Stability of Maxwell States in Thermo-Elasticity

Abstract

AbstractWe study the Cauchy problem for model equations of 1-D thermoelasticity that admit a solid-solid phase transition. The Maxwell states are defined to be two constant states such that the entropy is equal in the both states. A small perturbation of these Maxwell states will be our initial data. In the isothermal model, we shall show that: there exists a global in time admissible phase boundary satisfying the Abeyaratne-Knowles kinetic condition; as time goes to infinity, the strain and the velocity outside the phase boundary tend to the Maxwell states and the entropy tends to a certain limit function. In the polytropic case (the internal energy is proportional to the temperature), we shall show that there exist unique Maxwell states for given temperature and there exist at least two admissible solutions to the Riemann problem with certain initial data in a neighborhood of the Maxwell states.KeywordsPhase BoundaryIsothermal ModelRiemann ProblemAdmissible SolutionWeak Global SolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.