Pointwise Convergence of the Schrödinger Flow

2019/07/25 by E. Compaan, Renato Lucà, Compaan, E. +3 · 2 citations
Mathematics · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1907.11192

openalex publication_date 2019/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schrödinger flows to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain smoothing effects for the non-homogeneous part of the solution can be used to upgrade to a uniform convergence to zero of this part, and we discuss the sharpness of the results obtained. We also use randomization techniques to prove that with much less regularity of the initial data, both in continuous and the periodic settings, almost surely one obtains uniform convergence of the nonlinear solution to the initial data, hence showing how more generic results can be obtained.

Citations

• On nonlinear Schrödinger equations, II.H S -solutions and unconditional well-posedness

Cited by

• Almost sure global nonlinear smoothing for the 2D NLS
• Regularity and pointwise convergence for dispersive equations on Riemannian symmetric spaces of compact type

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