On the behavior in time of solutions to motion of Non-Newtonian fluids

We study the behavior on time of weak solutions to the non-stationary motion of an incompressible fluid with shear rate dependent viscosity in bounded domains when the initial velocity u0∈L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u}_0 {\in } {L}^2$$\end{document}. Our estimates show the different behavior of the solution as the growth condition of the stress tensor varies. In the “dilatant” or “shear thickening” case we prove that the decay rate does not depend on u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0$$\end{document}, then our estimates also apply for irregular initial velocity.


Introduction
Let Ω be a bounded open set of R N , N > 2. For 0 < T < +∞ we set Ω T = Ω × (0, T ). The unsteady motion of an incompressible fluid with sheardependent viscosity in a cylinder Ω T is governed by the conservation of volume and the conservation of momentum equations where ∇ · u ≡ ∂ xi u i and we use the notation that repeated subscripts imply summation over 1 to N . Here S = {S ij } is the deviatoric stress tensor , p is the pressure , u = {u 1 , . . . u N } is the velocity , f = {f i,j } is the external force tensor .
We assume that at the initial time t = 0, the initial velocity is a vector field u 0 u(0) = u 0 in Ω (1. 3) with ∇ · u 0 = 0 , and on the boundary the following aderence's condition holds The stress tensor S may depend on both (x, t) and the "rate of strain tensor" D = {D i,j }, which is defined by For a background on Non-Newtonian fluids we refer to [1,2,9] and the references therein. The model (1.1)-(1.4) has been considered by J. Wolf in [21] and by L. Diening, M. Ruzicka and J. Wolf in [5] to describe the motion of fluid u through Ω.
The fluids with "shear dependent viscosity" are given by the constitutive law S = ν(D II )D , and ν is the generalized viscosity function. As noticed in [5] the model (1.5) includes all power-law and Carreautype models, which are quite popular among rheologists. Such models are used in many areas of engineering sciences such as chemical engineering, colloidal mechanics, glaciology, geology and blood rheology (see [11] for a discussion of such models and further references).
We assume that the the deviatoric stress S : Ω T × M N sym → M N sym is a Caratheodory function satisfying for every ξ and η in M N sym and for almost every ( where ν 0 and c 0 are positive constants, and g i (i = 1, 2) are non negative functions satisfying g 1 ∈ L q (Ω T ) and Before recalling what we mean here by weak solutions to (1.1)-(1.4), we recall some function spaces we need.
Here, C ∞ 0 (Ω) denotes the space of all smooth functions having compact support in Ω, W k,q (Ω) (k ∈ N and 1 ≤ q ≤ +∞) are the usual Sobolev spaces and W k,q 0 (Ω) is the closure of C ∞ 0 (Ω) in W k,q (Ω). If (X, · X ) is a normed space, by L q (0, T ; X) we denote the space of all the Bochner measurable functions ϕ : (0, T ) → X such that Finally, we denote H q the closure of the set {ϕ ∈ C ∞ 0 (Ω) N : ∇ · ϕ = 0} respect to the norm of L q (Ω) N V q is the closure of the set {ϕ ∈ C ∞ 0 (Ω) N : ∇ · ϕ = 0} respect to the norm ϕ Vq = D(ϕ) L q (Ω) .
To simplify the notation, for q = 2 we use just notation H for the space H 2 and we will not distinguish between the norm of scalar-valued, vector-valued or tensor-valued versions of all the spaces defined above. Finally, |Ω| denotes the Lebesgue measure of Ω. We recall that since we are assuming Ω bounded, by means of Korn's inequality there exists a positive constant γ 0 , such that (1.10) (1.11) for every ϕ ∈ C ∞ (Ω T ) N with ∇ · ϕ = 0 and supp(ϕ) ⊂⊂ Ω × [0, T ). Thanks to the parabolic embedding, any weak solution u belongs to L q N +2 N (Ω T ). The assumption q ≥ 2N N +2 guarantees that u is in L 2 (Ω T ). Assuming that the external force verifies (1.12) and the monotonicity condition the first results on existence of solutions were proved by O. Ladyzhenskaya in [8] and by J.L. Lions in [10] under the restriction that The uniqueness holds if q > N +2 2 . For N = 3 and 2 ≤ q < 11 5 the existence of a weak solution was proved by J. Malek, J. Necas and M. Ruzicka in [12] under further regularity assumptions on the domain and on S.
