Damped Infinite Energy Solutions of the 3D Euler and Boussinesq Equations

TL;DR

This paper investigates how damping influences the finite-time blowup of infinite-energy solutions in 3D Euler and Boussinesq equations, revealing conditions under which damping can prevent or fail to prevent singularity formation.

Contribution

It demonstrates that damping can arrest blowup in 3D Euler solutions but may be insufficient for Boussinesq solutions, highlighting the nuanced role of damping in singularity development.

Findings

Damping can prevent blowup in certain 3D Euler solutions.

Infinite-energy solutions of Boussinesq can still develop singularities with insufficient damping.

Damping effects are critical in controlling finite-time singularities in fluid equations.

Abstract

We revisit a family of infinite-energy solutions of the 3D incompressible Euler equations proposed by Gibbon et al. [9] and shown to blowup in finite time by Constantin [6]. By adding a damping term to the momentum equation we examine how the damping coefficient can arrest this blowup. Further, we show that similar infinite-energy solutions of the inviscid 3D Boussinesq system with damping can develop a singularity in finite time as long as the damping effects are insufficient to arrest the (undamped) 3D Euler blowup in the associated damped 3D Euler system.