On the global well-posedness for the Boussinesq system with horizontal dissipation

In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is an axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between $\frac{u^{r}}{r}$ and $\frac{\omega_\theta}{r}$ by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of \eqref{eq1.1} entails the desired regularity.Comment: 32page

Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations

We review some results concerning the problem of global-in-time regularity for the initial boundary value problem for the Navier-Stokes equations in three-dimensional domains. In particular, we focus on sufficient conditions on the vorticity field which imply that strong (hence smooth) solutions exist on arbitrary time intervals