On the energy of inviscid singular flows

Journal of Mathematical Analysis and Applications

Volumn 349, Issue 2, 2009, Pages 583-595

  Shvydkoy, Roman ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

Energy conservation; Euler equation; Onsager conjecture; Vortex sheets; Weak solutions

Indexed keywords

EID: 53349162103     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2008.09.007     Document Type: Article

References (22)

1. Caflisch R.E. Long time existence and singularity formation for vortex sheets. Vortex Methods. Los Angeles, CA, 1987. Lecture Notes in Math. vol. 1360 (1988), Springer, Berlin 1-8
2. Caflisch R.E., Klapper I., and Steele G. Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Comm. Math. Phys. 184 2 (1997) 443-455
3. Chae D. On the conserved quantities for the weak solutions of the Euler equations and the quasi-geostrophic equations. Comm. Math. Phys. 266 1 (2006) 197-210
4. Cheskidov A., Constantin P., Friedlander S., and Shvydkoy R. Energy conservation and Onsager's conjecture for the Euler equations. Nonlinearity 21 6 (2008) 1233-1252
5. Constantin P., E W., and Titi E.S. Onsager's conjecture on the energy conservation for solutions of Euler's equation. Comm. Math. Phys. 165 1 (1994) 207-209
6. De Lellis C., and Székelyhidi L. The Euler equations as a differential inclusion. preprint. arXiv:math/0702079
7. Delort J.-M. Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 3 (1991) 553-586
8. Duchon J., and Robert R. Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13 1 (2000) 249-255
9. Eggers J., and Grossmann S. Origin of the Obukhov scaling relation in turbulence. Phys. Lett. A 153 1 (1991) 12-15
10. Eyink G.L. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78 3-4 (1994) 222-240
11. Frisch U., and Parisi G. On the singularity structure of fully developed turbulence. In: Ghil M., Benzi R., and Parisi G. (Eds). Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, Proc. International Summer School of Physics "Enrico Fermi" (1985), North-Holland, Amsterdam 84-87
12. Frisch U. Turbulence. The Legacy of A.N. Kolmogorov (1995), Cambridge University Press, Cambridge
13. Krasny R. A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167 (1986) 65-93
14. Kukavica I. Role of the pressure for validity of the energy equality for solutions of the Navier-Stokes equation. J. Dynam. Differential Equations 18 2 (2006) 461-482
15. Onsager L. Statistical hydrodynamics. Convegno Internazionale di Meccanica Statistica. Nuovo Cimento (9) 6 Suppl. 2 (1949) 279-287
16. Saffman P.G. Vortex Dynamics. Cambridge Monogr. Mech. Appl. Math. (1992), Cambridge University Press, New York
17. Scheffer V. An inviscid flow with compact support in space-time. J. Geom. Anal. 3 4 (1993) 343-401
18. Shnirelman A. On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. 50 12 (1997) 1261-1286
19. R. Shvydkoy, A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation, J. Dynam. Differential Equations (2008) doi:10.1007/s10884-008-9124-3, in press
20. Stefanov A., and Torres R. Calderón-Zygmund operators on mixed Lebesgue spaces and applications to null forms. J. London Math. Soc. 70 2 (2004) 447-462
21. Sein E.M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Monographs in Harmonic Analysis, III, Princeton Math. Ser. vol. 43 (1993), Princeton University Press, Princeton, NJ with the assistance of Timothy S. Murphy
22. Sulem C., Sulem P.-L., Bardos C., and Frisch U. Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability. Comm. Math. Phys. 80 4 (1981) 485-516