Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves

We study the existence, uniqueness and long time behaviour of a system consisting of the viscous Burgers' equation coupled to a kinetic equation. This system models the motion of a dispersed phase made of inertial particles immersed in a fluid modelled by the Burgers' equation. The initial conditions are in $L^\infty+W^{1,1}(\mathbb{R}_x)$ for the fluid and in the space $\mathcal {M}(\mathbb{R}_x\times\mathbb{R}_v\times\mathbb{R}_r)$ of bounded measures for the dispersed phase. This means that the limiting case where the particles are regarded as point particles is taken into account. First, we prove the existence and uniqueness of solutions to the system by using the regularizing properties of the viscous Burgers' equation. Then, we prove that the usual stability properties of travelling waves for the viscous Burgers' equation is not affected by the coupling with a small mass of inertial particles.