Global Bounded Weak Entropy Solutions to the Euler--Vlasov Equations in Fluid-Particle System

Abstract

In this paper, we consider a fluid-particle system which describes the evolution of a two-phase flow. The system consists of the compressible Euler equations for the fluid (fluid phase) coupled with the Vlasov equation for the particles (disperse phase) through the drag force. We obtain a global bounded weak entropy solution for such one-dimensional Euler--Vlasov equations with arbitrarily large initial data for the whole range of physical adiabatic exponents $\gamma>1$. To achieve this, we apply the vanishing viscosity method and compensated compactness theory. We construct globally defined approximate solutions by adding our novel viscosity terms to the Euler equations, which together with our key observation on relative velocity plays a fundamental role in the hardest part of our proof: the uniform $L^\infty$ estimate. After deriving an entropy dissipation estimate, we prove the convergence of approximate solutions by the compensated compactness argument.