Non-uniqueness for the forced one-dimensional Vlasov-Poisson equation

Mikaela Iacobelli

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arXiv · 2610.31629

Non-uniqueness for the forced one-dimensional Vlasov-Poisson equation

Mikaela Iacobelli

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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 $S\in C([0,1];L^1(\R^2))$ with $S(0)=0$, for which there exist two distinct nonnegative distributional solutions $f_\pm\in C([0,1];L^1(\R^2))$ 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.

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BibTeX RIS

Mikaela Iacobelli. 2026-10-22. Non-uniqueness for the forced one-dimensional Vlasov-Poisson equation. https://arxiv.org/abs/2610.31629

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