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PreprintarXiv licencearXiv

Unconditional uniqueness for the cubic nonlinear Schrödinger equation in Ḣ1/2(Bbb R3)

  • Yongming Luo
  • Jiayu Zheng
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Abstract

While unconditional uniqueness for the cubic nonlinear Schrödinger equation (NLS) on Bbb R3 is known in the scaling-subcritical Sobolev spaces Ḣs(Bbb R3) for 1/2<s<1, the critical case s=1/2 has remained open. We resolve this problem by adapting Kato's bootstrap argument to the critical space Ḃ−1∞,∞. The main difficulty at this endpoint is that the natural solution class does not provide the coefficient regularity required to close the bootstrap in Ḃ−1∞,∞. To overcome this issue, we perform a second Duhamel iteration of Born type and estimate the resulting double Duhamel integral as a whole. This yields improved estimates that allow the bootstrap to close in Ḃ−1∞,∞ using only the available regularity of the coefficients.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026unconditional,
  title = {{Unconditional uniqueness for the cubic nonlinear Schrödinger equation in $\dotH^\frac12(\Bbb R^3)$}},
  author = {Yongming Luo and Jiayu Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06501},
  eprint = {2610.06501},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06501},
}
RIS
TY  - UNPB
TI  - Unconditional uniqueness for the cubic nonlinear Schrödinger equation in Ḣ^1/2(Bbb R^3)
AU  - Yongming Luo
AU  - Jiayu Zheng
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.06501
UR  - https://arxiv.org/abs/2610.06501
AB  - While unconditional uniqueness for the cubic nonlinear Schrödinger equation (NLS) on Bbb R^3 is known in the scaling-subcritical Sobolev spaces Ḣ^s(Bbb R^3) for 1/2<s<1, the critical case s=1/2 has remained open. We resolve this problem by adapting Kato's bootstrap argument to the critical space Ḃ^−1_∞,∞. The main difficulty at this endpoint is that the natural solution class does not provide the coefficient regularity required to close the bootstrap in Ḃ^−1_∞,∞. To overcome this issue, we perform a second Duhamel iteration of Born type and estimate the resulting double Duhamel integral as a whole. This yields improved estimates that allow the bootstrap to close in Ḃ^−1_∞,∞ using only the available regularity of the coefficients.
ER  -