SMOOTH ATTRACTORS FOR THE BRINKMAN-FORCHHEIMER EQUATIONS WITH FAST GROWING NONLINEARITIES VARGA K. KALANTAROV AND SERGEY ZELIK 1 1 Abstract. We prove the existence of regular dissipative solutions and global attrac- 0 tors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrary 2 polynomial growth rate. In order to obtain this result, we prove the maximal regularity n estimate for the corresponding semi-linear stationary Stokes problem using some mod- a ification of the nonlinear localization technique. The applications of our results to the J Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also consid- 1 2 ered. ] P A 1. Introduction . h t We study the Brinkman-Forchheimer (BF) equations in the following form: a m ∂ u−∆u+f(u)+∇p = g, divu = 0, [ t (1.1) u = 0, u = u . 1 ( ∂Ω t=0 0 v 0 Here Ω ⊂ R3 is an open, b(cid:12)ounded do(cid:12)main with C2 boundary ∂Ω, g = g(x) = (g ,g ,g ) (cid:12) (cid:12) 1 2 3 7 is a given function, u = (u ,u ,u ) is the fluid velocity vector, p is the pressure and f is 0 1 2 3 4 a given nonlinearity. . The BF equations are used to describe the fluid flow in a saturated porous media, see 1 0 [16, 21] and references therein. The typical example for f is the following one: 1 1 f(u) = au+b|u|r−1u, r ∈ [1,∞), (1.2) : v where a ∈ R and b > 0 are the Darcy and Forchheimer coefficients respectively (the i X original Brinkman-Forchheimer model corresponds to the choice r = 2, more complicated r a nonlinear terms(r 6= 2)appear, e.g., inthetheoryofnon-Newtonianfluids, see[20]). Note also that the analogous equations are used in the study of tidal dynamics (see [6],[13]). Number of papers is devoted to the mathematical study of of the BF equations, for instance, continuous dependence on changes in Brinkman and Forchheimer coefficients and convergence of solutions of BF equations to the solution of the Forchheimer equation ∂ u+f(u)+∇p = g, divu = 0, t as the viscosity tends to zero have been established in [2, 3, 12, 18, 21] (see also references therein), and the long-time behavior of solutions for (1.1) has been studied in terms of global attractors in [17] [23] and [24]. However, to the best of our knowledge, only the 1991 Mathematics Subject Classification. 35B40,35B41, 35Q35. Key words and phrases. Brinkmann-Forchheimer equations, attractors, maximal regularity, nonlinear localization. 1 2 ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS case of the so-called subcritical growth rate of the nonlinearity f (r ≤ 3 in (1.2)) has been considered in the literature. The main aim of the present paper is to remove this growth restriction and verify the global existence, uniqueness and dissipativity of smooth solutions of the BF equations for the large class of nonlinearites f of the arbitrary growth exponent r ≥ 1. Namely, we assume that f ∈ C2(R3,R3) satisfies the following conditions: 1) f′(u)v.v ≥ (−K +κ|u|r−1)|v|2, ∀u,v ∈ R3, (1.3) 2) |f′(u)| ≤ C(1+|u|r−1), ∀u ∈ R3, ( where K,C,κ are some positive constants, r ≥ 1 and u.v stands for the standard inner product in R3. Our key technical tool is the maximal regularity result for the stationary problem −∆w +f(w)+∇p = g, divw = 0, u = 0 (1.4) ∂Ω which claims that the solution w belongs to H2 if g ∈ L2. T(cid:12)his result is straightforward (cid:12) for the case of periodic boundary conditions (it follows via the multiplication of the equation by ∆w and integrating by parts). However, for the case of Dirichlet boundary conditions it is far from being immediate since the additional uncontrollable boundary terms arise after the multiplication of the equation by ∆w and integrating by parts. Following the approach developed in [9], we overcome this problem using some kind of nonlinear localization technique, see Apendix below. In addition, we apply our maximal regularity result in order to establish the existence of smooth solutions for the so-called convective BF equations: ∂ u+(u,∇)u−∆u+f(u)+∇p = g, divu = 0, t (1.5) u = 0, u = u ( ∂Ω t=0 0 under the assumption(cid:12) (1.2) with(cid:12)r > 3. Note that the case f = 0 corresponds to the clas- (cid:12) (cid:12) sical Navier-Stokes problem where the existence of smooth solutions is an open problem. However, as also known (see [19]) the sufficiently strong nonlinearity f produces some kind of regularizing effect. Again, in contrast to the previous works, no upper bounds for the exponent r are posed here. The paper is organized as follows. A number of a priori estimates which are necessary to handle equation (1.1) is given in Section 2. Existence, uniqueness and regularity of solutions for the BF equation as well as the existence of the associated global attractor are established in Section 3. These results are extended to the case of convective BF equa- tions (1.5) in Section 4. Finally, the crucial maximal regularity results for the stationary equations (1.1) and (1.5) are obtained in Appendix. 2. A priori estimates. In this section, we obtain a number of a priori estimates for the solutions of the prob- lem (1.1) assuming that the sufficiently regular solution (u,p) of this equation is given. These estimates will be used in the next sections in order to establish the existence and uniqueness of solution, their regularity, etc. ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 3 We start with introducing the standard notations. As usual, we denote by Wl,p(Ω) the Sobolev space of all functions whose distributional derivatives up to order l belong to Lp(Ω). The Hilbert spaces Wl,2(Ω) will be also denoted by Hl(Ω). For the vector valued functions v = (v ,v ,v ), and u = (u ,u ,u ) we denote by (u,v) 1 2 3 1 2 3 the standard inner product in [L2(Ω)]3: 3 (u,v) := (v ,u ) , j j L2(Ω) j=1 X 3 and write k∇uk2 instead of k∇u k2 . In the sequel, where it does not lead to mis- L2 i L2 i=1 understandings, we will also uPse the notation Hl(Ω) and Wl,p(Ω) for the spaces of vector valued functions [Hl(Ω)]3 and [Wl,p(Ω)]3 respectively. As usual, we set V := v ∈ (C∞(Ω))3 : divv = 0 , 0 and denote by H and H = V the closure of V in L (Ω) and H1(Ω) topology respectively. 1 (cid:8) 2 (cid:9) And, more generally, Hs := D(As/2), where A := Π∆ and Π is the classical Helmholz- Leray orthogonal projection in L2(Ω) onto the space H. In particular, since Ω is smooth and bounded, we have H = {u ∈ L2(Ω), divu = 0,(u,n) = 0}, H1 := H ∩H1(Ω), H2 = H1 ∩H2(Ω), ∂Ω 0 see e.g. [10]. (cid:12) (cid:12) The next lemma gives the usual energy estimate for the BF equation. Lemma 2.1. Let (u,p) be a sufficiently smooth solution of problem (1.1). Then the following estimate holds: t+1 ku(t)k2 + k∇u(s)k2 +ku(s)kr+1 ds ≤ Cku(0)k2 e−αt +C(1+kgk2 ), (2.1) L2 L2 Lr+1 L2 L2 Zt where the positive c(cid:2)onstants C and α are in(cid:3)dependent of t and the concrete choice of the solution (u,p). Proof. Indeed, multiplying equation (1.1) by u, integrating over x ∈ Ω, using that f(u).u ≥ −C + κ|u|r+1 and (∇p,u) = (p,divu) = 0 and arguing in a standard way, we have 1 ∂ ku(t)k2 +αku(t)k2 +αku(t)kr+1 ≤ C(1+kgk2 ) (2.2) 2 t L2 H1 Lr+1 L2 for some positive α and C which are independent of u and t. Applying the Gronwall (cid:3) inequality to the last estimate, we derive (2.1) and finish the proof of the lemma. Remark 2.2. The standard (for the reaction-diffusion equations) next step in a priori estimates would be the multiplication of equation (1.1) by ∆u (or t∆u) and obtaining the dissipative estimate in H1 together with the L2 → H1 parabolic smoothing property. However, in our case, this scheme looks not applicable since ∆u 6= 0 in general and the ∂Ω term with pressure will not disappear. Multiplication by Π∆u (where Π is the Helmholz- (cid:12) Lerayprojectortothedivergent freevectorfields)alsodoesnotw(cid:12)orkduetothepresenceof the non-linearity f with arbitrary growth rate. So, we have to skip this step and estimate 4 ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS the L2-norm of ∂ u instead differentiating equation by t and using the quasi-monotonicity t of f. The H1 (and H2) estimate will be obtained after that using the maximal regularity theorem for the elliptic problem (5.1), see Appendix). The next simple corollary is, however, crucial for our method of proving the existence and dissipativity of the H2-solutions. Corollary 2.3. Let (u,p) be a sufficiently regular solution of problem (1.1). Then, the following estimate holds: k∂tukL1([t,t+1],H−2) ≤ Q(ku(0)kL2)e−αt +Q(kgkL2), (2.3) where the monotone function Q and the constant C are independent of t and u. Proof. Indeed, applying the Helmholz-Leray projector Π to both sides of equation (1.1) and using that div∂ u = 0, we arrive at t ∂ u = Au−Πf(u)+Πg. (2.4) t Thanks to the growth restriction on f and the control (2.1), we have kf(u)krLr∗([t,t+1],Lr∗) ≤ Cku(0)k2L2e−αt +C(1+kgk2L2) with r∗ := r+1. Using now that the Helmholz-Leray projector Π : Lr∗ → Lr∗ together r with the embedding Lr∗ ⊂ H−2 (recall that n = 3), we arrive at kΠf(u)kL1([t,t+1],H−2) ≤ Q(ku(0)kL2)e−αt +Q(kgkL2) for some monotone increasing function Q. This estimate, together with (2.4) and the control of u given by the energy estimate (2.1) give the desired estimate (2.3) and finish (cid:3) the proof of the corollary. Let us now differentiate (1.1) with respect to time and denote v = ∂ u. Then, this t function solves ∂ v = ∆v −f′(u)v+∇q, divv = 0, v(0) = Au(0)−Πf(u(0))+Πg. (2.5) t Moreover, using the embedding H2 ⊂ C, we see that kv(0)k ≤ Q(ku(0)k )+kgk (2.6) L2 H2 L2 and, therefore, the L2-norm of the initial data for v is under the control if u(0) ∈ H2. The next Lemma gives the control of v(t) for all t ≥ 0. Lemma 2.4. Let (u,p) be a sufficiently regular solution of problem (1.1). Then, the following estimate holds: t+1 kv(t)k2 + kv(s)k2 ds ≤ Q(ku(0)k )eKt +Q(kgk2 ) (2.7) L2 H1 H2 L2 Zt for some positive constant K and monotone function Q. Proof. Multiplying equation (2.5) by v(t), integrating over Ω and using that (f′(u)v)·v ≥ −K|v|2,∀u,v ∈ R3 (see the condition (1.3)), we arrive at ∂ kv(t)k2 +kv(t)k2 ≤ 2Kkv(t)k2 . (2.8) t L2 H1 L2 ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 5 Applying the Gronwall inequality to this estimate, we arrive at (2.7) and finish the proof (cid:3) of the lemma. Corollary 2.5. Let (u,p) be a sufficiently smooth solution of the problem (1.1). Then, the following estimate holds: ku(t)k +k∇p(t)k ≤ Q(ku(0)k )eKt +Q(kgk ) (2.9) H2 L2 H2 L2 for some positive constant K and monotone function Q independent of t and u . 