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

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Date
2012
Link to item file
http://hal.archives-ouvertes.fr/hal-00471646/fr/
Dewey
Probabilités et mathématiques appliquées
Sujet
Stochastic Calculus; Feynman-Kac Formula; Girsanov's Theorem; Quasi-linear Parabolic PDEs; Hyperbolic systems; Vanishing viscosity method; Smooth solutions
Journal issue
Stochastic Analysis and Applications
Volume
30
Number
1
Publication date
2012
Article pages
67-99
Publisher
Taylor & Francis
DOI
http://dx.doi.org/10.1080/07362994.2012.628914
URI
https://basepub.dauphine.fr/handle/123456789/4059
Collections
  • CEREMADE : Publications
Metadata
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Author
Lépinette, Emmanuel
Darses, Sébastien
Type
Article accepté pour publication ou publié
Abstract (EN)
We 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.

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