Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations

The dynamics of three dimensional Kelvin-Voigt-Brinkman- Forchheimer equations in bounded domains is considered in this work. The existence and uniqueness of strong solution to the system is obtained by exploiting the \begin{document}$ m $\end{document} -accretive quantization of the linear and nonlinear operators. The long-term behavior of solutions of such systems is also examined in this work. We first establish the existence of an absorbing ball in appropriate spaces for the semigroup associated with the solutions of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Then, we prove that the semigroup is asymptotically compact, which implies the existence of a global attractor for the system. Next, we show the differentiability of the semigroup with respect to the initial data and then establish that the global attractor has finite Hausdorff and fractal dimensions. Furthermore, we establish the existence of an exponential attractor and discuss about its fractal dimensions for the associated semigroup of such systems. Finally, we discuss about the inviscid limit of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations to the 3D Navier-Stokes-Voigt system and then to the simplified Bardina model.

2.2. Linear operator. Let P H : L 2 (Ω) → H denotes the Helmholtz-Hodge orthogonal projection (see [25,10]  It can be easily seen that the operator A is a non-negative self-adjoint operator in H with V = D(A 1/2 ) and Au, u = u 2 V , for all u ∈ V, so that Au V ≤ u V .
For a bounded domain Ω, the operator A is invertible and its inverse A −1 is bounded, self-adjoint and compact in H. Thus, the spectrum of A consists of an infinite sequence 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ k ≤ . . . , with λ k → ∞ as k → ∞ of eigenvalues. The behavior of these eigenvalues is well known in the literature (for example see Theorem 2.2, Corollary 2.2, [20] and for asymptotic behavior, see [13,34,21], etc). For all k ≥ 1 and n ∈ N, we have λ k ≥ Ck 2/n , where C = n 2 + n (2π) n ω n (n − 1)|Ω| 2/n , ω n = π n/2 Γ(1 + n/2), (3) and |Ω| is the n-dimensional Lebesgue measure of Ω. For n = 3, we find C = eigenvectors of A such that Ae k = λ k e k , for all k ∈ N. We know that u can be expressed as u = ∞ k=1 (u, e k )e k and Au = ∞ k=1 λ k (u, e k )e k . Thus, it is immediate that In this paper, we also need to define the fractional powers of A. For u ∈ H and α > 0, we define A α u = It can be easily seen that D(A 0 ) = H, D(A 1/2 ) = V. We set V α = D(A α/2 ) with u Vα = A α/2 u H . Using Rellich-Kondrachov compactness embedding theorem, we know that for any 0 ≤ s 1 < s 2 , the embedding D(A s2 ) ⊂ D(A s1 ) is also compact. Applying Hölder's inequality on the expression (5), one can obtain the following interpolation estimate: for any real s 1 ≤ s ≤ s 2 and θ is given by s = s 1 θ + s 2 (1 − θ). Let us denote by D(A −α ), the dual space of D(A α ) and we have the following dense and continuous inclusions, for α > 1/2, For negative powers, we also define (u, v) V−α = A −α/2 u, A −α/2 v and u V−α = A −α/2 u H .
Remark 1. 1. Note that V s is an algebra for s > 3/2, i.e., if u, v ∈ V s , then uv ∈ V s and uv Vs ≤ u Vs v Vs . 2. The space V s can be continuously embedded in L ∞ (Ω), for s > 3/2, i.e., u L ∞ ≤ C u Vs , for u ∈ V s , s > 3/2.
3. One can show that u V = A 1/2 u H = ∇u H and (I + µA) 1/2 u H are equivalent norms in V. It can be easily seen that using the Poincaré inequality. From (7), it is also clear that We combine (7) and (8) to obtain the required result.
If u, v are such that the linear map b(u, v, ·) is continuous on V, the corresponding element of V is denoted by B(u, v). We also denote (with an abuse of notation) B(u) = B(u, u) = P H (u · ∇)u. An integration by parts gives In the trilinear form, an application of Hölder's inequality yields and thus B(u, v) V ≤ u L 4 v L 4 . Hence, the trilinear map b : V × V × V → R has a unique extension to a bounded trilinear map from L 4 (Ω) × (L 4 (Ω) ∩ H) × V to R. It can also be seen that B maps L 4 (Ω) ∩ H (and so V) into V and for all v ∈ V, so that using the Poincaré inequality (see (4)). Note that B also maps L 6 (Ω) ∩ H into V and | B(u, u), v | = |b(u, v, u)| ≤ u L 3 u L 6 ∇v H ≤ C u 2 L 6 v V , so that once again, we get (11). Using (11), we also have and hence B(·) is a locally Lipschitz operator.

