Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs Via Stochastic Calculus
affiliation not provided to SSRN
CEREMADE, UMR CNRS 7534, Paris-Dauphine University.
April 12, 2012
Journal of Stochastic Analysis and Applications, Vol. 30, No. 1, 2012
We consider two quasi-linear initial-value Cauchy problems on Rd: a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u) · ∇u, a forcing term h(t, x, u) and an initial condition u0 ∈ L∞ (Rd ) ∩ C ∞ (Rd ), where φ (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.
Number of Pages in PDF File: 29
Keywords: quasi-linear parabolic PDEs, hyperbolic systems, vanishing viscosity method, smooth solutions, stochastic calculus, Feynman-Kac formula, Girsanov’s theoremAccepted Paper Series
Date posted: April 13, 2012
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