两相流体动力学是流体力学的一个重要分支，二相流包括气-液、气-固、不相溶的液-液、液-固等，它有重要的物理背景。两相流模型包括多种复杂的非线性偏微分方程组，关于其研究一直是偏微分方程的热点问题。本项目研究的数学模型主要由Vlasov-Fokker-Planck方程通过摩擦力耦合可压Navier-Stokes方程组成。对于无粘情形，法国数学家Desvillettes等证明了经典解的局部存在性，对于有粘情形，我们证明了小扰动强解的整体存在性（发表于SIAM J.Math.Anal.等）。我们将研究一般情况下解的适定性和相关的小Mach数极限。具体问题包括粒子-流体模型在非等熵、在半空间或有界域上解的适定性、可压缩模型在“好”或“坏”初始值情形下小Mach数极限的收敛性等。这些研究内容是我们已有工作的延续和深化，同时也希望通过这些研究，能够加深理解和建立更多可压缩N-S方程弱解和强解的性质。

Two-phase fluid dynamics is one of the important branches in fluid mechanics and admits a lot of interesting physics backgrounds. Two-phase flows include gas- liquid, gas-solid, non-soluble liquid-liquid, liquid -solid and so on. Two-phase fluid model includes many complicated systems of nonlinear partial differential equations. The mathematical studies of two-phase flow models are important areas in partial differential equations. Our concerned model is composed of the compressible Navier-Stokes equations coupled with the Vlasov-Fokker-Planck equations by friction forces. In the inviscid case, the local existence of classical solutions is established by France mathematician Desvillettes and so on; in the viscous case, we have proved the global existence of small perturbed strong solutions (see SIAM J.Math.Anal. and so on). For the general case, we will study the well-posedness and low Mach number limit of solutions. The concrete problems include: the well-posedness problems of nonisentropic case, half-space case and bounded domain case for two-phase fluid dynamics; the convergence problems of low Mach number limit under the “good” and “bad”initial data. These studies are the continuation and extension of our previous works. We hope that we can give deep comprehensions and establish more properties on the weak solutions or strong solutions of compressible Navier-Stokes equations.