Global Regularity for the Three‐Dimensional Boussinesq System Involving One Velocity Component

Abstract

ABSTRACT This article is devoted to the study of the Cauchy problem for the three‐dimensional Boussinesq system in Sobolev spaces. We first establish the global regularity of classical solutions to the fully dissipative system. Furthermore, we derive a new blow‐up criterion for the three‐dimensional Boussinesq system expressed solely in terms of one velocity component. This criterion is scaling‐invariant and involves only the vertical component , which extends the celebrated one‐component regularity results for the Navier–Stokes equations to the Boussinesq framework. The proof combines anisotropic energy estimates, vorticity decomposition, and refined bounds on in to close the nonlinear estimates.