Existence and Uniqueness of Global Strong Solutions for One-Dimensional Compressible Navier-Stokes Equations

We consider the Navier-Stokes equations for compressible viscous fluids in one dimension. It is a well known fact that if the initial data are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists for a small time. In this paper, we show that under the same hypothesis, the density remains bounded by below by a positive constant uniformly in time, and that strong solutions therefore exist globally in time. Moreover, while most existence results are obtained for positive viscosity coefficient, our result holds even if the viscosity coefficient vanishes with the density. Finally, we prove that our solution is unique in the class of weak solutions satisfying the usual entropy inequality. The key point of our paper is a new entropy-like inequality first introduced by Bresch and Desjardins for the shallow water system. This gives some regularity for the density (provided such regularity exists at initial time).