Long-time existence for the 2D ideal Boussinesq and the 2D density-dependent Euler equations

TL;DR

This paper proves long-time existence of smooth solutions for 2D ideal Boussinesq and density-dependent Euler equations with small perturbations, using an elementary technique with broad applicability.

Contribution

It introduces a simple, broadly applicable method to establish long-time existence of solutions for these fluid dynamics equations.

Findings

Proved long-time existence for 2D ideal Boussinesq equations.

Established long-time solutions for 2D density-dependent Euler equations.

Developed an elementary technique with potential for wider use.

Abstract

We establish long-time existence of smooth solutions to the 2D ideal Boussinesq equations and to the 2D non-homogeneous incompressible Euler equations for initial data consisting of small temperature perturbations, or small density perturbations, of smooth initial flows which are not necessarily small. Both results are known (see Danchin and Fanelli 2013, Danchin 2011 in the references) but the technique we develop to prove them is at the same time elementary and has broad potential applicability.