Regularity properties of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes initial value problem
Yosuke Asami, Toshiaki Hishida
TL;DR
The paper develops a comprehensive regularity theory for a generalized Oseen evolution operator in 3D exterior domains driven by time-dependent rigid motions. It constructs the evolution operator ${T}(t,s)$ on $L^q_\sigma(D)$, proves weighted $L^q$ and $L^q_\alpha$ smoothing estimates, and establishes temporal and Hölder regularity for both the linear and Duhamel problems. A key novelty is the weighted approach, enabling strong $L^q$-solutions and precise control of the Duhamel term, which then yields a local-in-time unique strong solution to the Navier–Stokes initial-value problem in exterior domains with moving obstacles. The results extend and complement prior mild-solution theories (HR14, Hi18–Hi20) by achieving strong regularity properties and applicability to nonlinear NS flow, including potential fluid–structure interaction scenarios. Overall, the work provides a robust linear framework that underpins local strong solvability for NS in exterior domains under time-dependent rigid motions, with explicit weighted and Hölder estimates that enhance understanding of temporal behavior and near-boundary dynamics.
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.