Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data

Abstract

We extend to general polytropic pressures P(ρ) = Kργ, γ > 1, the existence theory of [8] for isothermal (γ= 1) flows of Navier-Stokes fluids in two and three space dimensions, with fairly general initial data. Specifically, we require that the initial density be close to a constant in L2 and L∞, and that the initial velocity be small in L2 and bounded in L2n (in two dimensions the L2 norms must be weighted slightly). Solutions are obtained as limits of approximate solutions corresponding to mollified initial data. The key point is that the approximate densities are shown to converge strongly, so that nonlinear pressures can be accommodated, even in the absence of any uniform regularity information for the approximate densities.