PreprintCC BY 4.0HAL
Norm inflation and loss of regularity in negative Sobolev spaces for KdV and KP
Abstract
We prove norm inflation phenomena for KdV and KP equations in negative order Sobolev spaces, in the periodic case, as well as on the whole space, on an arbitrarily large scale of negative order Sobolev spaces as target spaces. The proof relies on WKB analysis for a semiclassical version of the equation, in a weakly nonlinear régime, and the creation of the zero Fourier mode by resonant interaction. Unlike in previous similar results, this average mode has a smaller order of magnitude than the initial data, which requires a more detailed WKB analysis.
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BibTeX
@misc{carles2026norm,
title = {{Norm inflation and loss of regularity in negative Sobolev spaces for KdV and KP}},
author = {Rémi Carles},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.23698},
eprint = {2605.23698},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05628744v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Norm inflation and loss of regularity in negative Sobolev spaces for KdV and KP
AU - Rémi Carles
PY - 2026
DA - 2026/10/06
PB - HAL
DO - 10.48550/arxiv.2605.23698
UR - https://hal.science/hal-05628744v2
AB - We prove norm inflation phenomena for KdV and KP equations in negative order Sobolev spaces, in the periodic case, as well as on the whole space, on an arbitrarily large scale of negative order Sobolev spaces as target spaces. The proof relies on WKB analysis for a semiclassical version of the equation, in a weakly nonlinear régime, and the creation of the zero Fourier mode by resonant interaction. Unlike in previous similar results, this average mode has a smaller order of magnitude than the initial data, which requires a more detailed WKB analysis.
ER -