On the well-posedness of the inviscid 2D Boussinesq equation

TL;DR

This paper investigates the mathematical properties of the inviscid 2D Boussinesq equation, demonstrating that its solution map lacks local uniform continuity in certain Sobolev spaces, which has implications for well-posedness.

Contribution

It provides a geometric proof that the solution map for the inviscid 2D Boussinesq equation is nowhere locally uniformly continuous in Sobolev spaces $H^s( ^2)$ for $s > 2$, highlighting ill-posedness issues.

Findings

Solution map is nowhere locally uniformly continuous

Ill-posedness in Sobolev spaces for the equation

Geometric approach used for proof

Abstract

In this paper we consider the inviscid 2D Boussinesq 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, $(u(0),\theta(0)) \mapsto (u(T),\theta(T))$, is nowhere locally uniformly continuous.