Blowup or no blowup? the interplay between theory and numerics. (English) Zbl 1143.76390
Summary: The question of whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data has been an outstanding open problem in fluid dynamics and mathematics. Recent studies indicate that the local geometric regularity of vortex lines can lead to dynamic depletion of vortex stretching. Guided by the local non-blowup theory, we have performed large scale computations of the 3D Euler equations on some of the most promising blowup candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. The local geometric regularity of vortex lines and the anisotropic solution structure play an important role in depleting the nonlinearity dynamically and thus prevents a finite time blowup.
MSC:
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
References:
| [1] | Beale, T. J.; Kato, T.; Majda, A. J., Remarks on the breakdown of smooth solutions of the 3-D Euler equations, Comm. Math. Phys., 96, 61-66 (1984) · Zbl 0573.76029 |
| [2] | Boratav, O. N.; Pelz, R. B.; Zabusky, N. J., Reconnection in orthogonally interacting vortex tubes: Direct numerical simulations and quantifications, Phys. Fluids A, 4, 581-605 (1992) · Zbl 0825.76662 |
| [3] | Boratav, O. N.; Pelz, R. B., Direct numerical simulation of transition to turbulence from a high-symmetry initial condition, Phys. Fluids, 6, 2757-2784 (1994) · Zbl 0845.76065 |
| [4] | Brachet, M. E.; Meiron, D. I.; Orszag, S. A.; Nickel, B. G.; Morf, R. H.; Frisch, U., Small-scale structure of the Taylor-Green vortex, J. Fluid Mech., 130, 411 (1983) · Zbl 0517.76033 |
| [5] | Chorin, A., The evolution of a turbulent vortex, Comm. Math. Phys., 83, 517 (1982) · Zbl 0494.76024 |
| [6] | Constantin, P., Geometric statistics in turbulence, SIAM Rev., 36, 73 (1994) · Zbl 0803.35106 |
| [7] | Constantin, P.; Fefferman, C.; Majda, A. J., Geometric constraints on potentially singular solutions for the 3-D Euler equation, Commun. PDEs, 21, 559-571 (1996) · Zbl 0853.35091 |
| [8] | Deng, J.; Hou, T. Y.; Yu, X., Geometric properties and non-blowup of 3-D incompressible Euler flow, Commun. PDEs, 30, 225-243 (2005) · Zbl 1142.35549 |
| [9] | Deng, J.; Hou, T. Y.; Yu, X., Improved geometric conditions for non-blowup of 3D incompressible Euler equation, Commun. PDEs, 31, 293-306 (2006) · Zbl 1158.76305 |
| [10] | Gibbon, J. D., The three-dimensional Euler equations: Where do we stand?, Physica D, 237, 14-17, 1894-1970 (2008) · Zbl 1143.76389 |
| [11] | Grauer, R.; Sideris, T., Numerical computation of three dimensional incompressible ideal fluids with swirl, Phys. Rev. Lett., 67, 3511 (1991) |
| [12] | Grauer, R.; Marliani, C.; Germaschewski, K., Adaptive mesh refinement for singular solutions of the incompressible Euler equations, Phys. Rev. Lett., 80, 19 (1998) |
| [13] | Grafke, T.; Homann, H.; Dreher, J.; Grauer, R., Numerical simulations of possible finite time singularities in the incompressible Euler equations: Comparison of numerical methods, Physica D, 237, 14-17, 1932-1936 (2008) · Zbl 1143.76515 |
| [14] | Hou, T. Y.; Li, R., Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci., 16, 639-664 (2006) · Zbl 1370.76015 |
| [15] | Hou, T. Y.; Li, R., Computing nearly singular solutions using pseudo-spectral methods, J. Comput. Phys., 226, 379-397 (2007) · Zbl 1310.76127 |
| [16] | T.Y. Hou, C. Li, Dynamic stability of the 3D axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., published online on August 27, 2007 (doi:10.1002/cpa.20213; T.Y. Hou, C. Li, Dynamic stability of the 3D axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., published online on August 27, 2007 (doi:10.1002/cpa.20213 |
| [17] | Kerr, R. M., Evidence for a singularity of the three dimensional, incompressible Euler equations, Phys. Fluids, 5, 1725-1746 (1993) · Zbl 0800.76083 |
| [18] | Kerr, R. M., Velocity and scaling of collapsing Euler vortices, Phys. Fluids, 17, 075103 (2005) · Zbl 1187.76264 |
| [19] | Kida, S., Three-dimensional periodic flows with high symmetry, J. Phys. Soc. Jpn., 54, 2132 (1985) |
| [20] | Majda, A. J.; Bertozzi, A. L., Vorticity and Incompressible Flow (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0983.76001 |
| [21] | Pelz, R. B., Locally self-similar, finite-time collapse in a high-symmetry vortex filament model, Phys. Rev. E, 55, 1617-1626 (1997) |
| [22] | Pumir, A.; Siggia, E. E., Collapsing solutions to the 3-D Euler equations, Phys. Fluids A, 2, 220-241 (1990) · Zbl 0696.76070 |
| [23] | Shelley, M. J.; Meiron, D. I.; Orszag, S. A., Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes, J. Fluid Mech., 246, 613 (1993) · Zbl 0781.76028 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
