Dynamics of 2D Stochastic non-Newtonian fluids driven by fractional Brownian motion

A 2D Stochastic incompressible non-Newtonian fluids driven by fractional Bronwnian motion with Hurst parameter $H \in (1/2,1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Four groups of assumptions, including the requirement of Nuclear operator or Hilbert-Schmidt operator, are discussed. The existence and regularity of stochastic convolution for the corresponding additive linear stochastic equation are obtained under each group of assumptions. Mild solution are then obtained for the non-Newtonian systems by the modified fix point theorem in the selected intersection space. When the domain is square, the random dynamical system generated by non-Newtonian systems has a random attractor under some condition on the spectrum distribution of the corresponding differential operator.


Introduction
In this paper, we consider the following 2D stochastic non-Newtonian fluids driven by fractional Brownian motion (fBm, for short) with Hurst parameter H ∈ ( 1 2 , 1): ∂u ∂t + (u · ∇)u + ∇p = ∇ · (µ(u)e − 2µ 1 ∆e) + Φ dB H (t) dt x ∈ O, t > 0 (1.1) where O is a smooth bounded domain of R 2 , τ ij is the components of the stress tensor, τ ijk is the components of the first multipolar stress tensor, and p is the pressure. e ij are the components of the rate of deformation tensor, i.e. e ij = 1 2 ǫ, µ 0 , µ 1 > 1 and α, 0 < α ≤ 1, are constitutive parameters. The noise, modeled by the formal derivative of fBm, enters linearly in the equation. This model describes the simplest case when a 2D isothermal, nonlinear, incompressible bipolar viscous fluids perturbed by the noise with long range dependence.
There are many works concerning the unique existence, regularity and asymptotic behavior of solution to the incompressible non-Newtonian fluids or its associated versions (see e.g. [3,5,16,23,35,36]). For instance, Zhao and Duan [35] established the existence of random attractor for some non-Newtonian model with additive noise.
Noise are intrinsic effects in a variety of settings and spatial scales. It could be mostly obviously influential at the microscopic and small scales but indirectly it plays a vital role in microscopic phenomena. We study fBm other than the standard commonly used Wiener process as the source of noise. The fBm is a family of Gaussian processes that is indexed by the Hurst parameter H ∈ (0, 1). These processes with values in R were introduced by Kolmogorov [22] and some useful properties of these process were given by Mandelbrot and Van Ness [28]. For H = 1 2 the fBm is not a semi-martingale and the increments of the process are not independent. These properties can be used in modeling "cluster" phenomena (system with memory and persistence) such as hydrology [19], economic data [27] and telecommunications [24]. Since there are limited publish works ([13, 14, 17, 29, 34]) on infinite-dimensional fBm-driven equations, we first introduce the Wiener-type stochastic integral with respect to one-dimensional fBm in section 2.
Then we discuss the existence and regularity of infinite-dimensional stochastic convolution under several existed assumptions (such as the requirement of Nuclear operator or Hilbert-Schmidt operator) for this model. We also study a fundamental example by some subtle calculus on the spectrum of differential operator without compactness assumption on other parameter. Inspired by [26] ,in section 3 we obtain the mild solution for (1.1) by a modified fix point theorem which needs careful estimation on the selected intersection space. Since the fractional noise is not Markovian, the solution to (1.1) can not be expected to define a Markov process. Therefore, the approach to study the longtime behavior of solution via invariant measure is not an option here. In particular, Corollary 4.4 in [9] can not be applied here. Consequently, our analysis in dynamics is instead based on the framework of random dynamical system(RDS), which more or less requires the driving process to have stationary increments. Since the foundational work in [9,8] the long time behavior of SPDE has been extensively investigated by means of proving the existence of a random attractor (e.g. [2,15,18,35]). In this work we use the stationary generalized stochastic integral (known as fractional Ornstein-Uhlenback process) to construct RDS associated to (1.1) and obtain a random attractor under certain condition in section 4.
Compared with [35], the difficulties in our work are: (i) Unlike the finite-dimensional white noise, handling infinite-dimensional fBm needs addition assumptions on parameter (such as the requirement of Nuclear operator or Hilbert-Schmidt operator) and subtle estimation on spectrum of differential operator. (ii) Unlike the classical Ito integral with Brownian motion, the stochastic integrals with fBm are more complicated.
