Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

Abstract

In this paper, we are concerned with the tri-dimensional anisotropic Boussinesq equations which can be described by{(∂t+u⋅∇)u−κΔhu+∇Π=ρe3,(t,x)∈R+×R3,(∂t+u⋅∇)ρ=0,divu=0. Under the assumption that the support of the axisymmetric initial data ρ0(r,z) does not intersect the axis (Oz), we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity ρr for large time by taking advantage of characteristic of transport equation. This growing property together with the horizontal smoothing effect enables us to establish H1-estimate of the velocity via the L2-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity ‖ω(t)‖L:=sup2⩽p<∞‖ω(t)‖Lp(R3)p<∞ which implies ‖∇u(t)‖L32:=sup2⩽p<∞‖∇u(t)‖Lp(R3)pp<∞. However, this regularity for the flow admits forbidden singularity since L (see (1.9) for the definition) seems to be the minimum space for the gradient vector field u(x,t) ensuring uniqueness of flow. To bridge this gap, we exploit the space–time estimate about sup2⩽p<∞∫0t‖∇u(τ)‖Lp(R3)pdτ<∞ by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space–time logarithmic inequality.