GEVREY CLASS REGULARITY FOR THE GLOBAL ATTRACTOR OF A TWO-DIMENSIONAL NON-NEWTONIAN FLUID*

Caidi ZHAO (趙才地) Zehan LIN (林澤瀚)

Department of Mathematics，Wenzhou University，Wenzhou 325035，China E-mail:zhaocaidi2013@163.com;zhaocaidi@wzu.edu.cn;linzehan97@163.com

T.Tachim MEDJO

Department of Mathematics，F(xiàn)lorida International University，DM413B University Park，Miami，F(xiàn)L 33199，USA E-mail:tachim@fiu.edu

Abstract This article investigates Gevrey class regularity for the global attractor of an incompressible non-Newtonian fluid in a two-dimensional domain with a periodic boundary condition.This Gevrey class regularity reveals that the solutions lying in the global attractor are analytic in time with values in a Gevrey class of analytic functions in space.

Key words non-Newtonian fluid;global attractor;Gevrey class regularity

1 Introduction

From the theory of fluid dynamics，the motion of an incompressible n-dimensional fluid can be described by the equations

Hereafter，the scalar function p is the pressure，δjkis the Kronecker delta，andis the viscous extra stress tensor，which is a function of the rate of the deformation tensor e (u)=(ejk(u))n×n，where

The incompressibility of the fluid implies that?·u=0.

The theory of multipolar material was first formulated by Green and Rivlin[1，2].Later，Bellout et al.[3]and Nec?s and?ilhavy[4]developed the mathematical theory of multipolar viscous fluids.The constitutive relations for an isothermal，nonlinear viscous bipolar fluid，which were introduced in[3]，have the form

where|e (u)|2=and Tjkmis the component of the first multipolar stress tensor.In addition，∈，μ0，μ1and α are constitutive parameters.For more details，we refer to[5，page 13]for the definition of a non-Newtonian fluid and to[6，page 8]for the definition of a bipolar fluid，as well as for the physical background of the non-Newtonian fluid.

For brevity，we set

μ(u)=2μ0(∈+|e (u)|2)-α/2，

then equations (1.1)-(1.3) and the relations (1.4) yield the following nonlinear partial differential equations (PDEs for short) describing the motions of an incompressible non-Newtonian fluid:

in Ω×[0，T)，Ω?Rn，T＞0，with initial value

One often assumes one of two boundary conditions.The first boundary condition is the spaceperiodic case，which is the one considered in this article.We assume that the fluid fills the entire space Rnbut with the condition that u，f and p are periodic in each direction 0xj，j=1，2，···，n，with corresponding periods Lj＞0.In this case，we use Ω to denote the period:

Ω=(-L1/2，L1/2)×(-L2/2，L2/2)×···(-Ln/2，Ln/2)，

and we consider the spatially periodic solutions of (1.5)-(1.7).Let{e1，e2，···，en}be the natural basis of Rn.The spatial periodic conditions associated to (1.5)-(1.7) are

The second boundary condition corresponds to the case where Ω?Rnis a bounded domain.In this situation，(1.5)-(1.7) is supplemented with the boundary conditions

There are several works pertaining to the existence，uniqueness，regularity and long-time behavior of solutions to equations (1.5)-(1.9) and (1.5)-(1.7) with (1.10)，or to related versions (see e.g.[3，5，7-22].For example，Bloom and Hao in[9]proved the existence of a maximal attractor for equations (1.5)-(1.7) with (1.10) in two-dimensional (2D)，unbounded，channel-like domains.Later，Zhao and Li in[23]proved that the global attractor obtained by[9]has H2-regularity.The existence of the H2-regular attractor implies the asymptotic smoothing effect of this non-Newtonian fluid in the sense that the solutions become eventually more regular than the initial data.

In the present article，we investigate equations (1.5)-(1.9) in a 2D periodic case，where the period is

Ω=(-L1/2，L1/2)×(-L2/2，L2/2).

By using the argument of[23，24]，one can easily prove the existence and H2-regularity of the global attractor (denoted by) A for the solution semigroup associated to equations (1.5)-(1.9) in the 2D periodic case.We will concentrate our attention on the properties of the solutions lying in the global attractor A.

The goal of this article is to prove a stronger regularity property for the solutions within the global attractor A.More precisely，we will prove the Gevrey class regularity for the global attractor A.Our main result shows that the solutions of equations (1.5)-(1.9) within A are analytic in time with values in a Gevrey class of analytic functions in space.This result also reveals the exponential decay rate of the Fourier coefficients to each solution with respect to the wavenumber.

