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Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system
Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation
1. | School of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China, China |
2. | Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616 |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
D. Blömker, M. Hairer and G. A. Pavliotis, Stochastic Swift-Hohenberg equaion near a change of stability, Proceedings of Equadiff, 11 (2005), 27-37. |
[3] |
D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity, J. Dyn. Differ. Equ., 22 (2010), 677-695.
doi: 10.1007/s10884-009-9145-6. |
[4] |
D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations, vol.3 of Interdisciplinary Mathematical Sciences. World Scientific Publishing, Singapore, 2007.
doi: 10.1142/9789812770608. |
[5] |
G. Chen, J. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations, J. Math. Phys., 52 (2011), 072702, 14pp.
doi: 10.1063/1.3614777. |
[6] |
M. D. Chekroun, H. H. Liu and S. H. Wang, Stochastic Parameterizing Manifolds and non-Markovian Reduced Equations-Stochastic Manifolds for Nonlinear SPDEs II, Springer Briefs in Mathematics, Springer-Verlag, New York, 2015.
doi: 10.1007/978-3-319-12520-6. |
[7] |
I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002.
doi: 10.1007/b83277. |
[8] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[9] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Rel. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[10] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, New York, 1992.
doi: 10.1017/CBO9780511666223. |
[11] |
J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic differential equations, Ann. Probab., 31 (2003), 2109-2135.
doi: 10.1214/aop/1068646380. |
[12] |
J. Duan and K. Lu, Smooth stable and unstable manifolds for stoahcstic partial differential equations, J. Dyn. Diff. Equ., 16 (2004), 949-972.
doi: 10.1007/s10884-004-7830-z. |
[13] |
J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, London, 2014. |
[14] |
J. P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem, Commun. Math. Phys, 136 (1991), 285-307.
doi: 10.1007/BF02100026. |
[15] |
H. Fu, X. Liu and J. Duan, Slow manifolds for multi-time-scale stochastic evolutionary systems, Commun. Math. Sci., 11 (2013), 141-162.
doi: 10.4310/CMS.2013.v11.n1.a5. |
[16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Varlag, Berlin, 1981. |
[17] |
D. Y. Hsieh, A. Q. Tang and X. P. Wang, On hydrodynamics instabilities, chaos, and phase transition, Acta Mech. Sin., 12 (1996), 1-14.
doi: 10.1007/BF02486757. |
[18] |
M. F. Hilali, S. Metens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg equation, Phys. Rev. E, 51 (1995), 2046-2052. |
[19] |
P. Imkeller and A. Monahan, Conceptual stochastic climate dynamics, Stoch. Dynam., 2 (2002), 311-326. |
[20] |
G. Lin, H. Gao, J. Duan and V. J. Ervin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, J. Math. Phys., 41 (2000), 2077-2089.
doi: 10.1063/1.533228. |
[21] |
A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains: Existence and comparison, Nonlinearity, 8 (1995), 734-768.
doi: 10.1088/0951-7715/8/5/006. |
[22] |
J. Oh, J. M. Ortiz de Zárate, J. V. Sengers and G. Ahlers, Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bnárd convection, Phys. Rev. E, 69 (2004), 021106. |
[23] |
J. Oh and G. Ahlers, Thermal-noise effect on the transition to Rayleigh-Bnárd convection, Phys. Rev. Lett., 91 (2003), 094501. |
[24] |
A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Varlag, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[25] |
I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers and H. R. Brand, Thermally induced hydrodynamic fluctuations below the onset of electroconvection, Phys. Rev. Lett., 67 (1991), 596-599.
doi: 10.1103/PhysRevLett.67.596. |
[26] |
X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical systems, J. Math. Phys., 51 (2010), 042702, 12pp.
doi: 10.1063/1.3371010. |
[27] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
[28] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York Berlin Heidelberg, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[29] |
W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701, 14 pp.
doi: 10.1063/1.2800164. |
[30] |
W. Wang, J. Sun and J. Duan, Ergodic dynamics of the stochastic Swift-Hohenberg system, Nonlinear Analysis: Real World Appl., 6 (2005), 273-295.
doi: 10.1016/j.nonrwa.2004.08.009. |
[31] |
E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics, IMA, Vol. 140, Springer, New York, 2005.
