Statistical properties of the burgers equation with brownian initial velocity

Journal of Statistical Physics

Volumn 134, Issue 3, 2009, Pages 589-640

  Valageas, Patrick ^a  

^a Centre National de la Recherche Scientifique   (France)

Author keywords

Cosmology: large scale structure of the universe; Inviscid Burgers equation; Turbulence

Indexed keywords

EID: 61349193753     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-009-9685-5     Document Type: Article

References (52)

1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1970)
2. E. Aurell S.N. Gurbatov I.I. Wertgeim 1993 Self-preservation of large-scale structures in Burgers' turbulence Phys. Lett. A 182 109 113
3. E. Aurell U. Frisch A. Noullez M. Blank 1997 Bifractality of the devil's staircase appearing in the Burgers equation with Brownian initial velocity J. Stat. Phys. 88 1151 1164
4. R. Balian R. Schaeffer 1989 Scale-invariant matter distribution in the universe. I-Counts in cells Astron. Astrophys. 220 1 29
5. R. Balian R. Schaeffer 1989 Scale-invariant matter distribution in the universe. II-Bifractal behaviour Astron. Astrophys. 226 373 414
6. J. Bec K. Khanin 2007 Burgers turbulence Phys. Rep. 447 1 66
7. F. Bernardeau 1994 The effects of smoothing on the statistical properties of large-scale cosmic fields Astron. Astrophys. 291 697 712
8. F. Bernardeau S. Colombi E. Gaztaaga R. Scoccimarro 2002 Large-scale structure of the universe and cosmological perturbation theory Phys. Rep. 367 1 248
9. J. Bertoin 1998 The inviscid Burgers equation with Brownian initial velocity Commun. Math. Phys. 193 397 406
10. Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974)
11. T.W. Burkhardt 1993 Semiflexible polymer in the half plane and statistics of the integral of a Brownian curve J. Phys. A 26 L1157 L1162
12. L. Carraro J. Duchon 1998 Equation de Burgers avec conditions initiales a accroissements independants et homogenes Ann. Inst. Henri Poincare 15 431 458
13. J.D. Cole 1951 On a quasi-linear parabolic equation occuring in aerodynamics Q. Appl. Math. 9 225 236
14. S. Colombi F.R. Bouchet R. Schaeffer 1994 Large scale structure statistics: Finite volume effects Astron. Astrophys. 281 301 313
15. M. Davis P.J.E. Peebles 1977 On the integration of the bbgky equations for the development of strongly nonlinear clustering in an expanding universe Astrophys. J. Suppl. S. 34 425 450
16. J.-D. Fournier U. Frisch 1983 L'equation de Burgers deterministe et statistique J. Mec. Theor. Appl. 2 699 750
17. L. Frachebourg Ph.A. Martin 2000 Exact statistical properties of the Burgers equation J. Fluid Mech. 417 323 349
18. Frisch, U., Bec, J.: "Burgulence". In: Lesieur, M., Yaglom, A., David, F. (eds.) Les Houches 2000: New trends in turbulence. Springer, Berlin (2001)
19. U. Frisch J. Bec E. Aurell 2005 "Locally homogeneous turbulence" is it an inconsistent framework Phys. Fluids 17 081706
20. J.N. Fry 1984 Galaxy n-point correlation functions-theoretical amplitudes for arbitrary n Astrophys. J. 277 L5 L8
21. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (1965)
22. Gurbatov, S., Malakhov, A., Saichev, A.: Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles. Manchester University Press, Manchester (1991)
23. S.N. Gurbatov G.V. Pasmanik 1999 Self-preservation of large-scale structures in a nonlinear viscous medium described by the Burgers equation Sov. Phys. JETP 88 309 319
24. S.N. Gurbatov A.I. Saichev 1981 Degeneracy of one-dimensional acoustic turbulence at large Reynolds numbers Sov. Phys. JETP 53 347 354
