Turbulence de Burgers en 1D : un cas modèle pour la théorie de Kolmogorov

Boritchev, Alexandre

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 40, 13 p.

Nous nous intéressons ici à la turbulence de Burgers 1D, ou « Burgulence ». Nous présentons des résultats valables pour l’équation de Burgers généralisée périodique stochastique 1D :

$$u_t+f^{\text{'}}\left(u\right)u_x=\nu u_{xx}+\eta ,t\ge 0,x\in S^1=ℝ/\mathbb{Z},$$

où $\eta (t,x)$ est une force de type bruit blanc en $t$ et lisse en $x$. Plus précisément, nous estimons les normes de Sobolev et les quantités à petite échelle analogues à celles qui sont intéressantes pour l’étude de la turbulence hydrodynamique, telles que les incréments et le spectre d’énergie. Les résultats exposés ici se trouvent dans l’article [7].

[1] E. Aurell, U. Frisch, J. Lutsko, M. Vergassola, On the multifractal properties of the energy dissipation derived from turbulence data, Journal of Fluid Mechanics 238, 1992, 467-486. | MR | Zbl

[2] G. Batchelor, The theory of homogeneous turbulence, Cambridge University Press, 1953. | MR | Zbl

[3] J. Bec, U. Frisch, “Burgulence”, Les Houches 2000 : New Trends in Turbulence ; M. Lesieur, A.Yaglom and F. David, eds., 341-383, Springer EDP-Sciences, 2001. | MR

[4] J. Bec, K. Khanin, Burgers turbulence, Physics Reports 447 (2007), 1-66. | MR

[5] A. Biryuk, Spectral properties of solutions of the Burgers equation with small dissipation, Functional Analysis and its Applications, 35 :1 (2001), 1-12. | MR | Zbl

[6] A. Boritchev, Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation, Prépublication (à paraître dans Proceedings of the Royal Society of Edinburgh A), arXiv :1107.4866v1.

[7] A. Boritchev, Sharp estimates for turbulence in white-forced generalised Burgers equation, Prépublication, arXiv :1201.5567.

[8] A. Boritchev, Note on Turbulence in a Generalised Burgers Equation, Prépublication disponible sur www.math.polytechnique.fr/$\sim$boritchev/ DeterministicBurgers.pdf.

[9] J. D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quarterly of Applied Mathematics, 1951 (9), 225-236. | MR | Zbl

[10] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopaedia of Mathematics and its Applications vol. 45, Cambridge University Press, 1992. | MR | Zbl

[11] Weinan E, K. Khanin, A. Mazel, Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Annals of Mathematics 151 (2000), 877-960. | EuDML | MR | Zbl

[12] U. Frisch, Turbulence : the legacy of A.N. Kolmogorov, Cambridge University Press, 1995. | MR | Zbl

[13] T. Gotoh, R. Kraichnan, Steady-state Burgers turbulence with large-scale forcing, Physics of Fluids 10 (1998), 2859-2866. | MR | Zbl

[14] E. Hopf, The partial differential equation $u_t+u{u}_x=\mu u_{xx}$, Communications in Pure and Applied Mathematics, 1950, 3 :3, 201-230. | MR | Zbl

[15] S. Kida, Asymptotic properties of Burgers turbulence, Journal of Fluid Mechanics, 93 (1979), no.2, 337-377. | MR | Zbl

[16] A. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Doklady Akademii Nauk SSSR 30 (1941), 9-13 (reprinted in Proceedings of the Royal Society of London A 434 (1991), 9-13). | MR | Zbl

[17] A. Kolmogorov, On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid, Doklady Akademii Nauk SSSR 31 (1941), 538-540. | MR | Zbl

[18] A. Kolmogorov, Dissipation of energy in locally isotropic turbulence, Doklady Akademii Nauk SSSR 32 (1941), 16-18 (reprinted in Proceedings of the Royal Society of London A 434 (1991), 15-17). | MR | Zbl

[19] A. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, Journal of Fluid Mechanics, 13 (1962), 82-85. | MR | Zbl

[20] R. H. Kraichnan, Lagrangian-history statistical theory for Burgers’ equation, Physics of Fluids 11(2), 1968, 265-277. | Zbl

[21] S. N. Kruzhkov, The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the first-order with several variables, Soviet Math. Doklady, 5 (1964), 493-496. | Zbl

[22] S. Kuksin, On turbulence in nonlinear Schrödinger equations, Geometric and Functional Analysis, 1997, vol. 7, 783-822. | MR | Zbl

[23] S. Kuksin, Spectral properties of solutions for nonlinear PDEs in the turbulent regime, Geometric and Functional Analysis, 1999, vol. 9, 141-184. | MR | Zbl

[24] A. Obukhov, On the distribution of energy in the spectrum of turbulent flow, Doklady Akademii Nauk SSSR, 32 (1), 22-24, 1941.

[25] A. Obukhov, Spectral energy distribution in a turbulent flow, Izvestiya Akademii Nauk SSSR, Seriya Geografii i Geofiziki, 5 (4-5), 453-466, 1941.

[26] G. Parisi, U. Frisch, Fully developed turbulence and intermittency, Proceedings of the International School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, ed. M. Ghil, R.Benzi, G. Parisi, 1985, 71-88, Amsterdam : North-Holland.

[27] S. Shandarin, Ya. Zeldovich, The large-scale structure of the universe : turbulence, intermittency, structures in a self-gravitating medium, Reviews of Modern Physics, 61, no. 2, 1989, 185-220. | MR

[28] A. Tsinober, An informal conceptual introduction to turbulence, Fluid Mechanics and its Applications, Springer, 2009. | MR | Zbl