Properties And Dimension Estimation Of Attractors For Three-Dimensional Brinkman-Forchheimer Equation

Brinkman-forchheimer equation is a kind of fluid dynamics equation used to describe the flow law in saturated porous media.It is very important in practical application and theory.At present,the three-dimensional Brinkman-Forchheimer equation has made a lot of progress in numerical solutions and applications in many other disciplines.However,there are still some unsolved problems in theory,such as properties of attractors and dimension estimation.Therefore,the dimension estimation of global attractors for strong solutions of BrinkmanForchheimer equations on three-dimensional bounded domains and the existence of pullback attractors for solutions are studied in this paper.The main work of this paper includes:Firstly,the Hausdorff dimension and the fractal dimension of the global attractor of the strong solution of the three-dimensional Brinkman-Forchheimer equation in a bounded region are studied.Based on the existence of the global attractor of the strong solution of the equation and the basic lemma of dimension estimation,the Frechet differentiability of the semigroup to the initial value u0 is proved,and the semigroup operator associated with the equation is derived.Then,the uniformly quasi differentiable method is applied for β>0.We prove that the global attractor of the equation has finite Hausdorff dimension and fractal dimension,and estimate the upper bounds of the Hausdorff dimension and fractal dimension of the global attractor of the strong solution.Then,we prove the existence of the pull-back attractor for the solution of the nonautonomous Brinkman-Forchheimer equation on a three-dimensional bounded region.The pullback attractor theory is used to study the long time behavior of the solution of the equation,and a series of uniform estimates are made for the solution of the equation under appropriate assumptions of the external force f（x,t）.We prove the existence of pullback D-absorption set and the pullback D-asymptotically compact of the process {U（t,τ）}_t≥τ in（H₀¹（Ω））³ and（H²（Ω））³ associated with the equation.Then we prove the existence of the pull-back attractors for solutions of the equations in and（H²（Ω））)³ when 0≤β≤4.