Statistical theory for the stochastic Burgers equation in the inviscid limit

Communications on Pure and Applied Mathematics

Volumn 53, Issue 7, 2000, Pages 852-901

  E, Weinan ^a   Vanden Eijnden, Eric ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001532759     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(200007)53:7<852::AID-CPA3>3.0.CO;2-5     Document Type: Article

References (42)

1. Avellaneda, M. Statistical properties of shocks in Burgers turbulence. II. Tail probabilities for velocities, shock-strengths and rarefaction intervals. Comm. Math. Phys. 169 (1995), no. 1, 45-59.
2. Avellaneda, M.; E, W. Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys. 172 (1995), no. 1, 13-38.
3. Avellaneda, M.; Ryan, R.; E, W. PDFs for velocity and velocity gradients in Burgers' turbulence. Phys. Fluids 7 (1994), no. 12, 3067-3071.
4. Balkovsky, E.; Falkovich, G.; Kolokolov, I.; Lebedev, V. Intermittency of Burgers' turbulence. Phys. Rev. Lett. 78 (1997), 1452-1455.
5. Blatter, G.; Feigelman, M. V.; Geshkenbein, V. B.; Larkin, A. I.; Vinokur, V. M. Vortices in high-temperature superconductors. Rev. Modern Phys. 66 (1994), 1125-1388.
6. Boldyrev, S. A. Velocity-difference probability density functions for Burgers turbulence. Phys. Rev. E 55 (1997), no. 6, part A, 6907-6910.
7. Boldyrev, S. A. Burgers turbulence, intermittency, and nonuniversality. Phys. Plasmas 5 (1998), no. 5, part 2, 1681-1687.
8. Bouchaud, J.-P.; Mézard, M.; Parisi, G. Scaling and intermittency in Burgers turbulence. Phys. Rev. E 52 (1995), no. 4, part A, 3656-3674.
9. Bouchaud, J.-P.; Mézard, M. Velocity fluctuations in forced Burgers turbulence. Phys. Rev. E 54 (1996), 5116-5121.
10. Chekhlov, A.; Yakhot, V. Kolmogorov turbulence in a random-force-driven Burgers equation. Phys. Rev. E 51 (1995), R2739-2742.
11. Chekhlov, A.; Yakhot, V. Kolmogorov turbulence in a random-force-driven Burgers equation: anomalous scaling and probability density functions. Phys. Rev. E(3) 52 (1995), no. 5, part B, R5681-5684.
12. Chekhlov, A.; Yakhot, V. Algebraic tails of probability density functions in the random-force-driven Burgers turbulence. Phys. Rev. Lett. 77 (1996), 3118-3121.
13. E, W.; Khanin, K.; Mazel, A.; Sinai, Ya. G. Probability distributions functions for the random forced Burgers equation. Phys. Rev. Lett. 78 (1997), 1904-1907.
14. E, W.; Khanin, K.; Mazel, A.; Sinai, Ya. G. Invariant measures for the random-forced Burgers equation. Preprint, 1999. Ann. Math., in press.
15. E, W.; Vanden Eijnden, E. On the statistical solution of Riemann equation and its implications for Burgers turbulence. Phys. Fluids 11 (1999), 2149-2153.
16. E, W.; Vanden Eijnden, E. Another note on forced Burgers turbulence. Phys. Fluids 12 (2000), 149-154.
17. E, W.; Vanden Eijnden, E. Asymptotic theory for the probability density functions in Burgers turbulence. Phys. Rev. Lett. 83 (1999), 2572-2575.
18. Falkovitch, G.; Kolokolov, I.; Lebedev, V.; Migdal, A. Instantons and intermittency. Phys. Rev. E 54 (1996), 4896-4907.
19. Feǐgel'man, M. V. One-dimensional periodic structures in a weak random potential. Sov. Phys. JETP 52 (1980), no. 3, 555-561; translated from Zh. Èksper. Teoret. Fiz. 79 (1980), no. 3, 1095-1107.
20. Feǐgel'man, M. V. One-dimensional periodic structures in a weak random potential. Sov. Phys. JETP 52 (1980), no. 3, 555-561; translated from Zh. Èksper. Teoret. Fiz. 79 (1980), no. 3, 1095-1107.
21. Fournier, J.-D.; Frisch, U. L'équation de Burgers déterministe et statistique. J. Méc. Théor. Appl. 2 (1983), no. 5, 699-750.
22. Goodman, J.; Xin, Z. P. Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rational Mech. Anal. 121 (1992), no. 3, 235-265.
23. Gotoh, T. Inertial range statistics of Burgers turbulence. Phys. Fluids 6 (1994), no. 12, 3985-3998.
24. Gotoh, T.; Kraichnan, R. H. Statistics of decaying turbulence. Phys. Fluids A 5 (1993), no. 2. 445-457.
25. Gotoh, T.; Kraichnan, R. H. Steady-state Burgers turbulence with large-scale forcing. Phys. Fluids 10 (1998), no. 11, 2859-2866.
26. Gurarie, V.; Migdal, A. Instantons in the Burgers equation. Phys. Rev. E 54 (1996), 4908-4914.
27. Gurbatov, S. N.; Simdyankin, S. I.; Aurell, E.; Frisch, U.; Tóth, G. On the decay of Burgers turbulence. J. Fluid Mech. 344 (1997), 339-374.
28. Kardar, M.; Parisi, G.; Zhang, Y. C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986), 889-892.
29. Kraichnan, R. H. Note on forced Burgers turbulence. Phys. Fluids 11 (1999), 3738-3742.
30. Krug, J.; Spohn, H. Kinetic roughening of growing surfaces. Solids far from equilibrium, 479-582. Edited by C. Godrèche. Cambridge University Press, Cambridge, 1992.
31. Lax, P. D. Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math. 10 (1957), 537-566.
32. Lax, P. D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. Society for Industrial and Applied Mathematics. Philadelphia, 1973.
33. Polyakov, A. M. Turbulence without pressure. Phys. Rev. E(3) 52 (1995), no. 6, part A, 6183-6188.
34. Polyakov, A. M. Personnal communication, 1999.
35. Ryan, R.; Avellaneda, M. The one-point statistics of viscous Burgers turbulence initialized with Gaussian data. Comm. Math. Phys. 200 (1999), no. 1, 1 1-23.
36. Shandarin, S. F.; Zel'dovich, Ya. B. The large-scale structure of the universe: turbulence, intermittency, structures in a self-graviting medium. Rev. Modern Phys. 61 (1989), no. 2, 185-220.
37. Sinai, Ya. Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Statist. Phys. 64 (1992), no. 1-2, 1-12.
38. Sinai, Ya. Burgers system driven by a periodic stochastic flow. Itô's stochastic calculus and probability theory, 347-353. Springer, Tokyo, 1996.
39. Smoller, J. Shock waves and reaction-diffusion equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 258. Springer, New York-Berlin, 1983.
40. Vergassola, M.; Dubrulle, B.; Frisch, U.; Noullez, A. Burgers equation, devil's staircases and the mass distribution for large-scale structures. Astron. Astrophys. 280 (1994), 325-356.
41. Vol'pert, A. I. Spaces BV and quasilinear equations. Mat. Sb. 73(115) (1967), 255-302; translated in Math. USSR-Sb. 2 (1967), 225-267.
42. Vol'pert, A. I. Spaces BV and quasilinear equations. Mat. Sb. 73(115) (1967), 255-302; translated in Math. USSR-Sb. 2 (1967), 225-267.