Global regularity and fast small-scale formation for Euler patch equation in a smooth domain

It is well known that the Euler vortex patch in R ² will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this article, we study Euler vortex patches in a general smooth bounded domain. We prove global in time regularity by providing an upper bound on the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.

Duke Scholars

Author Alexander A. Kiselev Mathematics