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Parabolic schemes for quasi-linear parabolic and hyperbolic PDEs via stochastic calculus

hal.structure.identifier
dc.contributor.authorLépinette, Emmanuel*
hal.structure.identifier
dc.contributor.authorDarses, Sébastien*
dc.date.accessioned2010-04-29T15:04:18Z
dc.date.available2010-04-29T15:04:18Z
dc.date.issued2012
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/4059
dc.language.isoenen
dc.subjectStochastic Calculusen
dc.subjectFeynman-Kac Formulaen
dc.subjectGirsanov's Theoremen
dc.subjectQuasi-linear Parabolic PDEsen
dc.subjectHyperbolic systemsen
dc.subjectVanishing viscosity methoden
dc.subjectSmooth solutionsen
dc.subject.ddc519en
dc.titleParabolic schemes for quasi-linear parabolic and hyperbolic PDEs via stochastic calculusen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenWe consider two quasi-linear initial-value Cauchy problems on Rd: a parabolic system and an hyperbolic one. They both have a rst order non-linearity of the form (t; x; u) ru, a forcing term h(t; x; u) and an initial condition u0 2 L1(Rd) \ C1(Rd), where (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t; x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but recursive parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.en
dc.relation.isversionofjnlnameStochastic Analysis and Applications
dc.relation.isversionofjnlvol30
dc.relation.isversionofjnlissue1
dc.relation.isversionofjnldate2012
dc.relation.isversionofjnlpages67-99
dc.relation.isversionofdoihttp://dx.doi.org/10.1080/07362994.2012.628914
dc.identifier.urlsitehttp://hal.archives-ouvertes.fr/hal-00471646/fr/en
dc.description.sponsorshipprivateouien
dc.relation.isversionofjnlpublisherTaylor & Francis
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
hal.author.functionaut
hal.author.functionaut


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