Geometric constraints on potentially singular solutions for the 3-D Euler equations

Journal Article · Mon Dec 30 23:00:00 EST 1996 · Communications in Partial Differential Equations

OSTI ID:441146

Constantin, P ^[1]; Fefferman, C; Majda, A J

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1. Univ. of Chicago, IL (United States)

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/t{r_arrow}T {integral}{sup t}{sub 0} {parallel}{Omega}({center_dot},s){parallel}{sub L}{sup {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_bar}{Omega}{vert_bar} plays an important role.

• OSTI ID: 441146
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 3-4 Vol. 21; ISSN 0360-5302; ISSN CPDIDZ
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ASYMPTOTIC SOLUTIONS
INCOMPRESSIBLE FLOW
PARTIAL DIFFERENTIAL EQUATIONS
THREE-DIMENSIONAL CALCULATIONS
VORTEX FLOW