Seminar Analysis and Mathematical Physics: Jaemin Park (Georgia Institute of Technology)

Radial symmetry in stationary/uniformly-rotating solutions to 2D Euler equation

In this talk, I will discuss whether all stationary/uniformly-rotating solutions of 2D Euler equation must be radially symmetric, if the vorticity is compactly supported. For a stationary solution that is either smooth or of patch type, we prove that if the vorticity does not change sign, it must be radially symmetric up to a translation. It turns out that the fixed-sign condition is necessary for radial symmetry result: indeed, we are able to find non-radial sign changing stationary solution with compact support. We have also obtained some sharp criteria on symmetry for uniformly-rotating solutions for 2D Euler equation and the SQG equation. The symmetry results are mainly obtained by calculus of variations and elliptic equation techniques, and the construction of non-radial solution is obtained from bifurcation theory. Part of this talk is based on joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.

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