0 Indeed, due to the control (2.7), we may rewrite equation (1.1) as an elliptic boundary value problem ∆w(t)−f(w(t))+∇p(t) = g (t) := −g +∂ u(t) (2.10) u t and apply the maximal regularity result of Theorem 5.2 (see Appendix) to that equation. Together with (2.7) this gives indeed estimate (2.9) and proves the corollary. We, however, note that the proved estimate (2.9) is divergent as t → ∞ and, by that reason, is not sufficient to verify the dissipativity of the problem (1.1) in H2. In order to overcome this drawback, we need the L2 → H2 smoothing property for the solutions of (1.1). This result will be obtained exploiting the parabolic smoothing for equation (2.5) together with the already established control (2.3) for v(t) = ∂ u(t). t Lemma 2.6. Let (u,p) be a sufficiently regular solution of the problem (1.1). Then, the following estimate holds: 1+t3 k∂ u(t)k ≤ Q(ku(0)k )e−αt +Q(kgk ) , t > 0, (2.11) t L2 t3 L2 L2 where the positive constant α and t(cid:0)he monotone function Q are (cid:1)independent of t and u. Proof. We first note that, due to the energy estimate (2.1), it is sufficient to verify (2.11) for t ∈ (0,1] only. To this end, we multiply (2.8) by tN (where the exponent N will be specified later) and integrate with respect to t. Then, we have t t sup sNkv(s)k2 + sNkv(s)k2 ds ≤ C sN−1kv(s)k2 := I(t), (2.12) L2 H1 L2 s∈[0,t] Z0 Z0 (cid:8) (cid:9) where C = C(N,K) is independent of t and u. 1/3 2/3 We estimate I(t) using (2.3) and the interpolation inequality kvk ≤ Ckvk kvk : L2 H−2 H1 t I(t) ≤ C sup sN/2kv(s)k sN/2−1kv(s)k ds ≤ L2 L2 s∈[0,t] Z0 (cid:8) (cid:9) t ≤ sup sN/2kv(s)kL2 (sN/2kv(s)kH1)2/3(sN/2−3kv(s)kH−2)1/3ds ≤ s∈[0,t] Z0 (cid:8) (cid:9) t ≤ 1/2 sup sNkv(s)k2 +1/2 sNkv(s)k2 ds+ L2 H1 s∈[0,t] Z0 (cid:8) (cid:9) t 2 +C′ sN/2−3kv(s)kH−2ds . (2.13) (cid:18)Z0 (cid:19) 6 ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS Fixing now N = 6, using the control (2.3) in order to estimate the right-hand side of (2.13) and inserting it into the right-hand side of (2.12), we see that sup s6kv(s)k2 ≤ 1/2 sup s6kv(s)k2 +Q(ku(0)k )+Q(kgk ) (2.14) L2 L2 L2 L2 s∈[0,t] s∈[0,t] (cid:8) (cid:9) (cid:8) (cid:9) (recall, we have assumed that t ≤ 1). It only remains to note that (2.14) immediately gives (2.11) for t ≤ 1. Lemma 2.6 is proved. (cid:3) We summarize the obtained estimates in the following theorem. Theorem 2.7. Let (u,p) be a sufficiently regular solution of the problem (1.1). Then, the following estimate holds: ku(t)k +k∇p(t)k ≤ Q(ku(0)k )e−αt +Q(kgk ), (2.15) H2 H1 H2 L2 where the positive constant α and a monotone function Q are independent of t and u. Moreover, the following smoothing property is valid: 1+t3 ku(t)k +k∇p(t)k ≤ Q ku(0)k e−αt +Q(kgk ), t > 0. (2.16) H2 L2 t3 L2 L2 (cid:18) (cid:19) Indeed, the estimate (2.16) is an immediate corollary of (2.11) and the maximal elliptic regularity of Theorem 5.2 applied to the elliptic equation (2.10). In order to verify (2.15), it is sufficient to use the divergent in time estimate (2.9) for t ≤ 1 and estimate (2.16) for t ≥ 1. 3. Well-posedness and attractors The estimates obtained in the previous section, allow us to prove the existence and uniqueness of a solution of the problem (1.1) as well as to establish existence of the global attractor for the associated semigroup. We start with the definition of a weak solution of that equation excluding the pressure in a standard way. Definition 3.1. A function u ∈ C([0,∞),H)∩L2 ([0,∞),H1)∩Lr+1([0,∞),Lr+1(Ω)) (3.1) loc loc is called a weak solution of (1.1) if it satisfies (2.4) in the sense of distributions, i.e., − (u(t),∂ ϕ(t))dt = − (∇u(t),∇ϕ(t))−(f(u(t)),ϕ(t))+(g,ϕ(t))dt t R R Z Z for all ϕ ∈ C∞(R ×Ω) such that divϕ(t) ≡ 0. 