MULTISCALE STEADY VORTEX PATCHES FOR 2D INCOMPRESSIBLE EULER EQUATIONS

In this paper, we study the existence and qualitative properties of multiscale steady vortex patches for Euler equations in a 2D bounded domain. By considering certain maximization problems for the vorticity, we obtain the existence of double vortex patches which are trapped in a neighborhood of two points. Limiting localizations of these two points are determined by the Robin function and the boundary of the domain, rather than critical points of the Kirchhoff-Routh function H₂, which is quite different from all the known results. Moreover, the strengths of two components of vorticity are of different order. Multiscale vortex patches concentrating near k points are also constructed for any integer k ≥ 2.