Desingularization of nondegenerate rotating vortex patches

We analyze the space of steady rotating solutions to the two-dimensional incompressible Euler equations nearby vortex patch solutions satisfying a nondegeneracy condition. We address the question of desingularization and prove that such vortex patch states are the limit of rotating Euler solutions that are smooth to infinite order, have compact vorticity support, and respect dihedral symmetry. Our nondegeneracy condition is proved to be satisfied by Kirchhoff ellipses and along the local bifurcation curves emanating from the Rankine vortex. The construction, that is based on a local stream function formulation in a tubular neighborhood of the patch boundary, is a synthesis of analysis on thin domains, nonlinear a priori estimates, and Newton's method. Our techniques additionally allow us to construct nearby exotic families of singular rotating vortex patch-like solutions. This is joint work with Razvan-Octavian Radu.