Maximal regularity of the time-periodic Navier-Stokes system

TL;DR

This paper establishes maximal regularity and a priori estimates for time-periodic solutions to the linearized Navier-Stokes system in n-dimensional space, providing a functional framework for analyzing such solutions.

Contribution

It introduces a Banach space framework where the linearized Navier-Stokes operator acts homeomorphically, advancing the understanding of time-periodic solutions in fluid dynamics.

Findings

Maximal regularity results for time-periodic solutions

Identification of a Banach space with specific mapping properties

A priori estimates for solutions in the linearized system

Abstract

Time-periodic solutions to the linearized Navier-Stokes system in the $n$-dimensional whole-space are investigated. For time-periodic data in $L^q$-spaces, maximal regularity and corresponding a priori estimates for the associated time-periodic solutions are established. More specifically, a Banach space of time-periodic vector fields is identified with the property that the linearized Navier-Stokes operator maps this space homeomorphically onto the $L^q$-space of time-periodic data.