Regularity issues in the problem of fluid structure interaction

TL;DR

This paper studies the evolution of rigid bodies in a viscous incompressible fluid governed by 2D Navier-Stokes equations, establishing existence, uniqueness, and collision criteria based on boundary regularity.

Contribution

It proves existence and uniqueness of solutions for fluid-structure interaction with non-smooth boundaries and identifies the critical regularity threshold for collision.

Findings

Collision can occur in finite time if boundary regularity exponent α < 1/2.

Existence and uniqueness of strong solutions are established up to collision.

A BMO bound on the velocity gradient replaces standard H^2 estimates for less regular domains.

Abstract

We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier-Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have H\"older regularity $C^{1, \alpha}$, $0 < \alpha \le 1$. First, we show the existence and uniqueness of strong solutions up to collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard $H^2$ estimate for smoother domains. Then, we study the asymptotic behaviour of one $C^{1, \alpha}$ body falling over a flat surface. We show that collision is possible in finite time if and only if $\alpha < 1/2$.