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

TL;DR

This paper establishes existence and uniqueness of Hölder continuous solutions for certain nonlinear transport equations with convolution-based velocity fields, encompassing models like aggregation, 3D quasi-geostrophic, and 2D Euler equations.

Contribution

It introduces a $C^{eta}$ well-posedness framework for nonlinear transport equations with specific convolution kernels, extending previous results to broader classes of equations.

Findings

Proves existence and uniqueness of Hölder solutions for the equations.

Includes classical models like aggregation and Euler equations as special cases.

Provides a unified approach to well-posedness for these nonlinear 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 H\"older 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 $k$.