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 \(C([s,t);H^1)\cap L^2([s,t);H^2)\), \(0\le s<t\le T\). 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).
- 3.L. Baňas, Z. Brzeźniak, M. Neklyudov, and A. Prohl, A convergent finite-element-based discretization of the stochastic Landau–Lifshitz–Gilbert equation, IMA Journal of Numerical Analysis, page drt020, 2013Google Scholar
- 4.D. Berkov, Magnetization dynamics including thermal fluctuations, In H. Kronmüller and S. Parkin (editors), Handbook of Magnetism and Advanced Magnetic Materials, Volume 2, pages 795–823, Wiley Online Library, 2007Google Scholar
- 9.J.-M. Coron, J.-M. Ghidaglia, and F. Hélein, Nematics, 1991Google Scholar
- 10.G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, 2008Google Scholar
- 11.R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 3: Spectral Theory and Applications, Springer Science & Business Media, 2012Google Scholar
- 14.A. Deya, M. Gubinelli, M. Hofmanova, and S. Tindel, A priori estimates for rough PDEs with application to rough conservation laws, 2016Google Scholar
- 17.J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, American Journal of Mathematics, pages 109–160, 1964Google Scholar
- 21.T. L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetization field, Physical Review, 100:1243, 1955Google Scholar
- 24.B. Guo and S. Ding, Landau–Lifshitz Equations, World Scientific Singapore, 2008Google Scholar
- 30.F. Hélein, Applications harmoniques, lois de conservation et repères mobiles, Diderot, 1996Google Scholar
- 31.A. Hocquet, The Landau-Lifshitz-Gilbert equation driven by Gaussian noise, PhD thesis, École Polytechnique, 2015Google Scholar
- 32.A. Hocquet, Finite time singularity of the stochastic harmonic map flow, arXiv preprint arXiv:1606.02939, 2016
- 35.O. Ladyzhenskaya, V. Solonnikov, and N. Uraltseva, Linear and quasilinear parabolic equations of second order, Translation of Mathematical Monographs, AMS, Rhode Island, 1968Google Scholar
- 37.J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Volume 2, Dunod, 1968Google Scholar
- 42.A. Skhorokhod, Studies in the theory of random processes, Volume 7021, Courier Corporation, 2014Google Scholar
- 44.L. Tartar, The compensated compactness method applied to systems of conservation laws, In Systems of nonlinear partial differential equations, pages 263–285, Springer, 1983Google Scholar
- 46.S. Watanabe and N. Ikeda, Stochastic differential equations and diffusion processes, Elsevier, 1981Google Scholar