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

Abstract

We prove the local well-posedness in Sobolev spaces and incompressible limit for the motion of a compressible isentropic inviscid elastic fluid with moving physical vacuum boundary provided that Rayleigh-Taylor sign condition initially holds. The deformation tensor satisfies the neo-Hookean linear elasticity and is tangential to the free boundary. We apply the tangential smoothing method to set the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou-Lindblad [13] elliptic estimates reduces the energy estimates to the control of tangentiallydifferentiated wave equations in spite of a potential loss of derivative in the source term. Then we use the hyperbolic approach instead of fixed-point argument to solve the linearized system and Picard iteration to establish the well-posedness. Our energy estimate is of no regularity loss and uniform in sound speed which yields the incompressible limit: The solutions and energy functionals of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. To the best of our knowledge, we first find out the suitable energy functionals without loss of regularity for free-boundary compressible elastodynamics system.It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to analyze the full time derivatives in boundary energy and higher order wave equations as in the previous works of compressible Euler equations even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad-Luo [62] and Luo [64] can still be recovered for a slightly compressible elastic fluid by further delicate analysis which is completely different from Euler equations.