Abstract
This paper surveys the theory and applications of solutions of the Burgers’ equation with random initial data which are often called statistical solution. It grew out of a series of lectures delivered by the author in the Summer of ′92 at Nagoya University, Japan. It is far from complete and reflects only the author’s own interests. A description of other recent developments can be found in the bibliography provided in the last section.
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Bibliography
Albeverio S., Molchanov S.A., Surgailis D. (1992), Stratified structure of the Universe and Burgers’ equation, preprint.
Bramson M., Liggett T.M. (1986), Lectures in Probability, The University of Nagoya Press.
Bulinski A.V., Limit theorems under weak dependence conditions, in Probability Theory and Math. Stat., Proc. Fourth Vilnius Conf. v.1, Utrecht VNP 1987, pp. 307–327.
Bulinski A.V., Molchanov. S.A. Asymptotic normality of solutions of the Burgers equation with random initial data. Teorya Veroyat Prim. 36 (1991) 217–235.
Burgers J.M., The Nonlinear Diffusion Equation, Reidel 1974.
Cole J.D. (1951), On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225.
Dobrushin (1980), Automodel generalized random fields and their renorm group, in Multicomponent Random Systems, R.L. Dobrushin and Ya. G. Sinai (Eds.), Dekker, pp. 153–198.
Giraitis L., Molchanov S.A., Surgailis D.(1992), Long memory shot noises and limit theorems with application to Burgers; equation, in New Directions in Time Series Analysis, IMA Volumes in Mathematics and Its Applications, Springer, to appear.
Gurbatov S., Saichev A., Shandarin S.F. (1989), The large-scale structure of the Universe in the frame of the model equation of nonlinear diffusion, Monthly Not. Royal Astronom. Soc. 236, 385–402.
Gurbatov S., Malakhov A., Saichev A. (1991), Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles, Manchester University Press.
Gutkin E., Kac M. (1983), Propagation of chaos and the Burgers’ equation, SIAM J. Appl. Math. 43, 971–980.
K. Handa, A remark on shocks in inviscid Burgers turbulence, in Nonlinear Waaves and Weak Turbulence, Birkhäuser-Boston 1993, pp. 333–339.
Hopf E. (1950), The partial differential equation u Pt + uu Pxx = u Pxx , Commun. Pure Appl. Math. 3, 201.
Hu Y., Woyczynski W.A. (1992), Limit behavior of quadratic forms of moving averages of i.i.d. random variables and statistical solutions of the Burgers’ equation, J. Multivariate Analysis, to appear.
Hu Y., Woyczynski W.A.(1993), An extremal rearrangement property of statistical solutions of the Burgers’ equation, Ann. Appl. Prob, to appear.
Hu Y., Woyczynski W.A.(1993a), A maximum principle for unimodal moving average data for Burgers’ equation, Prob. Math. Stat., to appear.
Ibragimov I.A., and Linnik Yu.V. (1965), Independent and Stationary Random Processes, Moscow, Nauka.
Kotani S., Osada H. (1985), Propagation of chaos for Burgers’ equation, J. Math. Soc. Japan, 37, 275–294.
Kwapien S., Woyczynski W.A. (1992), Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser-Boston.
Oeschläger K. (1985), A law of large numbers for moderately interacting diffusion processes, Z. Wahrscheinlichkeitstheorie venu. Geb. 69, 279–322.
Rosenblatt M. (1987), Scale renormalization and random solutions of the Burgers equation. J. Appl. Prob. 24. 328–338.
Rosenblatt M. (1978), Energy transfer for the Burgers’ equation, Physics of Fluids 21, 1694–1697.
Ruelle D. (1969), Statistical Mechanics, Benjamin.
Schur I. (1923), Über eine Klasse von Mittelbildungen mit Anwendungen zur die Determinanten-Theorie, Sitzungsber. Berlin Math. Gesellschaft 22, 9–20.
Shandarin S.F. and Zeldovich B.Z., Turbulence (1989), intermittency, structures in a self-gravitating medium: the large scale structure of the universe, Rev. Modern Phys. 61, 185–220.
She Z-S., Aurell E. and Frisch U. (1992), The inviscid Burgers equation with initial data of Brownian type, Commun. Math. Phys. 148, 623–641.
Sinai Ya.G. (1992), Statistics of shocks in solutions of inviscid Burgers equation, Commun. Math. Phys. 148, 601–621.
Spohn H. (1991), Large Scale Dynamics of Interacting Particles, Springer-Verlag.
Surgailis D., Woyczynski W.A. (1993), Long range prediction and scaling limit for statistical solutions of the Burgers’ equation, in Nonlinear Waaves and Weak Turbulence, Birkhäuser-Boston 1993, 23 pp., to appear.
Surgailis D., Woyczynski W.A. (1993b), Scaling limits of the solution of the Burgers’ equation with singular Gaussian initial data, in Proc. Mexico Conf on Multiple Stochastic Integration, Birkhauser-Boston, to appear.
Surgailis D., Woyczynski W.A. (1994), Burgers’ equation with nonlocal shot noise data, J. Appl. Prob., to appear.
Sznitman A. (1986), A propagation of chaos result for Burgers’ equation, Probab. Th. Rel. Fields 71, 581–613.
Weinberg D.H., Gunn J.E. (1990), Large-scale structure and the adhesion approximation, Monthly Not. Royal Astronom. Soc. 247, 260–286.
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Woyczyński, W.A. (1993). Stochastic Burgers’ Flows. In: Fitzmaurice, N., Gurarie, D., McCaughan, F., Woyczyński, W.A. (eds) Nonlinear Waves and Weak Turbulence. Progress in Nonlinear Differential Equations and Their Applications, vol 11. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0331-5_15
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DOI: https://doi.org/10.1007/978-1-4612-0331-5_15
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