Researches On The Well-posedness Of Solutions And Related Problems Of Attractors For Some Dynamical Models

In modern dynamics,the nonlinear partial differential equations are usually used to describe some physical phenomena in nature and describe the process of macro movement of matter,such as nonclassical reaction-diffusion equation,Prandtl equation and Prandtl-Hartmann equation.These equations can describe fluid mechanics,solid mechanics,non-Newtonian fluid,heat conduction process and aerodynamics,and have a wide range physical meaning.As is known to all,there are essential differences between the nonclassical reactiondiffusion equation and the classical reaction-diffusion equation,because the nonclassical reaction-diffusion equation contains dissipation term-?ut,such that the solution and the initial data have the same topological space.In other words,if the initial value belongs to the weak topological space,its solution can only belong to the corresponding weak topological space and cannot have higher regularity.Therefore,it is difficult to verify the compactness of the solution process for equations.For a nonclassical reaction-diffusion equation under the different assumptions condition,we use different techniques to verify the compactness of the solution process when proving the existence of the attractor.Another model,the Prandtl type equation is derived from fluid equation or coupled equation,including Prandtl equation and Prandtl-Hartmann equation.Prandtl equation is derived from two-dimensional incompressible Navier-Stokes equation by variable substitution.The Prandtl-Hartmann model is derived from the classical twodimensional incompressible MHD model.The same difficulty of the Prandtl type equation is the loss of x-derivative along the horizontal direction by the nonlinear term?(?)yu,so that estimates of some terms can not be effectively controlled.We will use the different techniques to overcome these difficulties.The mathematical theory of these models,especially the well-posedness and the attractors of solutions,lie in the hot topics in the international mathematics field.This dissertation is divided into two parts.The first part introduces the existence and finite fractal dimensional of the attractors of a nonclassical reaction-diffusion equation,the second part introduces the existence and uniqueness of solutions of the two dimensional Prandtl type equation in suitable topological spaces,and some meaningful new results are obtained.The research results of this paper are stated as follows:Chapter 1,we introduce the physical background of the nonclassical reactiondiffusion equation,Prandtl equation and Prandtl-Hartmann equation,the well-posedness of solutions and the research progress of the infinite dimensional dynamical system,some main results and the methods of research,and some inequalities.Chapter 2,we investigate the existence of pullback attractors for a nonclassical reaction-diffusion equation in the strong topological space H2(?)?H01(?).First,we use Galerkin methods to prove the existence and uniqueness of strong solutions for a nonclassical diffusion equation.Then we prove the existence of pullback attractors in H2(?)?H01(?)by applying asymptotic a priori estimate method.Chapter 3,we investigate pullback attractors for a nonclassical diffusion equation with memory in H2(?)?H01(?)×L?2(R+;H2(?)?H01(?)).First,we use Galerkin methods to prove the existence and uniqueness of the strong solutions for a nonclassical diffusion equation with memory.Then we prove the existence of pullback attractors in H2(?)?H01(?)×L?2(R+;H2(?)?H01(?))by the decomposition methods.Chapter 4,we study the finite fractal dimension of the pullback attractors for a nonclassical diffusion equation.First,we apply operator decomposition methods to prove the existence of pullback attractors for a nonclassical diffusion equation with arbitrary polynomial growth condition.Then,by the fractal dimension theorem of pullback attractors,we prove the finite fractal dimension of pullback attractors for a nonclassical diffusion equation in H01(?).Chapter 5,we use energy methods to study the existence of the solutions for the two-dimensional Prandtl equation in a weighted Sobolev space.The difficulty of the Prandtl equation is the loss of horizontal direction x-derivative by the term?(?)yu,so that estimates of some terms can not be effectively controlled.To overcome this difficulty,we construct a new unknown function gm=((?)xmu/(uys+uy))y(see(5.5.4)).Besides,we also give the relationship between the quantities gm and (?)xm? in L2norm.Chapter 6,we prove the global existence and uniqueness of solutions to 2D PrandtlHartmann equation by using classical energy methods in analytic framework.The difficulty of solving the Prandtl-Hartmann equation in the analytic framework is the loss of x-derivative in the term ?(?)yu.To overcome this difficulty,we introduce the Gaussian weighted Poincare inequality(see Lemma 6.2).Compared to the existence and uniqueness of solutions to the classical Prandtl equation where the monotonicity condition of the tangential velocity plays a key role,which is not needed for the 2D Prandtl-Hartmann equation in analytic framework.Besides,the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role,which is not needed for the 2D Prandtl-Hartmann equation in analytic framework,either.