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

Abstract

The inhomogeneous Navier-Stokes-Vlasov equations for fluid-particle flows are considered in the threedimensional space. The coupling in the fluid-particle system arises from the drag force in the fluid equations and the acceleration in the Vlasov equation. An initial-boundary value problem is studied in a bounded domain with large initial data. The existence of global weak solution is established through an approximation scheme, a fixed point argument, energy estimates, and a weak convergence method.

Existence of global weak solutions for the Navier–Stokes–Vlasov–Boltzmann equations

“…Thanks to (3.39) 2 and (3.40), the Aubin-Lions Lemma, u m = S(ũ) is compact in A. On the other hand, it is easy to verify that S is sequentially continuous, see [9,12] for the details. Thus Schauder's fixed point theorem gives that S has a fixed point u m in A.…”

“…The first work in this field was [8] where the author proved the global existence of weak solution and their largetime behavior for the Vlasov-Stokes equations. The existence theorem for weak solutions was extended in [1,3,10,12,13], where the authors did not neglect the convection term and considered the Navier-Stokes equations, including incompressible and compressible ones. In [5], the existence and uniqueness of global smooth solutions near an equilibrium was proved under smallness conditions for the Navier-Stokes system coupled with the Vlasov-Fokker-Planck equation in 3D.…”

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

“…To be more precise, we can not extract the desired dissipation rate of the density in D. On the other hand, if we consider the pressureless viscous fluid [25] or inhomogeneous fluid [14,27] instead of the compressible Navier-Stokes equations, we can show that the exponential alignment between particles and fluid velocities by using the Lyapunov functional L and the energy functional E without the terms related to the pressure. In fact, the lemma below shows the Lyapunov functional L is bounded by sum of the dissipation D and that term related to the pressure from above.…”

“…Some of the previous works on the existence theory for the coupled kinetic-fluid equations can be summarized as follows. 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].…”

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

“…Existence for fluid-kinetic systems is nowadays well-understood, and there is no exception for the Vlasov-Navier-Stokes system: global weak solutions are known to exist for periodic boundary conditions ( [5]), bounded domains with various boundary condition ( [21,20]) or even moving domains ( [7]) and if the initial data is small enough, the system is well-posed in a narrower set of solutions ( [11,18]). In contrast, the uniqueness issue has not been very much studied.…”