Stability for multiple Lamb dipoles

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.

39 pages, 7 figures. Corrected several typos and minor errors and expanded some of the proofs. To appear in Courant Journal of Pure and Applied Mathematics