Abstract: Two dimensional incompressible fluid flows in high Reynolds number regime behave quite differently from three dimensional flows. A distinct feature is the spontaneous formation and interaction of vortices, which, according to numerical simulation and physical experiments, undergoes two important processes: axi-symmetrization of small perturbations of a vortex, and merging of vortices of the same sign. In order to understand from PDE perspective these phenomenon, we study the two dimensional Euler equation in the perturbative regime near one vortex, and discuss recent results towards proving full nonlinear vortex symmetrization. We will explain the proof of nonlinear inviscid damping near general monotonic shear flows, their connection to the vortex symmetrization problem, some recent result on linear vortex symmetrization with precise description of the dynamics, and the remaining difficult in establishing full nonlinear vortex symmetrization which is primarily caused by the natural degenerate rotation near the center of the vortex.
