# 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

, Volume 40 (3) – Aug 1, 2091
14 pages

/lp/springer_journal/stability-of-solutions-of-parabolic-pdes-with-random-drift-and-00324w9o4U
Publisher
Springer-Verlag
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations