Continuity of the solution map of some active scalar equations in H\"older and Zygmund spaces

TL;DR

This paper establishes the continuity of the solution map for certain non-linear transport equations in specific function spaces, including cases that recover well-known fluid dynamics equations like 2D Euler and 3D quasi-geostrophic.

Contribution

It proves the solution map's continuity in H"older and Zygmund spaces for a class of active scalar equations with convolution-based velocity fields.

Findings

Solution map is continuous in little H"older and Zygmund spaces.

Includes special cases like 2D Euler and 3D quasi-geostrophic equations.

Provides a unified framework for continuity in these function spaces.

Abstract

We prove that the solution map for a family of non-linear transport equations in $\mathbb{R}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree $-(n-1)$, is continuous in both the little H\"older class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations.