Lagrangean Structure and Propagation of Singularities in Multidimensional Compressible Flow

David Hoff
Department of Mathematics
Indiana University

Abstract: We study the Lagrangean structure of solutions of the Navier-Stokes equations of multidimensional compressible flow in an intermediate regularity class in which initial energies are small and densities are essentially bounded. We prove that there is a unique particle trajectory emanating from each point of any open set in physical space in which the initial density is strictly positive and that such open sets are convected homeomorphically by the flow. As corollaries we show that Holder continuous surfaces are transported into Holder continuous surfaces, that sectional continuity of the density and the divergence of the velocity are preserved, that the Rankine-Hugoniot conditions hold in a strict, pointwise sense across such surfaces, and that the strengths of singularities decay exponentially in time when the pressure is a monotone function of density. These results require that initial velocities are in certain fractional Sobolev spaces depending on the dimension, and we display an example indicating that this requirement is necessary.