Local Well-posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics

TL;DR

This paper presents a new proof of local well-posedness and establishes the incompressible limit for a 3D free-boundary compressible elastodynamic system, using energy methods and hyperbolic approaches, with implications for Euler equations.

Contribution

It introduces a novel combination of classical energy and hyperbolic methods for free-boundary elastodynamics, simplifying previous approaches and extending to incompressible limits.

Findings

Established nonlinear energy estimates without loss of regularity.

Proved uniform energy estimates in sound speed, enabling the incompressible limit.

Method applicable to compressible Euler equations and elastic media.

Abstract

We consider 3D free-boundary compressible elastodynamic system under the Rayleigh-Taylor sign condition. It describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor satisfies the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin [84] by Nash-Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach and also establish the incompressible limit. We apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou-Lindblad elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations in spite of a potential loss of derivative in the source term. We first establish the nonlinear energy estimate without loss of regularity for the free-boundary compressible elastodynamic system. The energy estimate is also uniform in sound speed which yields the incompressible limit. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in boundary energy and analyze higher order wave equations as in the previous works of compressible Euler equations (cf. Lindblad-Luo [59] and Luo [61]) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in [59,61] can still be recovered for a slightly compressible elastic medium by further delicate analysis which is completely different from Euler equations.