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

Geometric properties and nonblowup of 3D incompressible Euler flow. (English) Zbl 1142.35549

Commun. Partial Differ. Equations 30, No. 2, 225-243 (2005).
Summary: By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding leads to an improved result of the global existence of the 3D Euler equation under mild assumptions.

Cited in 3 Reviews
Cited in 52 Documents

MSC:

35Q35 PDEs in connection with fluid mechanics
35L60 First-order nonlinear hyperbolic equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids

Keywords:

3D Euler equations; finite time blow-up; geometric properties; global existence
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Babin A., Indiana University Mathematics Journal 50 pp 1– (2001)
[2] DOI: 10.1007/BF01212349 · Zbl 0573.76029 · doi:10.1007/BF01212349
[3] DOI: 10.1016/0167-2789(93)90195-7 · Zbl 0789.76013 · doi:10.1016/0167-2789(93)90195-7
[4] Chorin A. J., 3rd ed., in: A Mathematical Introduction to Fluid Mechanics (1993)
[5] DOI: 10.1137/1036004 · Zbl 0803.35106 · doi:10.1137/1036004
[6] Constantin P., Commun. PDE 21 pp 559– (1996)
[7] DOI: 10.1007/s002200100502 · Zbl 0999.76020 · doi:10.1007/s002200100502
[8] Deng J., Comm. PDE.
[9] DOI: 10.2307/1970699 · Zbl 0211.57401 · doi:10.2307/1970699
[10] Ebin , D. G. , Fischer , A. E. , Marsden , J. E. ( 1970 ) . Diffeomorphism groups, hydrodynamics and relativity . In Proceedings of the 13th Biennial Seminar of the Canadian Mathematical Congress . ( Vanstone , J. ) Montreal . at pp. 135 – 279 . · Zbl 0284.58002
[11] Grauer R., Phys. Rev. Lett. 80
[12] DOI: 10.1063/1.858849 · Zbl 0800.76083 · doi:10.1063/1.858849
[13] Kerr R. M., Small-Scale Structure in Hydro and Magnetohydrodynamic Turbulence (1995)
[14] DOI: 10.1088/0951-7715/9/2/001 · Zbl 0899.76095 · doi:10.1088/0951-7715/9/2/001
[15] Kerr , R. M. ( 1997 ) . Euler singularities and turbulence . In 19th ICTAM Kyoto ’96 Tatsumi , T. Watanabe , E. , Kambe , T. , eds. Elsevier Science .
[16] Kerr R. M., Fundamental Problematic Issues in Turbulence (1998)
[17] DOI: 10.1007/s002200000267 · Zbl 0985.46015 · doi:10.1007/s002200000267
[18] Pelz R. B., Phys. Rev. 55 pp 1617– (1997) · doi:10.1103/PhysRevB.55.1617
[19] DOI: 10.1017/S0022112001005298 · Zbl 1002.76095 · doi:10.1017/S0022112001005298
[20] Scheffer V., J. Geom. Anal. 3 pp 343– (1993)
[21] DOI: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6 · Zbl 0909.35109 · doi:10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6
[22] DOI: 10.1088/0951-7715/14/3/305 · Zbl 0983.35100 · doi:10.1088/0951-7715/14/3/305
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.