Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems

TL;DR

This paper proves the global existence of weak solutions for zero Mach number fluid systems, revealing two-velocity hydrodynamics and extending models of gaseous mixtures and ghost effects.

Contribution

It relaxes previous algebraic constraints, introduces a new entropy, and advances understanding of two-velocity hydrodynamics in zero Mach number systems.

Findings

Proved global weak solutions for zero Mach number systems.

Extended gaseous mixture models to a weak solutions framework.

Analyzed the impact of density-dependent heat conductivity on solution existence.

Abstract

In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [D. Bresch, E.H. Essoufi, and M. Sy, J. Math. Fluid Mech. 2007] gives a more complete answer to an open question formulated in [P.-L. Lions, Oxford 1998]. A new mathematical entropy shows clearly the existence of two-velocity hydrodynamics with a fixed mixture ratio. As an application of our result we first discuss a model of gaseous mixture extending the results of [P. Embid, Comm. Partial Diff. Eqs. 1987] to the global weak solutions framework. Second, we present the ghost effect system studied by [C.D. Levermore, W. Sun, K. Trivisa, SIAM J. Math. Anal. 2012] and discuss a contribution of the density-dependent heat-conductivity coefficient to the issue of existence of weak solutions.