A regularity criterion for the 3D Boussinesq equations in homogeneous Besov spaces with negative indices
Mianlu Zou, Qiang Li
TL;DR
This work addresses the regularity problem for the 3D Boussinesq equations by establishing a criterion controlled by the vertical derivative of the velocity in negative-index homogeneous Besov spaces. The authors prove that if $(u_0,\\theta_0)\\in H^1(\\mathbb{R}^3)\\times H^1(\\mathbb{R}^3)$ and $\\int_{0}^{T} \\|\\partial_{3} u\\|_{\\dot{B}_{\\infty,\\infty}^{-r}}^{\\frac{2}{1-r}} \\mathrm{d}t < \infty$ for some $0\\le r < 1$, then the solution $(u,\\theta)$ is regular on $(0,T]$. The approach combines $L^2$ and $H^1$ energy estimates with a Gronwall argument, exploiting the Besov-control of $\\partial_{3} u$ to tame nonlinear terms. This generalizes earlier criteria (e.g., the $r=0$ Besov case) and extends Wu's result in the Besov scale to negative indices. The results deepen understanding of how vertical regularity in Besov spaces governs the buoyancy-driven flow stability and provide a flexible framework for regularity criteria in the Boussinesq system.
Abstract
In this paper, we study the regularity criteria for the 3D Boussinesq equations in terms of one partial derivative of the velocity in Besov spaces. More precisely, it is proved that if the velocity $u$ holds $\int_{0}^{T}\| \partial_{3} u\|_{\dot{B}_{\infty,\infty}^{-r}}^{\frac{2}{1-r}}\mbox{d}t<\infty,\ with\ \ 0\leq r<1$, then the solution $(u, θ)$ is regular on $[0,T]$.