Research On The Well-posedness Of Fractional Rayleigh-Stokes Problem And Related Problems

This thesis is concerned with the Rayleigh-Stokes problem for a generalized second grade fluid with Riemann-Liouville fractional derivative.This kind of problem is introduced from the problem of momentum conservation of non-Newtonian fluids over rectangular regions and it plays an important role in the development of chemistry,biological rheology,petrochemical industry,geophysics and other related fields.This thesis mainly studies the well-posedness of such problem,it includes six chapters:In Chapter 1,we introduce the background and development of fractional RayleighStokes problem,and give some preliminaries which will be used in this thesis.In Chapter 2,we mainly study the well-posedness of fractional Rayleigh-Stokes problem with initial value condition.Firstly,by using Laplace transform and eigenfunction expansion,we obtain the integral equation that corresponds to the fractional RayleighStokes problem,then we can give the definition of mild solutions.And then,we show that the solution operator of the problem is bounded,compact,and continuous in the uniform operator topology.Furthermore,we give the local and global existence of mild solutions for the initial value problem by using Schauder’s fixed point theorem,we also give the asymptotic property of global solution at initial time.In Chapter 3,we mainly discuss the existence and regularization of mild solutions of fractional nonlinear Rayleigh-Stokes problem with final value condition.From the methods used in Chapter 2,we can give the definition of mild solutions and the compactness of solution operators.And then,by using Schauder’s fixed point theorem,the existence of mild solutions is established under weak constraint on nonlinear source term.Finally,because of the ill-posedness of backward problem in the sense of Hadamard,the quasi-boundary value method is utilized to get the regularized solution,and the corresponding convergence rate is also obtained.In Chapter 4,we mainly discuss the existence and regularity of weak solutions of time-fractional Rayleigh-Stokes problem with linear source term.By virtue of the Galerkin method and some inequalities of fractional calculus,the existence,uniqueness and regularity of weak solutions of the proposed problem are obtained.Furthermore,if we improve the regularity conditions of initial value term and source term,we can verify that the spatial regularity of weak solutions is improved.In Chapter 5,we consider the well-posedness of initial value problem in the case that the nonlinearity term is an ?-regular mapping.The definition of ?-regular mild solutions is given at the beginning.And then,in virtue of skills of analysis,we discuss the properties of solution operators in interpolation spaces.By using Banach fixed point theorem,the local existence,uniqueness and continuous dependence upon the initial data of ?-regular mild solutions are obtained.Furthermore,by giving the definition of continuation of mild solutions,we obtain a unique continuation result and a blow-up alternative result of ?-regular mild solutions in virtue of the properties of solution operators and Banach fixed point theorem.In Chapter 6,we mainly study the well-posedness of initial value problem in Orlicz space.Firstly,from the fractional abstract evolution equation obtained from the fractional Rayleigh-Stokes problem,we get a new definition of mild solutions of such problem.And then,we discuss the properties of solution operator in Orlicz spaces.Finally,by using the H?lder’s inequality,H?lder’s interpolation inequality and other analytical skills,and in virtue of Banach fixed point theorem,we obtain the local and global existence and uniqueness of mild solutions in the whole space.