A Variational Family of Traveling Vortex Dipoles: Transition, Uniqueness, and Stability

RESUMO

We provide a unified description of a stable variational family of traveling vortex dipoles for the two-dimensional incompressible Euler equations, connecting the classical Lamb dipole to asymptotically radial dipoles in the large-impulse regime. We show that this family undergoes a sharp transition from the Lamb dipole, where the mass constraint is inactive, to non-explicit traveling waves for which the mass constraint becomes active, and determine the critical value exactly. We further prove that the variational solutions are unique up to translation both near the Lamb dipole and in the large-impulse regime. As a consequence, the corresponding individual traveling waves are orbitally stable. Our results provide a unified picture of these two distinct asymptotic regimes within a single variational family of traveling vortex dipoles.