Bifractality of the devil’s staircase appearing in the Burgers equation with Brownian initial velocity. (English) Zbl 0924.60044
The authors consider the Burgers equation \(\partial_tv+v\partial_xv=\nu\partial^2_x\nu\) with a Brownian (or fractional Brownian) motion function (of the space coordinate \(x\)) and the initial velocity \(v_0(x)\). In the paper [Z.-S. She, E. Aurell, and U. Frisch, Commun. Math. Phys. 148, No. 3, 623-641 (1992; Zbl 0755.60104)], it was conjectured that, in the inviscid limit, the inverse Lagrangian map for the solution of this equation has a bifractality (phase transition) similar to that known for the standard devil’s staircase associated to the triadic Cantor set. In this paper, the conjectured bifractality is demonstrated both heuristically and rigorously. Numerical simulations are presented. In particular, the authors explain why artifacts can easily hide the phenomenon and show how to overcome this.
Reviewer: V.Mackevičius (Vilnius)
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Citations:
Zbl 0755.60104References:
| [1] | Sinai, Ya. G., Statistics of shocks in solutions of inviscid Burgers equation, Commun. Math. Phys., 148, 601-622 (1992) · Zbl 0755.60105 · doi:10.1007/BF02096550 |
| [2] | She, Z. S.; Aurell, E.; Frisch, U., The inviscid Burgers equation with initial data of Brownian type, Commun. Math. Phys., 148, 623-641 (1992) · Zbl 0755.60104 · doi:10.1007/BF02096551 |
| [3] | Vergassola, M.; Dubrulle, B.; Frisch, U.; Noullez, A., Burgers’ equation, Devil’s staircases and the mass distribution for large-scale structures, Astron. Astrophys., 289, 325-356 (1994) |
| [4] | Artuso, R.; Cvitanović, P.; Kenny, B. G., Phase transitions on strange irrational sets, Phys. Rev. A, 39, 268-281 (1989) · doi:10.1103/PhysRevA.39.268 |
| [5] | G. M. Molchan, Burgers’ equation with self-similar Gaussian initial data: tail probabilities, preprint (1996). · Zbl 0944.60073 |
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