Pointwise Convergence of the Schrödinger Flow
Name
1907.11192.pdf
Description
Accepted version
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506.3 KB
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5719234c7ee6dae577f0ef2c93e41645
Author(s)
Compaan, Erin
Lucà, Renato
Staffilani, Gigliola
Date Issued
2020
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Version
Author's final manuscript
Abstract
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.
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1093/IMRN/RNAA036