doi: 10.1007/978-0-387-29371-4. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
D. Blömker, M. Hairer and G. A. Pavliotis, Stochastic Swift-Hohenberg equaion near a change of stability, Proceedings of Equadiff, 11 (2005), 27-37. |
[3] |
D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity, J. Dyn. Differ. Equ., 22 (2010), 677-695.
doi: 10.1007/s10884-009-9145-6. |
[4] |
D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations, vol.3 of Interdisciplinary Mathematical Sciences. World Scientific Publishing, Singapore, 2007.
doi: 10.1142/9789812770608. |
[5] |
G. Chen, J. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations, J. Math. Phys., 52 (2011), 072702, 14pp.
doi: 10.1063/1.3614777. |
[6] |
M. D. Chekroun, H. H. Liu and S. H. Wang, Stochastic Parameterizing Manifolds and non-Markovian Reduced Equations-Stochastic Manifolds for Nonlinear SPDEs II, Springer Briefs in Mathematics, Springer-Verlag, New York, 2015.
doi: 10.1007/978-3-319-12520-6. |
[7] |
I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002.
doi: 10.1007/b83277. |
[8] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[9] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Rel. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[10] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, New York, 1992.
doi: 10.1017/CBO9780511666223. |
[11] |
J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic differential equations, Ann. Probab., 31 (2003), 2109-2135.
doi: 10.1214/aop/1068646380. |
[12] |
J. Duan and K. Lu, Smooth stable and unstable manifolds for stoahcstic partial differential equations, J. Dyn. Diff. Equ., 16 (2004), 949-972.
doi: 10.1007/s10884-004-7830-z. |
[13] |
J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, London, 2014. |
[14] |
J. P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem, Commun. Math. Phys, 136 (1991), 285-307.
doi: 10.1007/BF02100026. |
[15] |
H. Fu, X. Liu and J. Duan, Slow manifolds for multi-time-scale stochastic evolutionary systems, Commun. Math. Sci., 11 (2013), 141-162.
doi: 10.4310/CMS.2013.v11.n1.a5. |
[16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Varlag, Berlin, 1981. |
[17] |
D. Y. Hsieh, A. Q. Tang and X. P. Wang, On hydrodynamics instabilities, chaos, and phase transition, Acta Mech. Sin., 12 (1996), 1-14.
doi: 10.1007/BF02486757. |
[18] |
M. F. Hilali, S. Metens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg equation, Phys. Rev. E, 51 (1995), 2046-2052. |
[19] |
P. Imkeller and A. Monahan, Conceptual stochastic climate dynamics, Stoch. Dynam., 2 (2002), 311-326. |
[20] |
G. Lin, H. Gao, J. Duan and V. J. Ervin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, J. Math. Phys., 41 (2000), 2077-2089.
doi: 10.1063/1.533228. |
[21] |
A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains: Existence and comparison, Nonlinearity, 8 (1995), 734-768.
doi: 10.1088/0951-7715/8/5/006. |
[22] |
J. Oh, J. M. Ortiz de Zárate, J. V. Sengers and G. Ahlers, Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bnárd convection, Phys. Rev. E, 69 (2004), 021106. |
[23] |
J. Oh and G. Ahlers, Thermal-noise effect on the transition to Rayleigh-Bnárd convection, Phys. Rev. Lett., 91 (2003), 094501. |
[24] |
A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Varlag, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[25] |
I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers and H. R. Brand, Thermally induced hydrodynamic fluctuations below the onset of electroconvection, Phys. Rev. Lett., 67 (1991), 596-599.
doi: 10.1103/PhysRevLett.67.596. |
[26] |
X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical systems, J. Math. Phys., 51 (2010), 042702, 12pp.
doi: 10.1063/1.3371010. |
[27] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
[28] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York Berlin Heidelberg, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[29] |
W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701, 14 pp.
doi: 10.1063/1.2800164. |
[30] |
W. Wang, J. Sun and J. Duan, Ergodic dynamics of the stochastic Swift-Hohenberg system, Nonlinear Analysis: Real World Appl., 6 (2005), 273-295.
doi: 10.1016/j.nonrwa.2004.08.009. |
[31] |
E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics, IMA, Vol. 140, Springer, New York, 2005.
doi: 10.1007/978-0-387-29371-4. |
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