Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows

In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero.

1. Introduction. Non-Newtonian (or complex) fluids are difficult to model and to analyze because they display essentially nonlinear and even discontinuous flow properties. In this paper we consider an evolution problem which appears in the investigation of the model of concentrated suspensions proposed by Hebraud and Lequex [11]. In this model the system is divided in mesoscopic blocks. Their size is large enough so that the stress and the strain tensors may be defined for each block, but small compared to the characteristic length of the macroscopic stress field. In the mathematical description each block carries a given shear stress x (σ in the original paper). The stress state in the material is described by the probability density p(x, t), which represents the description of shear stress in the assembly of blocks at time t and obeys the Fokker-Planck equation where α > 0 is a parameter, χ R\[− 1,1] is the characteristic function of the open set R \ [−1, 1], δ 0 is the Dirac delta function with support at the origin, Existence and uniqueness results for such model were proved in [4]. In [5] the authors investigated the qualitative behavior of a more general model, where the function b was not assumed to be homogeneous in space. Also in [5] it was shown for some values of the parameters that if b(t) → b ∞ , as t → ∞, then the corresponding stationary solution is (locally) asymptotically stable.
The theory of global attractors was applied first for (1) in Amigó et al. [1], where the authors proved under the assumption b(t) ≡ 0 the existence of global unbounded attractors with respect to the weak topology. Moreover, a lattice dynamical system generated by finite-difference numerical approximations was investigated in [2], [12], [16], whereas numerical simulations were provided in [9].
It is worth noting that in both [4] and [12] the key point was the analysis of the so-called vanishing viscosity approximation system, where the diffusion coefficient was everywhere positive.
In the present work we consider an evolution problem generated by vanishing viscosity approximations and prove the existence of a global attractor for the corresponding semiflow with respect to the strong topology of the phase space. Moreover, we prove the upper semicontinuity of these attractors with respect to the set of bounded complete trajectories of the original problem (1).
Suppose that b is an essentially bounded function, that is, there exists a constant B > 0 such that |b(t)| ≤ B for a.e. t > 0. (6) Further we will use the following notation: for each 1 ≤ p ≤ ∞. Let · , · be the pairing on H −1 × H 1 (on L q × L p respectively with p ≥ 1 and 1 < q ≤ ∞ such that 1 p + 1 q = 1) that coincides with the inner product on L 2 , that is, for each f ∈ L 2 and u ∈ H 1 (for each f ∈ L q and u ∈ L p , respectively).
Let 0 ≤ τ < T < ∞ be arbitrary fixed. Further, for the sake of simplicity, we will follow the conventions given in the following remark.
Remark 1. Let γ ≥ 1, E be a real separable Banach space and I ⊆ R + be a possibly infinite time interval. As L γ loc (I; E) we consider the Fréchet space of all locally integrable functions with values in E, that is, ϕ ∈ L γ loc (I; E) if and only if for any finite interval [τ, T ] ⊂ R + the restriction of ϕ on I τ,T := I ∩ [τ, T ] belongs to the space L γ (I τ,T ; E). To simplify further our arguments we denote the restriction of an E-valued function ϕ defined on time interval I to a time subinterval J ⊆ I by the same symbol ϕ. If, additionally, E ⊆ L 1 loc (R), then any function ϕ from L γ loc (I; E) can be considered as a measurable mapping that acts from R × I into R. Further, we write ϕ(x, t) when we consider this mapping as a function from R × I into R, and ϕ(t), if this mapping is considered as an element from L γ loc (I; E); see, for example, Gajewski et al. [8, Chapter III]; Temam [15]; Babin and Vishik [3]; Chepyzhov and Vishik [6]; Zgurovsky et at. [19] and the references therein. For a Banach space E the notation E w means that the vector space E is endowed with the standard topology of weak convergence.
A solution of equation (2) on a finite time interval [τ, T ] is defined as follows.
holds for each η ∈ L 2 (τ, T ; H 1 ). Remark 2. We note that the right hand-side of equality (7) is equal to Remark 3. Let 0 < ε 1, 0 ≤ τ < T < ∞, p τ ∈ L 1 ∩ L 2 , and p be a solution of equation (2) on [τ, T ]. Since p ∈ L 2 (τ, T ; H 1 ) and ∂p ∂t ∈ L 2 (τ, T ; H −1 ), then p ∈ C([τ, T ]; L 2 ); Gajewski et al. [8,Chapter III]. Therefore, the following initial condition p| t=τ = p τ (x), a.e. in R, makes sense. Let which is a Banach space with the norm is a Banach space with the following norm: Remark 5. Let I ⊆ R be either a finite or an infinite time interval and p be a measurable function on R × I. Further xp will denote the measurable function on R × I that equals to xp(x, t) for a.e. x ∈ R and t ∈ I.
