Smooth Attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities
aa r X i v : . [ m a t h . A P ] J a n SMOOTH ATTRACTORS FOR THE BRINKMAN-FORCHHEIMEREQUATIONS WITH FAST GROWING NONLINEARITIES
VARGA K. KALANTAROV AND SERGEY ZELIK
Abstract.
We prove the existence of regular dissipative solutions and global attrac-tors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrarypolynomial growth rate. In order to obtain this result, we prove the maximal regularityestimate for the corresponding semi-linear stationary Stokes problem using some mod-ification of the nonlinear localization technique. The applications of our results to theBrinkmann-Forchheimer equation with the Navier-Stokes inertial term are also consid-ered. Introduction
We study the Brinkman-Forchheimer (BF) equations in the following form: ( ∂ t u − ∆ u + f ( u ) + ∇ p = g, div u = 0 ,u (cid:12)(cid:12) ∂ Ω = 0 , u (cid:12)(cid:12) t =0 = u . (1.1)Here Ω ⊂ R is an open, bounded domain with C boundary ∂ Ω, g = g ( x ) = ( g , g , g )is a given function, u = ( u , u , u ) is the fluid velocity vector, p is the pressure and f isa given nonlinearity.The BF equations are used to describe the fluid flow in a saturated porous media, see[16, 21] and references therein. The typical example for f is the following one: f ( u ) = au + b | u | r − u, r ∈ [1 , ∞ ) , (1.2)where a ∈ R and b > r = 2, more complicatednonlinear terms ( r = 2) appear, e.g., in the theory of non-Newtonian fluids, see [20]). Notealso 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, forinstance, continuous dependence on changes in Brinkman and Forchheimer coefficientsand convergence of solutions of BF equations to the solution of the Forchheimer equation ∂ t u + f ( u ) + ∇ p = g, div u = 0 , as the viscosity tends to zero have been established in [2, 3, 12, 18, 21] (see also referencestherein), and the long-time behavior of solutions for (1.1) has been studied in terms ofglobal attractors in [17] [23] and [24]. However, to the best of our knowledge, only the Mathematics Subject Classification.
Key words and phrases.
Brinkmann-Forchheimer equations, attractors, maximal regularity, nonlinearlocalization. case of the so-called subcritical growth rate of the nonlinearity f ( r ≤ f of the arbitrary growth exponent r ≥ f ∈ C ( R , R ) satisfies the following conditions: ( f ′ ( u ) v.v ≥ ( − K + κ | u | r − ) | v | , ∀ u, v ∈ R , | f ′ ( u ) | ≤ C (1 + | u | r − ) , ∀ u ∈ R , (1.3)where K, C, κ are some positive constants, r ≥ u.v stands for the standard innerproduct in R .Our key technical tool is the maximal regularity result for the stationary problem − ∆ w + f ( w ) + ∇ p = g, div w = 0 , u (cid:12)(cid:12) ∂ Ω = 0 (1.4)which claims that the solution w belongs to H if g ∈ L . This result is straightforwardfor the case of periodic boundary conditions (it follows via the multiplication of theequation by ∆ w and integrating by parts). However, for the case of Dirichlet boundaryconditions it is far from being immediate since the additional uncontrollable boundaryterms 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 existenceof smooth solutions for the so-called convective BF equations: ( ∂ t u + ( u, ∇ ) u − ∆ u + f ( u ) + ∇ p = g, div u = 0 ,u (cid:12)(cid:12) ∂ Ω = 0 , u (cid:12)(cid:12) t =0 = u (1.5)under the assumption (1.2) with r >
3. Note that the case f = 0 corresponds to the clas-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 somekind of regularizing effect. Again, in contrast to the previous works, no upper bounds forthe exponent r are posed here.The paper is organized as follows. A number of a priori estimates which are necessaryto handle equation (1.1) is given in Section 2. Existence, uniqueness and regularity ofsolutions for the BF equation as well as the existence of the associated global attractorare 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 stationaryequations (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 anduniqueness of solution, their regularity, etc.
TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 3
We start with introducing the standard notations. As usual, we denote by W l,p (Ω)the Sobolev space of all functions whose distributional derivatives up to order l belong to L p (Ω). The Hilbert spaces W l, (Ω) will be also denoted by H l (Ω).For the vector valued functions v = ( v , v , v ) , and u = ( u , u , u ) we denote by ( u, v )the standard inner product in [ L (Ω)] :( u, v ) := X j =1 ( v j , u j ) L (Ω) , and write k∇ u k L instead of P i =1 k∇ u i k L . In the sequel, where it does not lead to mis-understandings, we will also use the notation H l (Ω) and W l,p (Ω) for the spaces of vectorvalued functions [ H l (Ω)] and [ W l,p (Ω)] respectively.As usual, we set V := (cid:8) v ∈ ( C ∞ (Ω)) : div v = 0 (cid:9) , and denote by H and H = V the closure of V in L (Ω) and H (Ω) topology respectively.And, more generally, H s := D ( A s/ ), where A := Π∆ and Π is the classical Helmholz-Leray orthogonal projection in L (Ω) onto the space H . In particular, since Ω is smoothand bounded, we have H = { u ∈ L (Ω) , div u = 0 , ( u, n ) (cid:12)(cid:12) ∂ Ω = 0 } , H := H ∩ H (Ω) , H = H ∩ H (Ω) , see e.g. [10].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 thefollowing estimate holds: k u ( t ) k L + Z t +1 t (cid:2) k∇ u ( s ) k L + k u ( s ) k r +1 L r +1 (cid:3) ds ≤ C k u (0) k L e − αt + C (1 + k g k L ) , (2.1) where the positive constants C and α are independent of t and the concrete choice of thesolution ( 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, div u ) = 0 and arguing in a standard way,we have 12 ∂ t k u ( t ) k L + α k u ( t ) k H + α k u ( t ) k r +1 L r +1 ≤ C (1 + k g k L ) (2.2)for some positive α and C which are independent of u and t . Applying the Gronwallinequality to the last estimate, we derive (2.1) and finish the proof of the lemma. (cid:3) Remark 2.2.
