The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: The non-autonomous case

TL;DR

This paper studies the non-autonomous Navier-Stokes flow around a rotating obstacle with time-dependent conditions, establishing linear evolution systems, smoothing properties, and local existence of solutions in an exterior domain.

Contribution

It introduces a framework for analyzing non-autonomous Navier-Stokes equations with unbounded drift in exterior domains, including evolution systems and solution existence results.

Findings

Established a strongly continuous evolution system for the linearized problem.

Derived $L^p$-$L^q$ smoothing and gradient estimates for the evolution system.

Proved local in time existence of mild solutions for the nonlinear problem in exterior domains.

Abstract

Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity -- sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to an non-autonomous problem with unbounded drift terms on a fixed exterior domain $\Omega\subset \R^d$. It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system $\{T_\Omega(t,s)\}_{t\geq s\geq0}$ on $L^p_\sigma(\Omega)$ for $1<p<\infty$. Moreover, $L^p$-$L^q$ smoothing properties and gradient estimates of $T_\Omega(t,s)$, $0\leq s \leq t$, are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for $p\geq d$ and initial value in $L^p_\sigma(\Omega)$.