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PreprintCC BY-NC-ND 4.0arXiv

Global theory for NLS in the weighted spaces I: finite pseudo conformal energy

  • Yujin Guo
  • Jia Shen
  • Changping Yang
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Abstract

We intend to prove that for defocusing nonlinear Schrödinger equations, solutions with finite pseudo conformal energy must be global and scatter. This fact is first proved by Bourgain when 0<sc<1: the equation is locally well-posed in Hxsc, and if one further assumes that the initial data satisfies xu0∈ Lx2, then the solution is global and scatters. The sc<0 case remains not studied, and the best result is given by Beceanu, Deng, Soffer, and Wu: If the initial data is in Hxsc, radial, and compactly supported, then the solution is globally well-posed. In this paper, we extend Bourgain's result to the sc<0 case. We prove that if the initial data satisfies |x|−scu0∈ Lx2 and xu0∈ Lx2, or if the radial initial data satisfies u0∈ Ḣxsc and xu0∈ Lx2, then the solution is global and scatters. Bourgain's result is based on the subcritical a priori control on Lxp+2, while our argument relies on the supercritical bound on ℱḢx1.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026global,
  title = {{Global theory for NLS in the weighted spaces I: finite pseudo conformal energy}},
  author = {Yujin Guo and Jia Shen and Changping Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37714},
  eprint = {2609.37714},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37714},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Global theory for NLS in the weighted spaces I: finite pseudo conformal energy
AU  - Yujin Guo
AU  - Jia Shen
AU  - Changping Yang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37714
UR  - https://arxiv.org/abs/2609.37714
AB  - We intend to prove that for defocusing nonlinear Schrödinger equations, solutions with finite pseudo conformal energy must be global and scatter. This fact is first proved by Bourgain when 0<s_c<1: the equation is locally well-posed in H_x^s_c, and if one further assumes that the initial data satisfies xu_0∈ L_x^2, then the solution is global and scatters. The s_c<0 case remains not studied, and the best result is given by Beceanu, Deng, Soffer, and Wu: If the initial data is in H_x^s_c, radial, and compactly supported, then the solution is globally well-posed. In this paper, we extend Bourgain's result to the s_c<0 case. We prove that if the initial data satisfies |x|^−s_cu_0∈ L_x^2 and xu_0∈ L_x^2, or if the radial initial data satisfies u_0∈ Ḣ_x^s_c and xu_0∈ L_x^2, then the solution is global and scatters. Bourgain's result is based on the subcritical a priori control on L_x^p+2, while our argument relies on the supercritical bound on ℱḢ_x^1.
ER  -