Global weak solutions to inviscid Burgers–Vlasov equations

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 to ingeniously add 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.

Keywords

weak solution, fluid-particles interaction, $L^\infty$ velocity, $L^1$ density, compensated compactness, Dunford–Pettis theorem

2010 Mathematics Subject Classification

35F20, 35Q35, 45K05, 76T10, 82D05

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Huimin Yu’s research is supported in part by the National Natural Science Foundation of China (Grant No. 11671237, 11501333), China Scholarship Council No. 201708370075. Wentao Cao’s research is supported by ERC Grant Agreement No. 724298.

Received 8 May 2019

Accepted 19 January 2020

Published 28 July 2020