A regularity criterion for the 3D axisymmetric Boussinesq equations with non-zero swirl

Abstract

In this paper, we are devoted to establishing a new regularity criterion of weak solutions to incompressible axisymmetric Boussinesq equations. More precisely, we prove that for a small ϵ > 0, if the angular component of velocity field uθ(t, r, z) satisfies sup0<t<T‖zuθ‖L∞(Ωδ)≤ϵ, where Ωδ≔x1,x2,z∈R3|x12+x22<δ denotes a thin cylinder with infinite height and radius δ > 0, which is independent of the initial data, then the weak solution (u, ρ) to 3D Boussinesq equations is regular in (0, T].