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Scalar conservation laws with white noise initial data

Abstract

The statistical description of the scalar conservation law of the form \(\rho _t=H(\rho )_x\) with \(H: {\mathbb {R}} \rightarrow {\mathbb {R}}\) a smooth convex function has been an object of interest when the initial profile \(\rho (\cdot ,0)\) is random. The special case when \(H(\rho )=\frac{\rho ^2}{2}\) (Burgers equation) has in particular received extensive interest in the past and is now understood for various random initial conditions. We prove in this paper a conjecture on the profile of the solution at any time \(t>0\) for a general class of Hamiltonians H and show that it is a stationary piecewise-smooth Feller process. Along the way, we study the excursion process of the two-sided linear Brownian motion W below any strictly convex function \(\phi \) with superlinear growth and derive a generalized Chernoff distribution of the random variable \(\text {argmax}_{z \in {\mathbb {R}}} (W(z)-\phi (z))\). Finally, when \(\rho (\cdot ,0)\) is a white noise derived from an abrupt Lévy process, we show that the structure of shocks of the solution is a.s discrete at any fixed time \(t>0\) under some mild assumptions on H.

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Notes

  1. Under some mild conditions on the Hamiltonian H, and a slight modification of the nature of the initial data.

  2. We write \(f=o(g)\) if \(\lim \frac{f}{g}=0\) and \(f=O(g)\) if \(\frac{f}{g}\) is bounded.

  3. There is a typo in the published paper, the term \(4^{\frac{2}{3}}\) in the denominator should be there instead of \(4^{\frac{1}{3}}\).

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Acknowledgements

I would like to thank my advisor Fraydoun Rezakhanlou for many fruitful discussions.

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The author has no relevant financial or non-financial interests to disclose. The author did not receive support from any organization for the submitted work.

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Correspondence to Mehdi Ouaki.

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Ouaki, M. Scalar conservation laws with white noise initial data. Probab. Theory Relat. Fields 182, 955–998 (2022). https://doi.org/10.1007/s00440-021-01083-z

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Keywords

  • Scalar conservation law
  • White noise
  • Random initial data
  • Path decomposition
  • Chernoff distribution
  • Abrupt process

Mathematics Subject Classification

  • 60G51
  • 60J65
  • 60J60
  • 60J75
  • 35L65