Abstract formulation and weak solution.
We take the Helmholtz-Hodge orthogonal projection P H in (1) to obtain the abstract formulation for t ∈ (0, T ) as: where f ∈ L 2 (0, T ; V ). Strictly speaking one should use P H f instead of f , for simplicity, we use f . The system (13) is also equivalent to the following system for t ∈ (0, T ): Let us now give the definition of weak solution of the system (13).

MANIL T. MOHAN
, is called a weak solution to the system (13), if for f ∈ L 2 (0, T ; V ), u 0 ∈ V and v ∈ V, u(·) satisfies: and the energy equality: Let us now state a result on existence and uniqueness of weak solution to the system (13). A proof using the Faedo-Galerkin method can be obtained from Theorem 3.1, [2]. Theorem 2.2 (Theorem 3.1, [2]). There exists a unique weak solution u(·) to the system (13) in the sense of Definition 2.1.
Let us now provide the definition of accretive operators and obtain the global solvability results of the system (13) using quasi-m-accretive property of linear and nonlinear operators. [4]). Let X be a real Hilbert space with inner product (·, ·) X . An operator F : , for some C > 0 and all x, y ∈ D, (iii) maximal accretive if there is no accretive operator that properly contains it, i.e., if for x ∈ X and w ∈ X, the inequality (w − F(x), x − y) X ≥ 0, for all y ∈ X implies w = F(x) (F is dissipative if −F is accretive), (iv) m-accretive (or hypermaximal accretive) if it is accretive and R(I + F) = X or equivalently R(I + λF) = X, for all λ > 0, (v) ω-accretive (respectively ω-m-accretive), where ω ∈ R, if F + ωI is accretive (respectively m-accretive), (vi) quasi-accretive (respectively quasi-m-accretive) if F is ω-accretive (respectively ω-m-accretive), for some ω ∈ R, (vii) demicontinuous if for all x ∈ D, the functional x → (F(x), y) X is continuous, or in other words, It should be noted that in Hilbert spaces, 'accretive' is also known as 'monotone'. It was established in [27] that in Hilbert spaces, the notions of 'maximal accretive' and 'm-accretive' are equivalent. Note also that every demicontinuous monotone operator with dense domain is maximal accretive (see [7]). Interested readers are referred to see [4,5], etc for more details on accretive operators.
Let us now present a proof of the existence of a strong solution to the system (13) by exploiting the m-accretive quantization of the linear and nonlinear operators. This method also avoids the tedious Galerkin approximation scheme used in Theorem 3.1, [2]. We apply the similar techniques used in [6] for the Navier-Stokes equations as well as in [29] for the Kelvin-Voigt fluid flow equations to obtain the well-posedness of the system (13).
Proof. We make use of the m-accretive quantization of the linear and nonlinear operators to get the required result. Since, the norms u V and (I + µA) 1/2 u H are equivalent (see Remark 1), we can define an another inner product on V as Let us first define the quantized nonlinearity B N (·) : V → V as We also define the following quantized nonlinearity B N (·) : V → V: We obtain the required result in the following steps.
Step (1). Γ N defines an operator on V, that is, Γ N : V → V. It can be easily seen that Next, we show that B N (·) : V → V. For u V ≤ N and all v ∈ V, using Hölder's, Poincaré and Ladyzhenskaya inequalities, we get Since v ∈ V is arbitrary, we obtain B N (u) V ≤ 2 A similar calculation holds for u > N also. Moreover, using Hölder's and Gagliardo-Nirenberg inequalities (see (A.6)), we get for all u, v ∈ V and r ∈ [1,5]. Thus, the operator β(I + µA) −1 P H (u|u| r−1 ) : V → V, for r ∈ [1,5]. Furthermore, comparing second and final inequalities, we have Step (2). Γ N + λI is strongly accretive, for some λ > 0. First note that Let us now consider the operator B N (·). Without loss of generality, we assume that v V ≥ u V . Let us take u V , v V ≤ N . Then using (9), (10), Ladyzhenskaya and Young's inequalities, we estimate ( Next, we consider the case, u V , u V > N . Using Hölder's, Ladyzhenskaya, Poincaré and Young's inequalities, we estimate ( Combining all the above cases, we have for some C N > 0. Since the function f (x) = x|x| r−1 is differentiable with derivative f (x) = r|x| r−1 , one can use Taylor's formula to obtain for some 0 < θ < 1. Finally, for any r ∈ [1, ∞), using (23), we have for all u, v ∈ L r (Ω) ∩ V. Combining (21)-(24), we infer that Γ N satisfies and λ ≥ C N , and hence Γ N + λI is an accretive operator.
