Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem

TL;DR

This paper constructs steady vortex solutions for 3D Euler and shallow water equations, showing they tend to singular vortex filaments, using asymptotic analysis of a semilinear elliptic problem.

Contribution

It introduces a novel method for desingularizing vortex rings and shallow water vortices via semilinear elliptic problem analysis.

Findings

Vortex solutions tend asymptotically to filament-like singularities.

The method applies to both Euler and shallow water equations.

Provides a new approach to vortex desingularization.

Abstract

Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on the asymptotic study of solutions to a semilinear elliptic problem.