Local well-posedness for the KdV equation on the half-line at the critical regularity H−3/4
Abstract
We prove local well-posedness for the initial-boundary-value problem for the Korteweg--de Vries equation on the right half-line at the critical regularity s=−3/4, with initial and boundary data in H−3/4(ℝ+) and H1/12(ℝ+), respectively. This reaches the endpoint left open in the previous half-line results of Holmer (2006) and Bona, Sun, and Zhang (2006), who established local well-posedness for s>−3/4. The proof is based on a contraction argument in the Besov-type Bourgain space introduced by Kishimoto. The main difficulty is to construct a boundary forcing operator compatible with the endpoint b=1/2 and the required ℓ1-summability in the modulation variable. We construct such an operator using a Laplace-transform representation of the linear half-line problem and a carefully designed extension to x<0. The extension combines the required regularity across the boundary with prescribed vanishing spatial moments, yielding favorable low-frequency behavior of the associated kernel. A pointwise kernel estimate then provides the crucial dyadic summability needed for the X−3/4,1/2,1 estimate. Together with a time-trace estimate for the Duhamel term at b=1/2, this yields the critical local well-posedness result. The ideas used here have prospects to be used in other nonlinear dispersive equations on the half-lines.
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BibTeX
@misc{cavalcante2026local,
title = {{Local well-posedness for the KdV equation on the half-line at the critical regularity $H^-\frac34$}},
author = {Márcio Cavalcante},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28009},
eprint = {2609.28009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28009},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local well-posedness for the KdV equation on the half-line at the critical regularity H^−3/4
AU - Márcio Cavalcante
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28009
UR - https://arxiv.org/abs/2609.28009
AB - We prove local well-posedness for the initial-boundary-value problem for the Korteweg--de Vries equation on the right half-line at the critical regularity s=−3/4, with initial and boundary data in H^−3/4(ℝ_+) and H^1/12(ℝ_+), respectively. This reaches the endpoint left open in the previous half-line results of Holmer (2006) and Bona, Sun, and Zhang (2006), who established local well-posedness for s>−3/4. The proof is based on a contraction argument in the Besov-type Bourgain space introduced by Kishimoto. The main difficulty is to construct a boundary forcing operator compatible with the endpoint b=1/2 and the required ℓ^1-summability in the modulation variable. We construct such an operator using a Laplace-transform representation of the linear half-line problem and a carefully designed extension to x<0. The extension combines the required regularity across the boundary with prescribed vanishing spatial moments, yielding favorable low-frequency behavior of the associated kernel. A pointwise kernel estimate then provides the crucial dyadic summability needed for the X^−3/4,1/2,1 estimate. Together with a time-trace estimate for the Duhamel term at b=1/2, this yields the critical local well-posedness result. The ideas used here have prospects to be used in other nonlinear dispersive equations on the half-lines.
ER -