Strong Solutions to the Compressible Navier--Stokes--Vlasov--Fokker--Planck Equations: Global Existence Near the Equilibrium and Large Time Behavior

Abstract

A kinetic-fluid model describing the evolutions of disperse twophase flows is considered. The model consists of the Vlasov-Fokker-Planck equation for the particles (disperse phase) coupled with the compressible Navier-Stokes equations for the fluid (fluid phase) through the friction force. The friction force depends on the density, which is different from many previous studies on kinetic-fluid models and is more physical in modeling but significantly more difficult in analysis. New approach and techniques are introduced to deal with the strong coupling of the fluid and the particles. The global well-posedness of strong solution in the three-dimensional whole space is established when the initial data is a small perturbation of some given equilibrium. Moreover, the algebraic rate of convergence of solution toward the equilibrium state is obtained. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential.

Asymptotic analysis for 1D compressible Navier-Stokes-Vlasov equations

“…For the global existence and large time behavior of strong solutions close to equilibrium of the above equations, we can see [18]. In recent years, there are many results in the direction of asymptotic analysis (hydrodynamic limit) of weak solutions to the fluid-particle system.…”

Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain

“…In the series of papers [1][2][3][4], the authors studied the well-posedness and large-time behavior of the Cucker-Smale-Navier-Stokes (CS-NS) systems in the three-dimensional periodic domain T 3 . For the other literature on the particle-fluid coupled model, we refer to [8,9,11,16,21,23,28,29]. In the same context, the first author and his collaborators presented the global existence of a weak and strong solution to (1.1) in [13], and the kinetic TCS equation coupled with compressible fluid [14], when the spatial domain is still the periodic box T 3 , as well as the emergent dynamics of it.…”

A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data

“…In [13], the authors proved the global existence of classical solutions to the coupled system of VFP equation and compressible Euler equations for small initial data in the whole space R 3 . In [27], the global well-posedness of a strong solution to compressible Navier-Stokes-Vlasov-Fokker-Planck system in the three-dimensional whole space is established when the initial data is a small perturbation of some given equilibrium. The global classical solutions to 1D and 2D coupled system of flocking particles and compressible fluids with large initial data have been obtained in [18,19].…”

“…Now we turn to the estimate of the third term in the RHS of (33). To the end, we first use Lemma 2.1 and (27) to have…”