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Hausdorff dimension of regular points in stochastic Burgers flows with Lévy α-stable initial data

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Abstract

This paper studies statistical properties of shocks for the inviscid Burgers equation with an α-stable Lévy motion initial data. In the absence of analytic results, numerical and computer simulation tools are utilized. Qualitative and quantitative information on the scaling properties of Lagrangian regular points of solutions is obtained and, in particular, their Hausdorff dimension is estimated to be 1/α. This suggestsa possible extension of Ya. Sinai's result for Brownian initial data.

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Janicki, A.W., Woyczynski, W.A. Hausdorff dimension of regular points in stochastic Burgers flows with Lévy α-stable initial data. J Stat Phys 86, 277–299 (1997). https://doi.org/10.1007/BF02180207

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Key Words

  • Burgers equation
  • shocks
  • Hausdorff dimension
  • α-stable process