Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

In this paper, we are concerned with the tri-dimensional anisotropic Boussinesq equations which can be described by [GRAPHICS] Under the assumption that the support of the axisymmetric initial data rho(0)(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 rho/r for large time by taking advantage of characteristic of transport equation. This growing property together with the horizontal smoothing effect enables us to establish H-1-estimate of the velocity via the L-2-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity parallel to omega(t)parallel to(root L):= sup(2 <= P<infinity) parallel to omega(t)parallel to(LP(R3))/root P < infinity which implies parallel to del u(t)parallel to(L3/2) := sup(2 <= P<infinity) parallel to del u(t)parallel to(LP(R3))/P root P < infinity. 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 sup(2 <= P<infinity) integral(t)(0) parallel to del u(tau)parallel to(LP(R3))/root P d tau < infinity 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. (C) 2013 Elsevier Masson SAS. All rights reserved.