Global weak solutions to inviscid Burgers-Vlasov equations

TL;DR

This paper proves the existence of global weak solutions for a one-dimensional fluid-particles interaction model, combining entropy solutions with artificial viscosity methods to ensure bounds independent of time.

Contribution

It introduces a novel approach using artificial viscosity to establish global weak solutions with uniform bounds for the inviscid Burgers-Vlasov equations.

Findings

Existence of global weak solutions established.

Fluid velocity and particle energy bounds are time-independent.

Solutions satisfy entropy conditions.

Abstract

In this paper, we consider the existence of global weak solutions to a one dimensional fluid-particles interaction model: inviscid Burgers-Vlasov equations with fluid velocity in $L^\infty$ and particles' probability density in $L^1$. Our weak solution is also an entropy solution to inviscid Burgers' equation. The approach is adding ingeniously artificial viscosity to construct approximate solutions satisfying $L^\infty$ compensated compactness framework and weak $L^1$ compactness framework. It is worthy to be pointed out that the bounds of fluid velocity and the kinetic energy of particles' probability density are both independent of time.