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Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping

In this paper we obtain the existence of global attractors for the dynamicalsystems generated by weak solution of the three-dimensional Navier-Stokesequations with damping. We consider two cases, depending on the values of the parameter β controlling the damping term. First, we prove that for β≥4 weaksolutions are unique and establish the existence of the global attractor forthe corresponding semigroup. Second, for 3≤β


1.
Introduction. The three-dimensional Navier-Stokes equations with damping have been studied intensively over the last years. They describe the situation where there exists resistance to the motion of a flow. One outstanding model in which a damping term appears comes from the flow of cerebrospinal fluid inside the porous brain tissues [12]. Such dissipative damping is also common in many different models. For example, compressible Euler equations with damping describe the flow of a compressible gas through a porous medium [8], whereas Saint-Venant equations are used in oceanography to describe the flow of viscous shallow water with friction [1].
In this paper we study the asymptotic behaviour of weak solutions of the following equation where Ω ⊂ R 3 , β ≥ 1, µ, α > 0, µ > 0 is the kinematic viscosity and u is the velocity vector of an incompressible fluid satisfying Dirichlet boundary conditions.
We would like to point out that the damping term is very helpful from the mathematical point of view, as it allows us to obtain solutions more regular than in the standard Navier-Stokes equations without damping (that is, when α = 0). For this reason it is possible to prove the existence of global attractors for (1), at least in a given range of the parameter β, whereas to date this problem remains open for the standard three-dimensional Navier-Stokes equations.
Existence of weak solutions for problem (1) with initial condition in the space of free-divergence square integrable functions was established at first in [2, Theorem 1] for β ≥ 1 and Ω = R 3 and in [15,Theorem 2.1] for bounded domains Ω. Also, in [2] and [15] existence of global strong solutions of (1) with more regular initial conditions for β ≥ 7/2 and uniqueness for 7/2 ≤ β ≤ 5 were proved. Existence and uniqueness of global strong solutions were extended for 3 < β < 7/2 in [23] and [17], for β = 3, α = µ = 1, Ω = R 3 in [25] and for β = 3, αµ > 1 4 , Ω bounded in [9]. On top of that, in [25] uniqueness of strong solutions was proved to be true for all β ≥ 1 when Ω = R 3 . If either the initial condition u 0 is small enough or the viscosity µ is large, existence of global strong solutions was established in [24] for 1 ≤ β < 3, Ω = R 3 . Also, existence of T -periodic solutions was studied in [9], and regularity of weak solutions in [25], [22]. Therefore, when 3 < β ≤ 5 strong solutions of (1) generate a dynamical system. When the domain Ω is bounded in the autonomous case the existence of a global attractor for 7/2 ≤ β ≤ 5 was proved in [15,Theorem 2.1], and this result was extended to the nonautonomous case in [16], where the existence of a uniform attractor was established. In addition, in [17] existence of pullback attractors in the nonautonomous case was studied in the more general situation where 3 < β ≤ 5.
It is worth noting that so far there are no results in the literature either on uniqueness or existence of attractors for weak solutions. In Section 2 we prove first that weak solutions are unique if β ≥ 4. In this way, in Section 3 we are able to define a semigroup of operators and prove that a global attractor exists. Moreover, if β ≥ 3 we obtain that every weak solution is continuous, and using this property we are able to prove for 3 ≤ β < 4 that the multivalued semiflow generated by the weak solutions possesses a global attractor.
In the case of strong solutions it would be desirable to get rid of the restriction β ≤ 5, which was assumed in all the previous papers. In Section 4 we obtain a conditional result stating that if every strong solution is continuous with values in the space L β+1 , then the global attractor exists for β > 5 as well.
In Section 5 we present some numerical simulations to illustrate some aspects about the asymptotic behaviour of model (1) for a particular flow domain and initial conditions. Specifically, we analyze how the input parameters α and β affect the behaviour of the fluid flow evolution over a sufficiently long term.
2. Uniqueness and regularity of weak solutions. Consider a bounded open set Ω ⊂ R 3 with smooth boundary ∂Ω. We study the three-dimensional Navier-Stokes equations with damping where µ > 0 is the kinematic viscosity and f is an external force. Also, β ≥ 1 and α > 0 are given constants. The functions u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)), p(x, t) stand for the velocity field and the pressure, respectively. Here and further, |·| denotes in general the norm in R d for any d ≥ 1.
