), \(\alpha \), \(\gamma \), r, R as in (32) and \(\alpha \ge 1/2\). Further fix \(\tau _1 >0\). Let
where \(\mathbf {C}\) is the constant appearing in Corollary 14. There exists \(\beta _*\) and \(T_*\) depending on \(\tau _1\), \(M_\mathrm {in}\), on \(\Vert \omega ^P\vert _{t=0}\Vert _{\mathcal {H}^s}\), on \(\sup _{[0,T_0]} |U^E|_{\gamma ,\tau _0,R}^2\) and on various Sobolev norms of \(U^E\), such that, for all \(\beta > \beta _*\) and for all \(T \le T_*\) with \(\tau (T) \ge \tau _1\): if \({\left| \left| \left| u \right| \right| \right| }^2 \le \frac{2M_\mathrm {in}}{\beta }\), then \({\left| \left| \left| u \right| \right| \right| }^2 \le \frac{3M_\mathrm {in}}{2\beta }\).
Proof
Let \(\beta , T\) such that \({\left| \left| \left| u \right| \right| \right| }^2 \le \frac{2M_\mathrm {in}}{\beta } \le 2 M_\mathrm {in}\) (assuming \(\beta \ge 1\)). We first apply Lemma 15, which yields
Integrating this differential inequality shows
for \(T \le T_1\), where \(T_1\) depends on \(M_\mathrm {in}\), \(\sup _{t \in [0,T_0]} \Vert U^e(t) \Vert _{H^{s+1}({\mathbb {T}})}\), \(\int _0^{T_0} \Vert \partial _t^l (\partial _t + U^E \partial _x U^E) \Vert ^2_{H^{s-2l}}\, \mathrm {d}t\) and on \(\Vert \omega ^P\vert _{t=0} \Vert _{\mathcal {H}^s}\).
Standard Sobolev imbeddings imply that
As regards the other terms defining \(\Vert (U^P,V^P)\Vert _{low}\), cf (19), they all involve \(\omega ^P\) and are controlled by \(\Vert \omega ^P\Vert _{\mathcal {H}^s}\) as soon as \(s \ge 5\). Hence, it follows from (39) that
for \(T \le T_1\) and for some K depending on \(M_\mathrm {in}\), \(\Vert \omega ^P\vert _{t=0} \Vert _{\mathcal {H}^s}\) and various norms of \(U^E\). If we now choose
where \(\mathcal {C}\) is the constant appearing in Corollary 14, we obtain for \(\beta \ge \beta _*\):
Taking \(\beta _*\) large enough so that
we get
If we take \(T_* \le T_1\) such that \(2 \mathbf {C} \int _0^{T_*} \Vert (1+y) f^e_j \Vert _{\gamma ,\tau ,r-2+\gamma }^2\, \mathrm {d}t \le \frac{1}{2} M_\mathrm {in}\), the result follows. \(\square \)
Existence and Uniqueness
On the basis of the previous a priori estimates, we now complete the proof of Theorem 1: we construct a unique solution of (1)–(2) with data \(U^P_\mathrm {in}\). This obviously amounts to constructing a unique solution of (13)–(15) with data \(u_\mathrm {in}:= U^P_\mathrm {in}- U^e\vert _{t=0}\).
We fix \(s=6\), \(\gamma =2\). We take \(m \ge s+2\) and \(\displaystyle \alpha \ge \tfrac{1}{2}\) that satisfy the inequalities in the first line of (32). Let \(0< \tau _1 < \tau _0\), \(r \in \mathbb {R}\), \(T_0 > 0\), and \(\displaystyle U^E\), \(U^P_\mathrm {in}= u_\mathrm {in}+ U^e\vert _{t=0}\) satisfying the assumptions of the theorem. Let now \((\tau '_0, \tau '_1)\) with \(\displaystyle 0< \tau _1< \tau '_1< \tau '_0 < \tau _0\). Let \(r'\) and \(R'\) as in the second line of (32). As \(\tau _0 > \tau '_0\), we have
while
and
for a constant C possibly depending on \(\tau _0, \tau _0', r, r', R'\).
