Large time behavior for a simplified N-dimensional model of fluid-solid interaction
Résumé
In this paper, we study the large time behavior of solutions of a parabolic equation coupled with an ordinary differential equation (ODE). This system can be seen as a simplified $N$-dimensional model for the interactive motion of a rigid body (a ball) immersed in a viscous fluid in which the pressure of the fluid is neglected. Consequently, the motion of the fluid is governed by the heat equation and the standard conservation law of linear momentum determines the dynamics of the rigid body. In addition, the velocity of the fluid and that of the rigid body coincide on its boundary. The time variation of the ball position, and consequently of the domain occupied by the fluid, are not known a priori, so we deal with a free boundary problem. After proving the existence and uniqueness of a strong global in time solution, we get its decay rate in $L^p$ ($1\leq p\leq \infty$), assuming the initial data to be integrable. Then, working in suitable weighted Sobolev spaces, and using the so-called similarity variables and scaling arguments, we compute the first term in the asymptotic development of solutions. We prove that the asymptotic profile of the fluid is the heat kernel with an appropriate total mass. The $L^\infty$ estimates we get allow us to describe the asymptotic trajectory of the center of mass of the rigid body as well. We compute also the second term in the asymptotic development in $L^2$ under further regularity assumptions on the initial data.