Long-time Behavior Of Solutions For The Three-dimensional Globally Modified Bénard Problem

In this paper,we mainly study the long time behavior of solutions of three-dimensional globally modified Bénard systems.Firstly,the existence of global at-tractors AO in H and global attractors A in V are proved;Secondly,the smoothness and time regularity of the solution operator semigroup generated by the system are proved,then we obtain the existence of exponential attractors.Finally,we mainly study the existence,uniqueness and asymptotic behavior of solutions of Bénard sys-tem with delays under the local Lipschitz condition.The paper is divided into five parts:The first chapter,we mainly introduces some problems which have studied of Bénard system in the other paper and the main conclusion of this paper.The second chapter,we mainly introduce the preparation knowledge to prove the existence of attractors.The third chapter,firstly,we study the existence of the global attractor A0 in H.We firstly make a priori estimation of the solution of the problem(1.1),and then obtain the existence of the global attractor from the compactness theorem.Secondly,we study the existence of the global attractor A in V.According to the literature[15],we firstly prove the property(C)of the problem(1.1)and then prove the existence of absorbing sets to explain the existence of the global attractor in V.Thirdly,the regularity of global attractors is illustrated by A=A0.The fourth chapter,we mainly study the existence of the exponential attractor in H.The existence of exponential attractors is illustrated by the smoothness and time regularity of semigroups generated by the system(1.1).The fifth chapter,we mainly study the existence,uniqueness and asymptotic behavior of solutions of Bénard system with delays under local Lipschitz conditions.