2002/10/31 by Jérémie Bec, J. Bec, Konstantin Khanin +1
Engineering · Mathematics · Physics and Astronomy · #Fluid Dynamics and Turbulent Flows #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #nlin.CD
paper · pdf · doi:10.1023/a:1027356518273
published as J. Stat. Phys. 113, 741–759 (2003) · 9 pages, 10 figures, revtex4, J. Stat. Phys, in press
arxiv created 2003/09/18 · openalex publication_date 2003/11/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time T we introduce the concept of T-global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904). In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as ρ(T) ∼ T-2/3 for large T. The probability density function p(A) of the age A of shocks behaves asymptotically as A-5/3. We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.