The geometric approaches to the possible singularities in the inviscid fluid flows

Journal of Physics A: Mathematical and Theoretical

Volumn 41, Issue 36, 2008, Pages

  Chae, Dongho ^a  

^a Sungkyunkwan University   (South Korea)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 54749092935     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/41/36/365501     Document Type: Article

References (34)

1. Arnold V I and Khesin B A 1998 Topological Methods in Hydrodynamics (Berlin: Springer)
2. Beale J T, Kato T and Majda A 1984 Remarks on the breakdown of smooth solutions for the 3-d Euler equations Commun. Math. Phys. 94 61-6
3. Brenier Y 2003 Topics on hydrodynamics and area preserving maps Handbook of Mathematical Fluid Dynamics vol II ed S Friedlander and D Serre (Amsterdam: North-Holland) pp 55-86
4. Chae D 2002 On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces Commun. Pure Appl. Math. 55 654-78
5. Chae D 2006 On the spectral dynamics of the deformation tensor and new a priori estimates for the 3D Euler equations Commun. Math. Phys. 263 789-801
6. Chae D 2006 On the Lagragian dynamics for the 3D incompressible Euler equations Commun. Math. Phys. 269 557-69
7. Chae D 2007 Nonexistence of self-similar singularities for the 3D incompressible Euler equations Commun. Math. Phys. 273 203-15
8. Chae D 2007 Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations Math. Ann. 338 435-49
9. Chorin A J and Marsden J E 1992 A Mathematical Introduction to Fluid Mechanics (Texts in Appl. Math. Ser. no 4) 3rd edn (Berlin: Springer)
10. Constantin P 1994 Geometric statistics in turbulence SIAM Rev. 36 73-98
11. Constantin P 2001 An Eulerian-Lagrangian approach for incompressible fluids: local theory J. AMS 14 263-78
12. Constantin P 1995 A few results and open problems regarding incompressible fluids Not. Am. Math. Soc. 42 658-63
13. Constantin P, Fefferman C and Majda A 1996 Geometric constraints on potential singularity formulation in the 3D Euler equations Comm. PDE 21 559-71
14. Constantin P, Majda A and Tabak E 1994 Formation of strong fronts in the 2-d quasi-geostrophic thermal active scalar Nonlinearity 7 1495-533
15. Córdoba D 1997 On the geometry of solutions of the quasi-geostrophic and Euler equations Proc. Natl Acad. Sci. 94 12769-70
16. Córdoba D 1998 Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation Ann. Math. 148 1135-52
17. Córdoba D and Fefferman C 2001 On the collapse of tubes carried by 3D incompressible flows Commun. Math. Phys. 222 293-8
18. Córdoba D, Fefferman C and de la Llave R 2004 On squirt singularities in hydrodynamics SIAM J. Math. Anal. 36 204-13
19. Deng J, Hou T Y and Yu X 2005 Geometric and nonblowup of 3D incompressible Euler flow Comm. PDE 30 225-43
20. Deng J, Hou T Y and Yu X 2006 Improved geometric conditions for non-blow upof the 3D incompressible Euler equations Comm. PDE 31 293-306
21. Euler L 1755 Principes généraux du mouvement des fluides Mémoires l'académie sciences Berlin 11 274-315
22. Frisch U, Matsumoto T and Bec J 2003 Singularities of Euler flow? Not out of blue! J. Stat. Phys. 113 761-81
23. Gibbon J D, Holm D D, Kerr R M and Roulstone I 2006 Quaternions and particle dynamics in Euler fluid flow Nonlinearity 19 1969-83
24. Helmholtz H 1867 On the integrals of the hydrodynamical equations which express vortex-motion Phil. Mag, Ser. 4 33 485-512
25. Horn R A and Johnson C R 1985 Matrix Analysis (Cambridge: Cambridge University Press)
26. Kerr R M 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations Phys. Fluids A 5 1725-46
27. Kerr R M 2006 Computational Euler history Preprint physics/0607148
28. Kozono H and Taniuchi Y 2000 Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations Commun. Math. Phys. 214 191-200
29. Kozono H, Ogawa T and Taniuchi T 2002 The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations Math Z. 242 251-78
30. Liu H and Tadmor E 2002 Spectral dyanamics of the velocity gradient field in restricted flows Commun. Math. Phys. 228 435-66
31. Majda A 1986 Vorticity and the mathematical theory of incompressible fluid flow Commun. Pure Appl. Math. 39 187-220
32. Majda A and Bertozzi A 2002 Vorticity and Incompressible Flow (Cambridge: Cambridge University Press)
33. Wu J 1997 Inviscid limits and regularity estimates for the solutions of the 2-d dissipative quasi-geostrophic equations Indiana Univ. Math. J. 46 1113-1124
34. Wu J 2002 The quasi-geostrophic equations and its two regularizations Comm. PDE 27 1161-81