Vortex axisymmetrization, inviscid damping, and vorticity depletion in 2D Euler

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We consider the incompressible 2D Euler equations linearized around a radially symmetric, strictly monotone decreasing vorticity distribution. For sufficiently regular data, we prove the inviscid damping of the $\theta$-dependent radial and angular velocity fields with the optimal rates. We moreover prove that the vorticity weakly converges back to radial symmetry as $t \to \infty$, a phenomenon known as vortex axisymmetrization. Furthermore, we prove that the $\theta$-dependent angular Fourier modes in the vorticity are ejected from the origin as $t \to \infty$, resulting in faster inviscid damping rates than those possible with passive scalar evolution. This non-local effect is called vorticity depletion. This is joint work with Jacob Bedrossian and Michele Coti Zelati.