Global solutions of the equations of one–dimensional, compressible flow with large data and forces, and with differing end states

Abstract.

We prove the global existence of solutions of the Navier–Stokes equations of compressible flow in one space dimension with minimal hypotheses on the initial data, the equation of state, and the external force. Specifically, we require of the initial data only that the density be bounded above and below away from zero, and that the density and velocity be in L ², modulo constant states at $x=\infty$ and $x=-\infty$, which may be different. There are no smallness hypotheses on either the data or on the external force. In particular, we include the important case that the initial data is piecewise constant with arbitrarily large jump discontinuities. Our results show that, even in this generality, neither vacuum states nor concentration states can form in finite time.