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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 |
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.
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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