We define the usual function spaces where cl X denotes the closure in the space X. It is well known that H, V are separable Hilbert spaces and identifying H and its dual we have V ⊂ H ⊂ V with dense and continuous injections. We denote by (·, ·), · H and ((·, ·)), · V the inner product and norm in H and V , respectively, and by ·, · duality between V and V . Let H w be the space H endowed with the weak topology. As usual, we define the continuous trilinear form The norm in the spaces L p (Ω), (L p (Ω)) 3 , p ≥ 1, will be denoted indistinctly by · p .
Let P be the orthogonal projection from (L 2 (Ω)) 3 onto H and Au = −P ∆u be the Stokes operator, defined by Au, v = ((u, v)) for u, v ∈ V. Since the boundary ∂Ω is smooth, D(A) = H 2 (Ω) 3 ∩ V and Au 2 defines a norm in D(A) which is equivalent to the norm in H 2 (Ω) 3 .
is said to be a weak solution to problem (2) on (0, T ) if u (0) = u 0 and for any v ∈ V ∩ L β+1 (Ω) 3 , in the sense of scalar distributions.
We recall the following well-known result on existence of weak solutions.  (2).
, the dual space of V ∩ L β+1 (Ω) 3 . We note that by standard estimates on B for any weak solution we have that It follows from equality (3) and a standard result [18, p.250, Lemma 1.1] that in the sense of Y -valued distributions. Hence, the derivate du dt belongs to the space (Ω) 3 and equality (4) is satisfied in the space Y for a.a. t ∈ (0, T ) .
In order to obtain good estimates of weak solutions we need du dt to be more regular. We can obtain such a result for β ≥ 3.
Lemma 2.2. Let u be a weak solution to (2) Then for any u 0 ∈ H there exists a unique weak solution u (·) to problem (2), which is continuous with respect to the initial datum u 0 .
Proof. Let u 1 , u 2 be two weak solutions such that u 1 (0) = u 2 (0) = u 0 . Denote w = u 1 − u 2 . We multiply equality (4) by w for each solution u 1 , u 2 . Hence, Taking the difference of these expressions and using (7) we get For the nonlinear terms we have where we have used the properties of the operator b and the fact that the function By Hölder's inequality we have where 1 r + 1 ρ = 1 2 , r ≥ 5, so that 2 ≤ ρ ≤ 10 3 . Now the interpolation inequality and the embedding H 1 (Ω) ⊂ L 6 (Ω) gives Then, by Young's inequality we have Since u 1 ∈ L r (0, T ; L r (Ω)) and r ≥ 5 imply that M (t) = u 1 (t) 2r r−3 r ∈ L 1 (0, T ), by Gronwall's lemma we obtain that Hence, both uniqueness and continuity of the solution u (·) with respect to u 0 follow.
3. Global attractor for weak solutions. Our aim now is to prove the existence of the global attractor for the weak solutions of problem (2).
We shall divide this section into two cases: a) β ≥ 4; b) 3 ≤ β < 4. In the first one uniqueness of weak solutions implies that we can define a semigroup of operators, to which we can apply the classical theory of attractors for semigroups. In the second one more than one solution can possibly exist for a given initial datum, and then we need to make use of the theory of attractors for multivalued semiflows.
3.1. Case β ≥ 4. In view of Theorem 2.4 we can define the semigroup of operators S : where u (·) is the unique solution to problem (2) with initial condition u 0 . It is straightforward to see that S satisfies the semigroup properties: S (0, u 0 ) = u 0 , for any u 0 ∈ H, and S (t + s, u 0 ) = S (t, S (s, u 0 )), for any u 0 ∈ H, t, s ≥ 0. Also, making use again of Theorem 2.4 we obtain that S(t, u 0 ) is continuous with respect to the initial condition u 0 for fixed t ≥ 0.
Lemma 3.1. If β ≥ 4, then any weak solution of (2) with initial data such that u 0 H ≤ R satisfies the estimate Proof. The following calculations are formal, but they can be justified using Galerkin approximations. Let R > 0 and u 0 ≤ R. We multiply the equation by u t and −∆ and integrate over Ω. Then µ 2 Summing up these expressions and using Hölder's and Young's inequalities we get For β ≥ 4 the term J = 1 + 1 µ |u·∇u| 2 2 was estimated in [2, p.806] as follows: for some constant R 1 > 0. If β ≥ 6, then 4(β+1) β−2 ≤ β + 1 and Young's inequality implies Joining these inequalities we get Thus,

DANIEL PARDO, JOSÉ VALERO ANDÁNGEL GIMÉNEZ
In view of Lemma 2.3, for any t ≥ 0, r > 0. By the uniform Gronwall lemma [19] we obtain As a consequence of this lemma and the compact embedding V ⊂ H we obtain the following result.
Corollary 2. For any r > 0 the map u 0 → S (r, u 0 ) maps bounded subsets of H onto bounded subsets of V ∩ L β+1 (Ω). Hence, S (r) is a compact operator, i.e., it maps bounded subsets of H onto relatively compact ones.
We recall that the set A is said to be a global attractor for S if it is invariant, i.e. S (t, A) = A, for all t ≥ 0, and it attracts every bounded subset B of the phase space H, which means that where dist X (C, A) = sup x∈C inf y∈A x − y X is the Hausdorff semidistance between subsets of the Banach space X.
Usually in the literature a global attractor is supposed to be compact as well. However, we prefer to use this more general definition and add compactness as an additional property, as generally speaking a global attractor does not have to be bounded (see [20] for a non-trivial example of an unbounded non-locally compact attractor).
A set B 0 is called absorbing if for any bounded set B there exists a time T (B) such that Proof. This result follows directly from (8).
The semigroup S is said to be asymptotically compact if for any bounded subset B every sequence y k ∈ S (t k , B), where t k → +∞, is relatively compact in H. Proof. Let K = S(1, B 0 ), which is a compact set in H due to Lemma 3.1 and the compact embedding V ⊂ H. Then since B 0 is absorbing, for any bounded set B there exists T (B) > 0 such that Corollary 3. If β ≥ 4, the semigroup S is asymptotically compact.
The following two conditions guarantee the existence of the global compact attractor [10]: 1. There exists a bounded absorbing set B 0 . 2. S is asymptotically compact. Proof. The existence of the global compact attractor is a consequence of Lemma 3.2 and Corollary 3. Since the space H is connected, the connectedness of A follows from [5, p.4].
It is possible to prove that the global attractor is more regular if 4 ≤ β < 5. Indeed, let us check that A is in fact bounded in the space H 2 (Ω) 3 , and then compact in V and L β+1 (Ω) 3 .
Lemma 3.5. If β ≥ 4, then any weak solution of (2) with initial data such that Proof. As before, the calculations here are formal, but they can be justified via Galerkin approximations. Integrating (13) over the interval t, t + r 2 , with t ≥ r 2 , and using inequality (14) and Lemmas 2.3, 3.1 we have where D 1 satisfies the above properties for D. Further, differentiating (4) with respect to the time variable and multiplying by du dt we obtain where J F is the Jacobian matrix of F (u) = α |u| β−1 u, which is definite positive [15,Lemma 2.4]. Hence, the last term is non-positive. By the standard estimate [18, p.297] Thus, from Lemma 3.1 we get where again D 2 satisfies the above properties for D. Now, integrating the last expression over (s, r), with r 2 ≤ s ≤ r, and using (16) Integrating again but now with respect to s over r 2 , r and applying again (16) we get Thus, the proof is finished.
Lemma 3.6. If 4 ≤ β < 5, then any weak solution of (2) with initial data such that u 0 H ≤ R satisfies the estimate Proof. We follow the same steps as in [15,Proposition 5]. By Proposition 9.2 in [14] we have whereas Gagliardo-Nirenberg inequality and β < 5 gives Hence, from (4) we get so the results follows by applying Lemmas 3.1, 3.5.
Let B 0 be the absorbing ball given in Lemma 3.2. By Lemma 3.6 the set B 1 = S(r, B 0 ) is bounded in H 2 (Ω) 3 and From here it is easy to deduce (17).
We know by Theorem 2.1 that for any u 0 ∈ H and T > 0 the set D T (u 0 ) is non-empty. We observe that as 3 ≤ β < 4, we have that  If u (·) ∈ D s (u 0 ) and w (·) ∈ D T −s (u (s)), then the function Proof. Let u (·) ∈ D T (u 0 ). Then it is obvious that so w satisfies (3) in the interval (0, T − s) . We infer that w ∈ D T −s (u (s)) .
Let now u (·) ∈ D s (u 0 ) and w (·) ∈ D T −s (u (s)). Arguing as in the previous case we obtain that w(t − s) satisfies equality (3) in the interval (s, T ). As the time derivative of a weak solution belongs to L proving that z is really a weak solution.
