Conditional global regularity of Schr "odinger maps: sub-threshold\n dispersed energy

2010/12/17 by Paul Smith, Smith, Paul
Mathematics · #35B33 #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1012.4048

openalex publication_date 2010/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Schr "odinger map initial value problem into the sphere in\n2+1 dimensions with smooth, decaying, subthreshold initial data. Assuming an a\npriori L⁴ boundedness condition on the solution, we prove that the\nSchr "odinger map system admits a unique global smooth solution provided that\nthe initial data is sufficiently energy-dispersed. Also shown are\nglobal-in-time bounds on certain Sobolev norms of the solution. Toward these\nends we establish improved local smoothing and bilinear Strichartz estimates,\nadapting the Planchon-Vega approach to such estimates to the nonlinear setting\nof Schr "odinger maps.\n

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