Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global $\kappa$-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities

TL;DR

This paper proves the global existence of $ abla$-entropy solutions for 3D compressible Navier-Stokes equations with degenerate viscosities, introducing a $ abla$-entropy involving a mixture of two velocities.

Contribution

It establishes the existence of $ abla$-entropy solutions for degenerate viscosity Navier-Stokes systems, extending the concept of BD-entropy with a new $ abla$-entropy involving two velocities.

Findings

Existence of $ abla$-entropy solutions in 3D with periodic boundary conditions.

Introduction of a $ abla$-entropy involving a mixture of velocities.

Connection between two-velocity hydrodynamics and degenerate viscosity models.

Abstract

This paper addresses the issue of global existence of so-called $\kappa$-entropy solutions to the Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. We consider the three dimensional space domain with periodic boundary conditions. Our solutions satisfy the weak formulation of the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: $\kappa$-entropy. This new entropy involves a mixture parameter $\kappa \in (0,1)$ between the two velocities $\bf{u}$ and $\bf{u} + 2\nabla\varphi(\varrho)$ (the latter was introduced by the first two authors in [C. R. Acad. Sci. Paris 2004]), where $\bf{u}$ is the velocity field and $\varphi$ is a function of the density $\varrho$ defined by $\varphi'(s)=\mu'(s)/s$. As a byproduct of the existence proof, we show that two-velocity hydrodynamics (in the spirit of S.C. Shugrin 1994) is a possible formulation of a model of barotropic compressible flow with degenerate viscosities. It is used in construction of approximate solutions based on a conservative augmented approximate scheme.