Desingularization of vortices for the Euler equation

TL;DR

This paper constructs smooth stationary solutions to the Euler equation that approximate singular vortex solutions, using asymptotic analysis and energy minimization techniques.

Contribution

It introduces a method to desingularize vortices in the Euler equation by analyzing asymptotics and minimal energy solutions, extending the understanding of vortex behavior.

Findings

Existence of smooth solutions approximating singular vortices.

Desingularization of vortex pairs via minimal energy solutions.

Construction of rotating vortex solutions.

Abstract

We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation $-\eps^2 \Delta u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p$ with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.