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

Sebastien Darses (1) , Emmanuel Denis (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Quasi-linear Parabolic PDEs; Hyperbolic systems; Vanishing viscosity method; Smooth solutions; Stochastic Calculus; Feynman-Kac Formula; Girsanov's Theorem

Domaines: Probabilités [math.PR]

Sebastien Darses, Emmanuel Denis. Parabolic schemes for quasi-linear parabolic and hyperbolic PDEs via stochastic calculus. 2010. ⟨hal-00471646v2⟩