Mathematical Aspects of Physics with Non-Self-Adjoint Operators: 10 Years After / Les aspects mathématiques de la physique avec les opérateurs non-auto-adjoints: 10 ans après

Collection Mathematical Aspects of Physics with Non-Self-Adjoint Operators: 10 Years After / Les aspects mathématiques de la physique avec les opérateurs non-auto-adjoints: 10 ans après

Organizer(s) Boulton, Lyonell ; Krejcirik, David ; Siegl, Petr

Date(s) 01/02/2021 - 05/02/2021

linked URL https://conferences.cirm-math.fr/2153.html

00:00:00 / 00:00:00

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Complications making amplitude equations in fluid mechanics

By Catherine Drysdale

Ginzburg-Landau equation

Owing to the interaction between modes, difficulties arise in creating amplitude equations where non-normality and nonlinearity is present in the original system. For example, if amplitude equations are made via weakly nonlinear analysis, then approximating via the critical mode only (least stable eigenvalue) does not work at higher orders where the mixing of the modes needs to be taken into consideration. However, using a different homogenisation technique, namely stochastic singular perturbation theory of authors like Papanicalaou , Blömker & al, where noise is applied to the stable modes only, then the linear operator in question is no longer non-self-adjoint. Although, the difficulty of the problem shifts to showing that we can use a Rigged Hilbert Space construction. If the original problem in a Hilbert space H. We force the main operator of our problem to be Hilbert-Schmidt by choosing our noise in a dense subspace S of H. We demonstrate this on the Complex-Ginsburg-Landau equation with cubic nonlinearity.

Information about the video

Date of recording 26/03/2020
Date of publication 26/02/2021
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19711503
Cite this video Drysdale, Catherine (26/03/2020). Complications making amplitude equations in fluid mechanics. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19711503
URL https://dx.doi.org/10.24350/CIRM.V.19711503

Domain(s)

Mathematical Physics

Bibliography

PAPANICOLAOU, George C. Some probabilistic problems and methods in singular perturbations. The Rocky Mountain Journal of Mathematics, 1976, p. 653-674. - https://www.jstor.org/stable/44240337
BLÖMKER, Dirk et MOHAMMED, Wael W. Amplitude equations for SPDEs with cubic nonlinearities. Stochastics An International Journal of Probability and Stochastic Processes, 2013, vol. 85, no 2, p. 181-215. - https://doi.org/10.1080/17442508.2011.624628

MSC codes

76E09 Stability and instability of nonparallel flows

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