On the existence and smoothness problem of the magnetohydrodynamics system

Fluid mechanics plays a pivotal role in engineering application to daily lives. The prominently famous fluid dynamics partial differential equations (PDE) due to its remarkable utility is the Navier-Stokes equations of which its mathematical and physical significance is so highly regarded that it has become one of the seven Millennium Prize problems declared by the Clay Research Institute. We study closely related systems of partial differential equations with focus on the magnetohydrodynamics system, of which its special case is the Navier-Stokes equations. Other systems of PDEs of our concern include the surface quasi-geostrophic equations, incompressible porous media equation governed by Darcy's law, Boussinesq system, Leray, Lans-alpha models, micropolar and magneto-micropolar fluid models. We discuss the properties of solutions to these systems such as the global regularity issue with fractional Laplacians, logarithmic supercriticality, component reduction results of regularity criteria

Global smooth solution for the 3D generalized tropical climate model with partial viscosity and damping

The three-dimensional generalized tropical climate model with partial viscosity and damping is considered in this paper. Global well-posedness of solutions of the three-dimensional generalized tropical climate model with partial viscosity and damping is proved for $\alpha\geq\frac{3}{2}$ and $\beta\geq4$. Global smooth solution of the three-dimensional generalized tropical climate model with partial viscosity and damping is proved in $H^s(\mathbb R^3)$ $(s>2)$ for $\alpha\geq\frac{3}{2}$ and $4\leq\beta\leq5$

Stability and well-posedness problems on the partially dissipated Boussinesq equations and the micropolar equations

Fluid Mechanics is a central theme of science concerned with the study of the behavior of fluids when they are in state of motion or rest. When the density of the fluid is constant or its change with the pressure is so small that can be neglected, the fluid is said to be incompressible. Examination of such fluid flow phenomena is carried out with the help of the incompressible Navier-Stokes equations. These fundamental equations provide a mathematical model of the motion of the fluid. In this direction, this thesis is concerned with the study of two closely associated systems, the micropolar equations and the Boussinesq equations. The work being conducted in this thesis includes four main chapters. In the first chapter, we give a small introduction to the concerned equations. The second chapter is devoted to show the existence and uniqueness of the weak solutions to the d-dimensional micropolar equation with general fractional dissipation. Additionally, in the third chapter, we focus first on the stability problem of the 2D Boussinesq equations with vertical dissipation and horizontal thermal diffusion in $\mathbb{R}^2$, then we present some decay properties of the corresponding linearized system. Lastly, the fourth chapter investigates the stability and large-time behavior of the solutions to the 2D Boussinesq equations with horizontal dissipation and vertical thermal diffusion in two different spatial domains

Initial-boundary value problem for 2D temperature-dependent tropical climate model

It is well known that the tropical climate model is an important model to describe the interaction of large scale flow fields and precipitation in the tropical atmosphere. In this paper, we address the issue of global well-posedness for 2D temperature-dependent tropical climate model in a smooth bounded domain. Through classical energy estimates and De Giorgi-Nash-Moser iteration method, we obtain the global existence and uniqueness of strong solution in classical energy spaces. Compared with Cauchy problem, we establish more delicate a priori estimates with exponential decay rates. To the best of our knowledge, this is the first result concerning the global well-posedness for the initial-boundary value problem in 2D tropical climate model.Comment: 20 page