There are several type of integration for fBm and each only preserve part of the properties of integrator. (iii) The principal method for creating weak solutions to stochastic differential equations is transformation of drift via the Girsanov theorem (cf. [20]).
However, significant difficulties arise when the application of infinite-dimensional Girsanov theorem is used for this nonlinear fBm-driven stochastic fluid equations. Thus, we have to use the mild solution. (iv) We emphasize that the cocycle property of RDS has to be satisfied for any ω ∈ Ω. It is not sufficient if this equation is only true almost surely where the exceptional set may depend on space variable or time varialbe. Thus, we have to switch fBm to its incomplete equivalent canonical realization and restrict the fBm parameter H > 1 2 which allow us to solve the equation using Wiener-type stochastic integral for deterministic integrands understood in a pathwise way.
We express this problem by the standard mathematical setting. We use lowercase c i , i ∈ N for global constants and capital C for local constants which may change value from line to line. Denote According to [5] Lemma 2.3, we can use Lax-milgram Theorem to define A ∈ L(V, V ′ ): And we have (ii) Operator A is self-adjoint positive with compact inverse. By Hilbert Theorem, form an orthonormal basis for H.
Define the trilinear form: (1.10) and the functional B(u, v) ∈ V ′ : For the properties of operator A, B and N we refer to [36]. Comprehensively, we have the following abstract evolution equation from problem (1.1)-(1.4): (1.13) Without loss of generality, we set µ 1 = 1 in the sequel.

FBm and random dynamical systems
Since the derivative of fBm exists almost nowhere, we seek the solution in the integral form. There are several approach to define an integral for one-dimensional fBm and each has its advantage (for a useful summary we refer to [4]). In this paper, we adopt the Wiener integrals since they deal with the simplest case of deterministic integrands.
However, the assumption on noise driven by infinite dimensional fBm varies. Therefore, we first introduce the general framework of the Wiener-type stochastic integral with respect to infinite dimensional fBm, then discuss three sets of assumptions and obtain the desired result.
Let β H (t) be the one-dimensional fBm with Hurst parameter H. Throughout this paper we only consider the case H ∈ ( 1 2 , 1). For a survey of Winer-type stochastic integral we refer to [4]. By definition β H is a centered Gaussian process with covariance where β is a Wiener process, and K H (t, s) is the kernel given by c H is a constant and Denote by E the linear space of step function of the form where n ∈ N, a i ∈ R and by H the closure of E with respect to the scalar product For φ ∈ E we define its Weiner integral with respect to the fBm as is an isometry between E and the linear space span{β H (t), 0 ≤ t ≤ T } viewed as a subspace of L 2 (0, T ) and it can be extended to an isometry between H and the The image on an element Ψ ∈ H by this isometry is called the Wiener integral of Ψ with respect to β H .
Next we give a characterization of the so-called reproducing kernel Hilbert space H.
Consider space L 2 (0, T ) equipped with the twisted scalar product We have that H can be represented by the closed of L 2 (0, T ) with respect to the twisted scalar product. Namely, H = (L 2 (0, T ), <, > H ) = (E, <, > H ). In [31] it is shown that the elements of H may not be functions but distributions of negative order. By [30] we have the following inclusion L 2 (0, T ) ⊂ L 1/H (0, T ) ⊂ H. We now introduce the linear operator K * H defined on φ ∈ E as follows: We refer to [1] for the proof of the fact that K * H is an isometry between the space E and L 2 (0, T ) that can be extended to the Hilbert space H and L 2 (0, T ), i.e., H = (K * H ) −1 (L 2 (0, T )). As a consequence, we have the following relationship between the Wiener integral with respect to fBm and the Wiener integral with respect to the Wiener process: (2.14) We formally define the infinite dimensional fBm on H with covariance operator Q as where {β H i (t)} i∈N is a sequence of real stochastically independent one-dimensional fBm's. This process, if convergence, is a H-valued Gaussian process, it starts from 0, has zaro mean and covariance where β i is the standard Brownian motion used to represent β H as in (2. Back to non-Newtonian systems. Our goal is to find a mild solution of problem Here S(t) := e −tA = ∞ 0 e −tλ dE λ , is an analytic semigroup generated by A Since A is a densely defined self-adjoint bounded-below operator in Hilbert space H and hence a sector operator (see, for instance, [12] section 1.3). The first and second integral are Bochner integral, the last integral is the Wiener-type stochastic integral defined by (2.17). By the form of integral equation, we care the stochastic convolution Then z, if it is well defined, is the unique mild solution of the following linear stochastic evolution equation In order to obtain the solution of non-Newtonian systems driven by fBm we only need to guarantee the existence and regularity of the stochastic Wiener-type convolution (see the proof of Theorem 3.4 ). There are three groups of assumption on stochastic convolution: Here we use notation: L 1 (H) the space of all nuclear operators on H; L 2 (H) the space of all Hilbert-Schmidt operators on H (for detailed see Appendix C in [10]).
These assumptions come from Maslowski, Schmalfuss [29] and Duncan, Maslowski, Pasik-Duncan [14] and Tindel, Tudor, Veins [34] respectively. We first give a general consequence about the existence and regularity of stochastic convolution under each of these assumptions, then remark it.
The assumption (A4) and the corresponding proof we will state later.
Under assumption (A1): is indeed a H-valued process with covariance Q since Q ∈ L 1 (H). In order to apply the theory in [29] we only need to verify that the semigroup S is analytic and exponentially stable, that is, And this is straightforward due to the properties of operator A. Thus we can use Proposition 3.1. in [29] to obtain that the stochastic convolution z is well defined by the variation of constants formula and it has a V -continuous modification.
Under assumption (A2): Sometimes the assumption that Q is nuclear is not convenient. In this case we consider the genuine cylindrical fBm (i.e. Q ≡ id H ). And this standard cylindrical fBm can be represented by the formal series that does not converge a.s. in H. Since Φ ∈ L 2 (H) and S is an analytic semigroup, we can use Proposition 2.6 in [14] to obtain a stronger conclusion. That is, let α < H − 1 2 then the stochastic convolution z has a C α ([0, T ]; D(A 1 2 )) version. In particular, there Under assumption (A3): Sometimes the assumption that Φ is Hilbert-Schmidt is not necessary. We can relax it to the case ΦΦ * ∈ L 1 (H). Since A −2H ∈ L(H), we have Thus, the assumptions in Theorem 1 [34] are fulfilled and we can deduce that z is well-defined and belongs to L 2 (Ω, H). For the regularity of z, we need to check the condition of Theorem 4 in [34].
Remark 2.3. No matter which assumption holds, the key point is the concept of compactness which guarantee us to handel the infinite-dimensional problem in a finitedimensional manner. Actually, the nuclear operators (elements of L 1 (H)) are compact.
The Hilbert-Schmidt operators (elements of L 2 (H)) are compact too.
Next we discuss the fundamental example in which the boundary is square and the parameter Q ≡ Φ ≡ id H (!they are not compact). That is Before proving the existence and regularity of stochastic convolution z, we state the following lemma in [25] (which is based on [21]) about the spectrum of operator A. proof of Proposition 2.2 under assumption (A4). The existence part is based on Theorem 1 in [34]. Let us estimate the mean square of z.
By the change of variable x = u − v and y = λ i x, we get where Γ(s) is Gamma function, β D (s) is the Dirichlet beta function and ζ(s) is the Riemann zeta function (for definition see [6]). This yields the existence of z. Since the proof of regularity is a modification of Lemma 5.13 in [11] and Proposition 3.1 in [29], we omit it.
A fundamental concept in the theory of random dynamical system is the notion of metric dynamical system. It is a model for a noise which is the source of perturbation of a dynamical system. We now recall the notions of RDS. For details we refer to [9].
A is said to absorb B if P-a.s. there exists an absorption time t B (ω) such that for The Ω-limit set of a random set K is defined by Definition 2.8. A random attractor for an RDS ϕ is a compact random set A satisfying P-a.s.: (i) A is invariant, i.e. ϕ(t, ω)A(ω) = A(θ t ω) for all t > 0.
