Time quasi-periodic vortex patches of Euler equation in the plane

Abstract

We prove the existence of time quasi-periodic vortex patch solutions of the 2$d$-Euler equations in ${\mathbb{R}}^2$, close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -- as well as the degeneracy of a normal frequency -- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper.

arXiv comment88 pages

References in corpus (14)

• Time quasi-periodic gravity water waves in finite depth
• KAM for autonomous quasi-linear perturbations of mKdV
• A KAM algorithm for the resonant non--linear Schrödinger equation
• Quasi-periodic solutions for fully nonlinear forced reversible Schroedinger equations
• KAM for reversible derivative wave equations
• Existence of corotating and counter-rotating vortex pairs for active scalar equations
• Boundary regularity of rotating vortex patches
• Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential
• Gravity capillary standing water waves
• Traveling quasi-periodic water waves with constant vorticity
• Global bifurcation of rotating vortex patches
• Normal Form for the Schrödinger equation with analytic non--linearities
• Reducible quasi-periodic solutions for the Non Linear Schrödinger equation
• Existence of small loops in the Bifurcation diagram near the degenerate eigenvalues

Cited by in corpus (3)

• Twisting in Hamiltonian Flows and Perfect Fluids
• Large amplitude traveling waves for the non-resistive MHD system
• Dynamic Behavior of a Multi-Layer Quasi-Geostrophic Model: Weak and Time-Periodic Solutions