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Hyperbolic asymptotics in Burgers’ turbulence and extremal processes. (English) Zbl 0818.60046

Summary: Large time asymptotics of statistical solution u(t,x),

u(t,x)= - [(x-y)/t]exp[(2μ) -1 (ξ(y)-(x-y) 2 /2t)]dy - exp[(2μ) -1 (ξ(y)-(x-y) 2 /2t)]dy,

of the Burgers’ equation

t u+u x u=μ x 2 u,

is considered, where ξ(x)=ξ L (x) is a stationary zero mean Gaussian process depending on a large parameter L>0 so that ξ L (x)σ L η(x/L) (L), where σ L =L 2 (2logL) 1/2 and η(x) is a given standardized stationary Gaussian process. We prove that as L the hyperbolicly scaled random fields u(L 2 t,L 2 x) converge in distribution to a random field with “saw-tooth” trajectories, defined by means of a Poisson process on the plane related to high fluctuations of ξ(x), which corresponds to the zero viscosity solutions. At the physical level of rigor, such asymptotics was considered before by S. N. Gurbatov, A. N. Malakhov and A. I. Saichev [Nonlinear random waves in media without dispersion (1990; Zbl 0753.76004)].

60H15Stochastic partial differential equations
35Q53KdV-like (Korteweg-de Vries) equations
[1]Albeverio, S., Molchanov, S.A., Surgailis, D.: Stratified structure of the Universe and Burgers’ equation: A probabilistic approach. Prob. Theory Rel. Fields (1995), to appear
[2]Bulinskii, A.V., Molchanov, S.A.: Asymptotic Gaussianness of solutions of the Burgers’ equation with random initial data. Teorya Veroyat. Prim.36, 271–235 (1991)
[3]Burgers, J.: The Nonlinear Diffusion Equation, Amsterdam: Dordrecht (1974)
[4]Chorin, A.J.: Lectures on Turbulence Theory, Berkeley CA: Publish or Perish (1975)
[5]Fournier, J.-D., Frisch, U.: L’équation de Burgers déterministe et statistique. J. Mec. Theor. Appl.2, 699–750 (1983)
[6]Funaki, T., Surgailis, D., Woyczynski, W. A.: Gibbs-Cox random fields and Burgers’ turbulence. Ann. Applied Probability5, to appear (1995)
[7]Gurbatov, S., Malakhov, A., Saichev, A.: Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles. Manchester: Manchester University Press (1991)
[8]Hopf, E.: The partial differential equationu t+uux=μuxx. Comm. Pure Appl. Math.3, 201 (1950) · Zbl 0039.10403 · doi:10.1002/cpa.3160030302
[9]Hu, Y., Woyczynski, W.A.: An extremal rearrangement property of statistical solutions of the Burgers’ equation. Ann. Applied Probability4, 838–858 (1994) · Zbl 0805.60053 · doi:10.1214/aoap/1177004974
[10]Kallenberg, O.: Random Measures. New York: Academic Press (1983)
[11]Kraichnan, R.H.: Lagrangian-history statistical theory for Burgers’ equation. Physics of Fluids11, 265–277 (1968) · Zbl 0153.57502 · doi:10.1063/1.1691900
[12]Kraichnan, R.H.: The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech.5, 497–543 (1959) · Zbl 0093.41202 · doi:10.1017/S0022112059000362
[13]Kwapien, S., Woyczynski, W.A.: Random Series and Stochastic Integrals: Single and Multiple, Boston: Birkhäuser (1992)
[14]Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Berlin, Heidelberg, New York: Springer (1983)
[15]Lindgren, G.: Some properties of a normal process mear a local maximum. Ann. Math. Stat.41, 1870–1883 (1970) · Zbl 0222.60027 · doi:10.1214/aoms/1177696688
[16]Rice S.O.: Mathematical analysis of random noise. Bell System Tech. J.24, 46–156 (1945)
[17]Shandarin, S.F., Zeldovich, Ya.B.: Turbulence, intermittency, structures in a self-gravitating medium: The large scale structure of the Universe. Rev. Modern Phys.61, 185–220 (1989) · doi:10.1103/RevModPhys.61.185
[18]Sinai, Ya.G.: Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Stat. Phys.64, 1–12 (1992) · Zbl 0978.35500 · doi:10.1007/BF01057866
[19]Sinai, Ya.G.: Statistics of shocks in solutions of inviscid Burgers’ equation. Commun. Math. Phys.148, 601–621 (1992) · Zbl 0755.60105 · doi:10.1007/BF02096550
[20]Surgailis, D., Woyczynski, W.A.: Scaling limits of solutions of the Burgers’ equation with singular Gaussian initial data, in Chaos Expansions, Multiple Wiener-Ito Integrals and Their Applications, C. Houdré and V. Pérez-Abreu, Eds, pp. 145–161, Boca Raton, Ann Arbor CRC Press (1994)
[21]Woyczynski, W.A.: Stochastic Burgers’ Flows. In: Nonlinear Waves and Weak Turbulence. Boston: Birkhäuser, pp. 279–311 (1993)