The Well-posedness Of Fractional Navier-stokes Equations With Caputo Derivatives

This paper is concerned with the time-fractional Navier-Stokes equations with the Caputo derivative of order ? ? (0,1). This type of equations can be used to simulate anomalous diffusion in fractal media. We focus our attention on the study of its well-posedness.This paper includes five main parts as follows: In Chapter 2, firstly, we get rid of the pressure term by applying the Helmholtz projector to the object equations,which converts them to evolution equations of the abstract form. Then by the theory of fractional evolution equations we obtain the corresponding integral equation as well as give the concept of mild solutions. Next, we make use of semigroup theory and fixed points techniques to establish the existence and uniqueness of local and global mild solutions, then proceed to show the regularity of classical solutions.In Chapter 3, we adopt the methods used in the second chapter to obtain the same abstract evolution equations, then we establish the existence and uniqueness of local mild solutions by using the iteration method and the Holder continuity of mild solutions. Next we use fixed point method to obtain the existence and uniqueness of mild solutions to each approximate equation (approximate solutions), as well as the convergence of the approximate solutions. Finally, some convergence results for the Faedo-Galerkin approximations are showed.In Chapter 4, we make use of energy methods to deal with the time-fractional Navier-Stokes equations. This chapter are divided into two parts. Firstly, we con-struct a regularized equation by using a smoothing process to transform unbounded differential operators into bounded operators, and then we obtain local approximate solutions (local solutions of regularized equations) and their properties. The sec-ond part addresses the procedure to take a limit in the approximation program to develop a new global result.In Chapter 5, then we give a new concept of mild solutions and study some properties of two new solution operators via the tools from harmonic analytic. Then we study the local existence of solutions in Sobolev spaces with the help of an assistant space, and two main results are presented.In Chapter 6, we use the classical Galerkin approximations and the theory of fractional calculus to establish the existence of weak solutions; then proceed to obtain the uniqueness of weak solutions in R2. Finally, we show an existence result of optimal control pairs.