2023/02/20 by Wang, Sheng, Zhou, Yi · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2302.09969
Wei-Yue Ding \citeDing 2002 proposeed a proposition about Schrödinger map flow in 2002 International Congress of Mathematicians in Beijing, which is called Wei-Yue Ding conjecture by Rodnianski-Rubinstein-Staffilani \citeRodnianski 2009. They proved \citeRodnianski 2009 that Schrödinger map flow for maps from the real line into Kähler 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 Schrödinger map flow into nonlinear Schrödinger-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 Schrödinger map flow for maps from the circle into Kähler manifolds is globally regular. So far, the Wei-Yue Ding's conjecture has been completely solved.