On the regularity of weak solutions to time-periodic Navier--Stokes equations in exterior domains

TL;DR

This paper establishes sufficient conditions under which weak solutions to time-periodic Navier-Stokes equations in exterior domains are smooth, focusing on the integrability of the purely periodic part of the velocity field.

Contribution

It provides a novel regularity criterion for weak solutions in exterior domains with time-periodic flow, addressing the challenge of infinite kinetic energy.

Findings

Weak solutions are smooth under certain integrability conditions.

Regularity depends on the purely periodic part of the velocity field.

Results extend regularity theory to time-periodic exterior domain flows.

Abstract

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.