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Regularity of almost
periodic modulo scaling solutions for mass-critical NLS and
applications
Dong Li and Xiaoyi Zhang |
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Vol. 3 (2010), No. 2, 175–195
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Abstract |
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We consider the solution to mass-critical NLS . We prove that in dimensions , if the solution is spherically symmetric and is almost periodic modulo scaling, then it must lie in for some . Moreover, the kinetic energy of the solution is localized uniformly in time. One important application of the theorem is a simplified proof of the scattering conjecture for mass-critical NLS without reducing to three enemies. As another important application, we establish a Liouville type result for initial data with ground state mass. We prove that if a radial solution to focusing mass-critical problem has the ground state mass and does not scatter in both time directions, then it must be global and coincide with the solitary wave up to symmetries. Here the ground state is the unique, positive, radial solution to elliptic equation . This is the first rigidity type result in scale invariant space . |
Keywords
Schrödinger equation, mass-critical
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Mathematical Subject Classification 2000
Primary: 35Q55
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Milestones
Received: 10 August 2009
Revised: 18 November 2009
Accepted: 17 December 2009
Published: 8 June 2010
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Authors |
| Dong Li | |
| Department of Mathematics University of Iowa 14 MacLean Hall Iowa City, IA 52240 United States |
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| Xiaoyi Zhang | |
| Academy of Mathematics and System
Sciences Beijing China |
Department of Mathematics University of Iowa 14 MacLean Hall Iowa City, IA 52240 United States |
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