Stability of monotone, non-negative, and compactly supported vorticities in the half cylinder and infinite perimeter growth for patches

TL;DR

This paper proves the nonlinear stability of certain stationary vorticity solutions in a half-cylinder domain for the Euler equations and demonstrates the potential for infinite perimeter growth of vortex patches over time.

Contribution

It establishes the nonlinear stability of non-negative, non-increasing vorticity profiles in a half-cylinder and links this to the existence of vortex patches with infinite perimeter growth.

Findings

Stationary solutions are stable under specific conditions.

Unique minimizer of horizontal impulse corresponds to these solutions.

Existence of vortex patches with infinite perimeter growth over time.

Abstract

We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of cylindrical patches $\{x_{1}<\alpha\},\; \alpha>0$. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.