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

Abstract

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 $C^2({\mathbb{R}}^n\setminus \{0\},{\mathbb{R}}^n).$ 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.

About the replacement: the main result has been significantly improved. The kernels to which the result applies are now of the more natural class conceivable. Furthermore, the proof has been adapted to the new context and an appendix has been added. About the replacement on September 2023: stylistic changes proposed by the journal

Cited by in corpus (2)

• The global regularity of vortex patches revisited
• Quantitative Hölder Estimates for Even Singular Integral Operators on Patches