Then, J. Wolf has completed the previous existence results in [21], covering the set Here < ·, · > H denotes the duality pairing between H and H. Solutions found in [5] and [21] are limit of functions solving approximating problems more regular than (1.1)-(1.4). Moreover these functions satisfy a balance energy equality (see also [4]). Motivated by this topic we state the following On the behavior in time of solutions Page 5 of 18 42 Let us remark that if a weak solution is bounded in Ω T , then it is a weak energy solution. Indeed (up to a regularization procedure) by using u as test function in (1.11), integration by parts gives that If (1.13) holds true, thanks to the results of [5] and [21], for Aim of this paper is to describe the behavior in time of weak energy solutions to (1. Schonbek in [19] and by S. Necasova and P. Penel in [14], respectively, when the spatial variable belongs to R N . Time decay estimates were also discussed in [4] when the spatial variable is in R N + and the Fourier splitting method does not apply. The solutions considered there are energy solutions in the sense of Definition 1.2. The model treated by the previous authors concerns the case in which the deviatoric stress tensor S is given by (1.7) with particular values of q .
Here we consider time decay estimates for a weak energy solution to general problem (1.1)-(1.4) having the spatial variable in a bounded domain Ω and in all cases in which this problem is known to be well posed.
Our estimates are based on an approach that highlights the different behavior as the exponent q varies.
To understand the large behavior of a viscous incompressible fluid we assume that conditions (1.8) and (1.9) hold for a.e. ( (1. 16) In this case we say that u ∈ L q loc (0, +∞; Our main result is the following. ; H) for any T > 0 and the following estimates hold true for any t > 0 An estimate of the constant B is given in formula (3.16) below. The existence of a global weak energy solution to (1. We point out that the previous result shows that although the global weak energy solution u decays in time for every value of q, there is a deep difference in the decay estimates between the "dilatant" or "shear thickening" case q > 2 and all the other cases ( "Newtonian" or "shear Thinning"). As a matter of fact, if q > 2, all the estimates in Theorem 1.1 are universal estimates, i.e., do not depend on the initial value u 0 . In other words, in the "dilatant" or "shear thickening" case the initial velocity does not influence at all the behavior in time of the velocity of the fluid.
Thanks to this argument, when q > 2, we are able to provide the existence of a suitable weak solution also when u 0 is just a summable function (see Sect. 4).
Moreover, we can show that, although the initial datum is not regular, as soon as t > 0 the solution we found has the same regularity and decay behavior of the case in which u 0 is regular.
In the particular case that the deviatoric stress tensor S is given by (1.7) we show that an exponential decay estimate holds true regardless of the growth exponent q (see Proposition 3.1 below).
We point out that in unbounded domains optimal algebraic estimates have been proved in [14] and [4]. Here we reach an exponential decay rate thanks to the boundedness of the domain Ω.
The paper is organized as follows: in Sect. 2 we recall some known results that will be an essential tool in proving our decay estimates (see Sect. 2.1) and we construct a weak global energy solution (see Sect. 2.2). In Sect. 3.1 we prove Theorem 1.1 and we study the particular case (1.7). Finally, in Sect. 4 we study the case of not regular initial data.

