On a modified shock front problem for the compressible Navier-Stokes equations

We discuss the possibility of considering the shock wave in a compressible viscous heat conducting gas as a strong discontinuity on which surface the generalized Rankine-Hugoniot conditions hold. The corresponding linearized stability problem for a planar shock lacks boundary conditions, i.e., the shock wave in a viscous gas viewed as a (fictitious) strong discontinuity is like undercompressive shock waves in ideal fluids and, therefore, it is unstable against small perturbations. We propose such additional jump conditions so that, the stability problem becomes well-posed and its trivial solution is asymptotically stable (by Lyapunov). The choice of additional boundary conditions is motivated by a priori information about steady-state solutions of the Navier-Stokes equations which can be calculated, for example, by the stabilization method. The established asymptotic stability of the trivial solution to the modified linearized shock front problem can allow us to justify, at least on the linearized level, the stabilization method that is often used, for example, for steady-state calculations for viscous blunt body flows.