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Gradient flow approach to Landau equation: A cross-product structure
Abstract
We introduce a cross-product Landau gradient that commutes with Gaussian mollification. For the hard-potential interaction kernels of the form A(|v−v*|)∼ ⟨ v−v*⟩γ|v−v*|2 with γ∈(−∞,1], this construction yields a variational characterisation of the gradient-flow structure of the spatially homogeneous Landau equation. The cross-product gradient induces the same gradient-flow geometry as that introduced by Carrillo, Delgadino, Desvillettes and Wu (2024), while providing a different representation of the Landau gradient. The main advantage of this formulation is that it requires only weighted L1-control. We also discuss a similar variational characterisation for the GENERIC structure of the fuzzy Landau equations.
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BibTeX
@misc{duong2026gradient,
title = {{Gradient flow approach to Landau equation: A cross-product structure}},
author = {Manh Hong Duong and Zihui He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07466},
eprint = {2609.07466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07466},
}
RIS
TY - UNPB
TI - Gradient flow approach to Landau equation: A cross-product structure
AU - Manh Hong Duong
AU - Zihui He
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07466
UR - https://arxiv.org/abs/2609.07466
AB - We introduce a cross-product Landau gradient that commutes with Gaussian mollification. For the hard-potential interaction kernels of the form A(|v−v_*|)∼ ⟨ v−v_*⟩^γ|v−v_*|^2 with γ∈(−∞,1], this construction yields a variational characterisation of the gradient-flow structure of the spatially homogeneous Landau equation. The cross-product gradient induces the same gradient-flow geometry as that introduced by Carrillo, Delgadino, Desvillettes and Wu (2024), while providing a different representation of the Landau gradient. The main advantage of this formulation is that it requires only weighted L^1-control. We also discuss a similar variational characterisation for the GENERIC structure of the fuzzy Landau equations.
ER -