American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On a reaction-diffusion model for sterile insect release method with release on the boundary
  • DCDS-B Home
  • This Issue
  • Next Article
    Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model
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

Jin Li 1, and  Jianhua Huang 2,

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.
Keywords: random attractor., stochastic non-Newtonian fluid, Infinite-dimensional fractional Brownian motion, stochastic convolu- tion.
Mathematics Subject Classification: Primary: 35Q35, 35R60; Secondary: 60G22, 37L5.
Citation: Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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. Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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. Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503
[2]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[3]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
[4]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084
[5]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[6]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197
[7]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[8]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010
[9]

Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379
[10]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066
[11]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
[12]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[13]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[14]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[15]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[16]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015
[17]

Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077
[18]

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068
[19]

Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231
[20]

Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy