Low Mach Number Limit of Non-isentropic Inviscid Elastodynamics with General Initial Data

TL;DR

This paper establishes the incompressible limit of 3D non-isentropic inviscid elastodynamic equations with general initial data, employing novel entropy regularity and wave equation structures to achieve uniform estimates and strong convergence.

Contribution

It provides the first rigorous proof of the incompressible limit for non-isentropic elastodynamics with general initial data in 3D half-space, introducing new analytical techniques.

Findings

Uniform estimates in Mach number are achieved.

Strong convergence of solutions in time is proven.

Entropy regularity and wave structure are key to analysis.

Abstract

We prove the incompressible limit of non-isentropic inviscid elastodynamic equations with general initial data in 3D half-space. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity and degenerates in the normal direction on the solid wall. The uniform estimates in Mach number are established based on two important observations. First, the entropy has enhanced regularity in the direction of each column of the deformation tensor, which exactly helps us avoid the loss of derivatives caused by the simultaneous appearance of elasticity and entropy in vorticity analysis. Second, a special structure of the wave equation of the pressure together with elliptic estimates helps us reduce the normal derivatives in the control of divergence and pressure. The strong convergence of solutions in time is obtained by proving local energy decay of the wave equation and using the technique of microlocal defect measure.