Work

Global Well-Posedness and Refined Regularity Criterion for the Uni-Directional Euler-Alignment System

Year: 2024

Type: article

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 $0<\alpha <1$.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnae246

Authors: Yatao Li, Qianyun Miao, Changhui Tan, Liutang Xue

Institutions: Jiangxi University of Finance and Economics, Beijing Institute of Technology, University of South Carolina, Beijing Normal University

Language: English

Topic: Navier-Stokes equation solutions

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, National Natural Science Foundation of China, National Key Research and Development Program of China