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.

中文翻译：

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无粘性 Burgers-Vlasov 方程的全局弱解

在本文中，我们考虑了一维流体-粒子相互作用模型的全局弱解的存在：无粘性 Burgers-Vlasov 方程，流体速度为 $L^\infty$，粒子概率密度为 $L^1$。我们的弱解也是无粘伯格斯方程的熵解。该方法巧妙地添加人工粘性来构建满足$L^\infty$补偿紧凑性框架和弱$L^1$紧凑性框架的近似解。值得指出的是，流体速度的界限和粒子概率密度的动能均与时间无关。