November  2018, 38(11): 5649-5684. doi: 10.3934/dcds.2018247

Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation

School of Mathematics and Statistics, and Center for Mathematics, and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

* Corresponding author: Peng Gao

Received  December 2017 Revised  May 2018 Published  August 2018

Fund Project: The first author is supported by NSFC Grant (11601073)

This work concerns the problem associated with averaging principle for a stochastic Kuramoto-Sivashinsky equation with slow and fast time-scales. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic Kuramoto-Sivashinsky equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single stochastic Kuramoto-Sivashinsky equation with a modified coefficient.

Citation: Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
References:
[1]

J. BaoG. Yin and C. Yuan, Two-time-scale stochastic partial differential equations driven by α-stable noises: Averaging principles, Bernoulli, 23 (2017), 645-669. doi: 10.3150/14-BEJ677. Google Scholar

[2]

M. Barton-Smith, Invariant measure for the stochastic Ginzburg Landau equation, Nonlinear Differential Equations and Applications NoDEA, 11 (2004), 29-52. doi: 10.1007/s00030-003-1040-y. Google Scholar

[3]

L. Bo and Y. Jiang, Large deviation for the nonlocal Kuramoto-Sivashinsky SPDE, Nonlinear Analysis: Theory, Methods & Applications, 82 (2013), 100-114. doi: 10.1016/j.na.2013.01.005. Google Scholar

[4]

L. BoK. Shi and Y. Wang, On a nonlocal stochastic Kuramoto-Sivashinsky equation with jumps, Stochastics and Dynamics, 7 (2007), 439-457. doi: 10.1142/S0219493707002104. Google Scholar

[5]

N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon & Breach Science Publishers, New York, 1961. Google Scholar

[6]

C. E. Bréhier, Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122 (2012), 2553-2593. doi: 10.1016/j.spa.2012.04.007. Google Scholar

[7]

S. Cerrai, A Khasminkii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948. doi: 10.1214/08-AAP560. Google Scholar

[8]

S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM Journal on Mathematical Analysis, 43 (2011), 2482-2518. doi: 10.1137/100806710. Google Scholar

[9]

S. Cerrai and M. I. Freidlin, Averaging principle for a class of stochastic reaction diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177. doi: 10.1007/s00440-008-0144-z. Google Scholar

[10]

L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surfaces-Ⅱ. Bifurcation analyses of the long-wave equation, Chem. Eng. Sci., 41 (1986), 2477-2486. doi: 10.1016/0009-2509(86)80033-1. Google Scholar

[11]

G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge university press, 2014. doi: 10.1017/CBO9781107295513. Google Scholar

[12]

Z. Dong, X. Sun, H. Xiao and J. Zhai, Averaging principle for one dimensional stochastic Burgers equation, arXiv: 1701.05920, 2018. doi: 10.1016/j.jde.2018.06.020. Google Scholar

[13]

J. Duan and V. J. Ervin, Dynamics of a nonlocal Kuramoto-Sivashinsky equation, Journal of Differential Equations, 143 (1998), 243-266. doi: 10.1006/jdeq.1997.3371. Google Scholar

[14]

J. Duan and V. J. Ervin, On the stochastic Kuramoto-Sivashinsky equation, Nonlinear Analysis: Theory, Methods & Applications, 44 (2001), 205-216. doi: 10.1016/S0362-546X(99)00259-X. Google Scholar

[15]

B. Ferrario, Invariant measures for a stochastic Kuramoto-Sivashinsky equation, Stochastic Analysis and Applications, 26 (2008), 379-407. doi: 10.1080/07362990701857335. Google Scholar

[16]

H. Fu and J. Duan, An averaging principle for two-scale stochastic partial differential equations, Stochastics and Dynamics, 11 (2011), 353-367. doi: 10.1142/S0219493711003346. Google Scholar

[17]

H. Fu and J. Liu, Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, Journal of Mathematical Analysis and Applications, 384 (2011), 70-86. doi: 10.1016/j.jmaa.2011.02.076. Google Scholar

[18]

