In this paper we rigorously derive stochastic amplitude equations for a rather general class of SPDEs with quadratic nonlinearities forced by small additive noise. Near a change of stability we use the natural separation of time-scales to show that the solution of the original SPDE is approximated by the solution of an amplitude equation, which describes the evolution of dominant modes. Our results significantly improve older results. We focus on equations with quadratic nonlinearities and give applications to the one-dimensional Burgers’ equation and a model from surface growth.
References
Blömker, Dirk. Approximation of the stochastic Rayleigh-Bénard problem near the onset of convection and related problems. Stoch. Dyn. 5 (2005), no. 3, 441–474.Blömker, Dirk. Approximation of the stochastic Rayleigh-Bénard problem near the onset of convection and related problems. Stoch. Dyn. 5 (2005), no. 3, 441–474.
Blömker, Dirk. Amplitude equations for stochastic partial differential equations. Interdisciplinary Mathematical Sciences, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. x+126 pp. ISBN: 978-981-270-637-9; 981-270-637-2 MR2352975Blömker, Dirk. Amplitude equations for stochastic partial differential equations. Interdisciplinary Mathematical Sciences, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. x+126 pp. ISBN: 978-981-270-637-9; 981-270-637-2 MR2352975
Blömker, Dirk; Hairer, Martin. Multiscale expansion of invariant measures for SPDEs. Comm. Math. Phys. 251 (2004), no. 3, 515–555. 1066.60057 10.1007/s00220-004-1130-7Blömker, Dirk; Hairer, Martin. Multiscale expansion of invariant measures for SPDEs. Comm. Math. Phys. 251 (2004), no. 3, 515–555. 1066.60057 10.1007/s00220-004-1130-7
Blömker, Dirk; Hairer, Martin. Amplitude equations for SPDEs: approximate centre manifolds and invariant measures. Probability and partial differential equations in modern applied mathematics, 41–59, IMA Vol. Math. Appl., 140, Springer, New York, 2005.Blömker, Dirk; Hairer, Martin. Amplitude equations for SPDEs: approximate centre manifolds and invariant measures. Probability and partial differential equations in modern applied mathematics, 41–59, IMA Vol. Math. Appl., 140, Springer, New York, 2005.
Blömker, D.; Hairer, M.; Pavliotis, G. A. Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities. Nonlinearity 20 (2007), no. 7, 1721–1744.Blömker, D.; Hairer, M.; Pavliotis, G. A. Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities. Nonlinearity 20 (2007), no. 7, 1721–1744.
Chekhlov, Alexei; Yakhot, Victor. Kolmogorov turbulence in a random-force-driven Burgers equation: anomalous scaling and probability density functions. Phys. Rev. E (3) 52 (1995), no. 5, part B, 5681–5684. MR1385911 10.1103/PhysRevE.52.5681Chekhlov, Alexei; Yakhot, Victor. Kolmogorov turbulence in a random-force-driven Burgers equation: anomalous scaling and probability density functions. Phys. Rev. E (3) 52 (1995), no. 5, part B, 5681–5684. MR1385911 10.1103/PhysRevE.52.5681
de Bouard, A.; Debussche, A. Random modulation of solitons for the stochastic Korteweg-de Vries equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 2, 251–278.de Bouard, A.; Debussche, A. Random modulation of solitons for the stochastic Korteweg-de Vries equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 2, 251–278.
Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 0761.60052Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 0761.60052
Doelman, Arjen; Sandstede, Björn; Scheel, Arnd; Schneider, Guido. The dynamics of modulated wave trains. Mem. Amer. Math. Soc. 199 (2009), no. 934, viii+105 pp. ISBN: 978-0-8218-4293-5Doelman, Arjen; Sandstede, Björn; Scheel, Arnd; Schneider, Guido. The dynamics of modulated wave trains. Mem. Amer. Math. Soc. 199 (2009), no. 934, viii+105 pp. ISBN: 978-0-8218-4293-5
A. Hutt; A. Longtin; L. Schimansky-Geier. Additive global noise delays Turing bifurcations. Physical Review Letters 98 (2007), 230601. 1152.92003 10.1016/j.physd.2007.10.013A. Hutt; A. Longtin; L. Schimansky-Geier. Additive global noise delays Turing bifurcations. Physical Review Letters 98 (2007), 230601. 1152.92003 10.1016/j.physd.2007.10.013
Hutt, Axel; Longtin, Andre; Schimansky-Geier, Lutz. Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation. Phys. D 237 (2008), no. 6, 755–773. 1152.92003 10.1016/j.physd.2007.10.013Hutt, Axel; Longtin, Andre; Schimansky-Geier, Lutz. Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation. Phys. D 237 (2008), no. 6, 755–773. 1152.92003 10.1016/j.physd.2007.10.013
Kirrmann, Pius; Schneider, Guido; Mielke, Alexander. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 1-2, 85–91. 0786.35122 10.1017/S0308210500020989Kirrmann, Pius; Schneider, Guido; Mielke, Alexander. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 1-2, 85–91. 0786.35122 10.1017/S0308210500020989
Liu, Kai. Stability of infinite dimensional stochastic differential equations with applications. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135. Chapman & Hall/CRC, Boca Raton, FL, 2006. xii+298 pp. ISBN: 978-1-58488-598-6; 1-58488-598-XLiu, Kai. Stability of infinite dimensional stochastic differential equations with applications. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135. Chapman & Hall/CRC, Boca Raton, FL, 2006. xii+298 pp. ISBN: 978-1-58488-598-6; 1-58488-598-X
M. Raible; S.G. Mayr; S.J. Linz; M. Moske; P. Hänggi; K. Samwer. Amorphous thin film growth: Theory compared with experiment. Europhysics Letters 50 (2000), 61–67.M. Raible; S.G. Mayr; S.J. Linz; M. Moske; P. Hänggi; K. Samwer. Amorphous thin film growth: Theory compared with experiment. Europhysics Letters 50 (2000), 61–67.
A.J. Roberts; Wei Wang. Macroscopic reduction for stochastic reaction-diffusion equations. Preprint (2008), arXiv:0812.1837v1 [math-ph]. 0812.1837v1 1284.60131 10.1093/imamat/hxs019A.J. Roberts; Wei Wang. Macroscopic reduction for stochastic reaction-diffusion equations. Preprint (2008), arXiv:0812.1837v1 [math-ph]. 0812.1837v1 1284.60131 10.1093/imamat/hxs019
Schneider, Guido. The validity of generalized Ginzburg-Landau equations. Math. Methods Appl. Sci. 19 (1996), no. 9, 717–736. 0874.35050 10.1002/(SICI)1099-1476(199606)19:9<717::AID-MMA792>3.0.CO;2-ZSchneider, Guido. The validity of generalized Ginzburg-Landau equations. Math. Methods Appl. Sci. 19 (1996), no. 9, 717–736. 0874.35050 10.1002/(SICI)1099-1476(199606)19:9<717::AID-MMA792>3.0.CO;2-Z
Schneider, Guido; Uecker, Hannes. The amplitude equations for the first instability of electro-convection in nematic liquid crystals in the case of two unbounded space directions. Nonlinearity 20 (2007), no. 6, 1361–1386. MR2327129 1116.76030 10.1088/0951-7715/20/6/003Schneider, Guido; Uecker, Hannes. The amplitude equations for the first instability of electro-convection in nematic liquid crystals in the case of two unbounded space directions. Nonlinearity 20 (2007), no. 6, 1361–1386. MR2327129 1116.76030 10.1088/0951-7715/20/6/003
