Displaying items by tag: unidirectional flow

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.

This work is supported by NSF grants DMS #2238219

Yatao Li, Qianyun Miao, Changhui Tan and Liutang Xue

International Mathematics Research Notices, Volume 2024, No. 23, pp. 14393-14422 (2024).

Abstract

We investigate global solutions to the Euler-alignment system in d dimensions with unidirectional flows and strongly singular communication protocols \(\phi(x)=|x|^{-d+\alpha}\) for \(\alpha\in(0,2)\). Our paper establishes global regularity results in both the subcritical regime \(1<\alpha<2\) and the critical regime \(\alpha=1\). Notably, when \(\alpha=1\), the system exhibits a critical scaling similar to the critical quasi-geostrophic equation. To achieve global well-posedness, we employ a novel method based on propagating the modulus of continuity. Our approach introduces the concept of simultaneously propagating multiple moduli of continuity, which allows us to effectively handle the system of two equations with critical scaling. Additionally, we improve the regularity criteria for solutions to this system in the supercritical regime \(0<\alpha<1\).

	doi:10.1093/imrn/rnae246
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	This work is supported by NSF grants DMS #2108264 and DMS #2238219