H. FuL. Wan and J. Liu, Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125 (2015), 3255-3279. doi: 10.1016/j.spa.2015.03.004. Google Scholar

[19]

H. FuL. WanJ. Liu and X. Liu, Weak order in averaging principle for stochastic wave equation with a fast oscillation, Stochastic Process. Appl., 128 (2018), 2557-2580. doi: 10.1016/j.spa.2017.09.021. Google Scholar

[20]

H. FuL. WanY. Wang and J. Liu, Strong convergence rate in averaging principle for stochastic FitzHugh-Nagumo system with two time-scales, Journal of Mathematical Analysis and Applications, 416 (2014), 609-628. doi: 10.1016/j.jmaa.2014.02.062. Google Scholar

[21]

P. Gao, The stochastic Swift-Hohenberg equation, Nonlinearity, 30 (2017), 3516-3559. doi: 10.1088/1361-6544/aa7e99. Google Scholar

[22]

P. GaoM. Chen and Y. Li, Observability Estimates and Null Controllability for Forward and Backward Linear Stochastic Kuramoto-Sivashinsky Equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500. doi: 10.1137/130943820. Google Scholar

[23]

P. Gao and Y. Li, Averaging principle for the Schrödinger equations, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 2147-2168. doi: 10.3934/dcdsb.2017089. Google Scholar

[24]

A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Phys. Rev. E., 53 (1996), 3573-3578. doi: 10.1103/PhysRevE.53.3573. Google Scholar

[25]

A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-45. doi: 10.1063/1.865160. Google Scholar

[26]

R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kibernetika, 4 (1968), 260-279. Google Scholar

[27]

Y. Kuramoto, Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys., 64 (1978), 346-367. doi: 10.1143/PTPS.64.346. Google Scholar

[28]

Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys., 54 (1975), 687-699. Google Scholar

[29]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356. Google Scholar

[30]

R. E. LaqueyS. M. MahajanP. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394. doi: 10.2172/4202869. Google Scholar

[31]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P. Kenneth, 1972. Google Scholar

[32]

B. A. Malomed, B. F. Feng and T. Kawahara, Stabilized Kuramoto-Sivashinsky system, Phys. Rev. E, 64 (2001), 046304. doi: 10.1103/PhysRevE.64.046304. Google Scholar

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[34]

B. PeiY. Xu and J. L. Wu, Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles, Journal of Mathematical Analysis and Applications, 447 (2017), 243-268. doi: 10.1016/j.jmaa.2016.10.010. Google Scholar

[35]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Berlin: Springer, 2007. Google Scholar

[36]

H. A. Simon and A. Ando, Aggregation of variables in dynamical systems, Econometrica, 29 (1961), 111-138. doi: 10.21236/AD0089516. Google Scholar

[37]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I Derivation of basic equations, Acta Astronaut., 4 (1977), 1177-1206. doi: 10.1016/0094-5765(77)90096-0. Google Scholar

[38]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286. doi: 10.1016/j.jde.2012.05.011. Google Scholar

[39]

J. Xu, Lp-strong convergence of the averaging principle for slow-fast SPDEs with jumps, Journal of Mathematical Analysis and Applications, 445 (2017), 342-373. doi: 10.1016/j.jmaa.2016.07.058. Google Scholar

[40]

J. XuY. Miao and J. Liu, Strong averaging principle for slow-fast SPDEs with poisson random measures, Discrete & Continuous Dynamical Systems-Series B, 20 (2015), 2233-2256. doi: 10.3934/dcdsb.2015.20.2233. Google Scholar

[41]

J. Xu, Y. Miao and J. Liu, Strong averaging principle for two-time-scale non-autonomous stochastic FitzHugh-Nagumo system with jumps, Journal of Mathematical Physics, 57 (2016), 092704, 21pp. doi: 10.1063/1.4963173. Google Scholar

[42]

D. Yang, Random attractors for the stochastic Kuramoto-Sivashinsky equation, Stochastic Analysis and Applications, 24 (2006), 1285-1303. doi: 10.1080/07362990600991300. Google Scholar

