Limit clusters in the inviscid Burgers turbulence with certain random initial velocities

Journal article

We study the infinite time shock limits given certain Markovian initial velocities to the inviscid Burgers turbulence. Specifically, we consider the one-sided case where initial velocities are zero on the negative half-line and follow a time-homogeneous nice Markov process X on the positive half-line. Finite shock limits occur if the Markov process is transient tending to infinity. They form a Poisson point process if X is spectrally negative. We give an explicit description when X is furthermore spatially homogeneous (a Lévy process) or a self-similar process on (0, ∞). We also consider the two-sided case where we suppose an independent dual process in the negative spatial direction. Both spatial homogeneity and an exponential Lévy condition lead to stationary shock limits.

Authors: Winkel, M

• Publication status: Published
• Journal: JOURNAL OF STATISTICAL PHYSICS
• Volume: 107
• Issue: 3-4
• Pages: 893-917
• Publication date: 2002-05-01
• DOI: 10.1023/A:1014598400004
• ISSN: 0022-4715
• Language: English
• Keywords: inviscid Burgers equation; random initial velocity; spectrally negative Levy processes; shock structure; self-similar processes; Markov processes