Step (3). Γ N is demicontinuous. Let {u n } be a sequence in V such that u n → u in V. In order to show Γ N is demicontinuous, we need to prove that (((Γ N +λI)u n − (Γ N + λI)u, v)) V → 0 as n → ∞. Let us consider where we used (7). Let us now consider ((B N (u n ) − B N (u), v)) V and estimate it using the definition of B N (·) by considering the following different cases.
Step (5). Global strong solution to the system (13). Finally, we show that the strong solution we obtained is global. Let us take inner product with u N (·) to the 3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 3405 first equation in (38) to obtain But, we know that Remark 2. Note that we have not used compactness arguments in the proof of Theorem 2.4. Thus, the well-posedness for the system (13) is valid even in unbounded domains like Poincaré domains.
Let us now discuss about the regularity of the weak solution to the system (13).
Remark 3. For f ∈ L 2 (0, T ; H) and u 0 ∈ D(A), we get further regularity of the weak solution of (13) as u ∈ C([0, T ]; D(A)). Let us take inner product with Au(·) to the first equation in (13) to obtain We use Hölder's, Agmon's (see (A.15)) and Young's inequalities to estimate the term |(B(u), Au)| as Using Cauchy-Schwarz and Young's inequalities, we estimate |(f , Au)| as

MANIL T. MOHAN
For r ∈ (3, 5), we use Hölder's, Gagliardo-Nirenberg (see (A.4)), Agmon's (see (A.15)) and Young's inequalities to estimate β|(P H (|u| r−1 u), Au)| as For r = 5, we estimate β|(P H (|u| 4 u), Au)| using Hölder's, Ladyzhenskaya and Agmon's inequalities as For r ∈ [1, 3], let us combine (42)-(44) and use it in (41) to find Integrate the inequality from 0 to t to obtain Since u ∈ C([0, T ]; V), u 0 ∈ D(A) and f ∈ L 2 (0, T ; H), one can easily see that the right hand side of the inequality (48) is finite and hence we easily obtain u ∈ C([0, T ]; D(A)), since u ∈ W 1,∞ (0, T ; D(A)). Similarly, for r ∈ [3, 5), we get and the result follows. Finally, for r = 5, combining (42), (43) and (46), substituting it in (41) and then integrating from 0 to t, we obtain An application for Gronwall's inequality in (50) yields For f (·) independent of t, using Theorem 2.2, we know that there exists a unique weak solution to the system (13) and the solution can be represented through a one parameter family of continuous semigroup. Thanks to Theorem 2.2, we can define a continuous semigroup where u(·) is the unique weak solution of the system (13) 3. Global attractor. This section is devoted for proving the existence of a global attractor for the semigroup S(t), t ≥ 0, for autonomous version of the 3D Kelvin-Voigt-Brinkman-Forchheimer system given in (13). We need the following notation also for the rest of the sections. For µ ≥ 0 and α ∈ R, we define the scale of Hilbert spaces, V µ α = D(A α/2 ) endowed with the scalar product so that both norms are equivalent on V α (see Remark 1 also). For r ∈ [1,4], our first aim is to establish the existence of an absorbing ball in V µ 1 for S(t), t ≥ 0 and then show that it is an asymptotically compact semigroup in V µ 1 . Using Theorem 2.3.5, [11], we show the existence of a global attractor S(t), t ≥ 0 in V µ 1 . Moreover, we establish that the semigroup S(t) defined on V µ 2 has an absorbing ball in V µ 2 . Let us prove the following lemma on Lipschitz continuity of the semigroup S(t).
