Global Attractor For A Discretized Bénard System

In this paper,we show the existence of global attractors in the phase space of the operator semigroup defined by the solution of the discretized modified three-dimensional Bénard system.First,we prove the existence of the solution to the discretized modified three-dimensional Bénard system in phase space,and then prove the existence of the global attractor in this phase space.Finally,we discuss the asymptotic behavior of the solution as N tends to infinity.The full paper is divided into five parts:The first chapter,we mainly introduce the preparation knowledge to prove the existence of attractors and the main results of this paper.The second chapter,we study the existence of solutions for a class of discretized modified three-dimensional Bénard system in bounded domain.First,we construct approximate solutions for the Bénard system and prove the existence of approxi-mate solutions.Then,we use prior estimation method to prove the boundedness of approximate solutions in D(A1)× D(A2).Finally,according to Sobolev’s tight embedding theorem,we obtain the existence of solutions for the discretized modified three-dimensional Bénard system.The third chapter,we study the boundedness of solutions of such discretized modified three-dimensional Bénard system in bounded domain.First,we construct the solution sequence of Bénard system according to the existence of the solution.Then we take the inner product of the test function with the two equations of the system.Finally,we get the boundedness of solutions of discrete modified three-dimensional Bénard system in H × L2(Ω),(?)× H01(Ω)and D(A1)× D(A2)through a series of estimates.The fourth chapter,we study the uniqueness of solutions of such discretized modified three-dimensional Bénard system in bounded domain.First,we give the solution sequence of two Bénard systems with different initial values and parameters N according to the existence of the solution.Then,we take the inner product of the test function with the two equations obtained by the system,and obtain the uniqueness of the solution of the discrete modified three-dimensional Bénard system through a series of estimates,and define an C0 semigroup Sm by the uniqueness of the solution.Therefore,we obtain that Sm has a bounded absorption set in phase space according to the boundedness of the solution.Finally,we prove the existence of global attractors for discrete modified three-dimensional Bénard system by Sobolev embedding theorem.The fifth chapter,we study the limit behavior as N→ ∞ of such discretized modified three-dimensional Bénard system in bounded domain.First,we construct a sequence of solutions to the Bénard system with respect to the parameter N based on the existence of the solution.Then,we obtain the strong convergence of the solution according to the uniform boundedness of the solution sequence about the parameter N and Sobolev’s tight embedding theorem.Finally,we get the limit behavior as N →∞ for the solution of the discrete modified three-dimensional Bénard system through a series of estimates.