This doctoral thesis studies the well-posedness theory of the initial (boundary) value problem for  incompressible Boussinesq equations and Oldroyd-B model with irregular dissipation, where the irregular dissipation includes anisotropic dissipation and fractional dissipation. The Boussinesq equations describe the convection phenomenon of geophysical flow, and play a crucial role in the study of atmosphere, oceanographic flows and Rayleigh-B\'enard convection. The Oldroyd-B model governs the motion of a class of complex non-Newtonian fluids, which obeys a constitutive law. Further, it can describe the viscoelastic properties of dilute polymer solutions under general flow conditions.
In Chapter 1, this thesis introduces the related background, the derivation of the incompressible Boussinesq equations and Oldroyd-B model, some known results and the main results obtained in this thesis.
In Chapter 2 and 3, this thesis studies the global stability and the large-time behavior of solutions of the initial problem for two kinds of two dimensional (2D) Boussinesq equations. The temperature equations of these two kinds of problems are the same: they both have a damping term and an external force term which is the second component of the velocity field. However, the momentum equations are different. For one kind of problem, the momentum equation only has dissipation in the vertical direction, and the other kind of momentum equation only has a damping term in the horizontal direction. Without the coupling, the corresponding velocity equation is the 2D Navier--Stokes equation with only vertical dissipation, and the vorticity equation is a 2D Euler-like equation with an extra Calderon--Zygmund type term, and their stability in the whole space are open problems. By constructing
a suitable Lyapunov functional, this thesis proves the buoyancy force exactly stabilizes the fluids by the coupling and interaction, and obtains large-time behavior of the correspondding solutions.
In Chapter 4, this thesis examines the large-time behavior for the two dimensional and three dimensional (3D) incompressible Oldroyd-B models without velocity dissipation and with only fractional diffusive stress. Since the Fourier-splitting method does not work here, this result offers a new frame and idea on how to obtain precise decay estimates on a fractional dissipated system. The discovery here is that the coupling and interaction of the velocity and the non-Newtonian stress actually
enhance the stability of the system. Without the stress, the Sobolev norms of the velocity could grow in time.  By constructing
a suitable Lyapunov functional, this thesis is able to control the growth in the derivatives and extract algebraic decay rates. The optimal decay rates are established by applying a bootstrapping argument to the integral equation.
In Chapter 5, this thesis investigates the 2D incompressible inviscid Boussinesq equations with fractional diffusion $(-\Delta)^{\beta}\theta\triangleq\Lambda^{2\beta}\theta$ on a bounded domain, equipped with the slip boundary condition for the velocity vector field and Dirichlet boundary condition for the temperature. This thesis defines the fractional Laplacian operator $(-\Delta)^{\beta}$ on bounded domain by the eigenfunction expansions of the $-\Delta$ operator; then through the Galerkin approximation method, this thesis obtains the local existence and uniqueness for $\beta\in[0,1)$. Since the fractional dissipation term $\Lambda^{\beta}\theta~(0<\beta<1)$ cannot balance the external force term of the vorticity equation $\partial_{x_1}\theta$, this thesis combines the energy inequalities for $\|\omega\|_{L^2}$ and $\|\Lambda^{1/4}\theta\|_{L^2}$ to show the global uniform bound of $\|\omega\|_{L^2}$. Finally, we obtain the global existence and uniqueness of classical solutions of the initial boundary value problem in the range of $\beta\in[3/4,1)$ via the Schauder fixed point theorem.