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Gevrey well-posedness of quasi-linear hyperbolic Prandtl equations

Wei-Xi Li, Tong Yang, Ping Zhang

Abstract

We study the hyperbolic version of the Prandtl system derived from the hyperbolic Navier-Stokes system with no-slip boundary condition. Compared to the classical Prandtl system, the quasi-linear terms in the hyperbolic Prandtl equation leads to an additional instability mechanism. To overcome the loss of derivatives in all directions in the quasi-linear term, we introduce a new auxiliary function for the well-posedness of the system in an anisotropic Gevrey space which is Gevrey class $\frac 32$ in the tangential variable and is analytic in the normal variable.

Gevrey well-posedness of quasi-linear hyperbolic Prandtl equations

Abstract

We study the hyperbolic version of the Prandtl system derived from the hyperbolic Navier-Stokes system with no-slip boundary condition. Compared to the classical Prandtl system, the quasi-linear terms in the hyperbolic Prandtl equation leads to an additional instability mechanism. To overcome the loss of derivatives in all directions in the quasi-linear term, we introduce a new auxiliary function for the well-posedness of the system in an anisotropic Gevrey space which is Gevrey class in the tangential variable and is analytic in the normal variable.
Paper Structure (10 sections, 8 theorems, 149 equations)

This paper contains 10 sections, 8 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. The a priori estimate in 2D
  3. Tangential derivatives of $u$
  4. Gevrey norm of $\varphi$ and $\partial_yu$
  5. Normal derivatives
  6. Tangential derivatives
  7. Proof of Proposition \ref{['prp:varu']}
  8. Proof of the a priori estimate
  9. The 3D hyperbolic Prandtl equation
  10. Proof of some estimates

Key Result

Theorem 1.2

If the initial data of the hyperbolic Prandtl system hypr satisfy $u_0\in G^{3/2,1}_{2\rho_0,\ell}$ and $u_1\in G^{3/2,1}_{2\rho_0,\ell+1}$ for some $\rho_0>0$ and are compatible to the boundary conditions in hypr. Then problem hypr admits a unique local solution $u\in L^\infty([0,T]; \ G^{3/2,1}_{\

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: A priori estimate
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Proposition 4.1
  • Lemma 4.2
  • ...and 16 more