GLOBAL-SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR MULTIDIMENSIONAL COMPRESSIBLE FLOW WITH DISCONTINUOUS INITIAL DATA

We prove the global existence of weak solutions of the Navier-Stokes e quations for compressible, isothermal flow in two and three space dime nsions when the initial density is close to a constant in L(2) and L(i nfinity), 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 gre at deal of qualitative information about the solution is obtained. For example, we show that the velocity and vorticity are relatively smoot h in positive time, as is the ''effective viscous flux'' F, which is t he divergence of the velocity minus a certain multiple of the pressure . We find that F plays a crucial role in the entire analysis, particul arly in closing the required energy estimates, understanding rates of regularization near the initial layer, and most important, obtaining t ime-independent pointwise bounds for the density. (C) 1995 Academic Pr ess, Inc.