In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is 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.

We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space.When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $\\infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.

In this paper, the authors investigate rigidity of geometric, topological and differentiable structures of compact submanifolds in a space form.