On the analysis of a coupled kinetic-fluid model with local alignment forces

Abstract

This paper studies global existence, hydrodynamic limit, and large-time behavior of weak solutions to a kinetic flocking model coupled to the incompressible Navier-Stokes equations. The model describes the motion of particles immersed in a Navier-Stokes fluid interacting through local alignment. We first prove the existence of weak solutions using energy and L p estimates together with the velocity averaging lemma. We also rigorously establish a hydrodynamic limit corresponding to strong noise and local alignment. In this limit, the dynamics can be totally described by a coupled compressible Euler -incompressible Navier-Stokes system. The proof is via relative entropy techniques. Finally, we show a conditional result on the large-time behavior of classical solutions. Specifically, if the mass-density satisfies a uniform in time integrability estimate, then particles align with the fluid velocity exponentially fast without any further assumption on the viscosity of the fluid.

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

“…Lin, Liu and Zhang [22] proved a global existence and uniqueness of classical solution to a micro-macro model for polymeric fluids with the initial data near the hydrodynamic equilibrium. Carrillo, Choi and Karper [6] proved the global existence, hydrodynamic limit, and large-time behavior of weak solution to a kinetic flocking model coupled with the incompressible Navier-Stokes equations.…”

Global existence of weak solutions for Navier–Stokes-BGK system

“…Along with that applicative interest, the mathematical analysis for various modelling is also emphasized. In the case when the direct particle-particle interactions are absent, there are a number of literature on the global existence of solutions; weak solutions for Vlasov or Vlasov-Fokker-Planck equation coupled with homogeneous/inhomogeneous fluids are studied in [7,13,22,28,41,45], strong solutions near a global Maxwellian for Vlasov-Fokker-Planck equation coupled with incompressible/compressible Euler system are obtained in [8,10,19]. We also refer to [11,12] for the large-time behavior of solutions and finite-time blow-up phenomena in kinetic-fluid systems.…”

Large-time behavior for the Vlasov/compressible Navier-Stokes equations

“…By employing a different Lyapunov functional from the one proposed in [3], we refine assumptions on the solutions and show that the particles will be aligned with the fluid velocity exponentially fast as time evolves. Our strategy can also be applied for the system (1.1) with nonlocal/local velocity alignment forces discussed in [3,9](see Remarks 1.2 and 1.3 below for details) and two-phase fluid models [13,15]. In particular, in [15], the a priori asymptotic behavior estimate plays an important role in constructing the global-in-time classical solutions.…”

“…On the other hand, without the diffusion term, it is not clear to find the nontrivial equilibria, and thus the previous arguments for the large-time behavior of solutions can not be applied. Recently, the author and his collaborators developed a new argument for that in [2,3,9,14]. In particular, in [3], the existence of strong solutions and large-time behavior are first established for the system (1.1) with nonlocal velocity alignment forces for particles.…”

“…The first result on the weak solutions to the Vlasov-Stokes equations is performed in [21], and later this work is extended to the Vlasov-Navier-Stokes equations in [1,7,16,27,29]. With a diffusion term in the Vlasov equation, i.e., VlasovFokker-Planck equations, there are a number of literature on the global existence of weak and strong solutions for the interactions with homogeneous or inhomogeneous fluids [2,5,9,10,11,12,17,23]. For those systems, we can easily find the equilibrium solutions, and this enables to show the large-time behavior of solutions converging to them.…”