Global classical solutions for a compressible fluid-particle interaction model (Q2857735)

It is a very interesting paper, concerning the motion of particles dispersed in a compressible viscous fluid, which can be described by coupled Navier-Stokes and Vlasov-Fokker-Plank equations. An important result was obtained in [\textit{A. Mellet} and \textit{A. Vasseur}, Math. Models Methods Appl. Sci. 17, No. 7, 1039--1063 (2007; Zbl 1136.76042)].NEWLINENEWLINENEWLINEIn the present paper, an important improvement is given: a global existence and uniqueness of a \textit{classical} solution for the considered model is proved. Moreover, an asymptotic stability of solution near an equilibrium point is obtained, in the case of initial data close enough to an equilibrium. The main tools are some energetic estimates, related with the partial coercive property of the linear Fokker-Plank operator \(L\), defined by the relation (1.11). The lack of coercivity of \(L\) is compensated by adapting the formulation of the compensating function given in [\textit{S. Kawashima}, Japan J. Appl. Math. 7, No. 2, 301--320 (1990; Zbl 0702.76090)]. It is pointed out that some regularity properties of the solution are controlled only by the initial data. A large list of related references is given in the last part.