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Struwe-like solutions for the Stochastic Harmonic Map flow

Struwe-like solutions for the Stochastic Harmonic Map flow We give new results on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau–Lifshitz–Gilbert model for thermal fluctuations in micromagnetics. We first construct strong solutions that belong locally to the spaces $$C([s,t);H^1)\cap L^2([s,t);H^2)$$ C ( [ s , t ) ; H 1 ) ∩ L 2 ( [ s , t ) ; H 2 ) , $$0\le s<t\le T$$ 0 ≤ s < t ≤ T . It that sense, these maps are a counterpart of the so-called “Struwe solutions” of the deterministic model. We then provide a natural criterion of uniqueness that extends A. Freire’s Theorem to the stochastic case. Both results are obtained under the condition that the noise term has a trace-class covariance in space. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Struwe-like solutions for the Stochastic Harmonic Map flow

Journal of Evolution Equations , Volume 18 (3) – Mar 14, 2018

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References (55)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-018-0437-3
Publisher site
See Article on Publisher Site

Abstract

We give new results on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau–Lifshitz–Gilbert model for thermal fluctuations in micromagnetics. We first construct strong solutions that belong locally to the spaces $$C([s,t);H^1)\cap L^2([s,t);H^2)$$ C ( [ s , t ) ; H 1 ) ∩ L 2 ( [ s , t ) ; H 2 ) , $$0\le s<t\le T$$ 0 ≤ s < t ≤ T . It that sense, these maps are a counterpart of the so-called “Struwe solutions” of the deterministic model. We then provide a natural criterion of uniqueness that extends A. Freire’s Theorem to the stochastic case. Both results are obtained under the condition that the noise term has a trace-class covariance in space.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Mar 14, 2018

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