Let 0 ≤ τ < T < ∞. We understand condition (5) in the sense of the following definition.
Definition 2.2. Let 0 ≤ ε 1. We recall that the solution p of equation (2) problem ( Therefore, the phase space for this problem is defined as follows: where cl X is the closure in the space X (see Amigó et al. [1]). Note that the convexity of E implies the equality H = cl Xw E. The phase space H is considered as a metric space with the distance ρ H endowed from the Banach space X, that is, The phase space H endowed with the induced topology from X w is denoted by H w .
Remark 7. Let 0 < ε 1. Conditions (10) imply that for each p τ ∈ E there exists a unique solution p of problem (2)-(5), (9) on [τ, T ]. According to the continuous embedding [14, Chapter III, Lemma 1.4, p. 263]. In particular, in light of (11) we have that p(t) ∈ E for each t ∈ [τ, T ]. The definition of the phase space H implies that for each p ∈ H the following two conditions hold: The main purpose of this paper is twofold: (i) to establish topological properties of solutions and their a priori estimates; and (ii) to prove in the autonomous case the existence of global attractors in the strong topology of the phase space H for the semiflow generated by the vanishing viscosity approximations (2)- (5). We also show the upper semicontinuity of these attractors with respect to the set of all complete bounded trajectories of the original problem (1) when ε → 0.
3. Existence and properties of solutions. In this section we provide existence of solutions for problem (2)-(5), (9) in the phase space H and study their topological properties. First, Lemma 3.1 establishes some a priori estimates for solutions. Then, the main result, concerning existence of solutions, is provided in Lemma 3.2, whereas in Lemma 3.3 we are dealing with convergence of solutions when we take approximations of the initial datum. Finally, Lemmas 3.4 and 3.5 prove that the dynamical system is dissipative.
For τ ≥ 0 and 0 ≤ ε 1 let K τ,ε denotes the family of all globally defined solutions of problem (2)-(4) on [τ, ∞) with p (τ ) ∈ H, that is, p ∈ K τ,ε if and only if for each T > τ the restriction of p on [τ, T ] is a solution of problem (2)-(4) on [τ, T ] and p (τ ) ∈ H. Similarly, let D τ,ε be the family of all globally defined solutions of problem (2)-(5) on [τ, ∞) with p (τ ) ∈ H. By definition, the following inclusion holds: for each τ ≥ 0 and 0 ≤ ε 1. The following Lemma 3.1 and its corollary establishes the converse inclusion in the sense that each globally defined solution p of problem (2)-(4) with initial datum from H is a globally defined solution of problem (2)-(5). Furthermore, Lemma 3.1 establishes a priori estimates for the solutions of our problem.
Proof. The proof of Lemma 3.1 is provided in Section 5.
Remark 8. The definition of the phase space H and Remark 4 yield that H ⊂ L 1 .
Remark 9. According to Lemma 3.1, each globally defined solution p of problem which implies that p is a globally defined solution of problem (2)-(5) on [τ, ∞).
The existence result has the following formulation.
The proof of Lemma 3.2 is provided in Section 5. Its statement follows from Lemma 3.3, that describes some convergence results, in particular, in the strong topology of the phase space H.
Proof. The proof of Corollary 2 is provided in Section 5.
The following lemmas provide dissipative results in the phase space H. Lemma 3.4. There exists R 0 > 0 such that for an arbitrary bounded (in L 2 ) set K ⊂ H and for arbitrary ε ∈ (0, 1) there exists a moment of time T = T (K, ε) such that for every τ ≥ 0 and p ∈ D τ,ε satisfying p(τ ) ∈ K the following inequality holds: Proof. The proof of Lemma 3.4 is provided in Section 5.
Proof. The proof of Lemma 3.5 is provided in Section 5.

Existence and properties of global attractors in the autonomous case.
