Stability of Multidimensional Thermoelastic Contact Discontinuities

TL;DR

This paper establishes the linear stability of multidimensional thermoelastic contact discontinuities in nonisentropic thermoelastic systems, providing a priori estimates that remain valid even as the discontinuity strength diminishes.

Contribution

It introduces a stability condition and proves linear stability for thermoelastic contact discontinuities, overcoming challenges posed by characteristic boundary conditions and zero-strength limits.

Findings

Established a stability condition for contact discontinuities.

Proved a priori tame estimates for the linearized problem.

Demonstrated estimates remain valid as discontinuity strength approaches zero.

Abstract

We study the system of nonisentropic thermoelasticity describing the motion of thermoelastic nonconductors of heat in two and three spatial dimensions, where the frame-indifferent constitutive relation generalizes that for compressible neo-Hookean materials. Thermoelastic contact discontinuities are characteristic discontinuities for which the velocity is continuous across the discontinuity interface. Mathematically, this renders a nonlinear multidimensional hyperbolic problem with a characteristic free boundary. We identify a stability condition on the piecewise constant background states and establish the linear stability of thermoelastic contact discontinuities in the sense that the variable coefficient linearized problem satisfies a priori tame estimates in the usual Sobolev spaces under small perturbations. Our tame estimates for the linearized problem do not break down when the strength of thermoelastic contact discontinuities tends to zero. The missing normal derivatives are recovered from the estimates of several quantities relating to physical involutions. In the estimate of tangential derivatives, there is a significant new difficulty, namely the presence of characteristic variables in the boundary conditions. To overcome this difficulty, we explore an intrinsic cancellation effect, which reduces the boundary terms to an instant integral. Then we can absorb the instant integral into the instant tangential energy by means of the interpolation argument and an explicit estimate for the traces on the hyperplane.