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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 738729, 6 pages

http://dx.doi.org/10.1155/2014/738729

## Optimal Convergence Rates for Solutions of the Monopolar Non-Newtonian Flows

Department of System Science and Applied Mathematics, Kunming University of Science and Technology, Kunming 650083, China

Received 22 May 2014; Accepted 1 July 2014; Published 9 July 2014

Academic Editor: Bo-Qing Dong

Copyright © 2014 Zhi-qiang Luo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the optimal convergence rates for solutions of the monopolar non-Newtonian flows. By using the energy methods, the perturbed weak solution of perturbed system asymptotically converges to the solution of the original system with the optimal rates .

#### 1. Introduction

In this paper we consider the monopolar non-Newtonian flows which is governed by the following system: Here and denote the unknown velocity vector and pressure of the fluids, while is the given initial velocity and is the external force. is the stress tensor specified in the following form: for the symmetric deformation velocity tensor with where , are viscous coefficients.

The system (1) was proposed for the first time by Ladyzhenskaya [1] which describes viscous non-Newtonian flows such as the molten plastics, dyes, adhesives, paints, and greases. The constitutive of those fluids is nonlinear and does not the Stokes laws. It should be mentioned that the fluid considered here is monopolar since only first order derivative of the velocity field is involved. When , the system becomes the classic Navier-Stokes equations [2].

Due to the importance in both mathematics and physics, the well-posedness and large time behavior of incompressible fluids have attracted more and more attention [3–6]. We refer to some classic results on the Navier-Stokes equations [3]. For the viscous non-Newtonian flows (1), there is also a large literature on the well-posedness and asymptotic behaviors [7–10]. In particular, Pokory [11] and Bae [12] recently studied the existence and uniqueness for the Cauchy problem of the system (1). When by developing the Fourier splitting method, Dong and Li [13] explored the optimal algebraic decay rate in as It should be mentioned that the time decay properties for weak solutions of the non-Newtonian fluids (1) essentially show that the trivial solution is stable. It is desirable to consider the stability of the nontrivial solutions of the non-Newtonian fluids (1) with the nonzero and nondecay external force. To do so, we consider the perturbed non-Newtonian fluids where is the initial perturbation. It is easy to check that if the external force satisfies , the non-Newtonian fluids (1) has nontrivial stationary solution and is not a solution of (1) any more. In the two-dimensional case, Dong and Chen [14] have considered the stability issue as for weak solutions of the non-Newtonian fluids (1) under any perturbation . It should be mentioned that Dong and Chen [14] cannot study the optimal convergence rates. One may also refer to some interesting stability results on the classic Navier-Stokes equations. When , () with sufficiently small; da Veiga and Secchi [15] proved that there is a unique global solution of perturbed Navier-Stokes equations satisfying for the weak solution of the original Navier-Stokes equations in the subcritical class When , Ponce et al. [16] further considered the weak solution of the original Navier-Stokes equations under the critical space and obtained that if for the sufficient small constant , then there is a unique solution of perturbed Navier-Stokes equations satisfying where is a constant satisfying . One may also refer to some related results on the Newtonian and viscous non-Newtonian flows [17, 18].

In this paper, we will consider the stability for weak solution of three-dimensional non-Newtonian fluids (1) and will derive the optimal convergence rates. More precisely, we will show that every perturbed solution of the non-Newtonian fluids (2) converges asymptotically to of the non-Newtonian fluids (1) as

#### 2. Preliminaries and Main Results

Throughout this paper, stands for a generic positive constant which may vary from line to line. with denotes the usual Lebesgue space [19] of all integral functions associated with the norm We denote the Fourier transformation or as

We recall the following Gronwall inequality which will be used in the following argument.

