2008/10/10 by Hailiang Liu, Eitan Tadmor, Liu, Hailiang +3
Mathematics · #35B35 #35P15 #35Q35 #76B03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #math.AP #msc:35B35 #msc:35P15 #msc:35Q35 #msc:76B03
paper · pdf · doi:10.48550/arxiv.0810.1964
arxiv created 2008/10/10 · openalex publication_date 2008/10/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are concerned with the critical threshold phenomena in the Restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for 3D and the 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic.