The regularity of the boundary of vortex patches for some non-linear transport equations

We prove the persistence of boundary smoothness of vortex patches for a non-linear transport equation in ${\mathbb{R}}^n$ with velocity field given by convolution of the density with an odd kernel, homogeneous of degree $-(n-1)$ and of class $C^2({\mathbb{R}}^n\setminus \{0\},{\mathbb{R}}^n)\mathrm{.}$ This allows the velocity field to have non-trivial divergence. The quasi-geostrophic equation in ${\mathbb{R}}^3$ and the Cauchy transport equation in the plane are examples.