On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data
For the focusing mass-critical nonlinear Schrodinger equation $iu_t+Delta u=-|u|^{4/d}u$, an important problem is to establish Liouville type results for solutions with ground state mass. Here the ground state is the positive solution to elliptic equation $Delta Q-Q+Q^{1+frac 4d}=0$. Previous r...
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Texas State University
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doaj-cf104c4de96f4006bb38805b62be3b7b2020-11-24T23:56:35ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912009-06-01200978,119On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial dataDong LiXiaoyi ZhangFor the focusing mass-critical nonlinear Schrodinger equation $iu_t+Delta u=-|u|^{4/d}u$, an important problem is to establish Liouville type results for solutions with ground state mass. Here the ground state is the positive solution to elliptic equation $Delta Q-Q+Q^{1+frac 4d}=0$. Previous results in this direction were established in [12, 16, 17, 29] and they all require $u_0in H_x^1(mathbb{R}^d)$. In this paper, we consider the rigidity results for rough initial data $u_0 in H_x^s(mathbb{R}^d)$ for any $s>0$. We show that in dimensions $dge 4$ and under the radial assumption, the only solution that does not scatter in both time directions (including the finite time blowup case) must be global and coincide with the solitary wave $e^{it}Q$ up to symmetries of the equation. The proof relies on a non-uniform local iteration scheme, the refined estimate involving the $P^{pm}$ operator and a new smoothing estimate for spherically symmetric solutions. http://ejde.math.txstate.edu/Volumes/2009/78/abstr.htmlMass-criticalnonlinear Schrodinger equation |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Dong Li Xiaoyi Zhang |
spellingShingle |
Dong Li Xiaoyi Zhang On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data Electronic Journal of Differential Equations Mass-critical nonlinear Schrodinger equation |
author_facet |
Dong Li Xiaoyi Zhang |
author_sort |
Dong Li |
title |
On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data |
title_short |
On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data |
title_full |
On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data |
title_fullStr |
On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data |
title_full_unstemmed |
On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data |
title_sort |
on the rigidity of minimal mass solutions to the focusing mass-critical nls for rough initial data |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2009-06-01 |
description |
For the focusing mass-critical nonlinear Schrodinger equation $iu_t+Delta u=-|u|^{4/d}u$, an important problem is to establish Liouville type results for solutions with ground state mass. Here the ground state is the positive solution to elliptic equation $Delta Q-Q+Q^{1+frac 4d}=0$. Previous results in this direction were established in [12, 16, 17, 29] and they all require $u_0in H_x^1(mathbb{R}^d)$. In this paper, we consider the rigidity results for rough initial data $u_0 in H_x^s(mathbb{R}^d)$ for any $s>0$. We show that in dimensions $dge 4$ and under the radial assumption, the only solution that does not scatter in both time directions (including the finite time blowup case) must be global and coincide with the solitary wave $e^{it}Q$ up to symmetries of the equation. The proof relies on a non-uniform local iteration scheme, the refined estimate involving the $P^{pm}$ operator and a new smoothing estimate for spherically symmetric solutions. |
topic |
Mass-critical nonlinear Schrodinger equation |
url |
http://ejde.math.txstate.edu/Volumes/2009/78/abstr.html |
work_keys_str_mv |
AT dongli ontherigidityofminimalmasssolutionstothefocusingmasscriticalnlsforroughinitialdata AT xiaoyizhang ontherigidityofminimalmasssolutionstothefocusingmasscriticalnlsforroughinitialdata |
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1725457704445542400 |