Let us consider the autonomous case, that is, when b(t) ≡ b. Then according to Lemma 3.2 for every ε ∈ (0, 1) problem (2)-(5) in the phase space H generates a classical semigroup where p is a global solution of (2)-(5) with p(0) = p 0 .
(21) We recall that a function ξ : R → H is a complete trajectory of V ε if ξ (t) = V ε (t − s, ξ (s)) for any s ≤ t. In the same way, ξ is a complete trajectory of the original problem (1) if for any τ ∈ R the map u (·) = ξ (·) | [τ,+∞) is a solution of (1).
Theorem 4.1. For each ε > 0 the semigroup (21) has the connected stable global attractor Θ ε in the phase space H. Moreover, Θ ε is bounded in H uniformly in ε, it consists of bounded complete trajectories and for any sequences ε n 0 and ξ n ∈ Θ εn there exists ξ ∈ Θ such that up to a subsequence ξ n → ξ in X w as n → ∞, where Θ = {y(0) : y (·) is a complete trajectory of problem (1)}.
Proof. Let us prove the first part of the theorem for fixed ε > 0. Due to Lemmas 3.1, 3.3, 3.4 and classical results about existence of global attractors (see [13], [15]) it will be enough to prove that V ε is asymptotically compact, i.e., Let ξ n = V ε (t n , p n 0 ). Then ξ n = p n (t n ), p n is a solution of (2)-(5) with p n (0) = p n 0 . Therefore from Lemma 3.5 So we can claim that {ξ n } is precompact in H w . Indeed, since ξ n L 2 ≤ R 0 + r then up to subsequence ξ n → ξ in L 2 w . Let us prove that up to a subsequence ξ n → ξ in L 1 w . Since ξ n = p n (t n ), then (13) yields that for each δ > 0 there exist k(δ) ≥ 1, According to Amigó et al. [1, Lemma 6.1] Thus, we set d n (x) = (1 + |x|)ξ n (x) and prove that {d n } is a Cauchy sequence in for each u ∈ L ∞ and n, m ≥ N = N (δ, k). Since the space L 1 is weakly complete, then up to a subsequence d n → d in L 1 w for some d ∈ L 1 . Thus ξ n →ξ = d 1+|x| in L 1 w . If we consider the restriction of ξ n to each interval [−k, k], then we deduce that ξ = ξ and up to a subsequence ξ n → ξ in H w . Now let us prove this convergence in the strong topology of H. Consider a smooth real function θ that satisfies the following three conditions: and define for k > 1 According to Amigó et al. [1, pp. 215-216] after multiplying (2) by ρ k (x)p n we obtain Then from (24) we have where β := max |s|∈ [1,2] {|θ (s)| + |θ (s)|}.
Combining (22) and (25) we deduce from Gronwall's Lemma that for some pos- On the other hand, for every solution of (2)-(5) we have the following energy equality (for details see the proof of Lemma 3.3): Let us consider the functions Thenp n is a solution of (2)-(5),p n (0) = p n (t n − 1),p n (1) = ξ n andp n satisfies (22), (24), (27). Moreover, analogously to the previous arguments we deduce that up to subsequencep n (0) = p n (t n − 1) →p 0 in H w . Hence, from Lemma 3.3 we obtain for every T > 1 that wherep is a solution of (2)-(5) withp(0) =p 0 . Since ε > 0 is fixed, we can derive from (22), (27) and the Aubin-Lions theorem [14] (see the proof of Lemma 3.3 for a more detailed explanation of this argument) that for every k > 1 up to subsequencē p n →p in L 2 (0, T ; L 2 (−k, k)).
In particular,p By a diagonal procedure we obtain that up to a subsequence and for some τ ∈ (0, 1), From (26) we get Combining ( In particular, ξ n =p n (1) →p(1) in H. Thus we obtain the required precompactness of {ξ n } and, therefore, the existence of the connected, stable global attractor Θ ε . Finally, let us prove that for any sequences ε n 0 and ξ n ∈ Θ εn there exists ξ ∈ Θ such that up to a subsequence ξ n → ξ in X w as n → ∞. Let us assume that ε n → 0. Denote V εn = V n , Θ εn = Θ n . From (12) and (18) we have where R 1 > 0 does not depend on ε > 0. Thus {Θ n } is uniformly bounded in H. Consider p n 0 ∈ Θ n . Let us prove that for some p 0 ∈ H up to a subsequence p n 0 → p 0 in H w . Indeed, since p n 0 L 2 ≤ R 1 , then up to a subsequence p n 0 → p 0 in L 2 w . Since p n 0 ∈ Θ εn = V εn (t, Θ εn ) for each t ≥ 0, we can repeat the previous arguments and obtain from [1, Lemma 6.1] that up to a subsequence p n 0 → p 0 in L 1 w . Since Θ n consists of complete trajectories, there exist p n : R → H such that p n (0) = p n 0 , p n (t) ∈ Θ n , for any t ∈ R, and p n | [0,+∞) is a solution of (2)-(5) with ε = ε n . Then due to Lemma 3.3 up to subsequence for each T > 0, where p is a solution of (2)-(5) with ε = 0.
After that we repeat the previous procedure for the same sequence {p n } n≥1 on [−1, +∞) and due to the weak convergence of {p n (−1)} n≥1 in X (because {Θ n } is uniformly bounded) we obtain that p is a solution of (2)-(5) whith ε = 0 on [−1, +∞). Repeating this argument inductively on each [−k, +∞), k ≥ 1, we obtain that the limit function p : R → H is a bounded complete trajectory of problem (2)-(5) with ε = 0. Proof of Lemma 3.1. We provide the proof in several steps.

From inequalities (37), (38) and (39) we have that
for a.e. t > τ. Let us consider the last term in the left hand-side of equality (33). The following equality holds: for a.e. t > τ, because (a) η(x) = 0 for each |x| ≤ k; and (b) k ≥ 1. Equalities (33), (34) and (45) and inequalities (36) and (44) give us the following inequality: for a.e. t > τ, where the positive constants c 2 , c 3 depend only on θ and d dt is the derivative in the sense of distributions. Gronwall's inequality implies that for each t ≥ τ, whenever k ≥ 2B and R ≥ 2k, because 0 ≤ ε 1. Since η(x)|x| ≤ |x| for each x ∈ R, according to the definition of the norm in L 1 , the following inequality holds: Moreover, for each t ≥ τ, whenever k ≥ 2B and R ≥ 2k, because (a) η is a nonnegative function; and (b) η(x) = 1, if 2k ≤ |x| ≤ R. Therefore, inequalities (47)-(49) yield that for each t ≥ τ, whenever k ≥ 2B and R ≥ 2k. If we set k = 2B in inequality (50) and pass in the obtained inequality to the limit as R → ∞, then we obtain the following inequality: for each t ≥ τ. Since, p(x, t) ≥ 0 for a.e. x ∈ R and t > τ, and R p(x, t)dx = 1 for a.e. t > τ, the following inequalities hold: for each t ≥ τ.
Proof of Lemma 3.3. We prove this lemma in several steps. We put τ = 0 for the sake of simplicity.
Proof of Lemma 3.4. Let K ⊂ H be a bounded set in L 2 and ε ∈ (0, 1) be arbitrary. Fix p ∈ D 0,ε with p(0) ∈ K. The following inequalities hold: where C = 2d α , and d is the constant from the embedding H 1 (−1, 1) ⊂ L ∞ (−1, 1). Further we shall use the Poincaré-Wirtinger inequality: where C 2 = C 1 + C 2 2 + C 2 2C1 . We assume that εC 1 < 1 and denote In all further arguments we will essentially use the fact that every open set in (0, +∞) is the union of no more than a countable number of open intervals.
(96) We can treat t 3 in the same way as t 1 . Repeating this procedure, we obtain (18) in this case.
If p 0 2 L 2 ≤ C 5 , then we can repeat the previous arguments with t 1 = 0 and obtain the required result.
Proof of Lemma 3.5. Let K = {p 0 ∈ H : p 0 X ≤ R} ⊂ H be a bounded set, where R > 0 be an arbitrary fixed. The first statement of Lemma 3.1 (see inequality (12)) yields that for each p ∈ D τ,ε with p(τ ) ∈ K the following inequality holds: for each t ≥ τ, where the positive constant C depends only on parameters of problem (2) and does not depend on K, τ, and ε. Therefore, it is sufficient to justify the a priori estimate in the space L 2 . Using (85) and the Poincaré-Wirtinger inequality we deduce that the following inequality holds in the sense of distributions on (τ, ∞): d dt R (p(x, t)) 2 dx + D(p( · , t))