Discontinuity of the Vlasov--Poisson Flow in LxpLv∞
Abstract
We prove that solution maps of the Vlasov--Poisson equation are discontinuous at the zero initial datum in Xp=LxpLv∞ for every 1≤ p<∞, in ℝd × ℝd with d ≤ 3, and for both attractive and repulsive interactions. For every T>0, the trajectory of the solution in L∞([0,T];Xp) is discontinuous at zero, and for every sufficiently small fixed t>0, the fixed-time map with values in Xp is also discontinuous at zero. The counterexamples are smooth, nonnegative, bounded by one, and supported in a common compact subset of phase space. Their mass and initial Xp norm tend to zero, whereas their solution norm at the observation time is at least one. The construction places narrow spatial packets on a lattice. Free transport allows a different velocity to select a packet at each spatial point, while a smallness estimate on the field ensures that the nonlinear characteristics are close to that of the free transport. For the same families, the phase-space Lq norms converge to zero uniformly in time for every 1≤ q<∞.
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BibTeX
@misc{chen2026discontinuity,
title = {{Discontinuity of the Vlasov--Poisson Flow in $L_x^pL_v^\infty$}},
author = {Ke Chen and In-Jee Jeong and Quoc-Hung Nguyen and Sangwook Tae},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20682},
eprint = {2609.20682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20682},
note = {CC0},
}
RIS
TY - UNPB
TI - Discontinuity of the Vlasov--Poisson Flow in L_x^pL_v^∞
AU - Ke Chen
AU - In-Jee Jeong
AU - Quoc-Hung Nguyen
AU - Sangwook Tae
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20682
UR - https://arxiv.org/abs/2609.20682
AB - We prove that solution maps of the Vlasov--Poisson equation are discontinuous at the zero initial datum in X_p=L_x^pL_v^∞ for every 1≤ p<∞, in ℝ^d × ℝ^d with d ≤ 3, and for both attractive and repulsive interactions. For every T>0, the trajectory of the solution in L^∞([0,T];X_p) is discontinuous at zero, and for every sufficiently small fixed t>0, the fixed-time map with values in X_p is also discontinuous at zero. The counterexamples are smooth, nonnegative, bounded by one, and supported in a common compact subset of phase space. Their mass and initial X_p norm tend to zero, whereas their solution norm at the observation time is at least one. The construction places narrow spatial packets on a lattice. Free transport allows a different velocity to select a packet at each spatial point, while a smallness estimate on the field ensures that the nonlinear characteristics are close to that of the free transport. For the same families, the phase-space L^q norms converge to zero uniformly in time for every 1≤ q<∞.
ER -