Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation

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

doi:10.1080/03605300500358152

Cited by 80 publications

(138 citation statements)

References 17 publications

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“…This is in contrast with the claim in [19] that the maximum velocity blows up like O(T − t) −1/2 with T = 18.7. With the velocity field being bounded, the local non-blowup criteria of Deng-Hou-Yu [10,11] can be applied, which implies that the solution of the 3D Euler equations remains smooth at least up to T = 19, see also [17].…”

Section: Does a Finite Time Singularity Develop?mentioning

confidence: 99%

Computing nearly singular solutions using pseudo-spectral methods

Hou

2007

Journal of Computational Physics

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In this paper, we investigate the performance of pseudo-spectral methods in computing nearly singular solutions of fluid dynamics equations. We consider two different ways of removing the aliasing errors in a pseudo-spectral method. The first one is the traditional 2/3 dealiasing rule. The second one is a high (36th) order Fourier smoothing which keeps a significant portion of the Fourier modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3 dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler equations are considered. We demonstrate that the pseudo-spectral method with the high order Fourier smoothing gives a much better performance than the pseudo-spectral method with the 2/3 dealiasing rule. Moreover, we show that the high order Fourier smoothing method captures about 12 ∼ 15% more effective Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D Euler equations, the gain in the effective Fourier codes for the high order Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method. Another interesting observation is that the error produced by the high order Fourier smoothing method is highly localized near the region where the solution is most singular, while the 2/3 dealiasing method tends to produce oscillations in the entire domain. The high order Fourier smoothing method is also found be very stable dynamically. No high frequency instability has been observed.

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Section: Does a Finite Time Singularity Develop?mentioning

confidence: 99%

Computing nearly singular solutions using pseudo-spectral methods

Hou

2007

Journal of Computational Physics

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203

194

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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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“…The existing literature, see [5,10,11,12] and references therein, is not decisive and evidence for both blowup and not blowup is presented. The study has further been limited in time to before blowup, discarding the highly relevant question of what happens after blowup.…”

Section: The Blowup Problem For Incompressible Flowmentioning

confidence: 99%

“…We show that the transition to turbulence in potential flow is driven by exponential perturbation growth in time with corresponding logarithmic growth in the mesh size of the effective time to transition. We thus study global blowup of EG2/viscosity solutions under decreasing mesh size/viscosity including wellposedness, and not as in [5,10,11,12] local blowup of exact Euler solutions without wellposedness.…”

Section: The Blowup Problem For Incompressible Flowmentioning

confidence: 99%

“…In the work cited in [5,6,10,11,12], blowup is identified by the development of an infinite velocity gradient of a specific initially smooth exact Euler solution as time approaches a blowup time. This form of blowup detection, which we refer to as local blowup, requires pointwise accurate information (analytically or computationally) of the blowup to infinity, which (so far) has shown to be impossible to obtain.…”

Section: The Blowup Problem For Incompressible Flowmentioning

confidence: 99%

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