Preliminary results
As recalled in the Introduction, for the convenience of the reader we recall here some known results that will be an essential tool in proving the decay estimates stated in the previous section.
for every 0 ≤ t 1 ≤ t 2 < T where M and ν are positive constants and g is a non negative function in L 1 loc ([0, T )). Then it results Moreover, the following estimate holds The previous Lemma can be easily proved by a straightforward modification of the proof of Proposition 3.1 of [13] and hence we omit it. Moreover, if T = +∞ and g belongs to L 1 ((0, +∞)) we have

Lemma 2.2. Assume T ∈ (0, +∞] and let φ(t) a continuous and non negative function defined in
In particular, we get that We omit the proof of the previous Lemma since it is an obvious modification of the proof of Proposition 3.2 of [13]. [13] and [7] to study asymptotic estimates for evolution problems. Analogous techniques, developed in [15] and in [17], have been useful also for other parabolic problems (see [16] and [18]).

Existence of a global weak energy solution
In this subsection we prove the existence of a global weak energy solution to problem (1.1)-(1.4). The existence of a weak energy solution can be deduced from [21] and [5]. Here, for convenience of the reader, we state the main tools of the construction in Ω T , in order to obtain a global weak energy solution.
Proof. As in [5] and [21] we consider for any ε > 0 and for any T > 0 the following problem Here, by a solution of (2.6) we mean a function u ε ∈ L ∞ (0, T ; H)∩L q (0, T ; V q ) satisfying By using fixed point Theorem in [21] it is proved that, for any ε > 0, problem (2.6) admits a unique solution u ε ∈ C([0, T ]; H) ∩ L q (0, T ; V q ). Moreover this solution satisfies the following energy equality where c here and in what follows denotes a positive constant, which may vary from line to line but does not depend on the parameter ε. By virtue of Korn's inequality and Sobolev's embedding Theorem we get u ε L q (0,T ;W 1,q (Ω)) + u ε L q (0,T ;L q * (Ω)) ≤ c (2.9) where q * = qN N −q . Then by Sobolev's inequalities and Holder's inequality from (2.8) to (2.9) we deduce that We recall that from the bound of q we have that Now we can pass to a subsequence, still denoted by u ε for simplicity, and find Arguing as in Section 4 of [5], it is possible to prove that It is now easy to prove (see [5] and [21]) that the function u constructed as above satisfies (1.11) and so it is a weak energy solution to (1.1)-(1.4) (i.e. a weak energy solution in Ω T ). Now we prove the existence of a global weak energy solution.

Decay estimates
Aim of this section is to prove Theorem 1.1 and to treat the special case (1.7).

Proof of
Hence, u belongs to C([0, T ]; Y * ) (see [6]). Now, to conclude that u is in C w ([0, T ]; H) we follow the reasoning in [21]. Let t 0 ∈ [0, T ] arbitrarily fixed. Since, u belongs to L ∞ (0, T ; H), there exists a sequence t n ∈ [0, T ] such that t n → t 0 with u(t n ) bounded in H. Hence, by the reflexivity, there exists a subsequence u(t n k ) and ξ ∈ H such that u(t n k ) ξ weakly in H as k → +∞ .
On the other hand, being u in C([0, T ]; Y * ), it results Recalling that H is continuously and densely imbedded into Y * it follows ξ = u(t 0 ). This implies that u(t n k ) converges weakly to u(t 0 ) in H as k → +∞. Whence u belongs to C w ([0, T ]; H). By definition, for every T > 0 there exist u ε ∈ L q (0, T ; V q ) ∩ L ∞ (0, T ; H) satisfying (1.14) and (1.15). To prove estimate (1.17) it is sufficient to show that this estimate is satisfied for every t ∈ (0, T ) by the approximating solutions u ε ∈ C([0, T ]; H) ∩ L q (0, T ; V q ) since this implies (thanks to the arbitrary choice of T ) that u satisfies (1.17) for almost every t > 0 and hence (thanks to the regularity C w ([0, +∞); H)) for any t > 0. Hence, choosing t = t 2 in (1.15) and then choosing t = t 1 again in (1.15) and subtracting the equations obtained in this way we deduce that for every We estimate the integrals in the previous equality. Thanks to assumption (1.9) we obtain 2) It remains to estimate the last integral in (3.1). By (1.10) Hence, using the previous estimates in (3.1) we obtain Notice that being Ω bounded by (1.10) we get Hence, we deduce (using Sobolev inequality 2 ) where c 1 = ν 0 γ −q 0 c q Sob . By (3.3) and (3.6) it follows We recall that Hence, since Sobolev inequality:
where we have set Now, to conclude the proof we distinguish the following three cases: q = 2, q > 2 and q < 2.
If q = 2, since we are assuming that f ∈ (L q (Ω ∞ )) N 2 and g 2 ∈ L 1 (Ω ∞ ) it follows that g belongs to L 1 ((0, +∞)). Hence, applying Lemma 2.2 we deduce that the following estimate holds true for every t > 0 By (3.11) it follows estimate (1.17) when q = 2. If q > 2 the inequality (3.9) allows to apply Lemma 2.1 and to conclude that the following estimate holds true for every t > 0 g(s) ds (3.13) where g is as in (3.10) and By (3.13) it follows estimate (1.17) when q > 2. Finally, let us consider the case q < 2. By (3.9) we get for every t ∈ [0, T ] from which we deduce, for every 0 ≤ t 1 < t 2 ≤ T , where Λ 0 is the constant defined in (3.12). By the previous inequality and (3.9) we have  Hence, applying again Lemma 2.2 we obtain that for every t > 0 u ε (t) 2 L 2 (Ω) ≤ Λ 0 e −Bt + with c Sob the Sobolev's constant defined in (3.5) and γ 0 as in (1.10).

Remark 3.2.
We notice that by (3.3) and (1.17) it follows that for every 0 < t 1 < t 2 < +∞ the following estimates hold true t2 t1 (3.17) We conclude this section studying the particular case (1.7).

The special case (1.7)
We prove the following result.