Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit

Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit Let u α be the solution of the Itô stochastic parabolic Cauchy problem $\partial u/\partial t-L_\a u = \xi \cdot \nabla u, \ u\v_{t=0} = f$ , where ξ is a space—time noise. We prove that u α depends continuously on α , when the coefficients in L α converge to those in L 0 . This result is used to study the diffusion limit for the Cauchy problem in the Stratonovich sense: when the coefficients of L α tend to 0 the corresponding solutions u α converge to the solution u 0 of the degenerate Cauchy problem $\partial u_0/ \partial t =\xi\circ \nabla u_0, \ u_0\v_{t=0} = f$ . These results are based on a criterion for the existence of strong limits in the space of Hida distributions (S) * . As a by-product it is proved that weak solutions of the above Cauchy problem are in fact strong solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit

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Publisher
Springer-Verlag
Copyright
Copyright © Inc. by 1999 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900132
Publisher site
See Article on Publisher Site

Abstract

Let u α be the solution of the Itô stochastic parabolic Cauchy problem $\partial u/\partial t-L_\a u = \xi \cdot \nabla u, \ u\v_{t=0} = f$ , where ξ is a space—time noise. We prove that u α depends continuously on α , when the coefficients in L α converge to those in L 0 . This result is used to study the diffusion limit for the Cauchy problem in the Stratonovich sense: when the coefficients of L α tend to 0 the corresponding solutions u α converge to the solution u 0 of the degenerate Cauchy problem $\partial u_0/ \partial t =\xi\circ \nabla u_0, \ u_0\v_{t=0} = f$ . These results are based on a criterion for the existence of strong limits in the space of Hida distributions (S) * . As a by-product it is proved that weak solutions of the above Cauchy problem are in fact strong solutions.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Aug 1, 2091

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