%0 Journal Article
%T Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs Via Stochastic Calculus
%+ CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
%+ Laboratoire d'Analyse, Topologie, Probabilités (LATP)
%A Lépinette, Emmanuel
%A Darses, Sebastien
%< avec comité de lecture
%@ 0736-2994
%J Stochastic Analysis and Applications
%I Taylor & Francis: STM, Behavioural Science and Public Health Titles
%V 30
%N 1
%P 67-99
%8 2012
%D 2012
%R 10.1080/07362994.2012.628914
%K quasi-linear parabolic PDEs
%K hyperbolic systems
%K vanishing viscosity method
%K smooth solutions
%K stochastic calculus
%K Feynman-Kac formula
%K Girsanov's theorem
%Z 60H30, 35K, 35L
%Z Mathematics [math]/Probability [math.PR]Journal articles
%X 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.
%G English
%L hal-00700872
%U https://hal.science/hal-00700872
%~ LATP
%~ CNRS
%~ UNIV-AMU
%~ UNIV-DAUPHINE
%~ EC-MARSEILLE
%~ CEREMADE
%~ I2M
%~ PSL
%~ UNIV-DAUPHINE-PSL