Desingularization of vortices for the Euler equation

Abstract

We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation $-\eps^2 Δu^\eps=(u^\eps-q-\fracκ{2π} \log \frac{1}{\eps})_+^p$ with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.

40 pages

Cited by in corpus (13)

• Existence of corotating and counter-rotating vortex pairs for active scalar equations
• Regularization of point vortices for the Euler equation in dimension two
• Gluing methods for vortex dynamics in Euler flows
• Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem
• Periodic solutions of the N-vortex Hamiltonian system in planar domains
• Vortex motion for the lake equations
• 2D point vortex dynamics in bounded domains: global existence for almost every initial data
• Steady vortex patch solutions to the vortex-wave system
• Hölder regularity for collapses of point vortices
• Vortex patches choreography for active scalar equations
• Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation
• On the uniqueness and monotonicity of solutions of free boundary problems
• Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation