Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1191.35258
Rodnianski, Igor; Rubinstein, Yanir A.; Staffilani, Gigliola
On the global well-posedness of the one-dimensional Schrödinger map flow. (English)
[J] Anal. PDE 2, No. 2, 187-209 (2009). ISSN 2157-5045; ISSN 1948-206X/e

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.
[Alina Stancu (Lowell)]

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
53C44 Geometric evolution equations (mean curvature flow)
35B10 Periodic solutions of PDE
32Q15 Kähler manifolds
42B35 Function spaces arising in harmonic analysis
15A23 Factorization of matrices
35B45 A priori estimates

Keywords: Schrödinger flow; periodic NLS; cubic NLS; Strichartz estimates; Kähler manifolds

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]