DISCONTINUOUS SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR MULTIDIMENSIONAL FLOWS OF HEAT-CONDUCTING FLUIDS

We prove the global existence of weak solutions of the Navier-Stokes e quations for compressible, heat-conducting fluids in two and three spa ce dimensions when the initial density is close to a constant in L-2 b oolean AND L-infinity, the initial temperature is close to a constant in L-2, and the initial velocity is small in H-s boolean AND L-4, wher e s = 0 when n = 2 and s > 1/3 when n = 3. (The L-p norms must be weig hted slightly when n = 2.) In particular, the initial data may be disc ontinuous across a hypersurface of R-n. A great deal of qualitative in formation about the solution is obtained. For example, we show that th e velocity, vorticity, and temperature are relatively smooth in positi ve time, as is the ''effective viscous flux'' F, which is the divergen ce of the velocity minus a certain multiple of the pressure. We find t hat F plays a central role in the entire analysis, particularly in clo sing the required energy estimates and in understanding rates of regul arization near the initial layer. Moreover, F is precisely the quantit y through which the hyperbolicity of the corresponding equations for i nviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density.