Lemma 1 (Gronwall inequality [2]). *Let , , be nonnegative continuous functions and satisfy the following inequality:
**
where . Then
*

*We now give the definition for weak solutions of three-dimensional non-Newtonian fluids (1) (see [11]).*

*Definition 2. *Suppose , , , and is called a weak solution of the non-Newtonian fluids (1) if the following conditions hold true.(i) and for all .(ii) is weakly continuous from to .(iii) satisfies (1) in the weak sense; that is, for all (iv) satisfies energy inequality; for all

*The following existence and regularity of weak solutions of the incompressible non-Newtonian fluids (1) are due to the work of Pokory [11].*

*Lemma 3 (existence and regularity of weak solutions). Suppose , , and . Then there exists a weak solution of the incompressible non-Newtonian fluids (1). Moreover, the solution is regular; that is,
*

*Our results now read.*

*Theorem 4. Suppose and is a weak solution of the incompressible non-Newtonian fluids (1) with the finite energy initial data and the nondecay external force . Then for any initial perturbation , the perturbed non-Newtonian fluids (2) has a weak solution which asymptotically converges with the optimal convergence rate:
*

*Remark 5. *Compared with some stability results of the classic Navier-Stokes equations such as da Veiga and Secchi [15] and Ponce et al. [16], we do not have any small assumptions on the initial perturbation. Moreover, our results also improve the previous results obtained by Dong and Chen [17] since we not only consider this problem in three-dimensional case instead of the two-dimensional case, but also derive the optimal convergence rates (20).

*Remark 6. *The proof of Theorem 4 is mainly based on Fourier splitting technique. The nonlinear system is essentially also a dissipative system which may allow us to divide the frequency domain into two time-dependent subdomains; the subdomain yields a first-order differential inequality for the spatial norm of the Fourier transform of the weak solutions of incompressible non-Newtonian fluids (1). Compared with the derivation of Dong and Chen [17], the estimates of the nonlinear term can be dealt with in a satisfied form, for the three-dimensional case; however, the bounds of the nonlinear term become difficult. In order to overcome, we make full use of the regular properties of weak solutions and energy methods.

*3. Proof of Theorem 4*

*3. Proof of Theorem 4**According to the condition in Theorem 4 and the existence results of non-Newtonian fluids (1) obtained in [11], the existence of weak solutions of the perturbation of the perturbed incompressible non-Newtonian fluids (5) can be obtained by the parallel methods in [11]. Therefore, what remains is to prove the optimal convergence rates (20).*

*Let and be weak solutions of the original non-Newtonian fluids (1) and the perturbed non-Newtonian fluids (5), respectively. We denote by and the difference between the two weak solutions of (1) and (2); then satisfies the following system in the weak sense:
*

*In order to investigate the optimal convergence rates, we now need some a priori estimates of the nonlinear system (21).*

*Lemma 7. Under the same condition in Theorem 4, then the difference satisfies the following inequality:
*

*Proof of Lemma 7. *We first formally derive inequality (22). The rigorous derivation is obtained by considering the approximated solutions of the following approximate system:
Here the retarded modification of is defined by
where is a positive modifier. is the zero extension of the function which is originally defined for . For the similar estimates (22) of the approximate solutions , we only need to take the limit as . Therefore we now deal with this settle in the following argument only for weak solution directly.

Taking Fourier transformation of (23), it follows that by solving an ordinary equation
where

We now compute , , one by one. For , applying the divergence free properties of the velocity fields and Hölder inequality, it follows that
where we have used the bounds of , , ; that is,

For , one shows that

In order to estimate , we act the divergence operator to both sides of (21); it follows that
Now together with the estimates of and , can be estimated as

Hence plugging the estimates (27)–(31) of , , into (26) and then (25), we may obtain that
Thanks to
for
and the following facts due to the energy inequality
thus (32) is written as
which completes the proof of this lemma.

