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Scale renormalization and random solutions of the Burgers equation

Published online by Cambridge University Press:  14 July 2016

M. Rosenblatt
M. Rosenblatt*
Affiliation:
University of California, San Diego
*
∗Postal address: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA.
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Abstract

Solutions of the Burgers equation with a stationary (spatially) stochastic initial condition are considered. A class of limit laws for the solution which correspond to a scale renormalization is considered.

Keywords

STATIONARY INITIAL CONDITIONSHORT-RANGE DEPENDENCELONG-RANGE DEPENDENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 2 , June 1987 , pp. 328 - 338
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

This research is supported in part by Office of Naval Research Contract N00014-81-K-003 and National Science Foundation Grant DMS 83-12106.

References

Batchelor, G. (1953) The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Burgers, J. M. (1948) A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199.Google Scholar
Cole, J. D. (1951) On a quasilinear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236.Google Scholar
Dehling, H. and Philipp, W. (1982) Almost sure invariance principles for weakly dependent vector-valued random variables. Ann. Prob. 10, 689–701.Google Scholar
Dobrushin, R. and Major, P. (1979) Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitsth. 50, 27–52.Google Scholar
Gikhman, I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. Saunders, Philadelphia, PA.Google Scholar
Hopf, E. (1950) The partial differential equation ut + uux = µuxx. Comm. Pure Appl. Math. 3, 201–230.Google Scholar
Ibragimov, I. (1975) A note on the central limit theorem for dependent random variables. Theory Prob. Appl. 20, 135–141.Google Scholar
Taqqu, M. (1981) Self-similar processes and related ultraviolet and infrared catastrophes. In Random Fields, ed. Fritz, J., Lebowitz, J., and Szasz, D., North-Holland, Amsterdam, 1057–1096.Google Scholar
Vishik, M., Komech, A. and Fursikov, A. (1979) Some mathematical problems of statistical hydrodynamics. Russian Math. Surveys 34, 149–234.Google Scholar
Whitham, G. B. (1974) Linear and Nonlinear Waves. Wiley, New York.Google Scholar