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Endpoint well-posedness for the Boussinesq equation

  • Mingjuan Chen
  • Minjie Shan
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Abstract

We prove small-data global well-posedness for the quadratic good Boussinesq equation on the real line and the circle at the endpoint H−1/2× H−3/2. On each domain, a small invariant spectral set supports a continuous global group, with uniqueness in a fixed local integral class. In the periodic case, both components have zero mean and the flow is Lipschitz on bounded time intervals. The a priori bounds follow from a balanced perturbation determinant for the third-order Lax operator. Averaging two real spectral branches yields coercivity and frequency-tail control. On the line, we construct the endpoint flow by commuting Hamiltonians and Green-field estimates. On the circle, discrete spectral bounds and Kishimoto's endpoint local theory give global well-posedness.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026endpoint,
  title = {{Endpoint well-posedness for the Boussinesq equation}},
  author = {Mingjuan Chen and Minjie Shan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34793},
  eprint = {2610.34793},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34793},
}
RIS
TY  - UNPB
TI  - Endpoint well-posedness for the Boussinesq equation
AU  - Mingjuan Chen
AU  - Minjie Shan
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34793
UR  - https://arxiv.org/abs/2610.34793
AB  - We prove small-data global well-posedness for the quadratic good Boussinesq equation on the real line and the circle at the endpoint H^−1/2× H^−3/2. On each domain, a small invariant spectral set supports a continuous global group, with uniqueness in a fixed local integral class. In the periodic case, both components have zero mean and the flow is Lipschitz on bounded time intervals. The a priori bounds follow from a balanced perturbation determinant for the third-order Lax operator. Averaging two real spectral branches yields coercivity and frequency-tail control. On the line, we construct the endpoint flow by commuting Hamiltonians and Green-field estimates. On the circle, discrete spectral bounds and Kishimoto's endpoint local theory give global well-posedness.
ER  -