25. S.N. Gurbatov A.I. Saichev S.F. Shandarin 1989 The large-scale structure of the universe in the frame of the model equation of non-linear diffusion Mon. Not. R. Astron. Soc. 236 385 402
26. S.N. Gurbatov S.I. Simdyankin E. Aurell U. Frisch G. Toth 1997 On the decay of Burgers turbulence J. Fluid Mech. 344 339 374
27. P.S. Hagan C.R. Doering C.D. Levermore 1989 The distribution of exit times for weakly colored noise J. Stat. Phys. 54 1321 1352
28. xx Commun. Pure Appl. Mech. 3 201 230
29. S. Kida 1979 Asymptotic properties of Burgers turbulence J. Fluid Mech. 93 337 377
30. Le Doussal, P.: Exact results and open questions in first principle functional rg. arXiv:0809.1192 (2008)
31. T.W. Marshall E.J. Watson 1985 A drop of ink falls from my pen... it comes to earth, I know not when J. Phys. A, Math. Gen. 18 3531 3559
32. T.W. Marshall E.J. Watson 1987 The analytic solutions of some boundary layer problems in the theory of Brownian motion J. Phys. A, Math. Gen. 20 1345 1354
33. A.L. Melott S.F. Shandarin D.H. Weinberg 1994 A test of the adhesion approximation for gravitational clustering Astrophys. J. 428 28 34
34. G.M. Molchan 1997 Burgers equation with self-similar Gaussian initial data: tail probabilities J. Stat. Phys. 88 1139 1150
35. S.A. Molchanov D. Surgailis W.A. Woyczynski 1995 Hyperbolic asymptotics in Burgers' turbulence and extremal processes Commun. Math. Phys. 168 209 226
36. A. Noullez S.N. Gurbatov E. Aurell S.I. Simdyankin 2005 Global picture of self-similar and non-self-similar decay in Burgers turbulence Phys. Rev. E 71 056305
37. Peebles, P.J.E.: The Large Scale Structure of the Universe. Princeton University Press, Princeton (1980)
38. W. Press P. Schechter 1974 Formation of galaxies and clusters of galaxies by self-similar gravitational condensation Astrophys. J. 187 425 438
39. H. Prodinger F.J. Urbanek 1983 On monotone functions of tree structures Discrete Appl. Math. 5 223 239
40. R. Schaeffer 1985 The probability generating function for galaxy clustering Astron. Astrophys. 144 L1 L4
41. Z.-S. She E. Aurell U. Frisch 1992 The inviscid Burgers equation with initial data of Brownian type Commun. Math. Phys. 148 623 641
42. R.K. Sheth G. Tormen 1999 Large-scale bias and the peak background split Mon. Not. R. Astron. Soc. 308 119 126
43. Ya.G. Sinai 1992 Statistics of shocks in solutions of inviscid Burgers equation Commun. Math. Phys. 148 601 621
44. P. Valageas 1999 Non-linear gravitational clustering: smooth halos, substructures and scaling exponents Astron. Astrophys. 347 757 768
45. P. Valageas 2002 Dynamics of gravitational clustering. II. Steepest-descent method for the quasi-linear regime Astron. Astrophys. 382 412 430
46. P. Valageas 2002 Dynamics of gravitational clustering. IV. The probability distribution of rare events Astron. Astrophys. 382 450 476
47. P. Valageas 2007 Using the Zeldovich dynamics to test expansion schemes Astron. Astrophys. 476 31 58
48. P. Valageas 2009 Ballistic aggregation for one-sided Brownian initial velocity Physica A 388 1031 1045 arXiv:0809.1192
49. P. Valageas D. Munshi 2004 Evolution of the cosmological density distribution function: a new analytical model Mon. Not. R. Astron. Soc. 354 1146 1158
50. Vallée, O., Soares, M.: Les Fonctions d'Airy pour la Physique. Diderot, Paris (1998)
51. M. Vergassola B. Dubrulle U. Frisch A. Noullez 1994 Burgers'equation, devil's staircases and the mass distribution for large-scale structures Astron. Astrophys. 289 325 356
52. Y.B. Zeldovich 1970 Gravitational instability: An approximate theory for large density perturbations Astron. Astrophys. 5 84 89