Attractors For The Brinkman-Forchheimer Equations

Based on the theory of dynamical system, we study the long-time be-havior for the solutions of the 3-dimensional Brinkman-Forchheimer equa-tions which describing the ?uid ?ows in a saturated porous. Since the?uid ?ows in a saturated porous is very important in the petroleum indus-try and designing of air-condition. Many authors are interested in themin the recent and have obtained some good results [1,2,23,24,25]. Let? is an open bounded domain of R3 with su?ciently smooth boundary. u = (u1,u2,u3) is the ?uid velocity vector, we study the followingBrinkm?an-Forchheimer equationsu(x,τ) = uτ(x),τ∈R,whereγ> 0 is the Brinkman coe?cient, a > 0 is the Darcy coe?cient,b > 0, c > 0 are Forchheimer coe?cients. f(x,t) is the external force, p isthe pressure,β∈(1,2] is a constant.Letwhere n is the normal vector to .In the present paper, we first prove the existence of global attractorfor autonomous system when 34 <β≤2, which was open in [2], and fur-thermore, we obtain the exponential attractors in H and V respectively.Finally when the external force f(x,t) in Ll2oc(R,H) is normal, we prove the existence of uniform attractor and the exponential attractors for thenon-autonomous system.