Existence Theory for the Kinetic-Fluid Coupling When Small Droplets Are Treated as Part of the Fluid

Abstract

We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incompressible Navier-Stokes equation and of the Vlasov-Boltzmann equation, together with a fragmentation kernel. We first show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna-Lions theory for singular transport equations, and a compactness lemma specifically tailored for our study.

Global weak solution to the inhomogeneous Navier–Stokes–Vlasov equations

“…We use f to denote the solution to (3.15). We follow the same idea in [4] for the compactness. Thanks to the maximum principle, we have…”

“…The uniqueness and existence of the Vlasov equation can be obtained when (ρ, u) is smooth, see [10,19]. The compactness of f will be obtained by an approach motivated by the recent work [4]. In fact, the DiPerna-Lions compactness in [10] will be helpful when we use the fixed point argument to solve our approximate system.…”

Uniqueness of the solution to the 2D Vlasov–Navier–Stokes system

“…Observe that the exponential factor appearing here comes from the dissipation term −v in the drag force (see (15) and (16)). (4) to hold is the following pointwise decay condition:…”

“…A paradigm of such models is the Vlasov-Navier-Stokes system: among other applications, it has been used to describe the transport of particles in the upperways for medical purposes (see for instance [6]) and at the same time it offers important mathematical issues to deal with: existence of solution [2,7], long-time behavior [11,14], asymptotic limit [4], controllability [18]...…”

The Vlasov–Navier–Stokes System in a 2D Pipe: Existence and Stability of Regular Equilibria

“…Typical examples are hydrodynamic limits for which the purpose is to replace the density function by averaged quantities (mass, momentum etc.) in order to recover, after a rigorous asymptotic, classical equations of fluid mechanics, see [25], [5] or [6] for an example involving a collision operator. Since kinetic equations are not "first principles" per se, a comprehensive derivation would suggest to obtain asymptotically fluid-kinetic systems starting from fluid-solid equations.…”