We are concerned with the global well-posedness of a two-phase flow system arising in the modelling of fluid-particle interactions. This sys- tem consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the incompressible Euler equations for a dense phase (fluid) through the friction forcing. Global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state, and moreover an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the macro-micro decomposition and Kawashima's hyperbolic-parabolic dissipation argument. This result is generalized to the periodic case, when particles are in the torus, improving the rate of convergence to exponential. © American Institute of Mathematical Sciences.
- Global well-posedness
- Rate of convergence
- Two-phase flow system
Carrillo, J. A., Duan, R., & Moussa, A. (2011). Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-euler system. Kinetic and Related Models, 4, 227-258. https://doi.org/10.3934/krm.2011.4.227