|Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs Via Stochastic Calculus |
Auteur(s): Lépinette Emmanuel, Darses Sébastien
(Article) Publié: -Journal Of Stochastic Analysis And Applications, vol. 30 p.67-99 (2012)
Ref HAL: hal-00700872_v1
Exporter : BibTex | endNote
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.