On flows generated by square-integrable vector fields
Abstract
For square-integrable divergence-free vector field v on ℝd we prove that the following properties are equivalent: 1) the operator A0 ρ= v · ∇ ρ (where ρ∈ C∞c(ℝd)) is essentially skew-adjoint on L2(ℝd); 2) square-integrable (with respect to spatial variables) generalized solutions of the continuity equation are renormalized; 3) generalized square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique both forward an backward in time. We also construct a compactly supported bounded divergence-free vector field v: ℝ3 → ℝ3 for which square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique forward, but not backward in time.
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BibTeX
@misc{gusev2026flows,
title = {{On flows generated by square-integrable vector fields}},
author = {Nikolay A. Gusev and Mikhail V. Korobkov and Evgeny Yu. Panov and Konstantin Yu. Zamana},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03831},
eprint = {2609.03831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03831},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On flows generated by square-integrable vector fields
AU - Nikolay A. Gusev
AU - Mikhail V. Korobkov
AU - Evgeny Yu. Panov
AU - Konstantin Yu. Zamana
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03831
UR - https://arxiv.org/abs/2609.03831
AB - For square-integrable divergence-free vector field v on ℝ^d we prove that the following properties are equivalent: 1) the operator A_0 ρ= v · ∇ ρ (where ρ∈ C^∞_c(ℝ^d)) is essentially skew-adjoint on L^2(ℝ^d); 2) square-integrable (with respect to spatial variables) generalized solutions of the continuity equation are renormalized; 3) generalized square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique both forward an backward in time. We also construct a compactly supported bounded divergence-free vector field v: ℝ^3 → ℝ^3 for which square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique forward, but not backward in time.
ER -