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Lévy-Process Intrinsic Statistical Solutions of a Randomly Forced Burgers Equation

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Abstract

We consider Burgers equation forced by a brownian in space and white noise in time process \(\partial_{t}u+\frac{1}{2}\partial_{x}(u)^{2}=f(x,t)\), with \(E(f(x,t)f(y,s))=\frac{1}{2}(|x|+|y|-|x-y|)\*\delta(t-s)\) and we show that there exist intrinsic statistical solutions that are Lévy processes at any given positive time. We give the evolution equation for the characteristic exponent of such solutions; in particular we give the explicit solution in the case u 0(x)=0.

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, UMR 5251 CNRS, Université Bordeaux1, 351 cours de la Libération, 33405, Talence Cedex, France

    Marie-Line Chabanol

  2. Institut Fourier, UMR 5582 CNRS-Université Joseph Fourier, 100 rue des Mathématiques, B.P. 74, 38041, Saint-Martin d’Hères Cedex, France

    Jean Duchon

Authors
  1. Marie-Line Chabanol
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  2. Jean Duchon
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Correspondence to Marie-Line Chabanol.

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Chabanol, ML., Duchon, J. Lévy-Process Intrinsic Statistical Solutions of a Randomly Forced Burgers Equation. J Stat Phys 136, 1095–1104 (2009). https://doi.org/10.1007/s10955-009-9824-z

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  • DOI: https://doi.org/10.1007/s10955-009-9824-z

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