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

TL;DR

This paper establishes the existence of global strong solutions for one-dimensional compressible Navier-Stokes equations with general viscosity coefficients by introducing a new effective pressure and extending existing techniques.

Contribution

It introduces a novel effective pressure concept enabling the proof of global strong solutions with degenerate viscosity in one dimension.

Findings

Proved existence of global strong solutions.

Developed a new effective pressure technique.

Extended Hoff's methods to variable viscosity case.

Abstract

In this paper we prove the existence of global strong solution for the 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 technics developed by Hoff for the constant viscosity case.