Well‐posedness for compressible Euler equations with physical vacuum singularity

Abstract

Abstract An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristic speeds u ± c coincide and have unbounded spatial derivative since c behaves like x 1/2 close to the boundary. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local‐in‐time well‐posedness of one‐dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity. © 2009 Wiley Periodicals, Inc.