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Stability of the travelling wave in a 2D weakly nonlinear Stefan problem

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorBRAUNER, Claude-Michel
hal.structure.identifierDepartment of Computer Science [Amsterdam]
dc.contributor.authorHULSHOF, Josephus
dc.contributor.authorLORENZI, Luca
dc.date.accessioned2024-04-04T02:38:20Z
dc.date.available2024-04-04T02:38:20Z
dc.date.created2008-09
dc.date.issued2009-03
dc.identifier.issn1937-5093
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190868
dc.description.abstractEnWe investigate the stability of the travelling wave (TW) solution in a 2D Stefan problem, a simplified version of a solid-liquid interface model. It is intended as a paradigm problem to present our method based on: (i) definition of a suitable linear one dimensional operator, (ii) projection with respect to the $x$ coordinate only; (iii) Lyapunov-Schmidt method. The main issue is that we are able to derive a parabolic equation for the corrugated front $\varphi$ near the TW as a solvability condition. This equation involves two linear pseudo-differential operators, one acting on $\varphi$, the other on $(\varphi_y)^2$ and clearly appears as a generalization of the Kuramoto-Sivashinsky equation related to turbulence phenomena in chemistry and combustion. A large part of the paper is devoted to study the properties of these operators in the context of functional spaces in the $y$ and $x,y$ coordinates with periodic boundary conditions. Technical results are deferred to the appendices.
dc.language.isoen
dc.publisherAIMS
dc.subject.enStefan problem
dc.subject.enstability
dc.subject.enfront dynamics
dc.subject.enKuramoto-Sivashinsky equation
dc.subject.enpseudo-differential operators
dc.subject.ensectorial operators
dc.title.enStability of the travelling wave in a 2D weakly nonlinear Stefan problem
dc.typeArticle de revue
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
bordeaux.journalKinetic and Related Models
bordeaux.page109-134
bordeaux.volume2
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue1
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00386986
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00386986v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Kinetic%20and%20Related%20Models&rft.date=2009-03&rft.volume=2&rft.issue=1&rft.spage=109-134&rft.epage=109-134&rft.eissn=1937-5093&rft.issn=1937-5093&rft.au=BRAUNER,%20Claude-Michel&HULSHOF,%20Josephus&LORENZI,%20Luca&rft.genre=article


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