The vanishing viscosity method in one-dimensional thermoelasticity

Abstract

The vanishing viscosity method is applied to the system of conservation laws of mass, momentum, and energy for a special class of one-dimensional thermoelastic media that do not conduct heat. Two types of vanishing "viscosity" are considered: Newtonian and artificial, in both cases accompanied by vanishing heat conductivity. It is shown that in either case one can pass to the zero viscosity limit by the method of compensated compactness, provided that velocity and pressure are uniformly bounded. Oscillations in the entropy field may propagate along the linearly degenerate characteristic field but do not affect the compactness of the velocity field or the pressure field. A priori bounds on velocity and pressure are established, albeit only for the case of artificial viscosity.