Abstract
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.
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Winkel, M. Limit Clusters in the Inviscid Burgers Turbulence with Certain Random Initial Velocities. Journal of Statistical Physics 107, 893–917 (2002). https://doi.org/10.1023/A:1014598400004
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DOI: https://doi.org/10.1023/A:1014598400004