Extension of the Hoff solutions framework to cover Navier-Stokes equations for a compressible fluid with anisotropic viscous-stress tensor

TL;DR

This paper extends Hoff's solutions framework for the Navier-Stokes equations of a compressible fluid to include anisotropic viscous-stress tensors, allowing for less restrictive initial density conditions and broader parameter ranges.

Contribution

It generalizes Hoff's intermediate regularity solutions to cover anisotropic viscous-stress tensors with variable coefficients, relaxing initial density integrability requirements.

Findings

Constructed global weak solutions under new conditions.

Relaxed the admissible range of the adiabatic coefficient b3.

Extended the framework to include general fourth order viscous-stress tensors.

Abstract

This paper deals with the Navier-Stokes system governing the evolution of a compressible barotropic fluid. We extend D. Hoff's intermediate regularity solutions framework by relaxing the integrability needed for the initial density which is usually assumed to be $L^{\infty}$. By achieving this, we are able to take into account general fourth order symmetric viscous-stress tensors with coefficients depending smoothly on the time-space variables. More precisely, in space dimensions $d=2,3$, under periodic boundary conditions, considering a pressure law $p(\rho)=a\rho^{\gamma}$ whith $a>0$ respectively $\gamma\geq d/(4-d)$) and under the assumption that the norms of the initial data $\left( \rho_{0}-M,u_{0}\right) \in L^{2\gamma}\left(\mathbb{T}^{d}\right) \times(H^{1}(\mathbb{T}^{d}))^{d}$ are sufficiently small, we are able to construct global weak solutions. Above, $M$ denotes the total mass of the fluid while $\mathbb{T}$ with $d=2,3$ stands for periodic box. When comparing to the results known for the global weak solutions \`{a} la Leray, i.e. constructed assuming only the basic energy bounds, we obtain a relaxed condition on the range of admissible adiabatic coefficients $\gamma$.