On the well-posedness of the inviscid SQG equation

TL;DR

This paper investigates the inviscid SQG equation's mathematical properties, demonstrating that its solution map lacks local uniform continuity in certain Sobolev spaces, which has implications for well-posedness.

Contribution

It establishes the non-uniform continuity of the solution map for the inviscid SQG equation in Sobolev spaces, using a geometric approach.

Findings

Solution map is nowhere locally uniformly continuous for any positive time.

Results hold in Sobolev spaces with s > 2.

Implications for the mathematical understanding of SQG dynamics.

Abstract

In this paper we consider the inviscid SQG equation on the Sobolev spaces $H^s(\R^2)$, $s > 2$. Using a geometric approach we show that for any $T > 0$ the corresponding solution map, $\theta(0) \mapsto \theta(T)$, is nowhere locally uniformly continuous.