Nonexistence of self-similar singularities for the 3D incompressible Euler equations

Communications in Mathematical Physics

Volumn 273, Issue 1, 2007, Pages 203-215

  Chae, Dongho ^a  

^a Sungkyunkwan University   (South Korea)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34248150423     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-007-0249-8     Document Type: Article

References (28)

1. ∞ solutions of the Euler equations for nonhomogeneous fluids. Comm. P.D.E. 5, 95-107 (1980)
2. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61-66 (1984)
3. Chae, D.: Remarks on the blow-up of the Euler Equations and the Related Equations. Commun. Math. Phys. 245(3), 539-550 (2003)
4. Chae, D., Lee, J.: Local existence and blow-up criterion of the inhomogeneous Euler equations. J. Math. Fluid Mech. 5, 144-165 (2003)
5. Chae, D., Nam, H-S.: Local existence and blow-up criterion for the Boussinesq equations. Proc. Roy. Soc. Edinburgh Sect. A 127(5), 935-946 (1997)
6. Constantin, P.: Private communication
7. Constantin, P.: An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14(2), 263-278 (2001)
8. Constantin, P.: A few results and open problems regarding incompressible fluids. Notices Amer. Math. Soc. 42(6), 658-663 (1995)
9. Constantin, P., Fefferman, C., Majda, A.: Geometric constraints on potential singularity formulation in the 3-D Euler equations. Comm. P.D.E 21(3-4), 559-571 (1996)
10. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495-1533 (1994)
11. Córdoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135-1152 (1998)
12. Córdoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Amer. Math. Soc. 15(3), 665-670 (2002)
13. Córdoba, D., Fefferman, C.: On the collapse of tubes carried by 3D incompressible flows. Commun. Math. Phys. 222(2), 293-298 (2001)
14. Córdoba, D., Fefferman, C., DeLa LLave, R.: On squirt singularities in hydrodynamics. SIAM J. Math. Anal. 36(1), 204-213 (2004)
15. Deng, J., Hou, T.Y., Yu, X.: Geometric and Nonblowup of 3D Incompressible Euler Flow. Comm. P.D.E 30, 225-243 (2005)
16. E, W., Shu, C.: Small scale structures in Boussinesq convection. Phys. Fluids 6, 48-54 (1994)
17. Greene, J.M., Pelz, R.B.: Stability of postulated, self-similar, hydrodynamic blowup solutions. Phys. Rev. E 62(6), 7982-7986 (2000)
18. 3. J. Funct. Anal. 9, 296-305 (1972)
19. 2). Duke Math. J. 55, 487-499 (1987)
20. Leray, J.: Essai sur le mouvement d'un fluide visqueux emplissant l'espace. Acta Math. 63, 193-248 (1934)
21. Lions, PL.: Mathematical Topics in Fluid Mechanics: Vol. 1 Incompressible Models. Oxford: Clarendon Press, 1996
22. Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39, 187-220 (1986)
23. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge: Cambridge Univ. Press, 2002
24. Nečas, J., Ružička, M., Šverak, V.: On Leray's self-similar solutions of the Navier-Stokes equations. Acta Math. 176, 283-294 (1996)
25. Pelz, R.B.: Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55, 1617-1626 (1997)
26. Pomeau, Y., Sciamarella, D.: An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow-up in finite time'?. Physica D 205, 215-221 (2005)
27. Saint Raymond, X.: Remarks on axisymmetric solutions of the incompressible Euler system. Comm. P. D.E. 19, 321-334 (1994)
28. Tsai, T-P.: On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rat. Mech. Anal. 143(1), 29-51 (1998)