New effective pressure and existence of global strong solution for compressible Navier–Stokes equations with general viscosity coefficient in one dimension

Abstract

In this paper we prove the existence of a unique global strong solution for the Cauchy problem associated to the one dimensional Navier–Stokes equations with general degenerate viscosity coefficients. The cornerstone of the proof is the introduction of a new effective pressure which allows to obtain an Oleinik-type estimate for the so called effective velocity. In our proof we make use of additional regularizing effects on the velocity which requires to extend the techniques developed by Hoff for the constant viscosity case.