Stability of Multidimensional Thermoelastic Contact Discontinuities

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