The standard (for the reaction-diffusion equations) next step in a prioriestimates would be the multiplication of equation (1.1) by ∆ u (or t ∆ u ) and obtainingthe dissipative estimate in H together with the L → H parabolic smoothing property.However, in our case, this scheme looks not applicable since ∆ u (cid:12)(cid:12) ∂ Ω = 0 in general and theterm with pressure will not disappear. Multiplication by Π∆ u (where Π is the Helmholz-Leray projector to the divergent free vector fields) also does not work due to the presence ofthe non-linearity f with arbitrary growth rate. So, we have to skip this step and estimate ATTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS the L -norm of ∂ t u instead differentiating equation by t and using the quasi-monotonicityof f . The H (and H ) estimate will be obtained after that using the maximal regularitytheorem for the elliptic problem (5.1), see Appendix).The next simple corollary is, however, crucial for our method of proving the existenceand dissipativity of the H -solutions. Corollary 2.3.
Let ( u, p ) be a sufficiently regular solution of problem (1.1) . Then, thefollowing estimate holds: k ∂ t u k L ([ t,t +1] ,H − ) ≤ Q ( k u (0) k L ) e − αt + Q ( k g k L ) , (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 ∂ t u = 0, we arrive at ∂ t u = Au − Π f ( u ) + Π g. (2.4)Thanks to the growth restriction on f and the control (2.1), we have k f ( u ) k rL r ∗ ([ t,t +1] ,L r ∗ ) ≤ C k u (0) k L e − αt + C (1 + k g k L )with r ∗ := r +1 r . Using now that the Helmholz-Leray projector Π : L r ∗ → L r ∗ togetherwith the embedding L r ∗ ⊂ H − (recall that n = 3), we arrive at k Π f ( u ) k L ([ t,t +1] ,H − ) ≤ Q ( k u (0) k L ) e − αt + Q ( k g k L )for some monotone increasing function Q . This estimate, together with (2.4) and thecontrol of u given by the energy estimate (2.1) give the desired estimate (2.3) and finishthe proof of the corollary. (cid:3) Let us now differentiate (1.1) with respect to time and denote v = ∂ t u . Then, thisfunction solves ∂ t v = ∆ v − f ′ ( u ) v + ∇ q, div v = 0 , v (0) = Au (0) − Π f ( u (0)) + Π g. (2.5)Moreover, using the embedding H ⊂ C , we see that k v (0) k L ≤ Q ( k u (0) k H ) + k g k L (2.6)and, therefore, the L -norm of the initial data for v is under the control if u (0) ∈ H .The next Lemma gives the control of v ( t ) for all t ≥ Lemma 2.4.
Let ( u, p ) be a sufficiently regular solution of problem (1.1) . Then, thefollowing estimate holds: k v ( t ) k L + Z t +1 t k v ( s ) k H ds ≤ Q ( k u (0) k H ) e Kt + Q ( k g k L ) (2.7) 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 | , ∀ u, v ∈ R (see the condition (1.3)), we arrive at ∂ t k v ( t ) k L + k v ( t ) k H ≤ K k v ( t ) k L . (2.8) TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 5
Applying the Gronwall inequality to this estimate, we arrive at (2.7) and finish the proofof the lemma. (cid:3)
Corollary 2.5.
Let ( u, p ) be a sufficiently smooth solution of the problem (1.1) . Then,the following estimate holds: k u ( t ) k H + k∇ p ( t ) k L ≤ Q ( k u (0) k H ) e Kt + Q ( k g k L ) (2.9) for some positive constant K and monotone function Q independent of t and u . Indeed, due to the control (2.7), we may rewrite equation (1.1) as an elliptic boundaryvalue problem ∆ w ( t ) − f ( w ( t )) + ∇ p ( t ) = g u ( t ) := − g + ∂ t u ( t ) (2.10)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 thatreason, is not sufficient to verify the dissipativity of the problem (1.1) in H . In order toovercome this drawback, we need the L → H 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 ) = ∂ t u ( t ). Lemma 2.6.
Let ( u, p ) be a sufficiently regular solution of the problem (1.1) . Then, thefollowing estimate holds: k ∂ t u ( t ) k L ≤ t t (cid:0) Q ( k u (0) k L ) e − αt + Q ( k g k L ) (cid:1) , t > , (2.11) where the positive constant α and the monotone function Q are 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 t N (where the exponent N will bespecified later) and integrate with respect to t . Then, we havesup s ∈ [0 ,t ] (cid:8) s N k v ( s ) k L (cid:9) + Z t s N k v ( s ) k H ds ≤ C Z t s N − k v ( s ) k L := I ( t ) , (2.12)where C = C ( N, K ) is independent of t and u .We estimate I ( t ) using (2.3) and the interpolation inequality k v k L ≤ C k v k / H − k v k / H : I ( t ) ≤ C sup s ∈ [0 ,t ] (cid:8) s N/ k v ( s ) k L (cid:9) Z t s N/ − k v ( s ) k L ds ≤≤ sup s ∈ [0 ,t ] (cid:8) s N/ k v ( s ) k L (cid:9) Z t ( s N/ k v ( s ) k H ) / ( s N/ − k v ( s ) k H − ) / ds ≤≤ / s ∈ [0 ,t ] (cid:8) s N k v ( s ) k L (cid:9) + 1 / Z t s N k v ( s ) k H ds ++ C ′ (cid:18)Z t s N/ − k v ( s ) k H − ds (cid:19) . (2.13) 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 thatsup s ∈ [0 ,t ] (cid:8) s k v ( s ) k L (cid:9) ≤ / s ∈ [0 ,t ] (cid:8) s k v ( s ) k L (cid:9) + Q ( k u (0) k L ) + Q ( k g k L ) (2.14)(recall, we have assumed that t ≤ 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: k u ( t ) k H + k∇ p ( t ) k H ≤ Q ( k u (0) k H ) e − αt + Q ( k g k L ) , (2.15) where the positive constant α and a monotone function Q are independent of t and u .Moreover, the following smoothing property is valid: k u ( t ) k H + k∇ p ( t ) k L ≤ Q (cid:18) t t k u (0) k L (cid:19) e − αt + Q ( k g k L ) , t > . (2.16)Indeed, the estimate (2.16) is an immediate corollary of (2.11) and the maximal ellipticregularity 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 ≤ t ≥
1. 3.
