Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs Via Stochastic Calculus - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Stochastic Analysis and Applications Année : 2012

Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs Via Stochastic Calculus

Résumé

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.

Dates et versions

hal-00700872 , version 1 (24-05-2012)

Identifiants

Citer

Emmanuel Lépinette, Sebastien Darses. Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs Via Stochastic Calculus. Stochastic Analysis and Applications, 2012, 30 (1), pp.67-99. ⟨10.1080/07362994.2012.628914⟩. ⟨hal-00700872⟩
105 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More