A new construction of weak solutions to compressible Navier-Stokes equations

TL;DR

This paper establishes the existence of weak solutions to the three-dimensional compressible Navier-Stokes equations with barotropic pressure, introducing a novel approximation scheme compatible with the Bresch-Jabin compactness criterion.

Contribution

It presents a new approximation method that directly truncates and regularizes nonlinear terms, differing from classical viscosity-based regularization, and rigorously applies the Bresch-Jabin criterion.

Findings

Existence of weak solutions for $ ho^ ext{gamma}$ pressure with $ ext{gamma} \\geq 9/5$

New approximation scheme compatible with compactness criteria

Rigorous validation of the Bresch-Jabin criterion in this context

Abstract

We prove the existence of the weak solutions to the compressible Navier--Stokes system with barotropic pressure $p(\varrho)=\varrho^\gamma$ for $\gamma\geq 9/5$ in three space dimension. The novelty of the paper is the approximation scheme that instead of the classical regularization of the continuity equation (based on the viscosity approximation $\ep \Delta \varrho$) uses more direct truncation and regularisation of nonlinear terms an the pressure. This scheme is compatible with the Bresch-Jabin compactness criterion for the density. We revisit this criterion and prove, in full rigour, that it can be applied in our approximation at any level.