Well-posedness and attractors
The estimates obtained in the previous section, allow us to prove the existence anduniqueness of a solution of the problem (1.1) as well as to establish existence of the globalattractor for the associated semigroup. We start with the definition of a weak solution ofthat equation excluding the pressure in a standard way.
Definition 3.1.
A function u ∈ C ([0 , ∞ ) , H ) ∩ L loc ([0 , ∞ ) , H ) ∩ L r +1 loc ([0 , ∞ ) , L r +1 (Ω)) (3.1)is called a weak solution of (1.1) if it satisfies (2.4) in the sense of distributions, i.e., − Z R ( u ( t ) , ∂ t ϕ ( t )) dt = − Z R ( ∇ u ( t ) , ∇ ϕ ( t )) − ( f ( u ( t )) , ϕ ( t )) + ( g, ϕ ( t )) dt for all ϕ ∈ C ∞ ( R + × Ω) such that div ϕ ( t ) ≡ Lemma 3.2.
Let the nonlinearity f satisfy assumptions (1.3) . Then, the weak solutionof problem (1.1) is unique. Moreover, for any two solutions u ( t ) and u ( t ) (with differentinitial data) of the equation (1.1) , the following estimate holds: k u ( t ) − u ( t ) k L ≤ e ( K − λ ) t k u (0) − u (0) k L , (3.2) where K is the same as in (1.3) and λ > is the first eigenvalue of the operator A . TTRACTOR 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 ) − u ( t ). Then, this function solves: ∂ t v = Av − Π( f ( u ) − f ( u )) , v (0) = u (0) − u (0) . (3.3)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 L ([0 , T ] , H − ) + L /r ([0 , T ] , L /r (Ω)) = [ L ([0 , T ] , H ) ∩ L r +1 ([0 , T ] , L r (Ω))] ∗ . In particular, the function t → k u ( t ) k H is absolutely continuous and ddt k u ( t ) k L = 2( ∂ t u ( t ) , u ( 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 | , ∀ u , u ∈ R (due to the first assumption of (1.3)), we arrive at1 / ddt k v ( t ) k L ≤ K k v ( t ) k L − ( Av ( t ) , v ( t )) ≤ ( K − λ ) k v ( t ) k L (3.4)and the Gronwall inequality now gives the uniqueness and estimate (3.2). Lemma 3.2 isproved. (cid:3) 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 ∈ L (Ω) . Then,for every u ∈ H , problem (1.1) possesses a unique weak solution u (in the sense ofDefinition (3.1) ). Moreover, u ( t ) ∈ H for all t > and the estimate (2.16) holds. Inaddition, if u ∈ H , the estimate (2.15) also holds.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 theestimates (2.7) and (2.11) for the differentiated equation (2.5) can be also first obtainedon 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 thedesired estimates (2.16) and (2.15). Thus, Theorem 3.3 is proved. (cid:3)
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)where u ( t ) solves (1.1) with u (0) = u . Our next task is to verify the existence of a globalattractor for that semigroup. For the convenience of the reader, we start with remindingthe definition of the attractor, see [1],[7],[11],[22] for more details. 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 thefollowing properties are satisfied:1) A is a compact subset of H ;2) A is strictly invariant: S ( t ) A = A for all t ≥ 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-Forchheimer equation (1.1) possesses a global attractor A (in the sense of the above definition) which is bounded in H and is generated by allcomplete bounded solutions of (1.1) defined for all t ∈ R : A = K (cid:12)(cid:12) t =0 , (3.6) where K := { u ∈ C b ( R , H ) , u solves (1.1) } . Indeed, according to the abstract attractor existence theorem (see e.g., [1],[22]), weonly need to check that the considered semigroup is continuous with respect to the initialdata (for every fixed t ) and it possesses a compact absorbing set in H . But the firstassertion is an immediate corollary of Lemma 3.2 and the second one follows from theestimate (2.16). Moreover, this estimate gives the absorbing set bounded in H . Sincethe attractor is always contained in an absorbing set, we have verified the existence ofa global attractor A which is bounded in H . Finally, the representation (3.6) of theattractor in terms of completer bounded trajectories is also a standard corollary of theattractor existence theorem mentioned above. Remark 3.6.
Although, we have stated only the H -regularity of the attractor A , it canbe further improved (if f , Ω and g are smooth enough) using the maximal regularity forthe linear Stokes 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 attractorhas 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 H ⊂ C , the nonlinearity f is subordinated tothe linear part of the equation (no matter how large is the growth exponent r ) and oneeven is able to reduce formally the problem considered to the case of abstract semilinearparabolic equations.To conclude this section, we discuss the particular case of (1.1) where f ( u ) = −∇ u F ( u ) , (3.7)for some scalar function F ∈ C ( R ). Note that this condition is satisfied for the ”mostnatural” nonlinearities f ( u ) = au | u | r − − bu. TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 9
In that case, multiplying the equation by ∂ t u and integrating over Ω, we get ddt L ( u ( t )) = −k ∂ t u ( t ) k L ≤ , where L ( u ) := 12 ( ∇ u, ∇ u ) + ( F ( u ) , . Thus, the solution semigroup S ( t ) possesses the global Lyapunov functional L ( u ) andapplying the standard arguments (see [7],[11]) to our problem, we obtain the followingresult. 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 ∈ H , Au − Π f ( u ) = Π g } . (3.8) 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. Remark 3.8.
Note that, for generic g ∈ L , 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 ofthe abstract theorem on regular attractors stated in [1]), the attractor A can be presentedas a finite union of finite-dimensional submanifolds of H (the unstable manifolds of allequilibria) 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 followingBrinkman-Forchheimer equation with the Navier-Stokes type inertial term: ∂ t u + ( u, ∇ ) u + ∇ p = ∆ u − f ( u ) + g, div u = 0 . (4.1)Note that the case f = 0 corresponds to the classical Navier-Stokes problem and thegeneral case f = 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 > u as a function of the class (3.1) satisfying (4.1) inthe sense of distributions, see Definition 3.1. In addition the assumption r ≥ u, ∇ ) u ∈ L / ⊂ L q , q := ( r + 1) ∗ ≤ / u with integration over Ω is justified for any weak energy solutionof that equation. Thus, we have verified that any weak energy solution of (4.1) satisfiesthe energy estimate (2.1). The existence of an energy solution can be then obtained in astandard way via the Galerkin approximation method.The next Lemma gives the uniqueness of the energy solution for the case r > Lemma 4.1.
Let the nonlinearity f satisfy (1.3) with r > and g ∈ L . Then, for every u ∈ H , the problem (4.1) possesses a unique weak solution u and this solution satisfiesthe energy estimate (2.1) .Proof. Indeed, let u and u be two solutions and let v = u − u . Then, this functionsolves ∂ t v + ( v, ∇ ) u + ( u , ∇ ) v + ∇ q = ∆ v − [ f ( u ) − f ( u )] , div v = 0 . (4.3)Multiplying this equation by v , integrating by parts and using that f satisfies (1.3), wewill have ddt k v k L + 2 k∇ v k L + α ( | u | r − + | u | r − , | v | ) ≤ C k v k L + 2 | (( v, ∇ ) u , v ) | for some positive α depending on κ from (1.3). Here we have implicitly used that the firstcondition of (1.3) implies that( f ( u ) − f ( u ) , u − u ) ≥ − C k u − u k L + α ( | u | r − + | u | r − , | u − u | ) , see [14] and [5] for the details.The last term in the above differential inequality can be estimated integrating by partsonce more and using that r − > | (( v, ∇ ) u , v ) | ≤ | u | · | v | , |∇ v | ) ≤ k∇ v k L + C ( | u | , | v | ) ≤≤ k∇ v k L + α ( | u | r − + | u | r − , | v | ) + C k v k L . (4.4)Thus, we have ddt k v k L + k∇ v k L ≤ C k v k L (4.5)and the uniqueness is proved. (cid:3) Remark 4.2.
As we see from the proof, the uniqueness holds for the case r = 3 ifthe coefficient κ in (1.3) is large enough. However, we do not know whether or not theuniqueness holds for any cubic nonlinearity (without this assumption).The next theorem is analogous to Theorem 3.3 and gives the regularity of solutions forproblem (4.1). Theorem 4.3.
Let the function f satisfy (1.3) with r > and let g ∈ L . Then, for any u ∈ H , the associated solution u ( t ) of (4.1) is more regular for t > ( u ( t ) ∈ H ) andestimate (2.16) holds. In addition, if u ∈ H then estimate (2.15) also holds.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 previouslemma, we conclude that the function v = ∂ t u satisfies the differential inequality (4.5).On the other hand, using (4.2) for the control of the inertial term and arguing as inCorollary 2.3, we derive estimate (2.3) and based on that estimate and inequality (4.5)for v = ∂ t u , one derives the controls (2.11) and (2.7) for the time derivative v = ∂ t u (allthese estimates can be justified via the Galerkin approximations).Finally, having the control of the L -norm of ∂ t u , one can treat problem (4.1) as anelliptic boundary value problem of the form (5.42) and apply Corollary 5.4 which givesthe desired estimate for the H -norm and finishes the proof of the theorem. (cid:3) TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 11
Remark 4.4.
Note that the nonlinear localization technique used in the proof of Corollary5.4 is not necessary if r ≤ Theorem 4.5.
Let the assumptions of Theorem 4.3 hold. Then the solution semigroup S ( t ) : H → H possesses a global attractor A which is a bounded subset of H and possessesthe standard description (3.6) . The proof of this theorem repeats word by word the proof of Theorem 3.5 and so isomitted.
Remark 4.6.
To conclude, we note that all assertions formulated in Remark 3.6 remaintrue for the convective case as well.5.
Appendix: Maximal regularity for semi-linear Stokes problem
The appendix is devoted to the stationary problem associated with the problem (1.1),that is the following semi-linear Stokes problem: ( − ∆ w + f ( w ) + ∇ p = g, div w = 0 , x ∈ Ω ,w = 0 , x ∈ ∂ Ω , R Ω p ( x ) dx = 0 . (5.1)Here w = ( w , w , w ), g ∈ L (Ω) is a given function and the nonlinearity f satisfiesassumptions (1.3) with arbitrary r > K = 0. Thus, we have assumed thatthe nonlinearity f is monotone f ′ ( u ) ≥ L -maximal regularity estimate for problem (5.1) (whichis the non-linear version of the classical L -regularity theorem for the Stokes operator).Before stating the main result, we first remind the straightforward L q -regularity resultwhere q = ( r + 1) ∗ = 1 + r . Lemma 5.1.