[43]

D. Yang, Dynamics for the stochastic nonlocal Kuramoto-Sivashinsky equation, Journal of Mathematical Analysis and Applications, 330 (2007), 550-570. doi: 10.1016/j.jmaa.2006.07.091. Google Scholar

show all references

References:
[1]

J. BaoG. Yin and C. Yuan, Two-time-scale stochastic partial differential equations driven by α-stable noises: Averaging principles, Bernoulli, 23 (2017), 645-669. doi: 10.3150/14-BEJ677. Google Scholar

[2]

M. Barton-Smith, Invariant measure for the stochastic Ginzburg Landau equation, Nonlinear Differential Equations and Applications NoDEA, 11 (2004), 29-52. doi: 10.1007/s00030-003-1040-y. Google Scholar

[3]

L. Bo and Y. Jiang, Large deviation for the nonlocal Kuramoto-Sivashinsky SPDE, Nonlinear Analysis: Theory, Methods & Applications, 82 (2013), 100-114. doi: 10.1016/j.na.2013.01.005. Google Scholar

[4]

L. BoK. Shi and Y. Wang, On a nonlocal stochastic Kuramoto-Sivashinsky equation with jumps, Stochastics and Dynamics, 7 (2007), 439-457. doi: 10.1142/S0219493707002104. Google Scholar

[5]

N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon & Breach Science Publishers, New York, 1961. Google Scholar

[6]

C. E. Bréhier, Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122 (2012), 2553-2593. doi: 10.1016/j.spa.2012.04.007. Google Scholar

[7]

S. Cerrai, A Khasminkii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948. doi: 10.1214/08-AAP560. Google Scholar

[8]

S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM Journal on Mathematical Analysis, 43 (2011), 2482-2518. doi: 10.1137/100806710. Google Scholar

[9]

S. Cerrai and M. I. Freidlin, Averaging principle for a class of stochastic reaction diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177. doi: 10.1007/s00440-008-0144-z. Google Scholar

[10]

L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surfaces-Ⅱ. Bifurcation analyses of the long-wave equation, Chem. Eng. Sci., 41 (1986), 2477-2486. doi: 10.1016/0009-2509(86)80033-1. Google Scholar

[11]

G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge university press, 2014. doi: 10.1017/CBO9781107295513. Google Scholar

[12]

Z. Dong, X. Sun, H. Xiao and J. Zhai, Averaging principle for one dimensional stochastic Burgers equation, arXiv: 1701.05920, 2018. doi: 10.1016/j.jde.2018.06.020. Google Scholar

[13]

J. Duan and V. J. Ervin, Dynamics of a nonlocal Kuramoto-Sivashinsky equation, Journal of Differential Equations, 143 (1998), 243-266. doi: 10.1006/jdeq.1997.3371. Google Scholar

[14]

J. Duan and V. J. Ervin, On the stochastic Kuramoto-Sivashinsky equation, Nonlinear Analysis: Theory, Methods & Applications, 44 (2001), 205-216. doi: 10.1016/S0362-546X(99)00259-X. Google Scholar

[15]

B. Ferrario, Invariant measures for a stochastic Kuramoto-Sivashinsky equation, Stochastic Analysis and Applications, 26 (2008), 379-407. doi: 10.1080/07362990701857335. Google Scholar

[16]

H. Fu and J. Duan, An averaging principle for two-scale stochastic partial differential equations, Stochastics and Dynamics, 11 (2011), 353-367. doi: 10.1142/S0219493711003346. Google Scholar

[17]

H. Fu and J. Liu, Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, Journal of Mathematical Analysis and Applications, 384 (2011), 70-86. doi: 10.1016/j.jmaa.2011.02.076. Google Scholar

[18]

H. FuL. Wan and J. Liu, Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125 (2015), 3255-3279. doi: 10.1016/j.spa.2015.03.004. Google Scholar

[19]

H. FuL. WanJ. Liu and X. Liu, Weak order in averaging principle for stochastic wave equation with a fast oscillation, Stochastic Process. Appl., 128 (2018), 2557-2580. doi: 10.1016/j.spa.2017.09.021. Google Scholar

