Struwe-like solutions for the Stochastic Harmonic Map flow
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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 \(C([s,t);H^1)\cap L^2([s,t);H^2)\), \(0\le s<t\le T\). 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.
KeywordsStochastic partial differential equation Harmonic Maps Nonlinear parabolic equations Uniqueness Landau–Lifshitz–Gilbert equation
Mathematics Subject Classification60H15 (35R60) 58E20 35K55 34A12
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This work was done in its majority while the Author was a Phd student at the École polytechnique. Anne De Bouard and François Alouges are gratefully acknowledged for numerous discussions and encouragements. Financial support was kindly provided by the ANR projects Micro-MANIP (ANR-08-BLAN-0199) and STOSYMAP (ANR-2011-BS01-015-03).
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