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

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 $\gamma$.