Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain

TL;DR

This paper investigates the long-term behavior of solutions to the 2D Navier-Stokes equations in exterior domains, showing that under certain conditions, solutions tend to a self-similar vortex as time progresses.

Contribution

It establishes the asymptotic convergence of infinite-energy solutions to a self-similar vortex in a 2D exterior domain, extending previous results to large initial perturbations.

Findings

Solutions approach a self-similar Oseen vortex as t → ∞

Uses logarithmic energy estimates and interpolation techniques

Applicable to large initial perturbations with finite energy

Abstract

We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations of a smooth vortex with small circulation at infinity, but are otherwise arbitrarily large. Using a logarithmic energy estimate and some interpolation arguments, we prove that the solution approaches a self-similar Oseen vortex as $t \to \infty$. This result was obtained in collaboration with Yasunori Maekawa (Kobe University).