$C^{\gamma }$ well-posedness of some non-linear transport equations

Abstract

Given $k:{\mathbb{R}}^n\setminus \{0\}\to {\mathbb{R}}^n$ a kernel of class $C^2$ and homogeneous of degree $1-n$, we prove existence and uniqueness of Hölder regular solutions for some non-linear transport equations with velocity fields given by convolution of the density with $k$. The Aggregation, the 3D quasi geostrophic, and the 2D Euler equations can be recovered for particular choices of $k$.

arXiv commentSome typos have been corrected. I have to thank to Joan Orobitg, Joan Mateu and Joan Verdera for their comments. In v3, we have updated the previous result to a wider class of equations. In v4, a problem with color has been solved