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On the global well-posedness of the one-dimensional Schrödinger map flow. (English) Zbl 1191.35258
In 2002, W.-Y. Ding conjectured that the Schrödinger map flow is globally well-posed for maps from one dimensional domains into compact Kähler manifolds. The present paper validates Ding’s conjecture for maps from the real line and for maps from the circle into Riemann surfaces. The general idea in both cases is to get a priori estimates on the short time solution to an equivalent system of nonlinear equations which, with non-trivial work, can be extended to stronger norms and these latter estimates imply global well-posedness to the map flow. Despite the similar approach, passing from the case when the domain is the real line to the domain being a circle one loses the simply connectedness, and the compactness, rendering the well-posedness for maps from the circle into Riemann surfaces more difficult in an essential way.
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
53C44Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
35B10Periodic solutions of PDE
32Q15Kähler manifolds
42B35Function spaces arising in harmonic analysis
15A23Factorization of matrices
35B45A priori estimates for solutions of PDE