On the Navier-Stokes equations with rotating effect and prescribed outflow velocity

TL;DR

This paper studies the Navier-Stokes equations with a rotating effect and a general outflow velocity, introducing a new coordinate system to handle non-autonomous PDEs with unbounded drift, and proves existence and uniqueness of solutions.

Contribution

It extends previous results by allowing a general outflow velocity and develops new $L^p$-techniques using a novel coordinate transformation.

Findings

Established an evolution system for the linearized problem in $L^p$ spaces.

Proved existence of a unique mild solution for the full Navier-Stokes system under certain conditions.

Utilized time-dependent Ornstein-Uhlenbeck operators for analysis.

Abstract

We consider the equations of Navier-Stokes modeling viscous fluid flow past a moving or rotating obstacle in $\mathbb{R}^d$ subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use $L^p$-techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in $\mathbb{R}^d$ is solved by an evolution system on $L^p_{\sigma}(\mathbb{R}^d)$ for $1<p<\infty$. For this we use results about time-dependent Ornstein-Uhlenbeck operators. Finally, we prove, for $p\geq d$ and initial data $u_0\in L^p_{\sigma}(\mathbb{R}^d)$, the existence of a unique mild solution to the full Navier-Stokes system.