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

Abstract

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.