Communications in Mathematical Sciences
Volume 18 (2020)
Global weak solutions to inviscid Burgers–Vlasov equations
Pages: 1087 – 1103
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.
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
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