Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data

Abstract

We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, isothermal flow in two and three space dimensions when the initial density is close to a constant in L 2 and L ∞, and the initial velocity is small in L 2 and bounded in L 2 n (in two dimensions the L 2 norms must be weighted slightly). A great deal of qualitative information about the solution is obtained. For example, we show that the velocity and vorticity are relatively smooth in positive time, as is the "effective viscous flux" F, which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a crucial role in the entire analysis, particularly in closing the required energy estimates, understanding rates of regularization near the initial layer, and most important, obtaining time-independent pointwise bounds for the density.