(ii) A attracts all deterministic bounded sets B ⊂ E.
The following proposition (cf. [9] Theorem 3.11) yields a sufficient criterion for the existence of a random attractor. We have following estimates.
Lemma 3.1. J 1 : X → X and for all u, v ∈ X, we have Proof. According to [5] Lemma 2.6, we have Then J 1 (u) is a weak solution of the following linear differential equation Integrating with respect to t over [0, t], we get we have Next we prove (3.4). For u, v ∈ X, let w = J 1 (u) − J 1 (v). Then w is the weak solution of By the same argument we have (3.14) Hereafter we estimate |B  Finally, (3.17) Lemma 3.2. J 2 : X → X and for all u, v ∈ X, we have Proof. For all u ∈ X, we have N (u) ∈ L 2 (0, T ; V ) (see [5] Lemma 2.6). As the proof of Lemma 3.1, we can prove that J 2 maps X into X, J 2 (u) is the weak solution of J(0) = 0 (3.21) and the following estimate holds: Therefore, Next we prove (3.19).
Similar to Lemma 3.1, we have For all α ∈ (0, 1) and φ ∈ V , we have Then the first order Fréchet derivative of F (s) is Consequently, Similarly, the second order Fréchet derivative of F (s) is a three-dimensional matrix where F i (s) = 2µ 0 (ǫ + |s| 2 ) −α/2 s i . By some computation we see that where c 4 is a positive constant depending on µ 0 , ǫ and α. For all a, b ∈ R 4 , Taking a = e(u) = (e ij (u)), b = e(v) = (e ij (v)), applying the integration by parts and the above inequality about F (s), we have (by the Sobolev interpolation theorem). (3.38) Finally, (3.39) We apply the following version of the contraction mapping theorem.
then the equation has a unique solution u ∈ E satisfying u ∈ B E (M ).
We now prove the main result of this paper. all ω ∈ Ω in the sense of (2.18).
In the rest part of this section we obtain a priori estimates and global existence.
Denote by u, the local solution of (2.18) over [0, is the mild solution of equation Therefore, v(t) is the weak solution of the following differential equation with random parameters: where c 5 and c 6 are positive constants depending on λ 1 and O, g 1 is an integrable function depending on z. (3.53) The above inequality take advantage of < N (v), v > ≥ 0 (see [5]) and orthogonality property of b (see [33] (2.21)). In the sequel we omit the time variable t. Firstly we where C 2 is a positive constant which will be specified later. Secondly we estimate nonlinear term N . For all r 1 > 0, we have (3.55) Comprehensively, where λ 1 is the first eigenvalue of operator A. Let g 1 = C 1 C 2 |z| 2 |z| 2 2C 1 and r 1 small enough such that By Gronwall Lemma we have and inequality (3.52) follows. The proof is complete.
Since z ∈ C([0, T ]; V ), the following theorem is an immediate consequence of theorem 3.4 and proposition 3.5.

Random attractor
In the sequel we aim to obtain a random attractor for RDS generated by equation (2.18) under assumption (A4). Denote To show that the limit exists, we have we have as H > 1 2 . Thus, Z (the so-called fractional Ornstein-Uhlenback process) is the unique stationary solution of the linear stochastic evolution equation (4.6) We need the stationary process Z to construct the RDS because when we investigate the long-time behavior of solution we encounter a generalized integration over time For all t 0 ∈ R, by Theorem 3.4, u(t, ω; t 0 , u 0 ) is the unique solution of the equation  In this section, let u(t, ω; t 0 ) = v(t, ω; t 0 ) + Z(t, ω), we have  Then v is the weak solution of the following differential equation We can now define an continuous mapping by setting (4.14) The measurability follows from the continuity dependence of solution with respect to initial value. the cocycle property follows from the uniqueness of solution for all noise path ω ∈ Ω. Thus, ϕ is a RDS associated with (2.18). In the rest of this section, we will compute some estimates in spaces H and V . Then we use these estimates and compactness of the embedding V ֒→ H to obtain the existence of a compact random attractor.
Thus, we have In the following we consider a bound of   such that for all M > 0 and |u 0 | < M , there exists t 2 (ω) < −1 such that P-a.s.