This work focus on the construction of weak solutions to a kinetic-fluid system of partial differential–integral equations modeling the evolution of particles droplets in a compressible fluid. The system is given by a coupling between the standard isentropic compressible Navier–Stokes equations for the macroscopic description of a gas fluid flow, and a Vlasov–Boltzmann type equation governing the evolution of spray droplets modeled as particles with varying radius. We establish the existence of global weak solutions with finite energy, whose density of gas satisfies the renormalized mass equation. The proof combines techniques inspired by the work of Feireisl et al. (J Math Fluid Mech 3:358–392, 2001) on the weak solutions of the compressible Navier–Stokes equations in a coupled system to the kinetic problem for the spray droplets by extending techniques of Leger and Vasseur (J Hyperbolic Differ Equ 6(1):185–206, 2009) developed for the incompressible fluid-kinetic system.

中文翻译：

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喷雾动力学的可压缩Navier–Stokes–Vlasov–Boltzmann系统的全球弱解决方案

这项工作的重点是建立偏微积分方程的动力学流体系统的弱解的构造，该方程对可压缩流体中液滴的演变进行建模。该系统由标准的等熵可压缩Navier–Stokes方程（用于宏观描述气体流动）与Vlasov–Boltzmann型方程之间的耦合给出，该方程控制建模为具有不同半径的粒子的液滴的演变。我们建立了具有有限能量的整体弱解的存在，其气体密度满足重新归一化的质量方程。该证明结合了Feireisl等人的工作启发的技术。（《数学流体力学》 3：358–392，