×

zbMATH — the first resource for mathematics

Energy dissipation without viscosity in ideal hydrodynamics. I: Fourier analysis and local energy transfer. (English) Zbl 0817.76011
We outline a proof and give a discussion at a physical level of an assertion of Onsager’s: namely, that a solution of incompressible Euler equations with Hölder continuous velocity of order \(h > 1/3\) conserves kinetic energy, but not necessarily if \(h \leq 1/3\). We prove the result under a “\(*\)-Hölder condition” which is somewhat stronger than usual Hölder continuity. Our argument establishes also the fundamental result that the instantaneous (sub-scale) energy transfer is dominated by local triadic interactions for a \(*\)-Hölder solution with exponent \(h\) in the range \(0 < h < 1\).

MSC:
76B99 Incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Onsager, L., Nuovo cim. suppl., 6, 279, (1949)
[2] Kolmogorov, A.N.; Kolmogorov, A.N.; Kolmogorov, A.N.; Kolmogorov, A.N., Dokl. akad. nauk. SSSR., Dokl. akad. nauk. SSSR., Dokl. akad. nauk. SSSR., Dokl. akad. nauk. SSSR., 32, 16, (1941)
[3] Daubechies, I., Ten lectures on wavelets, (1992), Society for Industrial and Applied Mathematics Philadelphia · Zbl 0776.42018
[4] Siggia, E.D., Phys. rev. A, 15, 1730, (1977)
[5] Zimin, V.D., Izv. atmospheric oceanic phys., 17, 941, (1981)
[6] Sulem, P.L.; Frisch, U., J. fluid mech., 72, 417, (1975)
[7] Frisch, U.; Sulem, P.-L., C.R. acad. sci. Paris, t. 280, 1117, (1975), Serie A
[8] Parisi, G.; Frisch, U., Turbulence and predictability in geophysical fluid dynamics, (), 84
[9] Kraichnan, R.H., J. fluid. mech., 5, 497, (1959)
[10] Katznelson, Y., Introduction to harmonic analysis, (1976), Dover New York · Zbl 0169.17902
[11] Zygmund, A., Trigonometric series., (1959), Cambridge University Press Cambridge · JFM 58.0280.01
[12] Kraichnan, R.H., J. fluid mech., 62, 305, (1974)
[13] Eggers, J.; Grossmann, S., Phys. fluids, 3A, 1958, (1991)
[14] Scheffer, V., An inviscid flow with compact support in space-time, J. geom. analysis, (1992), to appear in
[15] DiPerna, R.J.; Majda, A.J., Commun. math. phys., 108, 667, (1987)
[16] Scheffer, V., Commun. math. phys., 55, 97, (1977)
[17] Caffarelli, L.; Kohn, R.; Nirenberg, L., Commun. pure appl. math., 35, 771, (1982)
[18] Majda, A.J., Commun. pure appl. math., XXXIX, S187, (1986)
[19] Frisch, U., Proc. roy. soc. lond. A, 434, 89, (1991)
[20] Kolmogorov, A.N., J. fluid mech., 13, 82, (1962) · Zbl 0112.42003
[21] Obukhov, A.M., J. fluid mech., 13, 77, (1962)
[22] Mandelbrot, B., J. fluid mech., 62, 331, (1974)
[23] Frisch, U.; Sulem, P.L.; Nelkin, M., J. fluid. mech., 87, 719, (1978)
[24] Jaffard, S., Construction de fonctions multifractales ayant un spectre de singularités prescrit, (1992), preprint · Zbl 0780.28005
[25] Frisch, U.; Vergassola, M., Europhys. lett., 14, 439, (1991)
[26] Eyink, G.L., Besov spaces and the multifractal hypothesis, J. stat. phys., (Nov. 1993), submitted to
[27] P. Constantin, E. Weinan, and E.S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Commun. Math. Phys., to appear. · Zbl 0818.35085
[28] Eyink, G.L., Local energy flux and refined similarity hypothesis, J. stat. phys., (Dec. 1994), to appear
[29] Eyink, G.L., Large-eddy simulation and the ‘multifractal model’ of turbulence: a priori estimates on subgrid flux and locality of energy transfer, Phys. fluids, (Apr. 1994), submitted to
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.