Authors : Drysdale, Catherine (Author of the conference)
CIRM (Publisher )
Abstract :
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.
Keywords : Ginzburg-Landau equation; non-self-adjoint
MSC Codes :
76E09
- Stability and instability of nonparallel flows
Film maker : Hennenfent, Guillaume
Language : English
Available date : 26/02/2021
Conference Date : 26/03/2020
Subseries : Research talks
arXiv category : Mathematical Physics
Mathematical Area(s) : Mathematical Physics
Format : MP4 (.mp4) - HD
Video Time : 00:32:41
Targeted Audience : Researchers
Download : https://videos.cirm-math.fr/2021-02-02_Drysdale.mp4
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Event Title : 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 Event Organizers : Boulton, Lyonell ; Krejcirik, David ; Siegl, Petr Dates : 01/02/2021 - 05/02/2021
Event Year : 2021
Event URL : https://conferences.cirm-math.fr/2153.html
DOI : 10.24350/CIRM.V.19711503
Cite this video as:
Drysdale, Catherine (2021). Complications making amplitude equations in fluid mechanics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19711503
URI : http://dx.doi.org/10.24350/CIRM.V.19711503
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See Also
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