[20]

H. FuL. WanY. Wang and J. Liu, Strong convergence rate in averaging principle for stochastic FitzHugh-Nagumo system with two time-scales, Journal of Mathematical Analysis and Applications, 416 (2014), 609-628. doi: 10.1016/j.jmaa.2014.02.062. Google Scholar

[21]

P. Gao, The stochastic Swift-Hohenberg equation, Nonlinearity, 30 (2017), 3516-3559. doi: 10.1088/1361-6544/aa7e99. Google Scholar

[22]

P. GaoM. Chen and Y. Li, Observability Estimates and Null Controllability for Forward and Backward Linear Stochastic Kuramoto-Sivashinsky Equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500. doi: 10.1137/130943820. Google Scholar

[23]

P. Gao and Y. Li, Averaging principle for the Schrödinger equations, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 2147-2168. doi: 10.3934/dcdsb.2017089. Google Scholar

[24]

A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Phys. Rev. E., 53 (1996), 3573-3578. doi: 10.1103/PhysRevE.53.3573. Google Scholar

[25]

A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-45. doi: 10.1063/1.865160. Google Scholar

[26]

R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kibernetika, 4 (1968), 260-279. Google Scholar

[27]

Y. Kuramoto, Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys., 64 (1978), 346-367. doi: 10.1143/PTPS.64.346. Google Scholar

[28]

Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys., 54 (1975), 687-699. Google Scholar

[29]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356. Google Scholar

[30]

R. E. LaqueyS. M. MahajanP. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394. doi: 10.2172/4202869. Google Scholar

[31]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P. Kenneth, 1972. Google Scholar

[32]

B. A. Malomed, B. F. Feng and T. Kawahara, Stabilized Kuramoto-Sivashinsky system, Phys. Rev. E, 64 (2001), 046304. doi: 10.1103/PhysRevE.64.046304. Google Scholar

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[34]

B. PeiY. Xu and J. L. Wu, Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles, Journal of Mathematical Analysis and Applications, 447 (2017), 243-268. doi: 10.1016/j.jmaa.2016.10.010. Google Scholar

[35]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Berlin: Springer, 2007. Google Scholar

[36]

H. A. Simon and A. Ando, Aggregation of variables in dynamical systems, Econometrica, 29 (1961), 111-138. doi: 10.21236/AD0089516. Google Scholar

[37]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I Derivation of basic equations, Acta Astronaut., 4 (1977), 1177-1206. doi: 10.1016/0094-5765(77)90096-0. Google Scholar

[38]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286. doi: 10.1016/j.jde.2012.05.011. Google Scholar

[39]

J. Xu, Lp-strong convergence of the averaging principle for slow-fast SPDEs with jumps, Journal of Mathematical Analysis and Applications, 445 (2017), 342-373. doi: 10.1016/j.jmaa.2016.07.058. Google Scholar

[40]

J. XuY. Miao and J. Liu, Strong averaging principle for slow-fast SPDEs with poisson random measures, Discrete & Continuous Dynamical Systems-Series B, 20 (2015), 2233-2256. doi: 10.3934/dcdsb.2015.20.2233. Google Scholar

[41]

J. Xu, Y. Miao and J. Liu, Strong averaging principle for two-time-scale non-autonomous stochastic FitzHugh-Nagumo system with jumps, Journal of Mathematical Physics, 57 (2016), 092704, 21pp. doi: 10.1063/1.4963173. Google Scholar

[42]

D. Yang, Random attractors for the stochastic Kuramoto-Sivashinsky equation, Stochastic Analysis and Applications, 24 (2006), 1285-1303. doi: 10.1080/07362990600991300. Google Scholar

[43]

D. Yang, Dynamics for the stochastic nonlocal Kuramoto-Sivashinsky equation, Journal of Mathematical Analysis and Applications, 330 (2007), 550-570. doi: 10.1016/j.jmaa.2006.07.091. Google Scholar

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