Abstract
The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time T we introduce the concept of T-global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904). In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as ρ(T)∼T −2/3 for large T. The probability density function p(A) of the age A of shocks behaves asymptotically as A −5/3. We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.
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REFERENCES
S. N. Gurbatov, A. I. Saichev, and S. F. Shandarin, The large-scale structure of the Universe in the frame of the model equation of non-linear diffusion, Month. Not. Roy. Astron. Soc. 236:385-402 (1989).
D. Chowdhury, L. Santen, and A. Schadschneider, Statistical physics of vehicular traffic, Phys. Rep. 329:199-329 (2000).
J.-P. Bouchaud, M. MÉzard, and G. Parisi, Scaling and intermittency in Burgers turbulence, Phys. Rev. E 52:3656-3674 (2000).
M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56:889-892 (1986).
Ya. G. Sinai, Two results concerning asymptotic behavior of the solutions of the Burgers equation with force, J. Stat. Phys. 64:1-12 (1991).
A. Chekhlov and V. Yakhot, Kolomogorov turbulence in a random-force-driven Burgers equation, Phys. Rev. E 51:R2739-R2742 (1995).
W. E, K. Khanin, A. Mazel, and Ya.G. Sinai, Probability distribution functions for the random forced Burgers equation, Phys. Rev. Lett. 78:1904-1907 (1997).
W. E, K. Khanin, A. Mazel, and Ya. G. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. Math. 151:877-960 (2000).
R. Iturriaga and K. Khanin, Burgers turbulence and random Lagrangian systems, Comm. Math. Phys., in press (2002).
J. Bec, R. Iturriaga, and K. Khanin, Topological shocks in Burgers turbulence, Phys. Rev. Lett. 89:024501(2002).
A. Polyakov, Turbulence without pressure, Phys. Rev. E 52:6183-6188 (1995).
P. D. Lax, Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math. 10:537-566 (1957).
O. Oleinik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12:3-73 (1957). O. Oleinik(Russ. Math. Survey., Am. Math. Transl. Series 2 26:95-172.)
P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math., Vol. 69, Pitman Advanced Publishing Program, Boston (1982).
J. Bec, U. Frisch, and K. Khanin, Kicked Burgers turbulence, J. Fluid Mech. 416:239-267 (2000).
A. Noullez and M. Vergassola, A fast Legendre transform algorithm and applications to the adhesion model, J. Sci. Comp. 9:259-281 (1994).
P. H. Baxendale, Stability and Equilibrium Properties of Stochastic Flows of Diffeomorphisms. Diffusion processes and related problems in analysis, (Vol. II) (Charlotte, NC, 1990), pp. 3-35, P. H. BaxendaleProgr. Probab., Vol. 27 BirkhÄuser Boston, Boston, MA, (1992).
P. H. Baxendale, Asymptotic Behaviour of Stochastic Flows of Diffeomorphisms. Stochastic Processes and Their Applications (Nagoya, 1985), Lecture Notes in Math., Vol. 1203 (Springer, Berlin, 1986), pp. 1-19.
P. H. Baxendale, Asymptotic behaviour of stochastic flows of diffeomorphisms: Two case studies, Probab. Theory Related Fields 73:51-85 (1986).
A. Carverhill, Flows of stochastic dynamical systems: Ergodic theory, Stochastics 14:273-317 (1985).
H. V. Hoang and K. Khanin, Random Burgers equation and Lagrangian systems in non-compact domains, submitted to Nonlinearity 16:819-842 (2003).
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Bec, J., Khanin, K. Forced Burgers Equation in an Unbounded Domain. Journal of Statistical Physics 113, 741–759 (2003). https://doi.org/10.1023/A:1027356518273
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DOI: https://doi.org/10.1023/A:1027356518273