The Gevrey class regularity for solutions of the Navier-Stokes equations in two and three dimensional periodic cases was first investigated by Foias and Temam in[25]，and then was studied systematically by Foias et al.in their monograph[26].Nowadays，the ideas and methods of[25，26]have been widely used to study the Gevrey class regularity for the solutions of nonlinear PDEs;see for instance[27，28]for the 3-D Navier-Stokes equations，[29]for the time-dependent Ginzburg-Landau equations，[30]for the Navier-Stokes equations on the rotating 2-D sphere，[31]for the nonlinear analytic parabolic equations，[32]for a class of water-wave models，[33]for the Navier-Stokes-Voight equations，[34]for the Euler equations，[35，36]for the Kuramoto-Sivashinsky equation，[37]for the laser equations，[38，39]for the 3-D Bénard convection in porous medium，[40，41]for the second-grade fluid equations，[42]for the micropolar fluid equations，and[43，44]for the Navier-Stokes-α equations，etc..

We should remark that our idea concerning the Gevrey class regularity for the global attractor A originates from references[25，26].However，we want to point out that the nonlinear term involved in the non-Newtonian fluid equations is stronger than the one in the Navier-Stokes equations.Compared to the Navier-Stokes equations，the non-Newtonian fluid equations contain the additional nonlinear term?·(μ0(∈+|e (u)|2)-α/2e (u)).This nonlinear term leads to an additional difficulty in deriving the estimates of the corresponding nonlinear operator in the Gevrey class space.

The rest of this article is organized as follows:in Section 2，we first introduce some notations and operators.Then we rewrite (1.5)-(1.9) in an abstract form and show the well-posedness of this problem，as well as the existence and H2regularity of the global attractor A for the corresponding solution semigroup.In Section 3，we prove the Gevrey class regularity for the global attractor.

2 Notations and Preliminary Results

Throughout this article，we denote by R，R+，N and Z the sets of real，positive axis，natural numbers and integer numbers，respectively.Letbe the space of 2D vector functions u=u (x) which are defined for all x∈R2and are Ω-periodic in the sense that they are Lj-periodic in each direction 0xj(j=1，2)，and which belong to (L2(O))2for every bounded open set O?R2.Then we define the Sobolev space

For any u，v∈Vper，we set〈B (u，v)，w〉=b (u，v，w)，?w∈Vper，and

B (u)=B (u，u)，u∈Vper.

From the above definitions，one can check that the operators B (·) and N (·) are continuous from Vperto，and that A is a linear continuous operator both from Vpertoand from D (A) to Hper，where

Some useful estimations and properties for the operators A，b (·，·，·)，B (·) and N (·) can be established as those as in the works[8].For brevity，we present them as follows，and omit the proofs here:

Lemma 2.1There are some positive constants ci(i=1，2) depending only on Ω such that

In addition，if u∈D (A)，then N (u) can be extended to Hpervia

Using the notations and operators introduced above，we can express the weak version of equations (1.5)-(1.9) in the solenoidal vector field as follows:

We next specify the definition of solutions to equations (2.7)-(2.8).

Definition 2.2A global weak solution of equations (2.7)-(2.8) is a function

u∈L2(0，+∞;Hper)∩L2(0，+∞;Vper)∩L∞(0，+∞;Hper)，

with u (x，0)=u0，such that (2.7) holds in the distribution sense D′(0，+∞;).If u is a global weak solution and u∈L2(0，+∞;Vper)∩L2(0，+∞;D (A))∩L∞(0，+∞;Vper)，then u is called a global strong solution.

We end this section by recalling some results on the existence and uniqueness of global solutions to equations (2.7)-(2.8)，as well as the existence and H2regularity of the global attractors for the associated solution semigroup in spaces Hperand Vper，respectively.

Theorem 2.3Assume that∈＞0，μ0＞0，μ1＞0 and that α∈(0，1).Then，

(I) If f∈L2(0，+∞;Hper)，for any given u0∈Hper，equations (2.7)-(2.8) possess a unique global weak solution;for any given u0∈Vper，equations (2.7)-(2.8) possess a unique global strong solution.

(II) If f∈Hperis independent of time t，the solution operators

generate a continuous semigroup{S (t)}t≥0in spaces Hperand Vper，respectively，and the semigroup{S (t)}t≥0possesses a global attractor AHsatisfying

(a)(Compactness) AHis compact in Hper;

(b)(Invariance) S (t) AH=AH，?t∈R+;

(c)(Attractivity) for any bounded set BH?Hper，

Also{S (t)}t≥0possesses a global attractor AVsatisfying

(i)(Compactness) AVis compact in Vper;

(ii)(Invariance) S (t) AV=AV，?t∈R+;

(iii)(Attractivity) for any bounded set BV?Vper，

Furthermore，

ProofThe assertion (I) can be proven by arguments similar to those of Bloom and Hao[8，9]，and the assertion (II) can be established by the analogous approaches of Zhao and Li[23，24]，with the spaces H and V replaced by Hperand Vper，respectively.

From (2.9) we see that the global attractors AHand AVcoincide with each other.Thus we denote them by the same notation A for the rest of this paper.Note that (II)(a)，(II)(i) and (2.9) imply that there exists some fixed ρ＞0 such that

Indeed，(2.9) indicates that the global attractor A?Hperis a compact subset of the space Vper;this regularity result of the global attractor implies the asymptotic smoothing effect of the fluid in the sense that the solutions eventually become more regular (belonging to Vper) than the initial value (belonging to Hper).This asymptotic smoothing effect will play an important role when we investigate the Gevrey class regularity of the global attractor. □

3 Gevrey Class Regularity of the Global Attractor

The aim of this section is to investigate the Gevrey class regularity of the solutions lying on the global attractor A.We first introduce the Gevrey space in which we will work.Then we prove two estimates for the nonlinear terms B (u) and N (u) in the Gevrey space.These two estimates，especially the second one，play a key role in the proofs of the Gevrey class regularity.

To each index k=(k1，k2)∈Z2，we can associate wavenumbers k1/L1，k2/L2.For notational simplicity，we set.For the space-periodic problem under consideration here，the eigenvalues of the operator A have the form (see[46])

We will work with complex representations，for which we take i=.Then a square integrable and periodic vector function u=u (x) with x∈R2can be represented by the Fourier expansion

where the convergence of this expansion is in the L2norm and the amplitudeof each set of frequency k belongs to C2.By the Parseval identity，we have that

and hereinafter，|Ω|=L1L2denotes the“volume”of the domain Ω.Since we consider the vector field u=u (x) to be real-valued and the space to be divergence-free，we have that

Taking (3.1)-(3.2) into account，the spaces Hperand Vpercan be represented as

We can check that the operator A defined by (2.1) is elliptic，self-adjoint and positive definite.In fact，A is just the mapping

We endow the domain D (As) with the inner product

where we have used the Parseval identity and the orthogonality of the functionsin the second equality of (3.3).Obviously，D (A1/2)=Vperand‖A1/2·‖=.

For the Gevrey space，we consider the functions that can be represented locally by its Taylor series expansion.This motivates the definition of the analytic Gevrey class D (exp (σAs)，which consists of functions such that‖exp (σAs)) u‖＜+∞，where exp (σAs) is defined using the power series for exponentials

and in the Fourier space as

We endow the space D (exp (σAs)) with the following inner product:

Thus D (exp (σAs)) is actually a Hilbert space.In what follows，we will be mostly concerned with the space D (exp (σA1/4)).

Another Gevrey-type space that we will consider is D (A1/2exp (σA1/4)).For u，v∈D (A1/2exp (σA1/4))，the inner product and norm of D (A1/2exp (σA1/4)) are defined as

Then D (A1/2exp (σAs)) is also a Hilbert space.

Next we estimate the nonlinear terms N (u) and B (u) in the Gevrey space.

Lemma 3.1For some σ＞0，let u，v∈D (Aexp (σA1/4)).Then

We also denote that

Next we estimate I4.In a fashion similar to (3.6)，we have that

Then (3.7)，(3.9) and (3.10) give

where we have used the orthogonality of the functions.Analogously，we can obtain

It then follows from (3.8) and (3.11)-(3.14) that

Therefore，we have that

In fact，we can check that

u*(x)=exp (σA1/4) A1/2u (x)，v*(x)=exp (σA1/4) Av (x).

Therefore，(3.15) and H?lder’s inequality give (3.5).The proof is complete. □

For the estimate of the nonlinear term B (u，v) in the Gevrey space，we have

Lemma 3.2([36]) Let σ＞0 and let u，v，w∈D (Aexp (σA1/4)).Then B (u，v)∈D (exp (σA1/4)) and

We are going to investigate the Gevrey class regularity for the solutions lying in the global attractor A.

Theorem 3.3Assume that∈＞0，μ0＞0，μ1＞0，α∈(0，1) and f∈D (exp (σ1A1/4)) for some σ1＞0.Then，

(i) For any solution u lying in A，there exists some T*＞0 such that

is analytic on (0，T*)，with φ(t)=min{t，σ1，T*}.

(ii) All solutions u∈A are analytic on (T*，+∞) with values in D (A1/2exp (σ2A1/4)) for some σ2＞0 and T*，as before.

ProofWe first point out that the computations and estimates to follow are formal.However，as usual，these computations and estimates can be made rigorous by considering the Galerkin approximations of the solutions.For the given σ1＞0 such that f∈D (exp (σ1A1/4))，we denote ψ(t):=min{t，σ1}，and then have that

(3.16) indicates that ψ′(t) exists for a.e.t∈R+and|ψ′(t)|≤1.At time τ，we take the inner product of (2.7) with Au (τ) in D (exp (ψ(τ) A1/4)) to obtain

Using H?lder’s inequality，Young’s inequality，(3.16) and the Poincaré-type inequality

Now Lemma 3.1，Lemma 3.2 and Cauchy’s inequality give

Since f∈D (exp (σ1A1/4)) and ψ≤σ1，(3.17)-(3.23) yield

Notice that since u (t)∈A，we have by (2.10) that‖A1/2u (t)‖2=≤ρ2for t∈R+.In particular，‖A1/2u0‖2=‖A1/2u (0)‖2≤ρ2.Therefore，if we set

The assertion of Theorem 3.3(i) is proven.

Obviously，we can repeat the argument above at any initial time t＞0.More precisely，we choose the initial time t=T*and set φ(t)=min{σ1，T*}=σ2.Then，similarly to (3.25)，we have that

Repeating the above argument for all t＞T*，we obtain

u (t)∈D (A1/2exp (σ2A1/4)) for t＞T*.

The proof of Theorem 3.3 is now complete. □

Next we consider the complexified equations of equation (2.7) and extend the results of Theorem 3.3 to the case of a complex time variable z and a complex vector function u (z).To this end，we complexify the spaces Hper，Vper，and D (A)，and denote the complexified spaces as Hper，C，Vper，C，and D (A)C，respectively.We will keep the same notations for the inner product and norms and for the extensions of the operators A，B (·) and N (·) to these spaces.For notational simplicity，in the computations to come we drop the subscript C.

Equation (2.7) can be written for the complex time variable z∈C as

We consider the time z=seiθfor s＞0 and fix θ∈(-π/2，π/2).Later we will restrict θ to the interval[-π/4，π/4].At the initial time z=0，the solution u (x，0)=u0lies in Vperbut not necessarily for any Gevrey-type space.

Theorem 3.4Assume that∈＞0，μ0＞0，μ1＞0，α∈(0，1) and that f∈D (exp (σ2A1/4)) for some σ2＞0.Then there exists some T*＞0 such that

(I) all the solutions u∈A can be extended to a solution of the complexified non-Newtonian fluid equation (3.27)，and the function

is analytic in a neighborhood of (0，T*) in the complex plane.

(II) all the solutions u∈A belong toand are analytic in a neighborhood of (，+∞) in the complex plane.

ProofLet us first check that for a given u0∈Vper，the complex equation (3.27) possesses a unique complex-valued solution∈Vperwhich is analytic in the neighborhood of R+，and that the restriction ofto the positive half-axis is the strong solution of equation (2.7) in R+.The proof is composed of three steps:the construction of Galerkin approximation solutions，the showing of a priori estimates，and demonstration of the fact that the Galerkin approximation solutions are analytic and pass to the limit.We omit the details，and note that one can refer to[42]for a similar discussion regarding the micropolar fluid equations.Here we want to remark that estimates to follow are first obtained for the Galerkin approximations of the solutions.Then we pass to the limit and derive that the strong solutions satisfy the same estimates as their Galerkin approximations.

For a given σ2＞0 such that f∈D (exp (σ2A1/4))，we denote

φ(t)=min{t，σ2}for t＞0，

and then

(3.28) indicates that φ′(t) exists for a.e.t∈R+and

Note that z=seiθwith s＞0 and u=u (seiθ).We have

Now，taking the inner product of (3.27) with Au (seiθ) in D (exp (φ(scosθ) A1/4))，multiplying the resulting equality by eiθ，and then taking the real part，we get that

Now Lemma 3.1 and Young’s inequality yield

and Lemma 3.2 and Young’s inequality give that

We also have that

Moreover，H?lder’s inequality，(3.19)，(3.29) and Young’s inequality yield

It then follows from (3.30)-(3.34) that

We now restrict θ∈[-π/4，π/4].Note that f∈D (exp (σ2A1/4)) and φ≤σ2.Therefore，(3.35) gives

From (2.10) we see that for any solution u within the global attractor A，we have

‖A1/2u‖2≤ρ2，?t∈R+.

Thus (3.37) implies that

The assertion of Theorem 3.4(I) is proven.

If we start at time t0≥0，we can obtain the following estimate，which is similar to (3.38):

Now we set

Then，since t0≥0 is arbitrary，it follows that

The proof of Theorem 3.4 is complete. □

Corollary 3.5For any solution u∈A，the Fourier coefficient?ukof u has the following exponential decay rate:

ProofFrom the estimate (3.39) and the Fourier series characterization (3.4) of the space，we obtain

It is straightforward to obtain (3.40) from (3.41). □

Remark 3.6If α≥1，then for each u0∈H，the corresponding global weak solution to problem (2.3)-(2.4) is not unique.In this case，the addressed non-Newtonian fluid possesses a trajectory attractor Atr(cf.[47]).We can also discuss the Gevrey class regularity of Atr.Further more，we can use the translation semigroup and Atrto construct the trajectory statistical solutions and study their regularity (see e.g.[48-55]).