dc.contributor.author Demidov, Alexander dc.contributor.author Lohéac, Jean-Pierre dc.contributor.author Runge, Vincent dc.date.accessioned 2019-07-26T12:38:55Z dc.date.available 2019-07-26T12:38:55Z dc.date.issued 2016-01 dc.identifier.issn 1061-9208 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/19533 dc.language.iso en en dc.subject Mathematical Physic en dc.subject Free Boundary en dc.subject Discrete Model en dc.subject Geometrical Transformation en dc.subject Phase Field Model en dc.subject.ddc 515 en dc.title Stokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contours en dc.type Article accepté pour publication ou publié dc.description.abstracten The 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.isversionofjnlname Russian Journal of Mathematical Physics dc.relation.isversionofjnlvol 23 en dc.relation.isversionofjnlissue 1 en dc.relation.isversionofjnldate 2016-01 dc.relation.isversionofjnlpages 35-55 en dc.relation.isversionofdoi 10.1134/S1061920816010039 en dc.relation.isversionofjnlpublisher Springer en dc.subject.ddclabel Analyse en dc.relation.forthcoming non en dc.relation.forthcomingprint non en dc.description.ssrncandidate non en dc.description.halcandidate non en dc.description.readership recherche en dc.description.audience International en dc.relation.Isversionofjnlpeerreviewed non en dc.relation.Isversionofjnlpeerreviewed non en dc.date.updated 2019-07-26T12:33:36Z hal.person.labIds 88479 hal.person.labIds 193738 hal.person.labIds 60
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