Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data

TL;DR

This paper establishes the existence, uniqueness, and asymptotic behavior of small exterior Navier-Stokes flows with non-decaying boundary data, including time-periodic solutions and their stability.

Contribution

It proves the unique existence of solutions with small non-decaying boundary data and analyzes their long-term asymptotics, including periodic and discretely self-similar solutions.

Findings

Existence and uniqueness of solutions with small non-decaying boundary data.

Asymptotic description of periodic solutions by Landau solutions.

Stability of periodic solutions under certain initial conditions.

Abstract

We prove the unique existence of solutions of the 3D incompressible Navier-Stokes equations in an exterior domain with small non-decaying boundary data, for $t \in R$ or $t \in (0,\infty)$. In the latter case it is coupled with small initial data in weak $L^{3}$. As a corollary, the unique existence of time-periodic solutions is shown for the small periodic boundary data. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically discretely self-similar, then the solution is asymptotically the sum of a time-periodic vector field and a forward discretely self-similar vector field as time goes to infinity. It in particular shows the stability of periodic solutions in a local sense.