Geometric constraints on potentially

1996/01/01 by Charles Fefferman, Andrew J. Majda · 259 citations
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Fluid Dynamics and Turbulent Flows #Mathematics #Pure mathematics

paper · doi:10.1080/03605309608821197

published in Communications in Partial Differential Equations 21(3-4), 559-571 (Taylor & Francis)

openalex publication_date 1996/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

We discuss necessary and sufficient conditions for the formation of finite time singularities (blow up) in the incompressible three dimensional Euler equations. The well-known result of Beale, Kato and Majda states that these equations have smooth solutions on the time interval (0,t) if, and only if lim/tr_arrowT integralsup tsub 0 parallelOmega(center_dot,s)parallelsub Lsup infinity (dx)dx < infinity where Omega = triangledown X u is the vorticity of the fluid and u is its divergence=free velocity. In this paper we prove criteria in which the direction of vorticity xi = Omega/vert_barOmegavert_bar plays an important role.

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• Charles Fefferman [wikipedia]