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 $Γ=\nablaΛ<sup>{-α}u$, 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.