Regularity properties of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes initial value problem

TL;DR

This paper establishes regularity properties and weighted estimates for a generalized Oseen evolution operator in 3D exterior domains, and applies these results to prove local existence of unique strong solutions for the Navier-Stokes initial value problem.

Contribution

It provides new regularity estimates and weighted bounds for the generalized Oseen operator, enabling the proof of local strong solutions for Navier-Stokes equations in exterior domains.

Findings

Established temporal derivative and Hölder estimates for the operator

Derived weighted estimates of the evolution operator

Proved local existence of unique strong solutions for Navier-Stokes

Abstract

Consider a generalized Oseen evolution operator in 3D exterior domains, that is generated by a non-autonomous linearized system arising from time-dependent rigid motions. This was found by Hansel and Rhandi, and then the theory was developed by the second author, however, desired regularity properties such as estimate of the temporal derivative as well as the Hoelder estimate have remained open. The present paper provides us with those properties together with weighted estimates of the evolution operator. The results are then applied to the Navier-Stokes initial value problem, so that a new theorem on existence of a unique strong Lq-solution locally in time is proved.