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Non-uniqueness for the forced one-dimensional Vlasov-Poisson equation
Abstract
We prove non-uniqueness for the forced one-dimensional Vlasov-Poisson equation on the whole space. More precisely, for every finite p, we construct a nonnegative ion density, smooth for positive times, and a source S∈ C([0,1];L1(R2)) with S(0)=0, for which there exist two distinct nonnegative distributional solutions f±∈ C([0,1];L1(R2)) with the same zero initial datum and with spatial densities bounded in Lp(R) uniformly in time. The result is sharp in terms of the integrability of the spatial densities: our solutions have Lipschitz electric fields, and we show that in this class uniqueness holds as soon as the spatial density is bounded.
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BibTeX
@misc{iacobelli2026nonuniqueness,
title = {{Non-uniqueness for the forced one-dimensional Vlasov-Poisson equation}},
author = {Mikaela Iacobelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31629},
eprint = {2610.31629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31629},
}
RIS
TY - UNPB
TI - Non-uniqueness for the forced one-dimensional Vlasov-Poisson equation
AU - Mikaela Iacobelli
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31629
UR - https://arxiv.org/abs/2610.31629
AB - We prove non-uniqueness for the forced one-dimensional Vlasov-Poisson equation on the whole space. More precisely, for every finite p, we construct a nonnegative ion density, smooth for positive times, and a source S∈ C([0,1];L^1(R^2)) with S(0)=0, for which there exist two distinct nonnegative distributional solutions f_±∈ C([0,1];L^1(R^2)) with the same zero initial datum and with spatial densities bounded in L^p(R) uniformly in time. The result is sharp in terms of the integrability of the spatial densities: our solutions have Lipschitz electric fields, and we show that in this class uniqueness holds as soon as the spatial density is bounded.
ER -