Let the above assumptions on f hold and let g ∈ L q (Ω) . Then, problem (5.1) has a unique solution ( w, p ) ∈ F q where F q := { ( w, p ) , w ∈ W ,q (Ω) ∩ L r +1 (Ω) , p ∈ W ,q (Ω) } and the following estimate holds: k w k W ,q + k w k rL r +1 + k p k L / ε + k w k r/ ( r +1) H ≤ C (1 + k g k L q ) , (5.2) for some positive C independent of g and sufficiently small ε = ε ( q ) > .Proof. We give below only the derivation of the estimate in the space F q (the existenceand uniqueness of the solution can be obtained in a standard way, e.g., using the Galerkinapproximation method). Indeed, multiplying equation (5.1) by w , integrating by partsand using (1.3), we arrive at k w k H + k w k r +1 L r +1 ≤ C (1 + k g k qL q ) , q := ( r + 1) ∗ = 1 + 1 r , (5.3) where C is independent of g and w . Together with conditions on f , this gives, in particular,that k f ( w ) k L q ≤ C (1 + k g k L q ) , (5.4)for some (new) constant C .Rewriting now the problem (5.1) as a linear Stokes problem − ∆ w + ∇ p = h w := g − f ( w ) (5.5)and applying the maximal L q -regularity estimate for this linear Stokes problem, we have k w k W ,q + k p k W ,q ≤ C (1 + k g k L q ) . (5.6)In particular, due to Sobolev embedding theorem W ,q (Ω) ⊂ L s (Ω) with s := q − q > / r > q < k p k L / ε ≤ C (1 + k g k L q ) , (5.7)where ε = ε ( r ) > r . (cid:3) We are now ready to state the main result of this section.
Theorem 5.2.
Let w be an energy solution of problem (5.1) , g ∈ L (Ω) and the assump-tions (1.3) on f hold. Then, w ∈ H (Ω) and the following estimate is valid: k w k H (Ω) + k p k H (Ω) ≤ Q ( k ( w, p ) k F q )(1 + k g k − κL ) (5.8) for some monotone function Q and positive κ = κ ( r ) .Proof. As before, we restrict ourselves to the formal derivation of the regularity estimate(5.8). The existence of a solution can be verified in a standard way using, e.g., theLeray-Schauder fixed point theoremWe will use the so-called nonlinear localization method and split the derivation of thatestimate in several steps.
Step 1: Interior regularity.
At this stage, we obtain the interior H -regularity estimate for the solution w . Tothis end, we multiply equation (5.1) by P i ∂ x i ( φ∂ x i w ) where φ is a proper nonnegativecut-off function which vanishes near the boundary and equals one identically inside of thedomain. To be more precise, we assume that ϕ ∈ C ∞ ( R ) is such that 0 ≤ ϕ ( x ) ≤ ϕ ( x ) ≡ x ∈ R \ Ω and ϕ ( x ) ≡ x ∈ Ω ν whereΩ ν := { x ∈ Ω , dist( x, ∂ Ω) > ν } and ν > |∇ φ ( x ) | ≤ C ν,δ φ ( x ) − δ (5.9)for some δ > C ν,δ dependingonly on δ , ν and the shape of Ω. TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 13
Thus, we may estimate the term with the Laplacian as follows:(∆ w, X i ∂ x i ( ϕ∂ x i w )) = X i ( ∇ ∂ x i w, ∇ ( ϕ∂ x i w )) ≥ κ X i ( ϕ ∇ ∂ x i w, ∇ ∂ x i w ) −− ( ∇ ∂ x i w, ∇ ϕ · ∂ x i w ) ≥ κ ( ϕ, | D x w | ) − ( |∇ ϕ | ϕ − , |∇ w | ) ≥≥ κ k ϕ / w k H − C k w k H . (5.10)This, together with the energy estimate (5.3) for the subordinated terms, gives the fol-lowing estimate: k φ / w k H + ( φf ′ ( w ) ∇ w, ∇ w ) ≤ C (1 + k g k L ) + | ( p, ∂ x i ( ∇ φ · ∂ x i w ) | . (5.11)Using again the energy estimate to control the subordinated terms and (5.9) to control |∇ ϕ | , the last term can be estimated as follows: | ( p, ∂ x i ( ∇ φ · ∂ x i w ) | ≤ C (1 + k g k L ) + ε k φ / w k H + C ε k φ / − δ p k L , (5.12)where ε > α and − α ( α = 1 / − δ ) in thefollowing way: Z Ω ϕ α | u | dx = Z Ω ( ϕ | u | ) α | u | − α ) dx ≤ C k ϕu k αL k u k − α ) L − α )3 − α . (5.13)Since − α )3 − α = (1 + δ δ ) and 2 α = 1 − δ <
