Lack of collision in a simplified 1-d model for fluid-solid interaction

In this paper we consider a simplified model for fluid-solid interaction in one space dimension. The fluid is assumed to be governed by the viscous Burgers equation. It is coupled with a finite number of solid masses in the form of point particles, which share the velocity of the fluid and are accelerated by the pressure difference of the fluid on both sides. We prove global existence and uniqueness of solutions. This requires of proving that the solid particles never collide in finite time, a key fact that follows from suitable a priori estimates together with uniqueness results for ordinary differential equations. We also describe the asymptotic behavior of solutions as t → ∞, extending previous results established for a single solid mass. The possible 2-D analogues of these results in the context of Navier-Stokes equations are open problems.