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

Abstract

In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics with periodic boundary conditions. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [6] gives a more complete answer to an open question formulated in [23]. We introduce a new mathematical entropy which clearly shows 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 [10] to the global weak solutions framework. Second, we present the ghost effect system studied by C.D. Levermore, W. Sun and K. Trivisa [20] and discuss a contribution of the density-dependent heat-conductivity coefficient to the issue of existence of weak solutions.