Regularity Criterion for the Three-dimensional Boussinesq Equations

TL;DR

This paper establishes a regularity criterion for the 3D Boussinesq equations, showing that solutions remain smooth if certain low-mode velocity norms are integrable over time.

Contribution

It introduces a new regularity criterion based on low modes in Besov spaces, advancing understanding of solution blow-up conditions for the Boussinesq equations.

Findings

Solutions do not blow-up if low-mode velocity norms are integrable.

Provides a new criterion involving Littlewood-Paley projections.

Enhances criteria for regularity in fluid dynamics models.

Abstract

We prove that a solution to the three-dimensional Boussinesq equations does not blow-up at time T if $\| u_{\le Q}\|_{B^1_{\infty, \infty}}$ is integrable on $(0, T)$, where $u_{\le Q }$ represents the low modes of Littlewood-Paley projection of the velocity $u$.