Well-posedness And Asymptotic Behaviour For Several Kinds Of Fractional Partial Differential Equations

This paper concerns mainly about the well posedness for several kinds of fractional evolution partial differential equations and the asymptotic behavior of their solutions.In Chapter 1,We first briefly introduce the origin of the concept of fractional calculus,several influential definitions for fractional calculus in history,their derivations and give the Riemann-Liouville definition for fractional calculus which is most widely used in ba-sic mathematics today.Then we review some preliminary needed for the study of partial differential equations,including some classical assumptions,common symbols,the func-tion spaces,the definitions for semigroups,some properties and inequalities with respect to semigroups,and gather some concepts and inequalities about stochastic processes.In Chapter 2,we study a class of deterministic nonlocal particle diffusion systems rep-resented by fractional operators.At first,we carefully analyze the relevant results of the existing literature,and further mine the internal properties of the kernel function in the defi-nition of fractional operator,so as to make up some defects in the theoretical analysis.Then,according to the characteristics of the equation and the corresponding structure and proper-ties of its solution,we look for the corresponding classical equation and kernel function as its asymptotic equation and asymptotic kernel function,by the properties of the kernel func-tion of the classical equation,comparing the kernel function of the fractional equations here with the asymptotic kernel function by a proper third item and careful frequency division and spectral analysis techniques,we depicts the subtle differences between them.Finally,according to the convergence theory and analysis tools in classical mathematical analysis and real analysis,we obtain the asymptotic behavior of the solutions of the deterministic nonlocal particle diffusion system with operators with fractional powers.In Chapter 3,We study the well posedness of Log-Euler equation with random diffusion of white noise on the two-dimensional torus T~2.At first,through the existing classical methods,we transform the stochastic Log-Euler equation into a partial differential equation with random coefficients.Then,we determine the corresponding function space,construct a mapping coincide with the form of its mild solution in the function space,we prove that the mapping is contractive under certain assumptions by the use of a series of basic inequalities and finally come to the existence and uniqueness of the local solution for corresponding paths.At last,the existence of the global solution is obtained by the property that the norm of the solution is decreasing on the local time interval.Our method can be used to study the existence and uniqueness of global solutions to the?-generalized SQG equation and Loglog-Euler equation with singular velocity in the sense of probability.In Chapter 4,We consider the initial boundary problem of a heat equation with mixed boundary conditions and white noise initial value.First,by the use of the characteristic of its Green function and the technique of the convergence of series,we modify some limit formulas in the literature and simplify related proofs.Then,we discuss the blow up and rapid cooling behavior of the average heat in the initial boundary value problem of the heat equation with more general boundary conditions.The limit formulas and main estimates obtained in this chapter will lay the foundation for our further study for the time fractional equations and even the space-time fractional equations.In Chapter 5,we study the abstract Cauchy problem for a class of It?o type stochastic reaction diffusion equation in a bounded domain.Firstly,we use properties of fractional operators and operator semigroups to analyze the influence of nonlinear term and random coefficient on the well posedness of Cauchy problem of the abstract stochastic reaction diffusion equation.Then,when the nonlinear item and random coefficient are global Lips-chitz,we study the L~p-convergence of the the asymptotic solution for time-discretization semi-implicit scheme to the real solution of the abstract Cauchy problem,modify and complete the existing proof methods in the literature.