Xiaoxin Zheng

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 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

Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

{\frac{u^{r}}{r}}

Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation

and

Regularity Criteria of the 3D Boussinesq Equations in the Morrey-Campanato Space

{\frac{\omega_\theta}{r}}

Time-dependent singularities in the Navier-Stokes system

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 (1.2) entails the desired regularity.

On the well-posedness for Keller-Segel system with fractional diffusion

In this paper, we investigate the Cauchy problem for the two-dimensional incompressible chemotaxis-Navier-Stokes equations. By taking advantage of a coupling structure of the equations and using a scale decomposition technique, we explore a new estimate of solutions. This estimate together with a microlocal analysis entails the global existence and uniqueness of weak solutions to the chemotaxis-Navier--Stokes system for a large class of initial data.