Skip to main content Accessibility help
×
Home

Burgers' equation by non-local shot noise data

  • Donatas Surgailis and Wojbor A. Woyczynski

Abstract

We study the scaling limit of random fields which are solutions of a non-linear partial differential equation, known as the Burgers equation, under stochastic initial conditions. These are assumed to be of a non-local shot noise type and driven by a Cox process. Previous work by Bulinskii and Molchanov (1991), Surgailis and Woyczynski (1993a), and Funaki et al. (1994) concentrated on the case of local shot noise data which permitted use of techniques from the theory of random fields with finite range dependence. Those are not available for the non-local case being considered in this paper.

Burgers' equation is known to describe various physical phenomena such as non-linear and shock waves, distribution of self-gravitating matter in the universe, and other flow satisfying conservation laws (see e.g. Woyczynski (1993)).

Copyright

References

Hide All
Albeverio, S., Molchanov, S. A. and Surgailis, D. (1993) Stratified structure of the Universe and Burgers' equation: a probabilistic approach. Preprint.
Bulinskii, A. B. and Molchanov, S. A. (1991) Asymptotic Gaussianness of solutions of the Burgers' equation with random initial data. Teorya Veroyat. Prim. 36, 217235.
Burgers, J. (1974) The Nonlinear Diffusion Equation. Reidel, Dordrecht.
Dobrushin, R. L. (1979) Gaussian and their subordinated self-similar random generalized fields. Ann. Prob. 7, 128.
Dobrushin, R. L. (1980) Automodel generalized random fields and their renormgroup. In Multicomponent Random Systems , ed. Dobrushin, R. L. and Sinai, Ya. G., pp. 153198. Dekker, New York.
Funaki, T., Surgailis, D. and Woyczynski, W. A. (1994) Gibbs-Cox random fields and Burgers turbulence. Ann. Appl. Prob. To appear.
Giraitis, L., Molchanov, S. A. and Surgailis, D. (1992) Long memory shot noises and limit theorems with applications to Burgers' equation. In New Directions in Time Series Analysis, Part II , ed. Brillinger, D. et al. IMA Volumes in Mathematics and Its Applications, Springer-Verlag, Berlin.
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics 529, Springer-Verlag, Berlin.
Hopf, E. (1950), The partial differential equation ut + uux = uxx . Commun. Pure Appl. Maths. 3, 201.
Hu, Y. and Woyczynski, W. A. (1994) An extremal rearrangement property of statistical solutions of the Burgers' equation. Ann. Appl. Prob. To appear.
Rosenblatt, M. (1987) Scale renormalization and random solutions of the Burgers' equation. J. Appl. Prob. 24, 328338.
Shandarin, S. F. and Zeldovich, Ya. B. (1989) Turbulence, intermittency, structures in a self-gravitating medium: the large scale structure of the Universe. Rev. Modern Phys. 61, 185220.
Surgailis, D. and Woyczynski, W. A. (1993a) Long-range prediction and scaling limit for statistical solutions of the Burgers' equation. In Nonlinear Waves and Weak Turbulence , pp. 313338. Birkhäuser, Boston.
Surgailis, D. and Woyczynski, W. A. (1993b) Scaling limits of solutions of the Burgers equation with random Gaussian initial data. Proc. Workshop on Multiple Wiener-Itô Integrals and their Applications, Guanajuato, Mexico, 1992.
Woyczynski, W. A. (1993) Stochastic Burgers flows. In Nonlinear Waves and Weak Turbulence , pp. 279311. Birkhäuser, Boston.

Keywords

MSC classification

Related content

Powered by UNSILO

Burgers' equation by non-local shot noise data

  • Donatas Surgailis and Wojbor A. Woyczynski

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.