1, fixing δ > δ δ ≤ ε and using (5.7) for estimating the L / ε -norm of p , we arrive at k φ / − δ p k L ≤ C k φp k αL k p k − αL / ε ≤ Q ( k ( w, p ) k F q )(1 + k φp k L ) . (5.14)Thus, due to (5.12), k φ / w k H + ( φf ′ ( w ) ∇ w, ∇ w ) ≤ C (1 + k g k L ) + Q ( k ( w, p ) k F q )(1 + k φp k L ) . (5.15)In order to estimate the term in the right-hand side, we take the divergence from bothsides of (5.1) and write out ∆ p = − div f ( w ) + div g. (5.16)Multiplying this equation by φ , we have∆( φp ) = 2 ∇ φ · ∇ p + ∆ φp − φ div f ( w ) − φ div g := h. (5.17)Furthermore, due the growth assumptions (1.3) on f and the energy estimate (5.3), k φ div f ( w ) k qL q ≤ Z Ω φ q | f ′ ( w ) ∇ w | q dx ≤ Z Ω | φf ′ ( w ) ∇ w · ∇ w | q/ · | φf ′ ( w ) | q/ dx ≤≤ C ( φf ′ ( w ) ∇ w, ∇ w ) q/ k f ′ ( w ) k q/ L q − q = C ( φf ′ ( w ) ∇ w, ∇ w ) q/ k f ′ ( w ) k q/ L r +1 r − ≤≤ Q ( k ( w, p ) k F q )(1 + ( φf ′ ( w ) ∇ w, ∇ w )) q/ . (5.18)Using this estimate together with the energy estimate for the pressure, we arrive at k h k L q + H − ≤ β ( φf ′ ( w ) ∇ w, ∇ w ) + Q β ( k ( w, p ) k F q )(1 + k g k L ) , (5.19) where the positive constant β may be chosen arbitrarily small (the term H − appears dueto the term ϕ div g , other terms belong to L q ). Finally, due to the maximal regularity forthe Laplacian together with the Sobolev embedding W ,q ⊂ L s with s = q − q > H ⊂ L , we have k φp k W ,q + H + k φp k L ε ≤ β ( φf ′ ( w ) ∇ w, ∇ w ) + Q β ( k ( w, p k F q )(1 + k g k L ) , (5.20)for some small positive ε = ε ( r ) and β . Inserting this estimate into (5.15) and fixing β small enough, we conclude that k φ / w k H + ( φf ′ ( w ) ∇ w, ∇ w ) ≤ Q ( k ( w, p ) k F q )(1 + k g k L ) . (5.21)Using the embedding H ⊂ C and the fact that φ equals one identically inside of thedomain, we deduce the desired interior regularity k w k H (Ω ν ) + k p k H (Ω ν ) ≤ Q ν ( k ( w, p ) k F q )(1 + k g k L ) , (5.22)where ν > H -regularity estimate is proved. Step 2: Boundary regularity: tangent directions.
We now obtain the H -regularity in tangent directions near the boundary. The standardapproach here is to make the change of variables which straighten the boundary locallyin a small neighborhood of a boundary point x and then obtain the global estimateusing the proper partition of unity. However, in order to avoid the complicated notations,we will use the alternative equivalent approach working directly with the derivatives intangential directions. Namely, let τ = τ ( x ) and τ = τ ( x ) be two smooth vector fieldsin ¯Ω which are linear independent in a small neighborhood ¯Ω \ Ω ν and such that, for any x ∈ ∂ Ω ν , ν ∈ [0 , ν ] the vectors τ ( x ) and τ ( x ) generate the tangent plane to ∂ Ω ν .Being pedantic, such vector fields usually do not exist globally , but only locally (in aneighborhood of a fixed point x ∈ ∂ Ω). However, the plane-field spanned by the pairof vectors ( τ , τ ) is well-defined globally and the tangent gradient ∇ τ is also well-definedglobally. Nevertheless, in slight abuse of rigoricity and in order to avoid the completelystandard technicalities, we assume that the both vector fields τ and τ are globally definedin ¯Ω.Let z ( x ) := ∂ τ w ( x ) := P i =1 τ i ( x ) ∂ x i w (where τ = τ or τ = τ ). Then, this functionsolves − ∆ z + f ′ ( w ) z + ∇ ( ∂ τ p ) = h z ,h z := ∂ τ g + ∂ x i ( T i ( x ) p + K i ( x ) ∇ w ) + L ∇ w + M ( x ) p, div z = C ( x ) ∇ w, z (cid:12)(cid:12) ∂ Ω = 0 (5.23)for some smooth (matrix) functions T, K, L, M, C independent of p and w . Multiplyingthis equation by z and using the energy estimates (5.6) and (5.7) (in order to estimatethe subordinated terms) together with the facts that z (cid:12)(cid:12) ∂ Ω = 0 and div z = C ( x ) ∇ w , wededuce after the simple estimates that k z k H + ( f ′ ( w ) z, z ) ≤ Q ( k ( w, p ) k F q )(1 + k g k L ) + C k∇ w k L k ∂ τ p k L / + C k