$L^p$-$L^q$ Maximal Regularity for some Operators Associated with Linearized Incompressible Fluid-Rigid Body Problems

TL;DR

This paper establishes maximal regularity and exponential stability for an operator linked to fluid-rigid body interactions, enabling proof of global solutions for small initial data in an $L^p$-$L^q$ framework.

Contribution

It proves $ ext{R}$-sectoriality and maximal $L^p$-$L^q$ regularity of the linearized operator, and applies these results to the nonlinear problem.

Findings

Operator is $ ext{R}$-sectorial in $L^q$ for all $q extgreater 1$

Generated semigroup is exponentially stable in $L^q$

Global existence for small initial data in $L^p$-$L^q$ setting

Abstract

We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in (1,\infty)$, thus it has the maximal $L^p$-$L^q$ regularity property. Moreover, we show that the generated semigroup is exponentially stable with respect to the $L^q$ norm. Finally, we use the results to prove the global existence for small initial data, in an $L^p$-$L^q$ setting, for the original nonlinear problem.