Abstract
We consider scalar conservation laws with convex flux and random initial data. The Hopf–Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group; (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for N-dimensional approximations of the Lax equations, and yield formulas for the limit N → ∞. The main observation is that the Lax equations and zero-curvature equations are a Markovian analog of known integrable systems (geodesic flow on Lie groups and the N-wave model respectively). This allows us to introduce a variety of methods from the theory of integrable systems.
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References
Ablowitz M., Haberman R.: Resonantly coupled nonlinear evolution equations. J. Math. Phys. 16, 2301 (1975)
Ablowitz M., Kaup D., Newell A., Segur H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. App. Math. 53, 249–315 (1974)
Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society Lecture Note Series, Vol. 149. Cambridge University Press, 1991
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, Vol. 55. U.S. Government Printing Office, Washington, D.C., 1964
Adler M., van Moerbeke P.: Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. Math. 38(3), 318–379 (1980)
Adler M., Van Moerbeke P.: Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math. 38(3), 267–317 (1980)
Adler, M., Van Moerbeke, P., Vanhaecke, P.: Algebraic integrability, Painlevé geometry and Lie algebras. Springer, 2004
Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, 2004
Arnold V.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1989)
Arnold V., Khesin B.: Topological Methods in Hydrodynamics. Springer, Berlin (1998)
E. W., Avallaneda M.: Statistical properties of shocks in Burgers turbulence. Commun. Math. Phys. 172, 13–38 (1995)
Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems. Cambridge University Press, 2003
Beals R., Coifman R.: Scattering and inverse scattering for first order systems. Commun. Pure App. Math. 37(1), 39–90 (1984)
Beals R., Coifman R.: Inverse scattering and evolution equations. Commun. Pure App. Math. 38(1), 29–42 (1985)
Beals R., Sattinger D.: On the complete integrability of completely integrable systems. Commun. Math. Phys. 138(3), 409–436 (1991)
Bertoin J.: The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193, 397–406 (1998)
Bertoin, J.: Some aspects of additive coalescents. In: Proceedings of the International Congress of Mathematicians, Beijing 2002, Vol. III, 15–23. Higher Ed. Press, 2002
Burgers J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974)
Carraro L., Duchon J.: Solutions statistiques intrinsèques de l’équation de Burgers et processus de Lévy. C. R. Acad. Sci. Paris Sér. I Math. 319(8), 855–858 (1994)
Carraro L., Duchon J.: Équation de Burgers avec conditions initiales à accroissements indépendants et homogènes. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(4), 431–458 (1998)
Chabanol M.L., Duchon J.: Markovian solutions of inviscid Burgers equation. J. Stat. Phys. 114(1–2), 525–534 (2004)
Constantin P., E W., Titi E.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1), 207–209 (1994)
E. W., Sinai Y.: New results in mathematical and statistical hydrodynamics. Russian Math. Surv. 55(4), 635–666 (2000)
E. W., Vanden Eijnden E.: Statistical theory for the stochastic Burgers equation in the inviscid limit. Commun. Pure Appl. Math. 53(7), 852–901 (2000)
Eyink G.: Energy dissipation without viscosity in ideal hydrodynamics I Fourier analysis and local energy transfer* 1. Physica D 78(3–4), 222–240 (1994)
Faddeev D., Faddeeva V.: Computational Methods of Linear Algebra. WH Freeman, San Francisco (1963)
Frachebourg L., Martin P.A.: Exact statistical properties of the Burgers equation. J. Fluid Mech. 417, 323–349 (2000)
Golovin, A.M.: The solution of the coagulating equation for cloud droplets in a rising air current. Izv. Geophys. Ser. 482–487 (1963)
Groeneboom P.: Brownian motion with a parabolic drift and Airy functions. Probab. Theory Relat. Fields 81(1), 79–109 (1989)
Hopf E.: Statistical hydromechanics and functional calculus. J. Rational Mech. Anal. 1, 87–123 (1952)
Jimbo M., Miwa T., Môri Y., Sato M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica D 1(1), 80–158 (1980)
Johnson J.: Markov-type Lie groups in GL (n, R). J. Math. Phys. 26(2), 252 (1985)
Kerov, S.V.: Asymptotic representation theory of the symmetric group and its applications in analysis, Translations of Mathematical Monographs, Vol. 219. American Mathematical Society, Providence, RI, 2003
Kida S.: Asymptotic properties of Burgers turbulence. J. Fluid Mech. 93(2), 337–377 (1979)
Kolmogorov, A.N.: Dissipation of energy in the locally isotropic turbulence. Proc. R. Soc. London Ser. A 434(1890), 15–17 (1991). Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov’s ideas 50 years on
Manakov S.: Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body. Funct. Anal. Appl. 10(4), 328–329 (1976)
Mendoza R., Savin I., Thornton K., Voorhees P.: Topological complexity and the dynamics of coarsening. Nat. Mater. 3(6), 385–388 (2004)
Menon G., Srinivasan R.: Kinetic theory and Lax equations for shock clustering and Burgers turbulence. J. Stat. Phys. 140(6), 1195–1223 (2010)
Moser, J.: Geometry of quadrics and spectral theory. In: The Chern Symposium, 1979: Proceedings of the International Symposium on Differential Geometry in Honor of S.-S. Chern, held in Berkeley, California, June 1979, p. 147. Springer, 1980
Novikov, S.P. (ed.): Dynamical systems, VII. Encyclopaedia of Mathematical Sciences, Vol. 16. Springer, Berlin, 1994
Onsager L.: Statistical hydrodynamics. Il Nuovo Cimento (1943–1954) 6, 279–287 (1949)
Semenov-Tian-Shansky M.: What is a classical r-matrix?. Funct. Anal. Appl. 17(4), 259–272 (1983)
She Z.S., Aurell E., Frisch U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148(3), 623–641 (1992)
Sinai Y.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148(3), 601–621 (1992)
Terng C., Uhlenbeck K.: Bäcklund transformations and loop group actions. Commun. Pure Appl. Math. 53(1), 1–75 (2000)
Valageas P.: Statistical properties of the Burgers equation with Brownian initial velocity. J. Stat. Phys. 134(3), 589–640 (2009)
Zakharov V., Manakov S.: Resonant interaction of wave packets in nonlinear media. JETP Lett. (USSR) (Engl. Transl.) 18(7), 243–245 (1973)
Zakharov V., Manakov S., Novikov S., Pitaevskii L.: Theory of Solitons: The Inverse Scattering Method. Plenum, New York (1984)
Zakharov V., Shabat A.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. JETP 34, 62–69 (1972)
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Communicated by W. E
Supported by NSF grant DMS 07-48482.
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Menon, G. Complete Integrability of Shock Clustering and Burgers Turbulence. Arch Rational Mech Anal 203, 853–882 (2012). https://doi.org/10.1007/s00205-011-0461-8
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DOI: https://doi.org/10.1007/s00205-011-0461-8