THE REGULARITY OF THE BOUNDARY OF VORTEX PATCHES FOR SOME NONLINEAR TRANSPORT EQUATIONS

Abstract

We prove the persistence of boundary smoothness of vortex patches for a nonlinear transport equation in Rn. with velocity field given by convolution of the density with an odd kernel, homogeneous of degree −(n−1) and of class C2(Rn \{0},Rn). This allows the velocity field to have nontrivial divergence. The quasigeostrophic equation in R3 and the Cauchy transport equation in the plane are examples. © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.