On The Well-posedness For Two Kinds Of Nonlinear Diffusion Equations

There are lots of diffusion phenomena in nature, for instance, in the field of biochemistry, biological population dynamics and quantum mechanics. These diffusion phenonmena can be modelled by partial differential equations with diffusion terms. It is well known that the nonlinear diffusion equations are more authentic than the linear ones to describe some pritical phenomenona. Therefore, the study of nonlinear diffusion equations has attracted many mathematicians and physicists’ attention. In this thesis, we focous on two types of nonlinear diffusion equations: one is the Landau equation with fractional diffusion; the other is the chemotaxi-Stokes equation with nonlinear diffusion. This paper consists of the following contents:We review some background knowledges of two classes of the nonlinear diffusion equations(i.e., the fractional Landau equation and the chemotaxi-Stokes equation), and then summarize some related works and our main results.We devote to investigating the well-posedness of the fractional Landau equation. We state some definitions and a priori estimates about fractional operators. Next we establish the well-posedness of the fractional Landau equation. Finally, we will also investigate the long time asymptotic behavior of the non-negative global solutions.We study a boundary-value problem of the chemotaxi-Stokes equations with nonlinear diffusion in the three-dimensional bounded domain with smooth boundary. Due to the degeneracy of the equations, we first consider the regularization form of the equations and then establish some a priori estimates. Thus we can obtain the global existence of weak solutions under the constraint m?67.