Taking inner product with w(t) in the first equation in (54), we find where we used the fact that b(u, w, w) = 0. From (24), we know that |u| r−1 u − |v| r−1 v, w ≥ 0. Using Hölder's, Ladyzhenskaya and Poincaré inequalities, we estimate |b(w, u, w)| as Using (56) in (55) and then integrating the from 0 to t, we get An application of Gronwall's inequality in (57) yields Thus, we have and hence the map S(t) : 3.1. Absorbing ball in V µ 1 . Let u(t), t ≥ 0 be the unique weak solution of the system (13) with f (t) = f ∈ V . We show that S(t) has an absorbing ball in V µ 1 . Proposition 1. The set is a bounded absorbing set in V µ 1 for the semigroup S(t). That is, given a bounded Proof. Let us take inner product with u(·) to the first equation in (13) to obtain Using Cauchy-Schwarz and Young's inequalities, we estimate f , u(t) as Thus using (53) in (63), we get An application of Gronwall's inequality yields Taking limit as t → ∞, we find Integrating the inequality (63) from 0 to t, we also obtain The inequality (67) implies that the semigroup S(t) : V µ 1 → V µ 1 , t ≥ 0 associated with the weak solution to the problem (13) has an absorbing ball given in (60). Hence, the following uniform estimate is valid: for t large enough (t 1) depending on the initial data.

Asymptotic compactness.
Let us now show that the semigroup S(t), t ≥ 0 defined in V µ 1 is asymptotically compact. The following proposition can be proved using a Faedo-Galerkin approximation.
has a unique weak solution z ∈ C([0, T ]; V). The weak solution also belongs to C([0, T ]; V s ) and the following estimate holds: Proof. Using a Faedo-Galerkin approximation, the existence of a weak solution to the system (70) can be established. The uniqueness of weak solution follows from linearity. Let us now establish the regularity result. We take inner product with A s−1 z(·) in the first equation in (70) to obtain 1 2 Integrating the inequality (72) from 0 to t to get Taking supremum over t ∈ [0, T ], the estimate (71) follows.

12−2ε
). Thus the operator T(t) maps V to V 12−ε 12−2ε = V 1+ ε 12−2ε . Note that the embedding V 12−ε 12−2ε ⊂ V is compact and this implies that the operator T(t) is a compact operator for each t > 0. In fact, using equivalence of norms given in (53), one can also obtain that T(t) is a compact operator on V µ 1 . Combing this result with (79), we infer that the semigroup S(t) satisfies the conditions of the Theorem 3.3, [26] (see also Theorem 2.1, [22]), and hence S(t), t ≥ 0 is an asymptotically compact semigroup on V µ 1 , which completes the proof. Theorem 3.2. Let r ∈ [1, 6( 5−ε 6−ε )], for some ε > 0 and f ∈ H. Then the semigroup Moreover, the attractor A µ 1 is compact, connected and invariant.
Proof. An application of the Theorem 2.3.5, [11] gives the required result.
Our next aim is to establish that the global attractor A µ 1 is a bounded subset of V µ 2 . Due to technical reasons, we are able to establish the boundedness result for r ∈ [1,4] only, (i.e., we take ε = 3 in (80) and (81)).

Proposition 4.
For r ∈ [1,4], the global attractor A µ 1 is a bounded subset of V µ 2 . Proof. Remember that u(t) = y(t) + z(t) ∈ A µ 1 . Let us take inner product with A 1 2 z(·) to the first equation in (76) to obtain 1 2 The first term from the right hand side of the equality (82) can be estimated using Cauchy-Schwarz and Young's inequalities as Similarly, using (81), (for ε = 3) we estimate the term |(P H (|u| r−1 u), Combining (83)-(85) and then substituting it in (80), we find For large t, using (53) and (69) in (86), we further get , and hence we have and thanks to inequality (79), we have The inequality (87) reveals us that the sequence {z(t k )} belongs to a ball in V µ . Using (87) and the lowersemicontinuity property of the V µ 3 2 norm, that is, v V µ easily gives us that A µ 1 is bounded in V µ 3 2 .

