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January  2020, 19(1): 293-310. doi: 10.3934/cpaa.2020016

## Averaging principles for the Swift-Hohenberg equation

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

* Corresponding author

Received  December 2018 Revised  April 2019 Published  July 2019

Fund Project: This work is supported by NSFC Grant (11601073).

This work studies the effects of rapid oscillations (with respect to time) of the forcing term on the long-time behaviour of the solutions of the Swift-Hohenberg equation. More precisely, we establish three kinds of averaging principles for the Swift-Hohenberg equation, they are averaging principle in a time-periodic problem, averaging principle on a finite time interval and averaging principle on the entire axis.

Citation: Peng Gao. Averaging principles for the Swift-Hohenberg equation. Communications on Pure and Applied Analysis, 2020, 19 (1) : 293-310. doi: 10.3934/cpaa.2020016
##### References:
 [1] Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, (French) [Anosov flows with stable and unstable differentiable distributions], J. Amer. Math. Soc., 5 (1992), 33{74. doi: 10.2307/2152750. [2] N. N. Bogolyubov, On some statistical methods in mathematical physics, Izdat. Akad. Nauk Ukr. SSR, Kiev, (1945). [3] E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Bénard convection, Annual Review of Fluid Mechanics, 32 (2000), 709-778.  doi: 10.1146/annurev.fluid.32.1.709. [4] V. Burd, Method of averaging for differential equations on an infinite interval: theory and applications, CRC Press, (2007). doi: 10.1201/9781584888758. [5] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Fizmatgiz, Moscow 1963; English transl., Gordon and Breach, New York, (1962). [6] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Eviews of Modern Physics, 65 (1993), 851. [7] P. Collet and J. P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton Ser. Phys., Princeton University Press, 1990. doi: 10.1515/9781400861026. [8] D. Cheban, J. Duan and A. Gherco, Generalization of the second Bogolyubov's theorem for non-almost periodic systems, Nonlinear Analysis: Real World Applications, 4 (2003), 599-613.  doi: 10.1016/S1468-1218(02)00080-9. [9] V. P. Dymnikov and A. N. Filatov, Mathematics of Climate Modeling, Birkhauser, Boston, MA, 1997. [10] Y. L. Daletskii and M. G. Krein, Stability of solutions of differential equations in Banach space, Nauka, Moscow 1970; English transl., Araer. Math. Soc, Providence, RI, (1974). [11] A. N. Filatov, Asymptotic methods in the theory of differential and integrodifferential equations, Fan, Tashkent (Russian), (1974). [12] 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. [13] P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Analysis: Theory, Methods & Applications, 139 (2016), 169-195.  doi: 10.1016/j.na.2016.02.023. [14] P. Gao, Recurrent solutions of the derivative Ginzburg-Landau equation with boundary forces, Applicable Analysis, 97 (2018), 2743-2761.  doi: 10.1080/00036811.2017.1387250. [15] P. Gao, Recurrent solutions of the linearly coupled complex cubic-quintic Ginzburg-Landau equations, Mathematical Methods in the Applied Sciences, 41 (2018), 2769-2794.  doi: 10.1002/mma.4778. [16] A. Giorgini, On the Swift-Hohenberg equation with slow and fast dynamics: well posedness and long time behavior, Communications on Pure & Applied Analysis, 15 (2016), 219-241.  