The regularity of the boundary of vortex patches revisited

We provide a new argument to show the persistence of boundary smoothness of vortex patches for the vorticity form of Euler’s equation, quite in the spirit of the well-known Bertozzi-Constantin approach. Our argument avoids the use of defining functions and, surprisingly, yields persistence of boundary smoothness of vortex patches for transport equations in the plane given by velocity fields which are convolution of the density with an odd kernel, homogeneous of degree −1 and of class C2 off the origin. This allows the velocity field to have non-trivial divergence. AMS 2020 Mathematics Subject Classification: 35Q31,35Q35 (primary); 35Q49, 42B20 (secondary).