In view of this lemma every solution can be extended to a globally defined one, that is, a solution which exists for t ∈ [0, +∞). In this situation we denote by D (u 0 ) the set of all globally defined solutions with initial condition u 0 and observe that for any t ≥ 0 the following equality holds: Using Lemma 3.8 we can easily prove that G is a strict multivalued semiflow, that is, the following two properties hold: G (s, u 0 )), for all u 0 ∈ H, t, s ≥ 0. The set A is a global attractor for G if: • A is negatively invariant, i.e., A ⊂ G (t, A) for all t ≥ 0; • A attracts every bounded set of H, that is, dist (G (t, B) , A) → 0 as t → +∞.
It is invariant if, moreover, A = G (t, A) for all t ≥ 0. The next lemma is crucial for proving the existence of a global attractor. Lemma 3.9. Assume that 3 ≤ β < 4. Let u n 0 → u 0 weakly in H and let u n (·) ∈ D (u n 0 ). Then there exists a weak solution u (·) to (2) with u (0) = u 0 and a subsequence u n k (·) such that u n k → u in C ([ε, T ], H) for all 0 < ε < T.
If, moreover, u n 0 → u 0 strongly in H, then u n k → u in C ([0, T ], H) for all T > 0. Proof. We fix T > 0. We deduce from Lemma 2.3 that the sequence u n is bounded in Also, using (4) and standard estimates (see [18, p.297]) we obtain that du n dt is bounded in L 4 3 (0, T ; V ). Thus, making use of the compactness theorem [11] we obtain a function u (·) and a subsequence (denoted again by u n ) such that u n → u weakly star in L ∞ (0, T ; H) , Let us prove that u n (t n ) → u (t 0 ) weakly in H for any sequence {t n } such that t n → t 0 , where t n , t 0 ∈ [0, T ]. Denote q = β+1 β . The time derivatives are bounded in the space L 4 3 (0, T ; V ), which implies readily that the sequence u n (·) is equicontinuous in the space V . Moreover, u n (t n ) is bounded in H, and then the compact embedding H ⊂ V yields that it is relatively compact in V . Hence, by Ascoli-Arzelà's theorem we have u n → u in C ([0, T ], V ). Thus, by a contradiction argument we obtain that u n (t n ) → u (t 0 ) weakly in H. In particular, we have that u (0) = u 0 .
Further, we need to check that u (·) is a weak solution to problem (2).
In order to show that u is a weak solution it remains to pass to the limit in the term B. Since u n → u in L 2 (0, T ; H) implies that u ni u nj → u i u j in L 1 (0, T ; L 1 (Ω)), We conclude that equality (3) is satisfied for the function u for all ζ ∈ V, and by density of V in V we obtain that (3) holds true. Thus, u is a weak solution.
Finally, we will prove that u n → u in C ([ε, T ], H) for all 0 < ε < T . From (11) we get and the same inequality is true for u. Hence, the functions J n (t) = u n (t) It is a consequence of (19) that J n (t) → J (t) for a.a. t. Then we can choose t k < t 0 as close to t 0 as we wish such that J n (t k ) → J (t k ), and we can assume without loss of generality that t k < t n . Therefore, Since u (·) is continuous, for any δ > 0 there exists t k and N (t k ) such that |J (t k ) − J (t 0 )| ≤ δ/2 and |J n (t k ) − J (t k )| ≤ δ/2 for all n ≥ N . This implies that lim sup u (t n ) H ≤ u (t 0 ) H . (22) Joining (21) and (22) we deduce that u (t n ) H → u (t 0 ) H and then u (t n ) → u (t 0 ) in H.
Since T > 0 is arbitrary, by a diagonal arguments we obtain a common subsequence on an arbitrary interval [ε, T ].
The first part of the lemma is proved. For the second part, we need to prove only that u (t n ) → u (0) if t n → 0, t n ≥ 0. For this aim we repeat the above argument with t k = 0 = t 0 . Hence, Then we obtain the result arguing in the same way as above.  Proof. If not, there exist u 0 ∈ H, t > 0, sequences u 0 n → u 0 , y n ∈ G (t, u n 0 ) and a neighborhood O of G (t, u 0 ) such that y n ∈ O. Let y n = u n (t), where u n (·) ∈ D (u n 0 ). Then by Lemma 3.9 there is a subsequence y n k satisfying y n k → y ∈ G (t, u 0 ), which is a contradiction.
