The Attractors Of Some 2D Hydrodynamical Systems With Partial Dissipation

In this paper,we study the long-time behavior of solutions of the 2D Boussinesq equations and the 2D MHD equations with partial dissipation on a periodic domainFirst,we obtain the attraction properties for the velocity variable of the 2D Boussi-nesq equations with viscosity and without heat diffusion in the sense of the strong topology of V and prove that the weak sigma-attractor has a pancake-like structure,which answer partly some questions arising in Biswas et al.[Ann.Inst.H.Poincar'e Anal.Non Lin' eaire 34(2017),pp.381-405]and enrich the structure of the weak sigma-attractorSecond,we obtain the global well-posedness and the long-time behavior of solutions of the 2D Boussinesq equations with partial dissipation.We prove that this system is global well-posedness under the minimal assumptions on the initial data and has a weak sigma-attractor which retains some of the common properties of global attractors for the dissipative dynamical system,moreover,the local attractor which is the composition of the weak sigma-attractor is continuity under small viscosity perturbationsThen,we discuss the long-time behavior of solutions of nonautonomous 2D MHD equations with partial dissipation and time-dependent external forcing which is weakly normal.We get that the system is global well-posedness and exists uniform attractors Furthermore,the structure of uniform attractors is obtained and the fractal dimension is estimated for the kernel sections of the uniform attractors.