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

Abstract

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 paper, 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.

On nonexistence of splash singularities for the $α$-SQG patches

“…For small α > 0, finite time singularity formation has been proved for patches in the half-plane setting [17]. This singularity formation happens near the hyperbolic point of the flow on the boundary, and in a scenario Date: December 6, 2021. similar to very fast small scale growth in solutions to 2D Euler equation [18,16] and conjectured singularity formation in the 3D Euler Hou-Luo scenario [19]. On the other hand, there are also recent numerical simulations by Scott and Dritschel [22,23] which suggests a different pathway towards a singularity.…”

On vortex stretching for anti-parallel axisymmetric flows

“…Even for general non-negative vorticities on (R + ) 2 , a similar result is available ( [28]): the center of vorticity (R + ) 2 x 1 ω(t, x)dx grows linearly for all positive times. We note that this odd-odd scenario has been used to prove growth of |∇ω| in two-dimensional Euler flows ( [12,13,33,55,32,29]).…”

Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole

“…However, it is not clear to us whether such stability results from [26,42] are able to produce infinite in time growth of the vorticity gradient for nearby smooth solutions. For further results on small scale creation for patches, see [32,27,19,20,30,28].…”