doi: 10.3934/cpaa.2016.15.219. [17] H. Gao and J. Duan, Averaging principle for quasi-geostrophic motion under rapidly oscillating forcing, Applied Mathematics and Mechanics, 26 (2005), 108-120.  doi: 10.1007/BF02438372. [18] H. Gao and J. Duan, Dynamics of quasi-geostrophic fluid motion with rapidly oscillating Coriolis force, Nonlinear Analysis: Real World Applications, 4 (2003), 127-138.  doi: 10.1016/S1468-1218(02)00018-4. [19] P. C. Hohenberg and J. B. Swift, Effects of additive noise at the onset of Rayleigh-Bénard convection, Physical Review A, 46 (1992), 4773. [20] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, (1981). [21] A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sbornik: Mathematics, 187 (1996), 635. doi: 10.1070/SM1996v187n05ABEH000126. [22] A. A. Ilyin, Global averaging of dissipative dynamical systems, Rendiconti Academia Nazionale delle Scidetta dli XL. Memorie di Matematica e Applicazioni, 116 (1998), 165-191. [23] V. L. Khatskevich, On the homogenization principle in a time-periodic problem for the Navier-Stokes equations with rapidly oscillating mass force, Mathematical Notes, 99 (2016), 757-768.  doi: 10.4213/mzm10624. [24] M. B. Kania, A modified Swift-Hohenberg equation, Topological Methods in Nonlinear Analysis, 37 (2011), 165-176. [25] J. Lega, J. V. Moloney and A. C. Newell, Swift-Hohenberg equation for lasers, Physical review letters, 73 (1994), 2978. [26] D. J. B. Lloyd and A. Scheel, Continuation and bifurcation of grain boundaries in the Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems, 16 (2017), 252-293.  doi: 10.1137/16M1073212. [27] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol.Ⅰ, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P.Kenneth, (1972). [28] B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Izdat. Moskov. Gos. Univ., Moscow, (1978) English transl., Cambridge Univ. Press, Cambridge (1982). [29] Yu. A. Mitropolskii, The method of averaging in non-linear mechanics, Naukova Dumka, Kiev, (1971). [30] Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Lett. A, 75 (1980), 296-298. [31] L. A. Peletier and V. Rottschfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D: Nonlinear Phenomena, 194 (2004), 95-126.  doi: 10.1016/j.physd.2004.01.043. [32] L. A. Peletier and V. Rottschfer, Large time behaviour of solutions of the Swift-Hohenberg equation, Comptes Rendus Mathematique, 336 (2003), 225-230.  doi: 10.1016/S1631-073X(03)00021-9. [33] L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems, 6 (2007), 208-235.  doi: 10.1137/050647232. [34] M. Petcu, R. Temam and D. Wirosoetisno, Averaging method applied to the three-dimensional primitive equations, Discrete & Continuous Dynamical Systems-Series A, 36 (2016), 5681-5707.  doi: 10.3934/dcds.2016049. [35] J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Physical Review A, 15 (1977), 319. [36] I. B. Simonenko, A Justification of the method of the averaging for abstract parabolic equations, Mat. Sb., 81 (1970), 53-61; English transl. in Math. USSR-Sb., 10 (1970). [37] J. Zheng, Optimal controls of multidimensional modified Swift-Hohenberg equation, International Journal of Control, 88 (2015), 2117-2125.  doi: 10.1080/00207179.2015.1038587.