3D KELVIN-VOIGT-BRINKMAN-FORCHHEIMER EQUATIONS 3413
If we take inner product with A 2 3 z(·) to the first equation in (76), and use similar arguments as above, one can show that A µ 1 is bounded in V µ 5 3 . The estimate (84) needs to be replaced as One can estimate A for r ∈ [ 11 9 , 4]. Substituting (94) in (93), we find For r ∈ [1, 11 9 ), we use the embedding of L 2 ⊂ L 18r 11 and Poincaré inequality to obtain u r L 18r 11 u r V , and the result follows. Taking inner product with Az(·) to the first equation in (76), and following similarly as above, one can show that A µ 1 is bounded in V µ 2 also. In this case, one has to replace (84) with where we used the fact that V 5 3 is continuously embedded in L ∞ (Ω) (see Remark 1). Similarly, we estimate |(|u| r−1 u, Az)| as and this completes the proof.

Absorbing ball in
Let us now establish that the semigroup S(t) : V µ 2 → V µ 2 has an absorbing ball in V µ 2 . Proposition 5. For r ∈ [1, 3], the set is a bounded absorbing set in V µ 2 for the semigroup S(t). That is, given a bounded set B ⊂ V µ 2 , there exists an entering time t B > 0 such that For r ∈ (3, 4], the set where C r = C 5−r 4 r−1 ν r−1 5−r is a bounded absorbing set in V µ 2 for the semigroup S(t).
Proof. For r ∈ [1, 3], from (47), we have For large enough t, using (69) in (100), we deduce that An application of Gronwall's inequality in (100) yields That is, we have Hence, the inequality (102) assures the existence of an absorbing ball (98), so that for all t 1, we have u(t) V µ 2 ≤ M 2 . Similarly, for r ∈ (3, 5), we obtain so that B 2 given in (99) is an absorbing ball in V µ 2 for the semigroup S(t).

4.
Estimates of dimensions of the global attractor. The differentiability of the semigroup with respect to the initial data is established in this section. Then, we show that the global attractor for the 3D Kelvin-Voigt-Brinkman-Forchheimer system has finite Hausdorff and fractal dimensions. We apply similar techniques used in [22,37,30], etc to get required results. Let u(·) be the unique weak solution to the autonomous version of the system (13) belonging to the global attractor A µ 1 . Then, from the estimates (66)-(69), we have and lim sup Theorem 4.1. Let u 0 and v 0 be two members of V µ 1 . Then there exists a constant where the linear operator Λ(t) : V µ 1 → V µ 1 , for t > 0 is the solution operator of the problem: +βrP H (|u(t)| r−1 ξ(t)) = 0, t ∈ (0, T ), ξ 0 = u 0 − v 0 and u(t) = S(t)u 0 . That is, for every t > 0, the solution S(t)u 0 , as a map S(t) : V µ 1 → V µ 1 is Fréchet differentiable with respect to the initial data, and its Fréchet derivative D u0 (S(t)u 0 )w 0 = Λ(t)w 0 .
Let us now set G 2 = (I + µA), u = Gu, v = Gv. Thus, we rewrite the system (13) as where Gu 0 ∈ H. Note that the systems (132) and (13)  Since u(t) = Gu(t) and u 0 = Gu 0 , the semigroup S(t) is connected to the original semigroup S(t) through the relation Thus, it is immediate that the semigroup S(t) possesses the global attractor A µ 1 , where A µ 1 = GA µ 1 , and A µ 1 is the global attractor for S(t). Our first aim is to show a bound for the fractal dimension of A µ 1 in H. Furthermore, the same bound for fractal dimension of A µ 1 in V µ 1 follows easily using the following argument. From Proposition 3.1, Chapter VI, [35], we know that the fractal dimension estimates are preserved under Lipschitz maps. Moreover, form Remark 3.14, [30], we infer that Let us first consider the linear variations of the system (132). The equation of linear variations corresponding to (132) has of the form where The adjoint L * (t, u) of L(t, u) is given by Hence, L(t, u)w(t) = L(t, u)w(t) + L * (t, u)w(t) can be computed as Next, we derive the following important result.