0 + The next lemma establishes the uniqueness of a weak solution. Lemma 3.2. Let the nonlinearity f satisfy assumptions (1.3). Then, the weak solution of problem (1.1) is unique. Moreover, for any two solutions u (t) and u (t) (with different 1 2 initial data) of the equation (1.1), the following estimate holds: ku (t)−u (t)k ≤ e(K−λ1)tku (0)−u (0)k , (3.2) 1 2 L2 1 2 L2 where K is the same as in (1.3) and λ > 0 is the first eigenvalue of the operator A. 1 ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 7 Proof. Let u (t) and u (t) be two different energy solutions of (1.1) and let v(t) := u (t)− 1 2 1 u (t). Then, this function solves: 2 ∂ v = Av −Π(f(u )−f(u )), v(0) = u (0)−u (0). (3.3) t 1 2 1 2 Note that, due to the regularity (3.1) of a weak solution and the growth restrictions on f, all terms in equation (3.3) belong to the space L2([0,T],H−1)+L1+1/r([0,T],L1+1/r(Ω)) = [L2([0,T],H1)∩Lr+1([0,T],L1+r(Ω))]∗. In particular, the function t → ku(t)k2 is absolutely continuous and H d ku(t)k2 = 2(∂ u(t),u(t)). dt L2 t Multiplying now equation (3.3) by v(t), integrating over Ω and using the inequality (f(u )−f(u (t)).(u −u ) ≥ −K|u −u |2, ∀u ,u ∈ R3 1 2 1 2 1 2 1 2 (due to the first assumption of (1.3)), we arrive at d 1/2 kv(t)k2 ≤ Kkv(t)k2 −(Av(t),v(t)) ≤ (K −λ )kv(t)k2 (3.4) dt L2 L2 1 L2 and the Gronwall inequality now gives the uniqueness and estimate (3.2). Lemma 3.2 is (cid:3) proved. We are now able to state our main result on the well-posedness and regularity of solu- tions of problem (1.1). Theorem 3.3. Let the nonlinearity f satisfy assumptions (1.3) and let g ∈ L2(Ω). Then, for every u ∈ H, problem (1.1) possesses a unique weak solution u (in the sense of 0 Definition (3.1)). Moreover, u(t) ∈ H2 for all t > 0 and the estimate (2.16) holds. In addition, if u ∈ H2, the estimate (2.15) also holds. 0 Proof. Indeed, the existence of a weak solution can be obtained in a standard way using, say, the Galerkin approximation method. The uniqueness is proved in Lemma 3.2. Thus, we only need to justify the estimates (2.16) and (2.15). To this end, we note that the estimates (2.7) and (2.11) for the differentiated equation (2.5) can be also first obtained on the level of the Galerkin approximations and then justified by passing to the limit (remind that the uniqueness of a weak solution holds). Finally, rewriting the problem (1.1) in the form of elliptic problem (2.10) and using the Theorem 5.2, we justify the (cid:3) desired estimates (2.16) and (2.15). Thus, Theorem 3.3 is proved. Thus, under the assumptions of Theorem 3.3, the Brinkman-Forchheimer problem (1.1) generates a dissipative semigroup S(t) in the phase space H: S(t) : H → H, S(t)u := u(t), (3.5) 0 where u(t) solves (1.1) with u(0) = u . Our next task is to verify the existence of a global 0 attractor for that semigroup. For the convenience of the reader, we start with reminding the definition of the attractor, see [1],[7],[11],[22] for more details. 8 ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS Definition 3.4. A set A ⊂ H is a global attractor of a semigroup S(t) : H → H if the following properties are satisfied: 1) A is a compact subset of H; 2) A is strictly invariant: S(t)A = A for all t ≥ 0; 3) It attracts the images of all bounded sets as time goes to infinity, i.e., for every bounded subset B ⊂ H and every neighborhood O(A) of A, there exists T = T(B,O) such that S(t)B ⊂ O(A), ∀t ≥ T. The following theorem states the existence of the attractor for the problem considered. Theorem 3.5. Let the assumptions of Theorem 3.3 hold. Then the solution semigroup (3.5) associated with the Brinkman-Forchheimerequation (1.1) possesses a global attractor A (in the sense of the above definition) which is bounded in H2 and is generated by all complete bounded solutions of (1.1) defined for all t ∈ R: A = K , (3.6) t=0 where K := {u ∈ C (R,H2), u solves (1.1)}. (cid:12) b (cid:12) Indeed, according to the abstract attractor existence theorem (see e.g., [1],[22]), we only need to check that the considered semigroup is continuous with respect to the initial data (for every fixed t) and it possesses a compact absorbing set in H. But the first assertion is an immediate corollary of Lemma 3.2 and the second one follows from the estimate (2.16). Moreover, this estimate gives the absorbing set bounded in H2. Since the attractor is always contained in an absorbing set, we have verified the existence of a global attractor A which is bounded in H2. Finally, the representation (3.6) of the attractor in terms of completer bounded trajectories is also a standard corollary of the attractor existence theorem mentioned above. Remark 3.6. Although, we have stated only the H2-regularity of the attractor A, it can be further improved (if f, Ω and g are smooth enough) using the maximal regularity for the linearStokes equation and bootstrapping. In particular, if f, Ω and g are C∞ smooth, the attractor will be also C∞-smooth. Another standard corollary of the general theory is the fact that the obtained attractor has a finite Hausdorff and fractal dimension in H. The proof of this fact is a straight- forward implementation of the volume contraction technique to our equation (see e.g., [1, 22]). Indeed, due to the embedding H2 ⊂ C, the nonlinearity f is subordinated to the linear part of the equation (no matter how large is the growth exponent r) and one even is able to reduce formally the problem considered to the case of abstract semilinear parabolic equations. To conclude this section, we discuss the particular case of (1.1) where f(u) = −∇ F(u), (3.7) u for some scalar function F ∈ C2(R3). Note that this condition is satisfied for the ”most natural” nonlinearities f(u) = au|u|r−1−bu. ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 9 In that case, multiplying the equation by ∂ u and integrating over Ω, we get t d L(u(t)) = −k∂ u(t)k2 ≤ 0, dt t L2 where 1 L(u) := (∇u,∇u)+(F(u),1). 2 Thus, the solution semigroup S(t) possesses the global Lyapunov functional L(u) and applying the standard arguments (see [7],[11]) to our problem, we obtain the following result. Corollary 3.7. Let the assumptions of Theorem 3.5 and the condition (3.7) be satisfied. Then, every trajectory u(t) stabilizes as t → ∞ to the set of equilibria R := {u ∈ H2,Au −Πf(u ) = Πg}. (3.8) 0 0 0 Furthermore, if the set R is discrete, every trajectory u(t) converges to a single equilibrium u ∈ R and the rate of convergence is exponential if that equilibrium is hyperbolic. 0 Remark 3.8. Note that, for generic g ∈ L2, the set R will contain only hyperbolic equi- libria (see [1]). In that case, as it is not difficult to prove (again verifying the conditions of the abstract theorem on regular attractors stated in [1]), the attractor A can be presented as a finite union of finite-dimensional submanifolds of H (the unstable manifolds of all equilibria) and that the rate of attraction of any bounded subset B to the global attractor A is exponential. 