Geometric Properties and Nonblowup of 3D Incompressible Euler Flow

Deng, Jian; Hou, Thomas Y.; Yu, Xinwei

doi:10.1081/pde-200044488

Cited by 103 publications

(120 citation statements)

References 16 publications

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“…Initial tests show that these calculations have the resolution needed for quantifying the growth of the curvature and divergence on the vortex lines (Deng et al 2005). This should be on more than one line since the maximum curvature in figure 8 appears on lines that start near, but not at, ω ∞ .…”

Section: Discussionmentioning

confidence: 99%

“…The y > 0 half-domain highlights the growth of curvature in the bulge, then the growth in the divergence of the vortex lines. It is a kinematic requirement (Deng, Hou & Yu 2005) that the integral of the divergence of the vorticity direction ds∇ ·ω blows up if ω ∞ blows up. It is also possible that large values of ∇ ·ω, whereω = ω/|ω|, could be what inhibits the growth of ω ∞ .…”

Section: Three-dimensional Imagesmentioning

confidence: 99%

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Bounds for Euler from vorticity moments and line divergence

Kerr

2013

J. Fluid Mech.

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The inviscid growth of a range of vorticity moments is compared using Euler calculations of anti-parallel vortices with a new initial condition. The primary goal is to understand the role of nonlinearity in the generation of a new hierarchy of rescaled vorticity moments in Navier-Stokes calculations where the rescaled moments obey D m D m+1 , the reverse of the usual Ω m+1 Ω m Hölder ordering of the original moments. Two temporal phases have been identified for the Euler calculations. In the first phase the 1 < m < ∞ vorticity moments are ordered in a manner consistent with the new Navier-Stokes hierarchy and grow in a manner that skirts the lower edge of possible singular growth with D 2 m → sup |ω| ∼ A m (T c − t) −1 where the A m are nearly independent of m. In the second phase, the new D m ordering breaks down as the Ω m converge towards the same super-exponential growth for all m. The transition is identified using new inequalities for the upper bounds for the −dD −2 m /dt that are based solely upon the ratios D m+1 /D m , and the convergent super-exponential growth is shown by plotting log(d log Ω m /dt). Three-dimensional graphics show significant divergence of the vortex lines during the second phase, which could be what inhibits the initial power-law growth.

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Section: Discussionmentioning

confidence: 99%

Section: Three-dimensional Imagesmentioning

confidence: 99%

Bounds for Euler from vorticity moments and line divergence

Kerr

2013

J. Fluid Mech.

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“…On this direction there is a celebrated result on the blow-up criterion by Beale,Kato and Majda([2]). By geometric type of consideration some of the possible scenarios to the possible singularity has been excluded(see [8,12,14]. One of the main purposes of this paper is to exclude the possibility of self-similar type of singularities for the Euler system.…”

Section: Incompressible Euler Equationsmentioning

confidence: 99%

Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

Chae

2007

Commun. Math. Phys.

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We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in R n . This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.

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“…Recently Chae [5] and Hou and Li [14] proved independently the global existence of the two-dimensional viscous Boussinesq equations with viscosity entering only in the fluid equation, while the density equation remains inviscid. Recent studies by Constantin, Fefferman, and Majda [7] and Deng, Hou, and Yu [10,11] show that the local geometric regularity of the unit vorticity vector can play an important role in depleting vortex stretching dynamically.…”

Section: Introductionmentioning

confidence: 99%

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

Hou

2007

Comm Pure Appl Math

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In this paper, we study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the Navier-Stokes equations along the symmetry axis. An important property of this one-dimensional model is that one can construct from its solutions a family of exact solutions of the threedimensionaFinal Navier-Stokes equations. The nonlinear structure of the onedimensional model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the three-dimensional Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions.

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Geometric Properties and Nonblowup of 3D Incompressible Euler Flow

Cited by 103 publications

References 16 publications

Bounds for Euler from vorticity moments and line divergence

Bounds for Euler from vorticity moments and line divergence

Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

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