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October  2012, 17(7): 2483-2508. doi: 10.3934/dcdsb.2012.17.2483

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

1. 

Department of Mathematics, National University of Defense Technology, Changsha 410073, China

2. 

Department of Mathematics, National University of Defense Technology, Changsha, 410073

Received  July 2011 Revised  February 2012 Published  July 2012

A 2D Stochastic incompressible non-Newtonian fluid driven by fractional Brownian motion with Hurst index $H \in (\frac{1}{2},1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Including the requirements of Nuclear and Hilbert-Schmidt operators, three kinds of condition, which ensure the existence and regularity of infinite-dimensional stochastic convolution for the corresponding additive linear stochastic equation, are summarized. Without the requirements of compact parameters, another condition is proposed for the existence and regularity of stochastic convolution. By any of the four kinds of condition, the existence and uniqueness of mild solution are obtained for the stochastic non-Newtonian fluid through a modified fixed point theorem in the selected intersection space. Existence of a random attractor is then obtained for the random dynamical system generated by non-Newtonian fluid.
Citation: Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
References:
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E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801. doi: 10.1214/aop/1008956692.

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P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.

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H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Quart. Appl. Math., 50 (1992), 559-583.

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F. Biagini, Y. Hu, B. Øksendal and T. Zhang, "Stochastic Calculus for Fractional Brownian Motion and Applications," Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2008.

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F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3.

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J. M. Borwein and P. B. Borwein, "Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

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G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.

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T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion, SIAM J. Math. Anal., 40 (2009), 2286-2315. doi: 10.1137/08071764X.

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T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250.

[13]

L. Fang, "Stochastic Navier-Stokes Equations with Fractional Brownian Motions," Ph.D thesis, Louisiana State University, 2009.

[14]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493. doi: 10.3934/dcdsb.2010.14.473.

[15]

M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2761-2782.

[16]

B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013.

[17]

I. V. Girsanov, On transforming a class of stochastic processes by absolutely continuous substitution of measures, Teor. Verojatnost. i Primenen., 5 (1960), 314-330.

[18]

B. Guo and C. Guo, The convergence of non-Newtonian fluids to Navier-Stokes equations, J. Math. Anal. Appl., 357 (2009), 468-478. doi: 10.1016/j.jmaa.2009.04.027.

[19]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[20]

H. Hurst, Long term storage capacity of reservoirs, Trans. Am. Soc. Civ. Eng., 116 (1951), 770-799.

[21]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," 2nd edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.

[22]

J. P. Kelliher, Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane, Pacific J. Math., 244 (2010), 99-132. doi: 10.2140/pjm.2010.244.99.

[23]

A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 115-118.

[24]

O. A. Ladyžhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Revised English edition, Gordon and Breach Science Publishers, New York-London, 1963.

[25]

W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson, On the self-similar nature of ethernet traffic, in "Conference Proceedings on Communications Architectures, Protocols and Applications,'' ACM, (1993), 183-193.

[26]

J. Li and J. Huang, Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter $H \in (1/4,1/2)$, preprint, arXiv:1107.2706.

[27]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I," Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[28]

B. B. Mandelbrot, The variation of certain speculative prices, The Journal of Business, 36 (1963), 394-419. doi: 10.1086/294632.

[29]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437. doi: 10.1137/1010093.

[30]

B. Maslowski and B. Schmalfuss, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607. doi: 10.1081/SAP-200029498.

[31]

J. Mémin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett., 51 (2001), 197-206. doi: 10.1016/S0167-7152(00)00157-7.

[32]

V. Pipiras and M. S. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli, 7 (2001), 873-897.

[33]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[34]

S. Tindel, C. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields, 127 (2003), 186-204. doi: 10.1007/s00440-003-0282-2.

[35]

C. Zhao and J. Duan, Random attractor for the Ladyzhenskaya model with additive noise, J. Math. Anal. Appl., 362 (2010), 241-251. doi: 10.1016/j.jmaa.2009.08.050.

[36]

C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001.

show all references

References:
[1]

E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801. doi: 10.1214/aop/1008956692.

[2]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.

[3]

H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Quart. Appl. Math., 50 (1992), 559-583.

[4]

F. Biagini, Y. Hu, B. Øksendal and T. Zhang, "Stochastic Calculus for Fractional Brownian Motion and Applications," Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2008.

[5]

F. Bloom and W. Hao, Regularization of a non-Newtonian system in an unbounded channel: Existence and uniqueness of solutions, Nonlinear Anal., 44 (2001), 281-309. doi: 10.1016/S0362-546X(99)00264-3.

[6]

J. M. Borwein and P. B. Borwein, "Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.

[7]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.

[8]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[9]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.

[10]

G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridge, 1996.

[11]

T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion, SIAM J. Math. Anal., 40 (2009), 2286-2315. doi: 10.1137/08071764X.

[12]

T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250.

[13]

L. Fang, "Stochastic Navier-Stokes Equations with Fractional Brownian Motions," Ph.D thesis, Louisiana State University, 2009.

[14]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493. doi: 10.3934/dcdsb.2010.14.473.

[15]

M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2761-2782.

[16]

B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013.

[17]

I. V. Girsanov, On transforming a class of stochastic processes by absolutely continuous substitution of measures, Teor. Verojatnost. i Primenen., 5 (1960), 314-330.

[18]

B. Guo and C. Guo, The convergence of non-Newtonian fluids to Navier-Stokes equations, J. Math. Anal. Appl., 357 (2009), 468-478. doi: 10.1016/j.jmaa.2009.04.027.

[19]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[20]

H. Hurst, Long term storage capacity of reservoirs, Trans. Am. Soc. Civ. Eng., 116 (1951), 770-799.

[21]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," 2nd edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.

[22]

J. P. Kelliher, Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane, Pacific J. Math., 244 (2010), 99-132. doi: 10.2140/pjm.2010.244.99.

[23]

A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 115-118.

[24]

O. A. Ladyžhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Revised English edition, Gordon and Breach Science Publishers, New York-London, 1963.

[25]

W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson, On the self-similar nature of ethernet traffic, in "Conference Proceedings on Communications Architectures, Protocols and Applications,'' ACM, (1993), 183-193.

[26]

J. Li and J. Huang, Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter $H \in (1/4,1/2)$, preprint, arXiv:1107.2706.

[27]

J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I," Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.

[28]

B. B. Mandelbrot, The variation of certain speculative prices, The Journal of Business, 36 (1963), 394-419. doi: 10.1086/294632.

[29]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437. doi: 10.1137/1010093.

[30]

B. Maslowski and B. Schmalfuss, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607. doi: 10.1081/SAP-200029498.

[31]

J. Mémin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett., 51 (2001), 197-206. doi: 10.1016/S0167-7152(00)00157-7.

[32]

V. Pipiras and M. S. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli, 7 (2001), 873-897.

[33]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[34]

S. Tindel, C. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields, 127 (2003), 186-204. doi: 10.1007/s00440-003-0282-2.

[35]

C. Zhao and J. Duan, Random attractor for the Ladyzhenskaya model with additive noise, J. Math. Anal. Appl., 362 (2010), 241-251. doi: 10.1016/j.jmaa.2009.08.050.

[36]

C. Zhao and S. Zhou, Pullback attractors for a non-autonomous incompressible non-Newtonian fluid, J. Differential Equations, 238 (2007), 394-425. doi: 10.1016/j.jde.2007.04.001.

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