Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics

TL;DR

This paper establishes lower bounds for the lifespan of solutions to certain fluid mechanics models, showing in 2D that lifespan tends to infinity as initial temperature approaches zero, and extends methods to Euler equations with swirl.

Contribution

It provides the first known lower bounds for the lifespan of solutions to the inviscid Boussinesq system and develops continuation criteria for N-dimensional cases.

Findings

Lifespan tends to infinity as initial temperature approaches zero in 2D.

New continuation criteria for N-dimensional inviscid Boussinesq system.

Method adaptation to axisymmetric Euler equations with swirl.

Abstract

We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the $N$-dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl.