Parabolic schemes for quasi-linear parabolic and hyperbolic PDEs via stochastic calculus
Lépinette, Emmanuel; Darses, Sébastien (2012), Parabolic schemes for quasi-linear parabolic and hyperbolic PDEs via stochastic calculus, Stochastic Analysis and Applications, 30, 1, p. 67-99. http://dx.doi.org/10.1080/07362994.2012.628914
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00471646/fr/
Journal nameStochastic Analysis and Applications
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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.
Subjects / KeywordsStochastic Calculus; Feynman-Kac Formula; Girsanov's Theorem; Quasi-linear Parabolic PDEs; Hyperbolic systems; Vanishing viscosity method; Smooth solutions
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