For a multivalued semiflow G the concepts of absorbing set and asymptotically compactness are given in exactly the same way as for semigroups.
The following conditions are sufficient in order to obtain a global compact invariant minimal attractor A for a strict multivalued semiflow G [13, Theorem 3 and Remark 8]: 1. G possesses a bounded absorbing set B 0 ; 2. G is asymptotically compact; 3. G has closed values; 4. the map u 0 → G(t, u 0 ) is upper semicontinuous.
Theorem 3.11. Assume that 3 ≤ β < 4. Then G has a global invariant compact attractor A, which is minimal among all closed attracting sets.
Proof. We need to check the four aforementioned conditions. It follows from (8) that the ball B 0 = {u ∈ H : } is absorbing. In view of Corollary 4 and Lemma 3.10 G has compact (and then closed) values and the map u 0 → G(t, u 0 ) is upper semicontinuous.
Finally, again by Corollary 4 the operator G (1, ·) is compact. Hence, for any bounded set B an arbitrary sequence y n ∈ G (t n , B), which belongs to G (t n − 1, B)) ⊂ G (1, B 0 ) , for all n ≥ N, is relatively compact in H, so G is asymptotically compact.
We can give also some information about the structure of the global attractor in terms of bounded complete trajectories, which are continuous functions γ : R → H such that u (·) = γ (· + s) belongs to D (u 0 ) for all s ∈ R and satisfying that ∪ t∈R γ (t) is a bounded set. Indeed, by Lemmas 3.8, 3.9 we can apply Theorems 9, 10 from [6] and obtain that where is K the set of all bounded complete trajectories. Finally, in a similar way as in [7] let us prove that the global attractor is stable, which means that for any ε > 0 there is δ > 0 such that where O η (A) = {z ∈ H : dist(z, A) < η}.
Lemma 3.12. Assume that 3 ≤ β < 4. The global attractor A given in Theorem 3.11 is stable.
Proof. By contradiction if (24) does not hold, then there exist ε > 0 and sequences We consider two cases: 1) t k → +∞ for some subsequence; 2) t k ≤ C.
In the first situation, as the sequence {x k } belongs to a bounded set, by the definition of global attractor we get that dist (y k , A) → 0, which contradicts (25).
In the second one, up to a subsequence t k → t 0 , x k → x 0 ∈ A, so by Lemma 3.9 and the invariance of A we obtain that which is again a contradiction. Remark 1. Formula (23) and stability are also true for the global attractor A given in Theorem 3.4 for β ≥ 4.

4.
Global attractor for strong solutions. In this section we are going to define a semigroup generated by strong solutions of (2).
We recall that a function u : [0, T ] → V ∩ L β+1 (Ω) 3 is called a strong solution of (2) if u is a weak solution and Then there exists a unique strong solution of (2). Moreover, it satisfies Existence of strong solutions for β ≥ 7/2 and uniqueness for 7/2 ≤ β ≤ 5 were proved at first in [2] and [15]. In [23] and [17] existence and uniqueness were extended for 3 < β < 7/2. Finally, in [25] uniqueness of strong solutions was established for all β ≥ 1. We observe also that for β ≥ 4 uniqueness of strong solutions follows from Theorem 2.4.
From a standard result (see e.g. [14,Corollary 7.3]) and u ∈ L 2 (0, T ; H 2 (Ω) 3 ), du dt ∈ L 2 (0, T ; L 2 (Ω)) we obtain that u ∈ C([0, T ], V ). If 3 < β ≤ 5, then V ∩ where u (·) is the unique strong solution to problem (2) with initial condition u 0 , defines a semigroup having a global compact attractor in V . This result was proved in [15] for 7/2 ≤ β ≤ 5 and in [17] for 3 < β ≤ 5. Although in the last paper the nonautonomous case is studied, for a time-independent external force f the autonomous attractor is obtained as a particular case. In this section we will give an alternative proof of the existence of the strong attractor for β = 5. The reason is that in the above cited papers there is an argument which is correct for β < 5 but it is unclear when β = 5. Namely, if we take for example the paper [15], then in Proposition 5 inequality (28) in page 247 is correct for β < 5, but not for β = 5.
Assuming that any strong solution satisfies this property, we are able to obtain the result.
When β ≥ 5 we define the operator S X : R + × X → X, where X = V ∩ L β+1 (Ω) 3 , by where u (·) is the unique strong solution to problem (2) with initial condition u 0 . By the same argument given in Lemma 3.8 one can check that S is a semigroup.
Lemma 4.2. Let β ≥ 5. Assume additionally when β > 5 that every strong solution u (·) with initial condition in X satisfies that u ∈ C([0, T ], (L β+1 (Ω)) 3 ) for any T > 0. If u n 0 → u 0 weakly in X, then S X (t, u n 0 ) → S X (t, u 0 ) strongly in X for any t > 0. This implies, in particular, that the operator S X (t,·) is compact for t > 0 and continuous with respect to the initial condition for all t ≥ 0.

DANIEL PARDO, JOSÉ VALERO ANDÁNGEL GIMÉNEZ
Thus, taking into account that the space X is uniformly convex, we get that S X (t, u n 0 ) → S X (t, u 0 ) strongly in X. Proof. Let B 0 be the absorbing ball given in Lemma 3.2. By Lemma 3.1 the set where B is a bounded set in X. Hence, B 1 is a bounded absorbing set in X.
Let B be bounded in X and t n → ∞, y n ∈ S X (t n , B). By Lemma 4.2 the operator S X (1, ·) is compact, which implies using that y n is relatively compact in X.
The existence of the global compact attractor follows from [10] and its connectivity from [5, p.4].

Numerical simulations.
We shall now focus on solving numerically the equation (2) employing computational fluid dynamics (CFD) to visualize scenarios in which the evolution of the fluid flow converges to a steady state. It is important to stress that the examples reported below are only intended for showing the asymptotic behaviour of the fluid flow numerically when taking different values of the parameters α and β in the momentum equation of (2), but no conclusive results should be deduced from the numerical simulations.
The geometry of the flow domain used in all our numerical experiments is a sphere Ω of radius 6 centered at the origin. We also take the source term f in (2) as where C is a cylinder, with both radius and height of 4, within the flow domain symmetrically located at the center of the sphere Ω in such a way that the base of the cylinder is parallel to the xz-plane as in Figure 1. Observe that f (x, t) can be seen as a constant source force within the cylinder C propelling the fluid flow upwards. Numerical simulations were all performed using the CFD package OpenFOAM R , which is the acronym of Open Source Field Operation and Manipulation. It is an open-source CFD software based on C++ that contains a toolbox for tailored numerical solvers for a wide variety of problems relevant to the industry and scientific community. The solvers implemented in OpenFOAM R uses the Finite Volume Method (FVM) to discretize the governing equations on unstructured meshes (see [4,21]). The solver used to integrate our model numerically was pimpleFoam, which combines the two most common algorithms for solving the Navier-Stokes equations, namely, SIMPLE and PISO algorithms. The pimpleFoam code is inherently transient, requiring an initial condition and boundary conditions. OpenFOAM R includes pre-processing and post-processing capabilities such as snappyHexMesh and ParaFoam for meshing and visualization, respectively. In those experiments we have set the initial condition u 0 (x) = (0, 0, 0), and the images represent the velocity vector field in the xy-section at z = 0. The darker areas in the images are those where the magnitude for the velocity vector u is smaller, while the lighter ones represent the areas where the velocity is higher. According to the results of the experiments, when the magnitude of the velocity vector is greater than 1 and the parameter β becomes bigger, the medium provides increased resistance to movement, so the fluid flow slows down more quickly. On the contrary, when the magnitude of the velocity vector is less than 1 and the parameter β becomes smaller, then medium provides decreased resistance to movement, so the fluid flow spreads further through the medium. The effect of the parameter α does not depend of the u magnitude, acting proportionally, i.e., the larger the value of α, the higher resistance to motion of the fluid flow. It has also been observed that convergence speed to the steady state is higher as α and β increase. Therefore, when α and β are small, a higher period of time to get convergence to the steady state is required. In fact, we have also performed simulations (not shown here) for values of α close to 0 (also for α = 0) and a low value of β (for instance, β = 1), but we did not achieve convergence to a steady state for an approachable (from a computational point of view) time value. It is likely that for such values of the parameters the global attractor (if it exists!) is more complex than a fixed point.
Lastly, an experiment with a non-vanishing initial condition was carried out. The performance was made by taking u 0 (x) = (1, 0, 0), α = 0.2 and β = 1, and the results of the model is shown in Figure 5. As one might expect, the steady state does not depend of the initial condition, hence, it is the same as taking u 0 equal to zero (compare the right panel of Figure 5 to the left panel of Figure 2).