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

Joan Mateu (Universitat Autònoma de Barcelona)

Abstract.
In this talk we will discuss on the persistence of the boundary smoothness of vortex patches for some non-linear transport equations in $\mathbb R^n$, whose velocity field is given by the 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 case is much more complicated than the classical cases (Euler in the plane or aggregation equation in higher dimensions), because the divergence of the velocity field is not trivial. In fact, this divergence is given by the convolution of an even Calderon-Zygmund operator with the characteristic function of a domain. Two interesting examples of this type of equations are the quasi-geostrophic equation in $\mathbb  R^3$ and the Cauchy transport equation.