Discontinuous solutions of the Navier-Stokes equations for compressible flow

Abstract

We prove local existence and study properties of discontinuous solutions of the Navier-Stokes equations for one-dimensional, compressible, nonisentropic flow. We assume that, modulo a step function, the initial data is in L2 and the initial velocity and density are in the space BV. We show that the velocity and the temperature become smoothed out in positive time, and that discontinuities in the density, pressure, and gradients of the velocity and temperature persist for all time. We also show that for stable gases these discontinuities decay exponentially in time, more rapidly for smaller viscosities.