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:
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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.

[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.

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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.

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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.

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

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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.

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

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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.

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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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[42]

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

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[42]

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

[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.

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