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Statistics of shocks in solutions of inviscid Burgers equation
 Ya. G. Sinai
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The purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid Burgers equation having a typical realizationb(y) of the Brownian motion as an initial datum. This case was proposed and studied numerically in the companion paper by She, Aurell and Frisch. The description of the statistics is given in terms of the behavior of the convex hull of the random process \(w(y) = \int\limits_0^y {(b(\eta ) + \eta )} d\eta \) . The Hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied.
Communicated by A. Jaffe
 She, Z.S., Aurell, E., Frisch, U. (1992) The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148: pp. 623
 Hopf, E. (1950) The partial differential equationu t +uu x =μu xx +. Commun. Pure Appl. Math. 3: pp. 201230
 Burgers, J.M. (1974) The nonlinear diffusion equation. D. Reidel, Dordrecht
 Stroock, D.W., Varadhan, S.R.S. (1979) Multidimensional diffusion processes. Springer, Berlin, Heidelberg, New York
 Sinai, Ya. G.: Distribution of some functionals of the integral of the Brownian motion. Theor. Math. Phys. (in Russian) (1992 (in press))
 Falconer, K. (1985) The geometry of fractal sets. Cambridge University Press, Cambridge
 Title
 Statistics of shocks in solutions of inviscid Burgers equation
 Journal

Communications in Mathematical Physics
Volume 148, Issue 3 , pp 601621
 Cover Date
 19920901
 DOI
 10.1007/BF02096550
 Print ISSN
 00103616
 Online ISSN
 14320916
 Publisher
 SpringerVerlag
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 Authors

 Ya. G. Sinai ^{(1)} ^{(2)}
 Author Affiliations

 1. Landau Institute of Theoretical Physics, Moscow, Russia
 2. Mathematics Department, Princeton University, 08544, NJ, USA