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Periodic Schr\"odinger map flow on K\"ahler manifolds
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Wei-Yue Ding \cite{Ding 2002} proposeed a proposition about Schr\"odinger map flow in 2002 International Congress of Mathematicians in Beijing, which is called Wei-Yue Ding conjecture by Rodnianski-Rubinstein-Staffilani \cite{Rodnianski 2009}. They proved \cite{Rodnianski 2009} that Schr\"odinger map flow for maps from the real line into K\"ahler manifolds and for maps from the circle into Riemann surfaces is globally well-posed which is the first significant advance in this conjecture by translating the Schr\"odinger map flow into nonlinear Schr\"odinger-type equations or (systems) and partially solved this conjecture. In this article, we will derive a new div-curl type lemma and combined it with energy and ``momentum" balance law to get some space-time estimates. Based on this, we prove the Schr\"odinger map flow for maps from the circle into K\"ahler manifolds is globally regular. So far, the Wei-Yue Ding's conjecture has been completely solved.
Forward citations
Cited by 2 Pith papers
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Local well-posedness of the Schr\"odinger flow into $\mathbb{S}^2$ with natural boundary conditions
Initial-Neumann compatibility conditions are shown to be necessary and sufficient for local well-posedness in W^{2k+3,2} of the Schrödinger flow into S^2 for every k≥1.
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Physical Space Proof of Bilinear Estimates and Applications to Nonlinear Dispersive Equations (II)
A physical-space bilinear estimate method reproduces the sharpest known local well-posedness thresholds for the 2d and 3d Zakharov system without Bourgain spaces.
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