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The Inviscid Burgers Equation with Fractional Brownian Initial Data: The Dimension of Regular Lagrangian Points

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Abstract

Fractional Brownian motion, H-FBM, of index \(0<H<1\) is considered as initial velocity in the inviscid Burgers equation. It is shown that the Hausdorff dimension of regular Lagrangian points at any moment t is equal to H. This fact validates the Sinai-Frisch conjecture known since 1992. We find also that the integrated H-FBM does not exceed a fixed positive level in the interval \((-T, T)\) with probability having the log-asymptotics: \((H-1+o(1))\mathrm{log} T\).

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Acknowledgements

I am very grateful to the reviewers for their careful reading of the article and constructive comments. This research had been announced in the proposal that was supported by the Russian Science Foundation through the research project 17-11-01052.

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  1. Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Science, 84/32 Profsoyuznaya St., Moscow, Russian Federation, 117997

    G. Molchan

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Molchan, G. The Inviscid Burgers Equation with Fractional Brownian Initial Data: The Dimension of Regular Lagrangian Points. J Stat Phys 167, 1546–1554 (2017). https://doi.org/10.1007/s10955-017-1791-1

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