A non-autonomous model problem for the Oseen-Navier-Stokes flow with rotating effects

TL;DR

This paper develops a mathematical framework for analyzing the flow of fluids past rotating obstacles with time-dependent conditions, providing explicit formulas and estimates crucial for solving nonlinear Navier-Stokes problems.

Contribution

It introduces a non-autonomous model for Navier-Stokes flow with rotating effects, deriving explicit evolution formulas and estimates for the associated linear system.

Findings

Established a strongly continuous evolution system on L^p spaces

Derived explicit representation formulas similar to Ornstein-Uhlenbeck equations

Proved L^p-L^q and gradient estimates for the evolution system

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. After rewriting the problem on a fixed domain, one obtains a non-autonomous system of equations with unbounded drift terms. It is shown that the solution to a model problem in the whole space case $\R^d$ is governed by a strongly continuous evolution system on $L^p_\sigma(\R^d)$ for $1<p<\infty$. The strategy is to derive a representation formula, similar to the one known in the case of non-autonomous Ornstein-Uhlenbeck equations. This explicit formula allows to prove $L^p$-$L^q$ estimates and gradient estimates for the evolution system. These results are key ingredients to obtain (local) mild solutions to the full nonlinear problem by a version of Kato's iteration scheme.