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Global regularity for the unidirectional Euler-alignment system with supercritical dissipation
Abstract
We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order 0<α<1 and arbitrary non-vacuum initial data in the Sobolev class of the local theory. The key idea is to propagate simultaneously two moduli of continuity at critical scales for the density ρ and the potential gradient Γ=∇Λ−αu, both of which have scaling-invariant amplitudes. We also establish exponential alignment of the velocity and exponential convergence of the density to a traveling profile.
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BibTeX
@misc{tan2026global,
title = {{Global regularity for the unidirectional Euler-alignment system with supercritical dissipation}},
author = {Changhui Tan and Liutang Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25748},
eprint = {2609.25748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25748},
}
RIS
TY - UNPB
TI - Global regularity for the unidirectional Euler-alignment system with supercritical dissipation
AU - Changhui Tan
AU - Liutang Xue
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25748
UR - https://arxiv.org/abs/2609.25748
AB - We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order 0<α<1 and arbitrary non-vacuum initial data in the Sobolev class of the local theory. The key idea is to propagate simultaneously two moduli of continuity at critical scales for the density ρ and the potential gradient Γ=∇Λ^−αu, both of which have scaling-invariant amplitudes. We also establish exponential alignment of the velocity and exponential convergence of the density to a traveling profile.
ER -