Researches On The Well-posedness And Attractors For Some Classes Of Dynamic Equations

When studying atmospheric science,marine science,aerodynamics,ther-modynamics,life science,heat and electromagnetic radiation,quantum motion and other natural sciences,partial differential equations are often used,such as Navier-Stokes equations,wave equations and so on.These studies are widely ap-plied in many fields,like aviation,navigation,meteorology,material,biology,and control.As we all know,the dynamic systems are closely related to the differential equations,so it has important academic and applied value in the study of dynam-ic systems.In this dissertation,we mainly study the global well-posedness and attractors for several kinds of dynamical systems,including two-dimensional in-compressible Navier-Stokes-Vioght equations,a third order in time dynamics and radiation hydrodynamics models.Some meaningful results have been obtained.The results obtained in this dissertation are as follows:(1)The initial boundary value problem for two dimensional incompressible Navier-Stokes-Voight equations with three delays is studied.This model con-tains the distributed delay term g,and both the convective term and the exter-nal force term f contain a continuous delay p(t).By using the Faedo-Galerkin method,Lions-Aubin lemma and Arzela-Ascoli theorem,we establish the global well-posedness of solutions and the existence of pullback attractors in H1.(2)The initial boundary value problem for a third order(in time)partial d-ifferential equation in ?(?(?)Rn,n ? 1)is considered.This model contains a third order(in time)term ?uttt.The difficulty of this part is to construct the Lya-punov function equivalent to the energy function.The energy decay of the equa-tion is obtained by the decay of the Lyapunov function.In this dissertation,we use multiplier techniques to construct Lyapunov functionals and combine semi-group method to obtain the global existence,asymptotic behavior and uniform attractors in nonhomogeneous case.In addition,we also obtain the results of well-posedness in semilinear case.(3)The Cauchy problem for the 3D diffusion approximation model in radi-ation hydrodynamics is considered.By using the embedding theorem and inter-polation technique for careful energy estimation,and introducing the term 4?-?to overcome the difficulty come from the low order term of the radiation field n and the nonlinear term of temperature ? in estimation,we establish the global well-posedness of strong solutions in H2.