*Now we are in positive to prove (20). Taking the inner product of (21) with , it follows that
where we have used the following fact:
*

*Applying Hölder inequality, Gagliar-Nirenberg inequality, and Young inequality gives that
from which we rewrite (37) as
*

*Taking Parseval inequality into consideration
And then multiplying both sides by
Let
and then we divide the domain of the second integral in (37) into and ; that is to say,
Integrating in time from to gives
*

*By employing Lemma 7, we have after the direct computation
where we have used the following estimates of heat equations:
with .*

*Hence we insert (46) into (45) and take Gronwall inequality into consideration:
*

*Since
then by the interpolation inequality we have
Hence (48) implies
which completes the proof of Theorem 4.*

*Conflict of Interests*

*Conflict of Interests**The author declares that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgment*

*Acknowledgment**This work is partially supported by the Funds of Personnel Training of Kunming University of Science and Technology (no. KKSY201207019).*

*References*

*References*

- O. Ladyzhenskaya, “New equations for the description of the viscoue incompressible fluids and solvability in the large of the boundary value problems for them,” in
*Boundary Value Problem of Mathematical Physics V*, American Mathematical Society, Providence, RI, USA, 1970. View at Google Scholar - R. Temam,
*Navier-Stokes Equations*, North-Holland, Amsterdam, The Netherlands, 1977. - L. Caffarelli, R. Kohn, and L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier-Stokes equations,”
*Communications on Pure and Applied Mathematics*, vol. 35, no. 6, pp. 771–831, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Lin, “A new proof of the Caffarelli-Kohn-Nirenberg theorem,”
*Communications on Pure and Applied Mathematics*, vol. 51, no. 3, pp. 241–257, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Málek, J. Nečas, M. Rokyta, and M. Růžička,
*Weak and Measure-v alued Solutions to Evolutionary PDEs*, Chapman-Hall, New York, NY, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - B. Dong and Z. Zhang, “On the weak-strong uniqueness of Koch-Tataru's solution for the Navier-Stokes equations,”
*Journal of Differential Equations*, vol. 256, no. 7, pp. 2406–2422, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - B. Dong and Z. Chen, “Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows,”
*Discrete and Continuous Dynamical Systems A*, vol. 23, no. 3, pp. 765–784, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Dong and Z. Chen, “Regularity criteria of weak solutions to the three-dimensional micropolar flows,”
*Journal of Mathematical Physics*, vol. 50, no. 10, Article ID 103525, 13 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Dong and J. Song, “Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation,”
*Discrete and Continuous Dynamical System*, vol. 32, no. 1, pp. 57–79, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Qin, X. Liu, and X. Yang, “Global existence and exponential stability of solutions to the one-dimensional full non-Newtonian fluids,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 2, pp. 607–633, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Pokory, “Cauchy problem for the non-Newtonian viscous incompressible fluid,”
*Applications of Mathematics*, vol. 41, no. 3, pp. 169–201, 1996. View at Google Scholar · View at MathSciNet - H. Bae, “Existence, regularity, and decay rate of solutions of non-Newtonian flow,”
*Journal of Mathematical Analysis and Applications*, vol. 231, no. 2, pp. 467–491, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - B. Dong and Y. Li, “Large time behavior to the system of incompressible non-Newtonian fluids in R2,”
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 2, pp. 667–676, 2004. View at Google Scholar - B. Dong and Z. Chen, “Asymptotic stability of non-Newtonian flows with large perturbation in R
^{2},”*Applied Mathematics and Computation*, vol. 173, no. 1, pp. 243–250, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. B. da Veiga and P. Secchi, “${\text{L}}^{\text{p}}$-stability for the strong solutions of the Navier-Stokes equations in the whole space,”
*Archive for Rational Mechanics and Analysis*, vol. 98, no. 1, pp. 65–69, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - G. Ponce, R. Racke, T. C. Sideris, and E. S. Titi, “Global stability of large solutions to the $3$D Navier-Stokes equations,”
*Communications in Mathematical Physics*, vol. 159, no. 2, pp. 329–341, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Dong and Z. Chen, “Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation,”
*Nonlinearity*, vol. 19, no. 12, pp. 2919–2928, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Zhou, “Asymptotic stability for the 3D Navier-Stokes equations,”
*Communications in Partial Differential Equations*, vol. 30, no. 1-3, pp. 323–333, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Bahouri, J.-Y. Chemin, and R. Danchin,
*Fourier Analysis and Nonlinear Partial Differential Equations*, vol. 343 of*Fundamental Principles of Mathematical Sciences*, Springer, Berlin, Germany, 2011.

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