Stability for multiple Lamb dipoles

TL;DR

This paper proves the Lyapunov stability of finite sums of Lamb dipoles in the half-plane, using energy estimates and a Lagrangian scheme, under conditions of sufficient separation and specific positioning.

Contribution

It introduces a robust method combining energy estimates and Lagrangian bootstrapping to establish stability of multiple Lamb dipoles with potential for extensions.

Findings

Lyapunov stability of multiple Lamb dipoles established

Quantitative analysis of circulation, enstrophy, impulse, and energy exchanges

Method applicable to broader classes of vortex configurations

Abstract

In the class of nonnegative vorticities on the half-plane, we establish the Lyapunov stability of finite sums of Lamb dipoles under the initial assumptions that the dipoles are sufficiently separated and that the faster dipoles are positioned to the right of the slower ones. Our approach combines sharp energy estimates near the Lamb dipoles with a Lagrangian bootstrapping scheme, enabling us to quantify the exchanges of circulation, enstrophy, impulse, and energy between various parts of the solution. The strategy of the proof is robust, and we present several potential extensions of the result.