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

@article{Constantin1996GeometricCO,
title={Geometric constraints on potentially singular solutions for the 3-D Euler equations},
author={Peter Constantin and Charles Fefferman and Andrew J. Majda},
journal={Communications in Partial Differential Equations},
year={1996},
volume={21}
}
• Published 31 December 1996
• Mathematics
• Communications in Partial Differential Equations
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…
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## References

SHOWING 1-8 OF 8 REFERENCES

Three‐dimensional, incompressible Euler calculations of the interaction of perturbed antiparallel vortex tubes using smooth initial profiles in a bounded domain with bounded initial vorticity are
The author presents results concerning scaling exponents in turbulence and estimates the average dissipation rate, the average dimension of level sets, and a class of two-dimensional equations that are useful models of incompressible dynamics.
Theories of turbulence ([1, 2] as well as [3] and [4]) are statistical. There exists also a relevant mathematical framework [5-7]—that of statistical solutions of the Navier–CStokes equations.
• Physics, Environmental Science
• 1994
A two‐dimensional model for quasigeostrophic flow which exhibits an analogy with the three‐dimensional incompressible Euler equations is considered. Numerical experiments show that this model
This paper presents recent and ongoing research in mathematical fluid dynamics and emphasizes the interdisciplinary interaction of ideas from large-scale computation, asymptotic methods, and mathem...
• Mathematics
• 1984
The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially
• Physics, Environmental Science
• 1994
The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical