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dc.contributor.authorDemidov, Alexander
dc.contributor.authorLohéac, Jean-Pierre
dc.contributor.authorRunge, Vincent
dc.date.accessioned2019-07-26T12:38:55Z
dc.date.available2019-07-26T12:38:55Z
dc.date.issued2016-01
dc.identifier.issn1061-9208
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/19533
dc.language.isoenen
dc.subjectMathematical Physicen
dc.subjectFree Boundaryen
dc.subjectDiscrete Modelen
dc.subjectGeometrical Transformationen
dc.subjectPhase Field Modelen
dc.subject.ddc515en
dc.titleStokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contoursen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenThe Stokes–Leibenson problem for Hele-Shaw flow is reformulated as a Cauchy problem of a nonlinear integro-differential equation with respect to functions a and b, linked by the Hilbert transform. The function a expresses the evolution of the coefficient longitudinal strain of the free boundary and b is the evolution of the tangent tilt of this contour. These functions directly reflect changes of geometric characteristics of the free boundary of higher order than the evolution of the contour point obtained by the classical Galin–Kochina equation. That is why we managed to uncover the reason of the absence of solutions in the sink-case if the initial contour is not analytic at at least one point, to prove existence and uniqueness theorems, and also to reveal a certain critical set in the space of contours. This set contains one attractive point in the source-case corresponding to a circular contour centered at the source-point. The main object of this work is the analysis of the discrete model of the problem. This model, called quasi-contour, is formulated in terms of functions corresponding to a and b of our integro-differential equation. This quasi-contour model provides numerical experiments which confirm the theoretical properties mentioned above, especially the existence of a critical subset of co-dimension 1 in space of quasi-contours. This subset contains one attractive point in the source-case corresponding to a regular quasi-contour centered at the source-point. The main contribution of our quasi-contour model concerns the sink-case: numerical experiments show that the above subset is attractive. Furthermore, this discrete model allows to extend previous results obtained by using complex analysis. We also provide numerical experiments linked to fingering effects.en
dc.relation.isversionofjnlnameRussian Journal of Mathematical Physics
dc.relation.isversionofjnlvol23en
dc.relation.isversionofjnlissue1en
dc.relation.isversionofjnldate2016-01
dc.relation.isversionofjnlpages35-55en
dc.relation.isversionofdoi10.1134/S1061920816010039en
dc.relation.isversionofjnlpublisherSpringeren
dc.subject.ddclabelAnalyseen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewednonen
dc.relation.Isversionofjnlpeerreviewednonen
dc.date.updated2019-07-26T12:33:36Z
hal.person.labIds88479
hal.person.labIds193738
hal.person.labIds60


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