BRINKMAN-FORCHHEIMER EQUATIONS IN HOMOGENEOUS SOBOLEV SPACES

Abstract

Brinkman-Forchheimer Equations (BFE) are used to describe non-Darcy be havior of uid ow in a saturated porous medium. Researchers in this area have focused on the study of structural stability and long time behavior of solutions in square integrable space L 2 . However, problems on existence of weak solutions in the homogeneous Sobolev space H˙ s and stability of the system in the critical homo geneous Sobolev space H˙ 1 2 remain relatively unresolved. This study was therefore designed to obtain existence of weak solutions in H˙ s and stability of the system with respect to initial data, ϕn in H˙ 1 2 . Galerkin method was employed to obtain weak solutions in H˙ s = {u ∈ S 0 (R 3 ) : ||u||H˙ s(R3) < +∞} where S 0 is dual space of tempered distribution and ||u|| is the norm of u. BFE was approximated by nite-dimensional problem when the damp ing term is continuous, continuously di erentiable and satis es Lipschitz condi tion. Cauchy-Schwarz inequality and Parseval identity were used to obtain uni form bounds on the Fourier transforms of some terms in L 2 appearing in the weak formulation. The concept of pro le decomposition was employed to show that the sequence of solutions of BFE associated with sequence of initial data was bounded in E∞ = C 0 b ((R +, H˙ 1 2 (R 3 ))∩L 2 (R +, H˙ 3 2 (R 3 ))∩L 4 (R +, L4 (R 3 )) where H˙ 3 2 , is a space of vector elds whose rst derivative is in H˙ 1 2 , C 0 b , is a space of bounded continuous functions and L 4 is a space of four times integrable functions. By using the orthog onality property of sequence of scales and cores, (hj , xn) in (R+ \ {0} × R 3 ), the sequence of solutions was decomposed into a sum of orthogonal pro les in E∞ to generate a priori estimate. Finite time singularities of solutions was also obtained with respect to singularity generating initial data by pro le decomposition. The weak solution u(x, t) obtained belongs to L ∞((R+, H˙ s (R3 ))∩L 2 (R+, H˙ s+1(R3 ))∩ L r+1(R+, Lr+1(R3 )) for r ≥ 1 when the damping term is continuous and satis es the estimate Sup0≤t≤T ||u||H˙ s(R3 +2λ R T 0 ||5u||2 H˙ s(R3) dt+2β R T 0 ||u||r+1 Lr+1 dt ≤ ||u0||2 H˙ s v where λ and β are positive constants. Also for continuously di erentiable bi harmonic damping term, the solution is in L ∞((R+, L2 (R3 )) ∩ L 2 (R+, H˙ 1 (R3 )) ∩ L 2 (R+, H˙ 2 (R3 )) and satis es Sup0≤t≤T ||u||2 L2 + 2λ R T 0 || 5 u||2 L2 dt + 2β R T 0 ||u||2 H2 ≤ ||u0||2 L2 . The damping term with Lipschitz condition satis es the estimate Sup0≤t≤T ||u||2 H˙ s(R3) + 2λ R T 0 || 5 u||2 H˙ s(R3) dt + 2β R T 0 ||u||r+1 Lr+1 ≤ ||u0||2 H˙ s(R3) for u ∈ L ∞((R+, H˙ s (R3 )) ∩ L 2 (R+, H˙ s+1(R3 )) ∩ L r+1(R+, Lr+1(R3 )) and r ≥ 1 where L ∞ is a space of essentially bounded functions. The sequence of solutions, un obtained was bounded in E∞. The boundedness of the sequence of solution was due to the existence of a bound ||ϕn|| ≤ ρ on the sequence of initial data where ρ is any real number in [0, CA BF E]. Then a priori estimate ||BF E(ϕ)||E∞ ≤ B(||ϕ|| H˙ 1 2 (R3) , ||u||A) was obtained where B, BF E(ϕn), A, C A BF E are non-decreasing function from R + × [0, CA BF E] to R +, solution of BFE associated with initial data ϕ, admissi ble space, constant in R+ ∪ {+∞} respectively. Existence of singularity generating initial data gave rise to nite time singularities of solution that did not belong to E∞. The existence of weak solutions was obtained and the system was found to be stable with respect to initial data in the homogeneous Sobolev space