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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 |
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. |
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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