The Long-time Behavior Of Solutions For Several Types Of Hydrodynamic Equations

In this doctoral dissertation,we mainly studied the long-time behavior of solu-tions for 2D Navier-Stokes equations and 2D MHD equations defined on the time-varying domains,and the asymptotic dynamics of solutions to 2D MHD equations with delays on the cylindrical domains.Unlike the cylindrical domains,the mathematical processing of the problems defined on the time-varying domains often has special difficulties due to the spatial domains changing with time,especially on the selection of the functional spaces?the definition of solutions?the proof of the well-posedness of solutions and the depiction of long-time behavior.For 2D Navier-Stokes equations and 2D MHD equations de-fined on the time-varying domains with homogeneous Dirichlet boundary conditions,in addition to the above difficulties,the changing of the spatial domains also brings essential problems to the processing of the pressure p.In the paper,when the spatial domains Ot in R2 are obtained from a bounded base domain O by a diffeomorphism r(·,t),first of all,we introduced a special coordinate transformation to establish the invariance of the divergence operators and some equivalent estimates of the vec-tors on the time-varying domains and the cylindrical domains,and then we proved the well-posedness of weak solutions.Secondly,under the appropriate assumptions about the spatial domains(change with respect to time)and the external forces,we constructed the pullback absorbing sets,and then we used the method of the finite?-net to prove that the pullback absorbing sets are relatively compact.Finally we obtained the existence of the pullback attractors for the two types of systems(see Chapter 3 and Chapter 5).In this part,we used the hydrodynamic equations as the background model,and then we studied the effect brought by the change of the spatial domains to the specific equations,which not only provides a new perspective for revealing the essential difficulties of the problems defined on the time-varying domains,but also provides an effective entry point for the structure and complexity description of the attractors on the time-varying domains.In order to more realistically describe certain phenomena in the natural sci-ences such as physics and biology,in the paper we introduced the delays for the 2D MHD equations defined on the cylindrical domains.Compared with 2D MHD equa-tions without delays,the model contains the delay terms g1(t,ut)?g2(t,Bt),which leads to some mathematical difficulties,especially the compactness.In this paper,we established the well-posedness of strong and weak solutions by the method of Faedo-Galerkin,and proved the compactness of the solution process,furthermore we obtained the existence of pullback attractors(see Chapter 4).The introduction of this model not only can enrich the related content of infinite dimensional dynamical systems,but also plays a positive role in promoting for the hydrodynamic models.