The time periodic problem for the Navier-Stokes equations in exterior domains in weighted spaces

TL;DR

This paper establishes the existence of time periodic solutions to the Navier-Stokes equations in exterior domains using novel weighted space methods, expanding the mathematical tools available for analyzing fluid dynamics problems.

Contribution

It introduces a new approach employing radially symmetric Muckenhoupt weights in space to solve the time periodic Navier-Stokes problem in exterior domains, avoiding divergence form reliance.

Findings

Proves existence of periodic solutions in weighted spaces for exterior domain Navier-Stokes.

Reestablishes weighted decay estimates for the Stokes semigroup with rigorous proof.

Applies new weighted space techniques to solve classical fluid dynamics problems.

Abstract

The paper considers the time periodic problem of the Navier-Stokes system in an exterior domain under time periodic external forces. Existence of periodic mild solutions is obtained in the critical scale invariant space $C(\mathbb{R};L^n)$ if the external force is small without exploiting any divergence form as in the study of Okabe and Tsutsui (2017) for the whole space case in Lorentz spaces. Previous studies mainly rely on either potential theoretical estimates or time-space integral estimates in Lorentz spaces introduced by Yamazaki (Math. Ann.(2000)). To the best of our knowledge, there are no results using Muckenhoupt weights in $L^q$ class for $1< q <\infty$ to construct time periodic solutions of the Navier-Stokes equations in the exterior domain case. In this article, a new method based on radially symmetric Muckenhoupt weights in space is used. To apply these weights, we reconsider weighted $L^p$-$L^q$ decay estimates for the Stokes semigroup. This important result was announced by Kobayashi and Kubo (2012-2015) about ten years ago with a sketch of the proof by Kubo. In this paper, we give a rigorous proof of the result and, as an important application, solve the time periodic problem for the Navier-Stokes equations on an exterior domain.