no. 3-4
Partial Differential Equations/Mathematical Physics
Dissipation anomaly and energy cascade in 3D incompressible flows
[Dissipation anormale et cascade énergétique pour des fluides incompressibles tridimensionnels]

Présenté par : the Editorial Board

Dascaliuc, Radu1 ; Grujić, Zoran2
1 Department of Mathematics, Oregon State University, Corvallis, OR 97332, USA
2 Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA
Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 199-202.

Le but de cette Note est de présenter une mise en évidence mathématique de la dissipation anormale pour des flots turbulents tridimensionnels, dans un cadre général récemment introduit par les auteurs pour lʼétude de la cascade énergétique aux échelles physiques dans les fluides incompressibles en trois dimensions.

The purpose of this Note is to present a mathematical evidence of the dissipation anomaly in 3D turbulent flows within a general setting for the study of energy cascade in physical scales of 3D incompressible flows recently introduced by the authors.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.01.009
Dascaliuc, Radu 1 ; Grujić, Zoran 2

1 Department of Mathematics, Oregon State University, Corvallis, OR 97332, USA
2 Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA 
@article{CRMATH_2012__350_3-4_199_0,
     author = {Dascaliuc, Radu and Gruji\'c, Zoran},
     title = {Dissipation anomaly and energy cascade in {3D} incompressible flows},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {199--202},
     publisher = {Elsevier},
     volume = {350},
     number = {3-4},
     year = {2012},
     doi = {10.1016/j.crma.2012.01.009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.01.009/}
}
TY  - JOUR
AU  - Dascaliuc, Radu
AU  - Grujić, Zoran
TI  - Dissipation anomaly and energy cascade in 3D incompressible flows
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 199
EP  - 202
VL  - 350
IS  - 3-4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.01.009/
DO  - 10.1016/j.crma.2012.01.009
LA  - en
ID  - CRMATH_2012__350_3-4_199_0
ER  - 
%0 Journal Article
%A Dascaliuc, Radu
%A Grujić, Zoran
%T Dissipation anomaly and energy cascade in 3D incompressible flows
%J Comptes Rendus. Mathématique
%D 2012
%P 199-202
%V 350
%N 3-4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.01.009/
%R 10.1016/j.crma.2012.01.009
%G en
%F CRMATH_2012__350_3-4_199_0
Dascaliuc, Radu; Grujić, Zoran. Dissipation anomaly and energy cascade in 3D incompressible flows. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 199-202. doi : 10.1016/j.crma.2012.01.009. http://www.numdam.org/articles/10.1016/j.crma.2012.01.009/

[1] Cheskidov, A.; Constantin, P.; Friedlander, S.; Shvydkoy, R. Energy conservation and Onsagerʼs conjecture for the Euler equations, Nonlinearity, Volume 21 (2008), p. 1233

[2] Constantin, P.; E, W.; Titi, E. Onsagerʼs conjecture on the energy conservation for solutions of Eulerʼs equation, Comm. Math. Phys., Volume 165 (1994), p. 207

[3] Dascaliuc, R.; Grujić, Z. Energy cascades and flux locality in physical scales of the 3D NSE, Comm. Math. Phys., Volume 305 (2011), p. 199

[4] Dascaliuc, R.; Grujić, Z. Anomalous dissipation and energy cascade in 3D inviscid flows, Comm. Math. Phys., Volume 309 (2012), p. 757

[5] Duchon, J.; Robert, R. Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations, Nonlinearity, Volume 13 (2000), p. 249

[6] Eyink, G. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Phys. D, Volume 78 (1994), p. 222

[7] Eyink, G. Dissipative anomalies in singular Euler flows, Phys. D, Volume 237 (2008), p. 1956

[8] Eyink, G.; Sreenivasan, K. Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phys., Volume 78 (2006), p. 87

[9] Foias, C.; Manley, O.; Rosa, R.; Temam, R. Estimates for the energy cascade in three-dimensional turbulent flows, C. R. Acad. Sci. Paris, Sér. I, Volume 333 (2001), p. 499

[10] Frisch, U. Turbulence, Cambridge University Press, 1995 (The legacy of A.N. Kolmogorov)

[11] Kolmogorov, A.N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR, Volume 30 (1941), p. 9

[12] Onsager, L. Statistical hydrodynamics, Nuovo Cimento (9), Volume 6 (1949) no. Supplemento, 2 (Convegno Internazionale di Meccanica Statistica), p. 279

[13] Shvydkoy, R. Lectures on the Onsager conjecture, Discrete Contin. Dynam. Systems S, Volume 3 (2010), p. 473

Cité par Sources :