Proposition 6. Let w ∈ H. Then, we have where Proof. Let us take inner product with w(·) to the first equation in (136) to find Remember that Using Hölder's, Ladyzhenskaya and Young's inequalities, we estimate the term Let us use (103) and (53) to estimate u 4 V as
Proposition 7. The global attractor A µ 1 has the finite fractal dimension in H, with Proof. Let w 1,0 , . . . , w n,0 , for some n ≥ 1, be an initial orthogonal set of infinitesimal displacements. The volume of the parallelopiped spanned by w 1,0 , . . . , w n,0 , is given by where ∧ denotes the exterior product. The evolution of such displacements satisfies the following evolution equation: for all i = 1, . . . , n. Using Lemma 3.5, [12] (see [13] also), we know that the volume elements where P n (s) is the orthogonal projection onto the linear span of {w 1 (t), . . . , w n (t)} in H. We also know that Tr(P n (s)L(s, u)) = From (146), we have for all t ≥ 0, where the supremum over P n (0) is a supremum over all choices of initial n orthogonal set of infinitesimal displacements that have taken around u. Let us now show that the volume elements V n (t) decays exponentially in time whenever n ≥ N , with N > 0 to be determined later. Let us use Proposition 6 and Lemma 6.2, Chapter VI, [35] to estimate 1 T T 0 Tr(P n (t)L( u, t))dt as 1 T T 0 Tr(P n (t)L( u, t))dt where C is defined in (3). One can estimate 1 T T 0 h 1 (t)dt using (138), (103) and (104) as Substituting (138) and (149) in (148), we find We need the right hand side of the above inequality must be negative, therefore, we require n ≥ N := 2 where C is defined in (3). Applying the definition of M 1 and K 1 given in (103) and (104), we obtain which completes the proof.
Since A µ 1 has finite fractal dimension in H with the bound (144), using (134), one can easily prove the following Theorem.
Theorem 4.2. The global attractor A µ 1 obtained in Theorem 3.2 has finite Hausdorff and fractal dimensions, which can be estimated by Remark 4. In terms of the three-dimensional Grashof number the fractal dimension bound given in (153) can be written as where C is defined in (3).

Exponential attractors.
In this section, we establish the existence of an exponential attractor in V µ 1 for the semigroup S(t) associated with the Kelvin-Voigt-Brinkman-Forchheimer equations. The results obtained in this section are true for r = 1, 2 and r ∈ [3,4] only, due to technical reasons. Similar result for 3D Navier-Stokes-Voigt models is obtained in [37] and for 3D Kelvin-Voigt fluid flow equations with "fading memory" is established in [30]. Let us first provide the definition of exponential attractors.
Definition 5.1 (Definition 3.1, [17,14]). A set M µ is an exponential attractor for S(t) in V µ 1 if (i) it is compact in V µ 1 , (ii) it is positive invariant, i.e., S(t)M µ ⊂ M µ , for all t ≥ 0, (iii) it has finite fractal dimension, (iv) it attracts exponentially fast the bounded sets of initial data, i.e., there exists a monotone function Q and a constant α > 0 such that for all t ≥ 0. We use Proposition 1, [16], Theorem 3.2, [17] (see Theorem A1, [38] also) to obtain the existence of an exponential attractor of our system. From the estimate (66) (see Proposition 1), it is immediate that where B 2 is defined in (99). Let us first define D := B 1 ∩ B 2 . Now, we take t e > 0 as the entering time of D in the absorbing set B 1 . Moreover, we define (158) We prove that K µ is invariant and compact. In order to establish that D is invariant, we use semigroup property to obtain Our next aim is to show that K µ is compact set in V µ 1 and bounded set in V µ 2 . In order to do this, it is enough to show the boundedness of K µ in V µ 2 as V µ 2 is compactly embedded in V µ 1 and K µ is a closed set in V µ 1 . Let us first take w ∈ K µ . Then, there exists a sequence t n ≥ t e and w n ∈ S(t)D, such that w n → w strongly in V µ 1 . Thanks to the estimate (156) and hence we have w n V µ 2 ≤ M 2 . Using the weak compactness and uniqueness of limits, along with the Banach-Alaoglu theorem, we can extract a subsequence of {w n } (still denoted by {w n }) such that w n w − → w weakly in V µ 2 . Since the norm is weakly lower semicontinuous, we finally arrive at and hence the boundedness in V µ 2 follows. The exponentially attracting property of K µ can be established similarly as in Lemma 4.6, [37].
