Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids

Abstract

We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial density is close to a constant in L ²∩L ^∞, the initial temperature is close to a constant in L ², and the initial velocity is small in H ^s∩L ⁴, where s=0 when n=2 and when n=3. (The L ^p norms must be weighted slightly when n=2.) In particular, the initial data may be discontinuous across a hypersurface of ⁿ. A great deal of qualitative information about the solution is obtained. For example, we show that the velocity, vorticity, and temperature 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 central role in the entire analysis, particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer. Moreover, F is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density.