show all references

##### References:
 [1] Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, (French) [Anosov flows with stable and unstable differentiable distributions], J. Amer. Math. Soc., 5 (1992), 33{74. doi: 10.2307/2152750. [2] N. N. Bogolyubov, On some statistical methods in mathematical physics, Izdat. Akad. Nauk Ukr. SSR, Kiev, (1945). [3] E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Bénard convection, Annual Review of Fluid Mechanics, 32 (2000), 709-778.  doi: 10.1146/annurev.fluid.32.1.709. [4] V. Burd, Method of averaging for differential equations on an infinite interval: theory and applications, CRC Press, (2007). doi: 10.1201/9781584888758. [5] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Fizmatgiz, Moscow 1963; English transl., Gordon and Breach, New York, (1962). [6] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Eviews of Modern Physics, 65 (1993), 851. [7] P. Collet and J. P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton Ser. Phys., Princeton University Press, 1990. doi: 10.1515/9781400861026. [8] D. Cheban, J. Duan and A. Gherco, Generalization of the second Bogolyubov's theorem for non-almost periodic systems, Nonlinear Analysis: Real World Applications, 4 (2003), 599-613.  doi: 10.1016/S1468-1218(02)00080-9. [9] V. P. Dymnikov and A. N. Filatov, Mathematics of Climate Modeling, Birkhauser, Boston, MA, 1997. [10] Y. L. Daletskii and M. G. Krein, Stability of solutions of differential equations in Banach space, Nauka, Moscow 1970; English transl., Araer. Math. Soc, Providence, RI, (1974). [11] A. N. Filatov, Asymptotic methods in the theory of differential and integrodifferential equations, Fan, Tashkent (Russian), (1974). [12] 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. [13] P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Analysis: Theory, Methods & Applications, 139 (2016), 169-195.  doi: 10.1016/j.na.2016.02.023. [14] P. Gao, Recurrent solutions of the derivative Ginzburg-Landau equation with boundary forces, Applicable Analysis, 97 (2018), 2743-2761.  doi: 10.1080/00036811.2017.1387250. [15] P. Gao, Recurrent solutions of the linearly coupled complex cubic-quintic Ginzburg-Landau equations, Mathematical Methods in the Applied Sciences, 41 (2018), 2769-2794.  doi: 10.1002/mma.4778. [16] A. Giorgini, On the Swift-Hohenberg equation with slow and fast dynamics: well posedness and long time behavior, Communications on Pure & Applied Analysis, 15 (2016), 219-241.  doi: 10.3934/cpaa.2016.15.219. [17] H. Gao and J. Duan, Averaging principle for quasi-geostrophic motion under rapidly oscillating forcing, Applied Mathematics and Mechanics, 26 (2005), 108-120.  doi: 10.1007/BF02438372. [18] H. Gao and J. Duan, Dynamics of quasi-geostrophic fluid motion with rapidly oscillating Coriolis force, Nonlinear Analysis: Real World Applications, 4 (2003), 127-138.  doi: 10.1016/S1468-1218(02)00018-4. [19] P. C. Hohenberg and J. B. Swift, Effects of additive noise at the onset of Rayleigh-Bénard convection, Physical Review A, 46 (1992), 4773. [20] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, (1981). [21] A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sbornik: Mathematics, 187 (1996), 635. doi: 10.1070/SM1996v187n05ABEH000126. [22] A. A. Ilyin, Global averaging of dissipative dynamical systems, Rendiconti Academia Nazionale delle Scidetta dli XL. Memorie di Matematica e Applicazioni, 116 (1998), 165-191. [23] V. L. Khatskevich, On the homogenization principle in a time-periodic problem for the Navier-Stokes equations with rapidly oscillating mass force, Mathematical Notes, 99 (2016), 757-768.  doi: 10.4213/mzm10624. [24] M. B. Kania, A modified Swift-Hohenberg equation, Topological Methods in Nonlinear Analysis, 37 (2011), 165-176. [25] J. Lega, J. V. Moloney and A. C. Newell, Swift-Hohenberg equation for lasers, Physical review letters, 73 (1994), 2978. [26] D. J. B. Lloyd and A. Scheel, Continuation and bifurcation of grain boundaries in the Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems, 16 (2017), 252-293.  doi: 10.1137/16M1073212. [27] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol.Ⅰ, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P.Kenneth, (1972). [28] B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Izdat. Moskov. Gos. Univ., Moscow, (1978) English transl., Cambridge Univ. Press, Cambridge (1982). [29] Yu. A. Mitropolskii, The method of averaging in non-linear mechanics, Naukova Dumka, Kiev, (1971). [30] Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Lett. A, 75 (1980), 296-298. [31] L. A. Peletier and V. Rottschfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D: Nonlinear Phenomena, 194 (2004), 95-126.  doi: 10.1016/j.physd.2004.01.043. [32] L. A. Peletier and V. Rottschfer, Large time behaviour of solutions of the Swift-Hohenberg equation, Comptes Rendus Mathematique, 336 (2003), 225-230.  doi: 10.1016/S1631-073X(03)00021-9. [33] L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems, 6 (2007), 208-235.  doi: 10.1137/050647232. [34] M. Petcu, R. Temam and D. Wirosoetisno, Averaging method applied to the three-dimensional primitive equations, Discrete & Continuous Dynamical Systems-Series A, 36 (2016), 5681-5707.  doi: 10.3934/dcds.2016049. [35] J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Physical Review A, 15 (1977), 319. [36] I. B. Simonenko, A Justification of the method of the averaging for abstract parabolic equations, Mat. Sb., 81 (1970), 53-61; English transl. in Math. USSR-Sb., 10 (1970). [37] J. Zheng, Optimal controls of multidimensional modified Swift-Hohenberg equation, International Journal of Control, 88 (2015), 2117-2125.  doi: 10.1080/00207179.2015.1038587.
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