clopen.solutions

Polynomial decay for the Korteweg--de Vries equation at length $14\pi$

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Kaiwen Jin

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 quadratic response is made periodic only on the exponentially stable subspace. Resonant terms in the undamped subspace may grow, but their boundary observation vanishes. An explicit nonzero double-frequency observation then gives a uniform quartic energy loss, which yields the all-time decay estimate.

Claims

SOLUTION CLAIMED OP-00784 — Polynomial decay for KdV with three critical linear modes
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Files and sources

Paper
AIM @ 24edaad, paper/manuscript.pdf ↗
Posted on GitHub
14 October 2026

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