Remarks on type I blow up for the 3D Euler equations and the 2D Boussinesq equations

TL;DR

This paper derives new criteria for the formation of singularities in 3D Euler and 2D Boussinesq equations, improving understanding of when solutions remain regular or blow up.

Contribution

It introduces novel kinematic relations and establishes improved blow up criteria and conditions for the absence of type I singularities in these fluid dynamics equations.

Findings

New blow up criteria for 3D Euler and 2D Boussinesq equations.

Conditions for preventing type I singularities.

Enhanced results over previous studies.

Abstract

In this paper we derive kinematic relations for quantities involving the rate of strain tensor and the Hessian of the pressure for solutions of the 3D Euler equations and the 2D Boussinesq equations. Using these kinematic relations, we prove new blow up criteria and obtain conditions for the absence of type I singularity for these equations. We obtain both global and localized versions of the results. Some of the new blow up criteria and type I conditions improve previous results of [3].