The idea is then to apply Proposition 16 to a solution of an approximate system, for which well-posedness is granted. Inspired by [27], we consider the regularized equation
adding a tangential diffusion \(- \epsilon \partial _x^2 u\). The modified vector field \((U^e_\epsilon , V^{e}_\epsilon )\) takes the form
where \(U^E_\epsilon \) is an analytic approximation of \(U^E\), converging to \(U^E\) in the norm \(| \ |_{2,\tau _0, r}\) as \(\epsilon \rightarrow 0\). Note that \((U^e_\epsilon , V^e_\epsilon )\) is still divergence-free, but has now fast decay in y, so that all difficulties generated by the linear growth of \(V^e\) vanish. Accordingly, the right-hand side \(f^e\) is modified into \(f^e_\epsilon \) replacing \(U^E\) by \(U^E_\epsilon \), resp. \((U^e, V^e)\) by \((U^e_\epsilon , V^e_\epsilon )\) in (14). Similarly, one regularizes the initial data to obtain some \(u_{\mathrm {in},\epsilon }\) real analytic in x, y, with fast decay at infinity in y (and obeying suitable compatibility conditions).
One can show that system (40) is well-posed following classical methods for fully parabolic equations. For instance, for \(T_{\epsilon ,max}\) small enough, one can prove the existence of a Sobolev solution \(u_\epsilon \) on \((0, T_{\epsilon ,max})\) through a fixed point argument applied to
with \(F_\epsilon [u] = f^e_\epsilon - (u \partial _x + v \partial _y) u - (U^e_\epsilon \partial _x + V^e_\epsilon \partial _y) u - (u \partial _x + v \partial _y) U^e_\epsilon \). Moreover, \(u^\epsilon \) remains (real) analytic in (x, y) as long as the Sobolev norm of \(u_\epsilon \) does not blow up, that is on \((0,T_{\epsilon ,max})\). This property, related to the analytic regularization of the heat kernel is well-known, even in the more difficult context of the Navier-Stokes equation: see [2, 7, 25] and references therein.
We now claim that all a priori estimates obtained for a solution u of (13) can be established for \(u_\epsilon \) solution of (40), uniformly in \(\epsilon \). For this, one just needs to adapt the definitions of the auxiliary quantities \(H_j\) and \(\phi _j\): we rather consider
and
The additional good terms coming from \(-\epsilon \partial ^2_x\) allow to control the extra commutator terms that it generates. Hence, we can apply Proposition 16 with \(\tau '_0, \tau '_1\), \(r'\) and \(R'\) instead of \(\tau _0, \tau _1, r\), and R. Let \(\beta _*\) and \(T_*\) given by the proposition (note that they are independent of \(\epsilon \)). We then introduce
where \(\beta > \beta _*\) is fixed, and \({\left| \left| \left| u \right| \right| \right| }\) is defined in (35). Note that \({\left| \left| \left| u \right| \right| \right| }\) implicitly depends on T. By continuity in time of \(u^\epsilon \), one has \(T_{\epsilon , *} > 0\). But from Proposition 16, one deduces easily that for any \(T \le T_*\), \(\, T_{\epsilon , max} \ge T_{\epsilon , *} \ge T\).
From there, by standard compactness arguments, one obtains a solution to the Prandtl system over [0, T], with the regularity properties stated in the theorem. It remains to show uniqueness. For this, we take two solutions \(u^1\) and \(u^2\) up to time T. The difference \(u^d\) then satisfies (from (13))
We then find for \(u^d_j\) that
where again \(F^d_j\) consists of \(f_j^{d,e}\) and mixed terms with less than j derivatives on \(u^1\), \(u^2\) or \(u^d\). Comparing with (16), we see that the only difference is the replacement of \((U^P, V^P)\) by \((U^{1,P}, V^{2,P}\)). Let us stress that the latter field not being divergence-free is not an issue: none of the a priori estimates carried in Sect. 4 and Sect. 5 were using the fact that \((U^P, V^P)\) was divergence-free. One can therefore obtain a similar Gevrey bound on \(u^d\), under a lower bound on \(\beta \) (involving the low norms of \((U^{1,P}, V^{1,P})\) and \((U^{2,P}, V^{2,P})\)). This provides a stability estimate which shows uniqueness. These considerations now finish the proof of our main result Theorem 1.
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Acknowledgements
The authors thank Weiren Zhao for useful remarks on the first version of this manuscript. They acknowledge the support of the Université Sorbonne Paris Cité, through the funding “Investissements d’Avenir”, convention ANR-11-IDEX-0005. This work was also supported by the SingFlows Project, Grant ANR-18-CE40-0027 of the French National Research Agency (ANR). H.D. is grateful to the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. PCOFUND-GA-2013-609102, through the PRESTIGE programme coordinated by Campus France. D.G.-V. acknowledges the support of the Institut Universitaire de France.
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Dietert, H., Gérard-Varet, D. Well-Posedness of the Prandtl Equations Without Any Structural Assumption. Ann. PDE 5, 8 (2019). https://doi.org/10.1007/s40818-019-0063-6
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DOI: https://doi.org/10.1007/s40818-019-0063-6