p k L . (5.24)Let us first estimate the L -norm of ∇ w . To this end, we need to use the interpolationin the spaces with different regularity in tangent ( τ and τ ) and normal ( n ) directions(here we need also the boundary ∂ Ω to be smooth enough). Indeed, since we control the W ,q -norm of w (due to estimate (5.6)), from the Sobolev’s trace theorem, we have the TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 15 control of the traces of ∇ w on the surfaces ∂ Ω ν in the W /r,q -norm (remind that for small ν , ∂ Ω ν are uniformly smooth if the initial ∂ Ω = ∂ Ω is smooth and the Sobolev tracetheorem as well as the interpolation theorems below work): k∇ w k C ( ν ∈ [0 ,ν ] ,W /r, /r ( ∂ Ω ν )) ≤ Q ( k ( w, p ) k F q ) . (5.25)On the other hand, using the estimate (5.24) for ∇ ∂ τ w , and the embedding H ⊂ L s forevery s < ∞ for 2D domains Ω ν , we conclude that k∇ w k L ( ν ∈ [0 ,ν ] ,L s ( ∂ Ω ν )) ≤ C s k z k H , (5.26)where the constant C s depends only on s .Using now the standard interpolation( L /r ( ∂ Ω ν ) , L s ( ∂ Ω ν )) / = L ( ∂ Ω ν )if =
13 11+1 /r +
23 1 s , i.e., with s r = 2(1 + r ), we see that k∇ w k L (Ω \ Ω ν ) = k∇ w k L ( ν ∈ [0 ,ν ] ,L ( ∂ Ω ν )) ≤≤ C k∇ w k / L ∞ ( ν ∈ [0 ,ν ] ,L /r ( ∂ Ω ν )) k∇ w k / L ( ν ∈ [0 ,ν ] ,L sr ( ∂ Ω ν )) ≤ Q ( k ( w, p ) k F q ) k z k / H . (5.27)Using also the interior regularity (5.22), we conclude that k∇ w k L (Ω) ≤ Q ( k ( w, p ) k F q )(1 + k z k / H (Ω) ) . (5.28)Let us now estimate the L / -norm of ∂ τ p . To this end, we rewrite equation (5.23) in theform ∆ z − ∇ ( ∂ τ p ) = h, z | ∂ Ω = 0 , div z = C ( x ) ∇ w (5.29)with h = f ′ ( w ) z + ∂ τ g + ∂ x i ( T i ( x ) p + K i ( x ) ∇ w ) + L ( x ) ∇ w + M ( x ) p (5.30)and note that, due to the energy estimates (5.6), (5.7) and similar to (5.18), k h k H − + L q ≤ Q ( k ( w, p ) k F q )(1 + ( f ′ ( w ) z, z ) / ) + C k g k L . (5.31)Applying the H − → H regularity and L q → W ,q regularity estimates for the linearnon-homogeneous Stokes equation, similar to (5.20), we conclude that k ∂ τ p k L + W ,q + k ∂ τ p k L / ε ≤ Q ( k ( w, p ) k F q )(1 + ( f ′ ( w ) z, z ) / ) + C k g k L , (5.32)where ε = ε ( r ) > L -norm of p . To this end, using the fact that we havethe control of the W ,q -norm of the pressure p and Sobolev trace theorems, similar to(5.25), we have k p k L ∞ ( ν ∈ [0 ,ν ] ,L ( ∂ Ω ν )) ≤ Q ( k ( w, p ) k F q ) . (5.33)On the other hand, due to the embedding W , / ⊂ L for the 2D domains Ω ν , k p k L / ( ν ∈ [0 ,ν ] ,L ( ∂ Ω ν )) ≤ C k ∂ τ p k L / (5.34) and, consequently, due to interpolation with exponent ( L / = ( L , L ) / = ( L ∞ , L / ) / )together with already proved interior regularity (and the fact that 9 / > k p k L (Ω) ≤ k p k L (Ω \ Ω ν ) + k p k L (Ω ν ) ≤≤ Q ( k ( w, p ) k F q )(1 + k g k L + k p k L / ( ν ∈ [0 ,ν ] ,L / ( ∂ Ω ν )) ) ≤≤ Q ( k ( w, p ) k F q )(1 + k g k L + k p k / L ∞ ( ν ∈ [0 ,ν ] ,L ( ∂ Ω ν )) k p k / L / ( ν ∈ [0 ,ν ] ,L ( ∂ Ω ν )) ) ≤≤ Q ( k ( w, p ) k F q )(1 + k g k L + k ∂ τ p k / L / ) ≤≤ Q ( k ( w, p ) k F q )(1 + k g k L + ( f ′ ( w ) z, z ) / ) . (5.35)Inserting now estimates (5.28),(5.32) and (5.35) into the right-hand side of ((5.24), wefinally arrive at k z k H + k∇ w k L + ( f ′ ( w ) z, z ) + k p k L ≤ Q ( k ( w, p ) k F q )(1 + k g k L ) (5.36)and the H -regularity of w in tangent directions is verified. Now, we note that k f ′ ( w ) z k H − ≤ C k f ′ ( w ) z k L / ≤ C ( | f ′ ( w ) | z, z )) / k f ′ ( w ) k / L / and, therefore, applying the H − -regularity theorem to the linear Stokes problem (5.29)and using (5.36), we arrive at k ∂ τ p k L ≤ Q ( k ( w, p ) k F q )(1 + k g k L )(1 + k f ′ ( w ) k / L / ) . (5.37) Step 3: Regularity in normal direction and the final estimate.
