Lp-strong solution to fluid-rigid body interaction system with Navier slip boundary condition

TL;DR

This paper proves the existence of strong solutions for fluid-rigid body interaction systems with Navier slip boundary conditions, covering both Newtonian and non-Newtonian fluids, and demonstrates exponential stability in the Newtonian case.

Contribution

It establishes global-in-time strong solutions for Newtonian fluids and local-in-time solutions for non-Newtonian fluids under slip boundary conditions, using maximal regularity and sectoriality analysis.

Findings

Existence of global strong solutions for Newtonian fluids with small data.

Local strong solutions for non-Newtonian fluids.

Exponential stability of the system in the Newtonian case.

Abstract

We study a fluid-structure interaction problem describing movement of a rigid body inside a bounded domain filled by a viscous fluid. The fluid is modelled by the generalized incompressible Naiver-Stokes equations which include cases of Newtonian and non-Newtonian fluids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and balance of forces at the fluid-rigid body interface. Our analysis also includes the case of the nonlinear slip condition. The main results assert the existence of strong solutions, in an $L^p-L^q$ setting, globally in time, for small data in the Newtonian case, while existence of strong solutions in $L^p$-spaces, locally in time, is obtained for non-Newtonian case. The proof for the Newtonian fluid essentially uses the maximal regularity property of the associated linear system which is obtained by proving the $\mathcal{R}$-sectoriality of the corresponding operator. The existence and regularity result for the general non-Newtonian fluid-solid system then relies upon the previous case. Moreover, we also prove the exponential stability of the system in the Newtonian case.