Nonexistence of a shock layer in gas dynamics with a nonconvex equation of state

Abstract

A classical result of Gilbarg states that a simple shock wave solution of Euler's equations is compressive if and only if a corresponding shock layer solution of the Navier-Stokes equations exists, assuming, among other things, that the equation of state is convex. An “entropy condition” appropriate for weeding out “unphysical” shocks in the nonconvex case has been introduced by T.-P. Liu. For shocks satisfying his entropy condition, Liu showed that purely viscous shock layers exist (with zero heat conduction). Dropping the convexity assumption, but retaining many other reasonable restrictions on the equation of state, we construct an example of a (large amplitude) shock which satisfies Liu's entropy condition but for which a shock layer does not exist if heat conduction dominates viscosity. We also give a simple restriction, weaker than convexity, which does guarantee that shocks which satisfy Liu's entropy condition always admit shock layers.