Let us nowmultiply equation (5.1) by ∆ w . Then, after integration by parts, we get k w k H + ( | f ′ ( w ) |∇ w, ∇ w ) ≤ C (1 + k g k ) + C | ( ∇ p, ∆ w ) | . (5.38)In addition, the second term in the right-hand side gives( f ′ ( w ) ∇ w, ∇ w ) ≥ κ ( | w | r − , |∇ w | ) ≥ κ k∇ ( | w | ( r +1) / ) k L ≥≥ κ k w k r +1 L r +1) ≥ κ k f ( w ) k r/ ( r +1) L − C and, consequently, using the energy estimate and the interpolation k f ( w ) k L ≤ C k f ( w ) k / L k f ( w ) k / L , we will have k w k H + k f ( w ) k r +1) / (3 r ) L ≤ C (1 + k g k ) + C | ( ∇ p, ∆ w ) | . (5.39)Thus, we only need to estimate the last term in the right-hand side of this inequality. Tothis end, we split the tangential and normal derivatives in that term and use (5.36) and TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 17 (5.37) for estimating the tangential derivatives: | ( ∇ p, ∆ w ) | ≤ | ( ∂ τ p, (∆ w ) τ ) | + | ( ∂ n p, (∆ w ) n ) | ≤ k w k H ++ C k ∂ τ p k L + | ( ∂ n p, ∂ n w n ) | + C ( |∇ p | , |∇ ∂ τ w | + |∇ w | ) ≤ k w k H ++ Q ( k ( w, p ) k F q )[1 + (1 + k g k L ) k f ( w ) k L / + k∇ p k L (1 + k g k L )]++ C k∇ p k L k ∂ n w n k L . (5.40)To estimate the last term in the right-hand side of (5.40), we use that w is divergent freeand, therefore, ∂ n w n + ∂ τ w τ + ∂ τ w τ = C ( x ) w. Differentiating that equation in the direction of the normal vector field, we obtain theestimate k ∂ n w n k L ≤ C k∇ ∂ τ w k L + C k∇ w k L . Inserting that estimate to the right-hand side of (5.40) and (5.39) and using (5.36) togetherwith the obvious estimate k∇ p k L ≤ C k f ( w ) k L + C k g k L (which follows from the L -maximal regularity for the linear Stokes equation) and theinterpolation k f ( w ) k L / ≤ C k f ( w ) k / L k f ( w ) k / L , we arrive at k w k H + k f ( w ) k r +1) / (3 r ) L ≤≤ Q ( k ( w, p ) k F q )[1 + (1 + k g k L ) k f ( w ) k / L + k f ( w ) k L (1 + k g k L )] . (5.41)Finally, thanks to Young inequality, we derive from (5.41) that k w k H ≤ Q ( k ( w, p ) k F q )(1 + k g k − κL ) , where κ = κ ( r ) > (cid:3) Remark 5.3.
Clearly, when the nonlinear terms has the growth | f ( u ) | ≤ C (1 + | u | ) , ∀ u ∈ R , the maximal regularity estimate (5.8) follows directly from the energy estimate (5.3) andthe regularity estimate for the linear Stokes problem. Note also that, in the case of periodicboundary conditions the simple multiplication of the initial problem by ∆ w gives betterestimate k w k H ≤ C (1 + k g k L ) . However, for the case of Dirichlet boundary conditions, the additional (uncontrollable)boundary terms appear under the integration by parts, and we unable to obtain the H -regularity estimate which is linear with respect to the L -norm of g . However, as we willsee below, the sub-quadratic growth rate of that estimate with respect to g is enough tobe able to apply it for the Navier-Stokes-type problem. To be more precise, we want to apply the above result to the following analogue ofproblem (5.1) perturbed by the Navier-Stokes inertial term: ( − ∆ w + ( w, ∇ ) w + f ( w ) + ∇ p = g, div w = 0 , x ∈ Ω ,w = 0 , x ∈ ∂ Ω , R Ω p ( x ) dx = 0 . (5.42)Indeed, since the inertial term vanishes after the multiplication the equation by w andintegrating over x , arguing as in Lemma 5.1, we have: k w k H + k w k r +1 L r +1 ≤ C (1 + k g k qL q ) . In addition, k ( w, ∇ ) w k L / ≤ C k w k L k∇ w k L ≤ Q ( k g k L ) , (5.43)for some monotone function Q . Thus, the L q -norm of the inertial term is under thecontrol if q ≤ / r ≥
2) and, applying the L q -regularity estimate for the linear Stokesproblem, we also have the control of the W ,q -norm of p . So, we have proved that k ( w, p ) k F q ≤ Q ( k g k L q ) (5.44)if r ≥
2. The next Corollary gives the H -regularity estimate for the problem(5.42). Corollary 5.4.
Let f satisfy (1.3) with r ≥ and g ∈ L . Then, any energy solution ( w, p ) ∈ F q of problem (5.42) belongs to H × H and the following estimate holds: k w k H + k p k H ≤ Q ( k g k L ) (5.45) for some monotone function Q .Proof. Let us first formally deduce a priori estimate (5.45). To this end, we interpret theinertial term as an external force and apply estimate (5.8). Then, using (5.44), we have k w k H ≤ Q ( k g k L )(1 + k ( w, ∇ ) w k − κL ) (5.46)for some monotone Q and some positive κ = κ ( r ). Thus, we only need to estimate the L -norm of the inertial term. To this end, we use (5.43) together with the interpolationinequalities and the fact that H ⊂ W , : k ( w, ∇ ) w k L ≤ k ( w, ∇ ) w k / L / k ( w, ∇ ) w k / L ≤≤ Q ( k g k L ) k w k / L ∞ k w k / H ≤ Q ( k g k L ) k w k / H k w k / H ≤ Q ( k g k L ) k w k / H . (5.47)Inserting this estimate in the right-hand side of (5.46), we deduce the desired a prioriestimate (5.45).The existence of a solution ( w, p ) ∈ H × H can be obtained in a standard way basedon that estimate and approximating, for instance, the growing non-linearity f ( w ) by asequence f n ( w ) of globally bounded ones (see, e.g., [9] for the details). However, since thesolution of (5.42) may be not unique, we still need to verify that any energy solution ofthat equation satisfies estimate (5.45).Indeed, let w be an energy solution of (5.42). Let us consider the following modifiedequation (5.42): ( − ∆ v + ( v, ∇ ) v + f ( v ) + Rv + ∇ p = g w , div v = 0 , x ∈ Ω ,v = 0 , x ∈ ∂ Ω , R Ω p ( x ) dx = 0 . (5.48) TTRACTOR FOR THE BRINKMAN-FORCHHEIMER EQUATIONS 19 with g w := g + Rw . We claim that the solution v = w of that equation is unique in theclass of energy solutions if R ≫ v, p ) ∈ H × H of that equation satisfying (5.45). The uniquenessguarantees then that the initial solution u is also regular and satisfies this estimate. Thus,the corollary is proved. (cid:3) Remark 5.5.
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