Let us now establish our main goal of this section, that is, to show that the semigroup S(t) on V µ 1 associated with the system (13) admits an exponential attractor M µ contained and bounded in V µ 2 . Our major idea behind this is to use Proposition 1, [16], Theorem 3.2, [17] (or Theorem A1, [38]). As discussed in the previous sections, we start by decomposing the solution semigroup S(t) in the following way. For u 0 ∈ K µ , let us split the solution of (13) as (see section 3 also) where v(t) = R(t)u 0 and w(t) = T(t)u 0 , solves the following systems for t ∈ (0, T ): Using Cauchy-Schwarz and Young's inequalities, we estimate f ,u as Once again using Cauchy-Schwarz and Young's inequalities, we estimate Au,u as Using Hölder's, Ladyzhenskaya, Young's and Poincaré inequalities, we estimate B(u),u as Finally, we estimate β P H (|u| r−1 u),u = β |u| r−1 u,u using (A.6), (A.2), Hölder's and Young's inequalities as Combining (181)-(184) and using it in (180), we find Since u 0 ∈ K µ , using (155) in (185), we also infer that which implies the result (179).
Our next aim is to establish that the map S(t) is Lipschitz continuous.
Lemma 5.7. Let T > 0 be arbitrary and fixed. Then the map is Lipschitz continuous.
We are now ready to prove the existence of an exponential attractor for the nonautonomous version of the system (13). Let us apply Proposition 1, [16], Theorem 3.2, [17] (see Theorem A1, [38] also) to obtain such an exponential attractor.
Theorem 5.8. For r = 1, 2 and r ∈ [3, 4], the dynamical system S(t) on V µ 1 associated with the system (13) admits an exponential attractor M µ contained and bounded in V µ 2 in the sense of Definition 5.1. Proof. We have already shown that K µ is compact and invariant. From Lemma 5.7, we know that the operator S(t) is Lipschitz continuous for any t * ∈ [0, T ]. Let us fix t * according to the estimate given in Lemma 5.3 (see for example (163)), i.e., we fix Thus, we have for all t ≥ 0. From Lemma 5.5, we infer that where C 0 = e 2µ 3/2 λ 1/4 . Thus, one can apply Theorem A1, [38] with L = R(t * ) and K = T(t * ) to get an exponential attractor for the system (13).
Let us now discuss a result on the fractal dimension of exponential attractors associated with the Kelvin-Voigt-Brinkman-Forchheimer model (13).
Theorem 5.9. For r = 1, 2 and r ∈ [3,4], the semigroup S(t) admits an exponential attractor M µ whose fractal dimension satisfies the estimate: where A µ 1 is the global attractor for S(t). Proof. From Theorem 4.1 (see (126), (127) and (130)), we know that S(t) is differentiable. Moreover, from the Lemma 5.7, we have (see (188) and (189)) where Since a solution to the system (162) is unique and T(t) is the semigroup associated with the system (162), using uniqueness, one can obtain that the whole sequence {T(t)w n } n∈N converges strongly to T(t)w in V µ 1 . Hence, the operator T(t) is completely continuous for sufficiently large t. Using Remark 4.6.2, [19], it is immediate that S(t) is an αcontraction. Using Theorem 2.2, [15], one can assure the existence of an exponential attractor M µ for S(t) satisfying (192).
(208) Remark 6. One can also study the convergence (as β → 0) of global and exponential attractors for the autonomous version of the Kelvin-Voigt-Brinkman-Forchheimer equations (see (13)) towards global and exponential attractors for the autonomous version of the Navier-Stokes-Voigt equations (see (195)) obtained in [22,37].
Let Ω ⊂ R n , u ∈ W m,p (Ω; R n ), p ≥ 1 and fix 1 ≤ p, q ≤ ∞ and a natural number m. Suppose also that a real number η and a natural number j are such that and j m ≤ η ≤ 1. Then for any u ∈ W m,p (Ω; R n ), we have where s > 0 is arbitrary and the constant C depends upon the domain Ω, m, n,.
If 1 < p < ∞ and m − j − n p is a non-negative integer, then it is necessary to assume also that η = 1. Note also that (A.8) can be written as ∇ j u L r ≤ C u η W m,p u 1−η L q , (A.9) for any u ∈ W m,p (Ω; R n ).