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 [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]. 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 [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.

Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global κ-entropy solutions to the compressible Navier–Stokes systems with degenerate viscosities

“…Each of the velocity vector fields u and 2∇ϕ( ) is associated with different density (1 − κ) and κ , respectively. The main objective is not to prove global existence of solutions rigorously but to show that the two-velocity hydrodynamics is consistent with the study performed in the first part of the present series [11] for the zero Mach number system. Our formulation uses the generalized κ-temperature which is not a priori the usual temperature.…”

“…Here we construct the weak solution to the system similar to the considered in [39] modulo capillarity terms. It was however shown in [11] that this construction is compatible with the quantum capillary term of the form κ ∇(∆ √ / √ ) appearing in the ghost system [28], see also [21] for the study on quantum viscous Navier-Stokes system and [4] for a full range of compatible capillary terms. Concerning construction of approximate solutions with singular pressure, drag terms or capillarity terms, the authors gave some hints in [8] for the general setting λ( ) = 2(µ ( ) − µ( )).…”

“…Our construction of the approximate solutions is based on an augmented approximate scheme using this two-velocity structure. Similar approximate scheme has been introduced for the zero Mach number system in the first part of the present series [11]. For reader's convenience, since lack of divergence-free condition gives rise to several new terms and since the κ-entropy and the limit process changes, we repeat the whole construction from [11].…”

Measure-valued solutions to the complete Euler system revisited

“…[19], Cai, Cao, Sun [10]. Recently, the interest in "two velocity models" has been revived in Bresch et al [7], [8].…”

A Singular Limit Problem for Rotating Capillary Fluids with Variable Rotation Axis

“…Remark A. 10. The rotation term in [13] was dealt with in a slightly different way, exploiting the special law of the classical component of the pressure.…”