Polynomial decay for KdV with three critical linear modes
Set $L = 14\pi$ and consider the real-valued Korteweg–de Vries problem $$y_t + y_x + y_{xxx} + y y_x = 0, \quad 0 < x < L, \qquad y(t, 0) = y(t, L) = y_x(t, L) = 0,$$ with solutions in the usual energy class $C([0, \infty); L^2(0, L)) \cap L^2_{\mathrm{loc}}([0, \infty); H^1_0(0, L))$. Its $L^2$ norm is nonincreasing. Rosier showed that for each critical length $L = 2\pi\sqrt{(k^2 + k\ell + \ell^2)/3}$, $k, \ell \in \mathbb{N}^*$, the linearized problem has a nontrivial finite-dimensional space $M \subset L^2(0, L)$ of initial data whose solutions have no boundary dissipation, $\int_0^t |y_x(s, 0)|^2 \, ds = 0$. Nguyen proved that small solutions decay at least like $t^{-1/2}$ when $\dim M = 1$, and when a nondegeneracy condition holds, which requires $\dim M$ to be even. For $L = 14\pi$ the pairs $(k, \ell) = (7, 7), (2, 11), (11, 2)$ give $\dim M = 3$.
Problem. Do there exist $\delta, C, \alpha > 0$ such that every solution with $\|y(0)\|_{L^2(0, L)} \le \delta$ satisfies $$\|y(t)\|_{L^2(0, L)} \le C (1 + t)^{-\alpha} \qquad (t \ge 0)?$$
Source: Niu and Xiang (2025), §2.4.5, for every critical length whose undamped space has odd dimension greater than one. Stated for L = 14π by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, critical lengths, decay rates, Korteweg–de Vries equation
References
@misc{niu2025,
author = {Niu, Jingrui and Xiang, Shengquan},
title = {Small-time local controllability of a {K}d{V} system for all critical lengths},
year = {2025},
eprint = {2501.13640},
archivePrefix = {arXiv},
note = {Section 2.4.5},
}
@article{nguyen2021,
author = {Nguyen, Hoai-Minh},
title = {Decay for the nonlinear {K}d{V} equations at critical lengths},
journal = {J. Differential Equations},
volume = {295},
year = {2021},
pages = {249--291},
doi = {10.1016/j.jde.2021.05.057},
eprint = {2012.08792},
archivePrefix = {arXiv},
}
@article{rosier1997,
author = {Rosier, Lionel},
title = {Exact boundary controllability for the {K}orteweg--de {V}ries equation on a bounded domain},
journal = {ESAIM Control Optim. Calc. Var.},
volume = {2},
year = {1997},
pages = {33--55},
doi = {10.1051/cocv:1997102},
}
@misc{aim-critical-kdv-odd-mode-polynomial-decay,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Polynomial decay for KdV with three critical linear modes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/396-critical-kdv-odd-mode-polynomial-decay.md},
note = {Entry 396 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove uniform small-data polynomial decay for the real Korteweg--de Vries equation on an interval of length $14\pi$, with homogeneous Dirichlet boundary conditions and zero Neumann datum at the right endpoint. This critical length has three undamped linear modes, including a zero frequency. Every sufficiently small real $L^2$ energy solution satisfies $\|y(t)\|_2\le C(1+t)^{-1/2}$, with constants independent of the datum. The …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00784, "Polynomial decay for KdV with three critical linear modes", clopen.solutions.