星期二, 22 9月 2026 09:58

Global regularity for the unidirectional Euler-alignment system with supercritical dissipation

Changhui Tan, Liutang Xue

Abstract

We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order \(0<\alpha<1\) 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 \(\rho\) and the potential gradient \(\Gamma=\nabla\Lambda^{-\alpha}u\), 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.