4. The convective Brinkman-Forchheimer equations In this section, we extend the results of the previous section to the case of the following Brinkman-Forchheimer equation with the Navier-Stokes type inertial term: ∂ u+(u,∇)u+∇p = ∆u−f(u)+g, divu = 0. (4.1) t Note that the case f = 0 corresponds to the classical Navier-Stokes problem and the general case f 6= 0 can be also considered as the so-called tamed Navier-Stokes equation, see [19]. As before, the nonlinearity f is assumed to satisfy conditions (1.3) but with the ad- ditional lower bound r > 3 which is necessary for the uniqueness. Note that no upper bounds for the growth exponent is posed. As before, we define a weak solution u as a function of the class (3.1) satisfying (4.1) in the sense of distributions, see Definition 3.1. In addition the assumption r ≥ 3 guarantees that (u,∇)u ∈ L4/3 ⊂ Lq, q := (r +1)∗ ≤ 4/3 (4.2) and, therefore, in contrast to the case of the classical Navier-Stokes equations, the mul- tiplication of (4.1) by u with integration over Ω is justified for any weak energy solution of that equation. Thus, we have verified that any weak energy solution of (4.1) satisfies the energy estimate (2.1). The existence of an energy solution can be then obtained in a standard way via the Galerkin approximation method. The next Lemma gives the uniqueness of the energy solution for the case r > 3. 10 ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS Lemma 4.1. Let the nonlinearity f satisfy (1.3) with r > 3 and g ∈ L2. Then, for every u ∈ H, the problem (4.1) possesses a unique weak solution u and this solution satisfies 0 the energy estimate (2.1). Proof. Indeed, let u and u be two solutions and let v = u − u . Then, this function 1 2 1 2 solves ∂ v +(v,∇)u +(u ,∇)v +∇q = ∆v −[f(u )−f(u )], divv = 0. (4.3) t 1 2 1 2 Multiplying this equation by v, integrating by parts and using that f satisfies (1.3), we will have d kvk2 +2k∇vk2 +α(|u |r−1 +|u |r−1,|v|2) ≤ Ckvk2 +2|((v,∇)u ,v)| dt L2 L2 1 2 L2 1 for some positive α depending on κ from (1.3). Here we have implicitly used that the first condition of (1.3) implies that (f(u )−f(u ),u −u ) ≥ −Cku −u k2 +α(|u |r−1 +|u |r−1,|u −u |2), 1 2 1 2 1 2 L2 1 2 1 2 see [14] and [5] for the details. The last term in the above differential inequality can be estimated integrating by parts once more and using that r −1 > 2: 2|((v,∇)u ,v)| ≤ 2(|u |·|v|,|∇v|) ≤ k∇vk2 +C(|u |2,|v|2) ≤ 1 1 L2 1 ≤ k∇vk2 +α(|u |r−1 +|u |r−1,|v|2)+Ckvk2 . (4.4) L2 1 2 L2 Thus, we have d kvk2 +k∇vk2 ≤ Ckvk2 (4.5) dt L2 L2 L2 (cid:3) and the uniqueness is proved. Remark 4.2. As we see from the proof, the uniqueness holds for the case r = 3 if the coefficient κ in (1.3) is large enough. However, we do not know whether or not the uniqueness holds for any cubic nonlinearity (without this assumption). The next theorem is analogous to Theorem 3.3 and gives the regularity of solutions for problem (4.1). Theorem 4.3. Let the function f satisfy (1.3) with r > 3 and let g ∈ L2. Then, for any u ∈ H, the associated solution u(t) of (4.1) is more regular for t > 0 (u(t) ∈ H2) and 0 estimate (2.16) holds. In addition, if u ∈ H2 then estimate (2.15) also holds. 0 Proof. The proof of this theorem is also analogous to the proof of Theorem 3.3. Indeed, differentiating equation (4.1) with respect to t and arguing as in the proof of the previous lemma, we conclude that the function v = ∂ u satisfies the differential inequality (4.5). t On the other hand, using (4.2) for the control of the inertial term and arguing as in Corollary 2.3, we derive estimate (2.3) and based on that estimate and inequality (4.5) for v = ∂ u, one derives the controls (2.11) and (2.7) for the time derivative v = ∂ u (all t t these estimates can be justified via the Galerkin approximations). Finally, having the control of the L2-norm of ∂ u, one can treat problem (4.1) as an t elliptic boundary value problem of the form (5.42) and apply Corollary 5.4 which gives the desired estimate for the H2-norm and finishes the proof of the theorem. (cid:3)