Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D

摘要

We consider Navier-Stokes equations for compressible viscous fluids in onedimension. We prove the existence of global strong solution with large initialdata for the shallow water system. The key ingredient of the proof relies to anew formulation of the compressible equations involving a new effectivevelocity $v$ (see \cite{cras,para,CPAM,CPAM1}) such that the density verifies aparabolic equation. We estimate $v$ in $L^\infty$ norm which enables us tocontrol the vacuum on the density.