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Title: Stochastic Modulation Equations on Unbounded Domains (English)
Author: Bianchi, Luigi A.
Author: Blömker, Dirk
Language: English
Journal: Proceedings of Equadiff 14
Volume: Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017
Issue: 2017
Pages: 295-304
Category: math
Summary: We study the impact of small additive space-time white noise on nonlinear stochastic partial differential equations (SPDEs) on unbounded domains close to a bifurcation, where an infinite band of eigenvalues changes stability due to the unboundedness of the underlying domain. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, and we rely on the approximation via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. One technical problem for establishing error estimates in the stochastic case rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time the error is always very large somewhere far out in space. Thus we have to work in weighted spaces that allow for growth at infinity. As a first example we study the stochastic one-dimensional Swift-Hohenberg equation on the whole real line [1, 2]. In this setting, because of the weak regularity of solutions, the standard methods for deterministic modulation equations fail, and we need to develop new tools to treat the approximation. Using energy estimates we are only able to show that solutions of the Ginzburg-Landau equation are Hölder continuous in spaces with a very weak weight, which provides just enough regularity to proceed with the error estimates. (English)
Keyword: Modulation equations, amplitude equations, convolution operator, regularity, Rayleigh-Benard, Swift-Hohenberg, Ginzburg-Landau
MSC: 60H10
MSC: 60H15
Date available: 2019-09-27T08:14:47Z
Last updated: 2019-09-27
Stable URL:
Reference: [1] Bianchi, L. A., Bl\"omker., D.: Modulation equation for SPDEs in unbounded domains with space–time white noise—linear theory.. Stochastic Processes and their Applications, 126(10):3171–3201, (2016). MR 3542631, 10.1016/
Reference: [2] HASH(0x2fa9550): .[2] L. A. Bianchi, D. Bl\"omker, and G. Schneider. //Modulation equation and SPDEs on unboundeddomains. Preprint, arXiv, (2017).
Reference: [3] Bl\"omker, D., Hairer, M., HASH(0x2fbf640), Pavliotis., G. A.: Modulation equations: Stochastic bifurcation in large domains.. Commun. Math. Physics., 258(2):479–512, (2005). MR 2171705
Reference: [4] Collet, P., Eckmann., J.-P.: The time dependent amplitude equation for the Swift-Hohenberg problem.. Comm. Math. Phys., 132(1):139–153, (1990). MR 1069205, 10.1007/BF02278004
Reference: [5] Prato, G. Da, Zabczyk., J.: Stochastic equations in infinite dimensions., 2nd Edition, vol. 152 of Encyclopedia of of Mathematics and its Applications. Cambridge University Press, Cambridge, 2014. MR 3236753
Reference: [6] D\"ull, W.-P., Kashani, K. S., Schneider, G., HASH(0x2fc45b0), Zimmermann., D.: Attractivity of the Ginzburg Landau mode distribution for a pattern forming system with marginally stable long modes.. J. Differ. Equations, 261(1):319–339, (2016). MR 3487261
Reference: [7] Hairer, M., Ryser, M. D., HASH(0x2fc4fd0), Weber., H.: Triviality of the 2D stochastic Allen-Cahn equation.. Electron. J. Probab. 17, Paper No. 39, 14 p. (2012). MR 2928722, 10.1214/EJP.v17-1731
Reference: [8] Hohenberg, P. C., Swift., J. B.: Effects of additive noise at the onset of Rayleigh-Bénard convection.. Physical Review A, 46:4773–4785, (1992). 10.1103/PhysRevA.46.4773
Reference: [9] Hutt., A.: Additive noise may change the stability of nonlinear systems.. Europhys. Lett. 84, 34003:1–4, (2008). 10.1209/0295-5075/84/34003
Reference: [10] Hutt, A., Longtin, A., HASH(0x2fc8ee0), Schimansky-Geier., L.: Additive global noise delays Turing bifurcations.. Physical Review Letters, 98, 230601, (2007). 10.1103/PhysRevLett.98.230601
Reference: [11] Kirrmann, P., Schneider, G., HASH(0x2fcd418), Mielke., A.: The validity of modulation equations for extended systems with cubic nonlinearities.. Proc. R. Soc. Edinb., Sect. A 122(1-2):85–91, (1992). MR 1190233
Reference: [12] Klepel, K., Bl\"omker, D., HASH(0x2fcde38), Mohammed., W. W.: Amplitude equation for the generalized Swift-Hohenberg equation with noise.. Z. Angew. Math. Phys. 65(6):1107–1126, (2014). MR 3279520, 10.1007/s00033-013-0371-8
Reference: [13] Melbourne., I.: Derivation of the time-dependent Ginzburg-Landau equation on the line.. J. Nonlinear Sci., 8(1):1–15, (1998). MR 1604558, 10.1007/s003329900041
Reference: [14] Mielke, A., Schneider., G.: Attractors for modulation equations on unbounded domains – existence and comparison.. Nonlinearity, 8(5):743–768, (1995). MR 1355041, 10.1088/0951-7715/8/5/006
Reference: [15] Mohammed, W. W., Bl\"omker, D., HASH(0x2fd3418), Klepel., K.: Modulation equation for stochastic Swift–Hohenberg equation.. SIAM Journal on Mathematical Analysis, 45(1):14–30, (2013). MR 3032967, 10.1137/110846336
Reference: [16] Oh, J., Ahlers., G.: Thermal-Noise Effect on the Transition to Rayleigh-Bénard Convection., Phys. Rev. Lett. 91, 094501 (2003). 10.1103/PhysRevLett.91.094501
Reference: [17] Oh, J., Zarate, J. Ortiz de, Sengers, J., Ahlers., G.: Dynamics of fluctuations in a fluid below the onset of Rayleigh-Benard convection., Phys. Rev. E 69 , 021106, (2004). 10.1103/PhysRevE.69.021106
Reference: [18] Rehberg, I., Rasenat, S., Torre, J. M. de la, Brand., H. R.: Thermally induced hydrodynamic fluctuations below the onset of electroconvection., Physical Review Letters 67(5):596–599,(1991)
Reference: [19] Roberts., A.: Planform evolution in convection–an embedded centre manifold.. J. Austral. Math.Soc. B., 34(2), 174–198, (1992). MR 1181572, 10.1017/S0334270000008717
Reference: [20] Schneider, G., Uecker., H.: The amplitude equations for the first instability of electroconvection in nematic liquid crystals in the case of two unbounded space directions.. Nonlinearity, 20(6):1361–1386, (2007). MR 2327129, 10.1088/0951-7715/20/6/003


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