Viscous evolution of a point vortex in a half-plane

TL;DR

This paper proves the global existence, uniqueness, and long-term decay of a viscous vortex in a half-plane, modeling vortex-wall interactions with no smallness restrictions on circulation.

Contribution

It introduces a novel decomposition approach that allows analysis of large circulation vortices in bounded domains without smallness assumptions.

Findings

Unique global solution for all Reynolds numbers.

Solution converges to zero energy as time progresses.

Decomposition method handles large circulation vortices near boundaries.

Abstract

As a model for vortex-wall interactions, we consider the two-dimensional incompressible Navier--Stokes equations in the half-plane $R^2_+$ with no-slip boundary condition and point vortices as initial data. We focus on the paradigmatic example of a single vortex in an otherwise stagnant fluid, which is already quite challenging from a mathematical point of view. We prove that this system has a unique global solution for all values of the Reynolds number $|\Gamma|/\nu$, where $\Gamma$ is the circulation of the vortex and $\nu$ the kinematic viscosity of the fluid. The solution we construct has finite energy for positive times and converges to zero in energy norm as $t \to +\infty$. Uniqueness holds under the assumption that the solution is close to a Lamb--Oseen vortex for small times. To our knowledge, all previous results in domains with boundaries assume that the initial vorticity has small or zero atomic part. In our particular situation, we remove the smallness condition by decomposing the solution into a vortex and a boundary layer term, so that we can apply the techniques developed in the whole plane $R^2$ to avoid the difficulties related to the large circulation of the vortex.