Global Stability and Energy Growth in the Sheared Vlasov--Poisson--Boltzmann System
Abstract
We study the Vlasov--Poisson--Boltzmann system on the three-dimensional torus under uniform shear flow. For Maxwell molecules with Grad's angular cutoff and sufficiently small shear rate, we consider perturbations around the spatially homogeneous self-similar profile of the sheared Boltzmann equation. In self-similar variables, we prove the global stability of this profile, including global existence and uniqueness, the exponential decay of nonzero-spatial derivatives in weighted L∞ spaces, and the uniform boundedness of the renormalized perturbation. The analysis combines a Caflisch's decomposition, Guo's L∞--L2 estimates, macro--micro analysis, and a spectral study of the zero-frequency mode. We further derive a closed zero-frequency system for a renormalized total energy and suitable second-order moments, which yields a precise large-time description of the shear-induced energy growth.
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BibTeX
@misc{chang2026global,
title = {{Global Stability and Energy Growth in the Sheared Vlasov--Poisson--Boltzmann System}},
author = {Zhida Chang and Mingying Zhong},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.13356},
eprint = {2608.13356},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13356},
}
RIS
TY - UNPB
TI - Global Stability and Energy Growth in the Sheared Vlasov--Poisson--Boltzmann System
AU - Zhida Chang
AU - Mingying Zhong
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.13356
UR - https://arxiv.org/abs/2608.13356
AB - We study the Vlasov--Poisson--Boltzmann system on the three-dimensional torus under uniform shear flow. For Maxwell molecules with Grad's angular cutoff and sufficiently small shear rate, we consider perturbations around the spatially homogeneous self-similar profile of the sheared Boltzmann equation. In self-similar variables, we prove the global stability of this profile, including global existence and uniqueness, the exponential decay of nonzero-spatial derivatives in weighted L^∞ spaces, and the uniform boundedness of the renormalized perturbation. The analysis combines a Caflisch's decomposition, Guo's L^∞--L^2 estimates, macro--micro analysis, and a spectral study of the zero-frequency mode. We further derive a closed zero-frequency system for a renormalized total energy and suitable second-order moments, which yields a precise large-time description of the shear-induced energy growth.
ER -