Regularity of solution maps of the generalized surface quasi-geostrophic equations

TL;DR

This paper investigates the regularity of solution maps for generalized surface quasi-geostrophic equations, revealing a dichotomy: Lagrangian maps are real analytic and well-posed, while Eulerian maps lack local uniform continuity.

Contribution

It proves the real analyticity of Lagrangian solution maps and demonstrates the discontinuity of Eulerian maps, sharpening previous results and highlighting differences between formulations.

Findings

Lagrangian maps are real analytic and locally well-posed.

Eulerian maps are nowhere locally uniformly continuous.

Eulerian maps fail to be continuous in standard H"older topologies.

Abstract

We study regularity properties of the data-to-solution maps of the family of generalized surface quasi-geostrophic equations which includes both the 2D incompressible Euler and the standard surface quasi-geostrophic equations. We prove that the Lagrangian solution maps, interpreted as Riemannian exponential maps on the group of exact Sobolev class diffeomorphisms, are real analytic and, consequently, the Cauchy problems are locally well-posed in the sense of Hadamard. On the other hand, we also show that the corresponding Eulerian solution maps are nowhere locally uniformly continuous on bounded subsets in the Sobolev topology and fail to be continuous in the standard (large-) H\"older topologies. These results sharpen earlier theorems and further highlight the striking dichotomy between regularity properties of the solution maps in the Lagrangian and Eulerian formulations.