The Research Of Solutions For Some Fractional Differential Equations

In recent years,with the rapid development of science and technology,fractional partial differential equations have been used widely in various fields includ-ing quantum mechanics,geophysical fluid dynamics and plasma physics.The research on fractional partial differential equations will not only help us to explore more tech-nical methods to solve the equations,which in turn contribute to the development of mathematical theories,but also provide an effective way to understand the physical phenomena deeply from the mathematical point of view.So the implementation of this project is very important for both mathematical theories and real applications.In this dissertation,we will study the following space or time fractional differen-tial equations:space fractional Ginzburg-Landau equation,space fractional modified Zakharov equation,Quasi-geostrophic equation and time fractional diffusion equation.This thesis consists of six chapters and the main contents are divided into four parts.In the first part,we consider the following space fractional Ginzburg-Landau equa-tion-iduf =(dg2+1/U+a)u+g[a+d(2v-2?)]?-c/4m?2au—g/4m(c-d)?2??-b?u+g??2(u+g?)-idf(x),i?t=i??-g/Uu+(2v-2?)?+1/4m?2??+ih(x)u(x^,0)= u0(x),?(x,0)= ?0(x),x ?Rn,u(x + 2?ei,t)=u(x,t),?(x + 2?ei,t)=?(x,t),x?Rn,t?0,By using Galerkin method and a priori estimates,together with the properties of Sobolev space,we first establish the existence of global weak solutions.Then we prove the existence of global attractor.Furthermore,we consider the space fractional Ginzburg-Landau equation with multiplicative noises and show the existence of random global attractor.In the second part,we consider the following space fractional modified Zakharov equation with quantum correction i(?)tE +(?)xxE-H2?2?E = nE(?)ttn-(?)xxn + H2?2?n =(?)xx(?E?2),x ? ? 0,E(x,0)= EO(x),n(x,0)= nO(x),(?)tn(x,0)= ni(x),E(x + 2?,t)= E(x,t),n(x + 2?,t)= n(x,t),(?)tn(x + 2?,t)=(?)tn(x,t),By Galerkin method and a priori estimates,we consider the existence and uniqueness of global weak solution and then its regularity.Then by using Strichartz estimates and fixed point theory,we establish the local existence of the strong solution,together with the a priori estimates before,we could extend the solution to[0,T],for any T>0.In the third part,we consider the following dissipative Quasi-geostrophic equation with additive noise ut + v · ?u + K,?2?u + ?u = f + m?j=1?jd?,x ?T2 with initial condition u(tO,x)= uo(x).and?·u = 0.By using Ornstein-Uhlenbeck process,we first transform the dissipative Quasi-geostrophic equation with additive noise into the Quasi-geostrophic equation with random coeffi-cients.Then by using a priori estimates and compact embedding theory,we obtain the compact attracting set at time zero,then we can conclude the existence of random attractors on the periodic domains.In the fourth part,we consider the following coupled time fractional diffusion equation (?)?tu(x,t)= Lu(x,t)+ F1(u,u),(?)?tu(x,t)= Lu(x,t)+ F2(u,u),u = v = 0onx?(?)?,t ?(0,T],u?t=0 = ?1(x),u?t=0 = ?2(x),By using eigenfunction expansion,we first express the solution by Mittag-Lefiler func?tions.Then by the properties of Mittag-Lefiler functions and energy method,we estab-lish the existence and uniqueness of the weak solution.Then we study the regularity of solution.