Large deviations for the stochastic shell model of turbulence
Tóm tắt
In this work, we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for solutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell–Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc.
Tài liệu tham khảo
Amirdjanova A., Xiong J.: Large deviation principle for a stochastic Navier–Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete Contin. Dyn. Syst. Ser. B 6(4), 651–666 (2006)
Barbato D., Barsanti M., Bessaih H., Flandoli F.: Some rigorous results on a stochastic Goy model. J. Stat. Phys. 125(3), 677–716 (2006)
Bensoussan A., Temam R.: Equations aux dérivées partielles stochastiques non linéaries(1). Isr. J. Math. 11(1), 95–129 (1972)
Budhiraja A., Dupuis P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat. 20, 39–61 (2000)
Capinsky M., Gatarek D.: Stochastic equations in Hilbert space with application to Navier–Stokes equations in any dimension. J. Funct. Anal. 126, 26–35 (1994)
Chang M-H.: Large deviation for Navier–Stokes equations with small stochastic perturbation. Appl. Math. Comp. 76, 65–93 (1996)
Chow P-L.: Large deviation problem for some parabolic Îto equations. Comm. Pure Appl. Math. 45, 97–120 (1992)
Constantin P., Levant B., Titi E.S.: Analytic study of shell models of turbulence. Phys. D 219(2), 120–141 (2006)
Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Dembo A., Zeitouni O.: Large Deviations Techniques and Applications. Springer, New York (2000)
Deuschel J-D., Stroock D.W.: Large Deviations. Academic Press, San Diego (1989)
Dunford N., Schwartz J.: Linear Operators Interscience Publishers. Wiley, New York (1958)
Dupuis P., Ellis R.S.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley-Interscience, New York (1997)
Ellis R.S.: Entropy, Large Deviations and Statistical Mechanics. Springer, New York (1985)
Flandoli F., Gatarek D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Prob. Theory Rel. Fields. 102, 367–391 (1995)
Fleming W.H.: A stochastic control approach to some large deviations problems. In: RecentMathematicalMethods. Dolcetta, C., Fleming, W.H., Zolezzi, T. (eds) Dynamic Programming. Lecture Notes in Math., vol 1119., pp. 52–66. Springer, Heidelberg (1985)
Freidlin M.I., Wentzell A.D.: Random Pertubations of Dynamical Systems. Springer, New York (1984)
Frisch U.: Turbulence. Cambridge University Press, Cambridge (1995)
Kadanoff L., Lohse D., Wang J., Benzi R.: Scaling and dissipation in the GOY shell model. Phys. Fluids 7(3), 617–629 (1995)
Kallianpur, G., Xiong, J.: Stochastic Differential Equations in Infinite Dimensional Spaces. Institute of Math. Stat. (1996)
Karatzas I., Shreve S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)
Krylov N.V., Rozovskii B.L.: Stochastic evolution equations. J. Sov. Math. 16, 1233–1277 (1981)
Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)
L’vov V.S., Podivilov E., Pomyalov A., Procaccia I., Vandembroucq D.: Improved shell model of turbulence. Phys. Rev. E(3) 58(2), 1811–1822 (1998)
Manna, U., Menaldi, J.L., Sritharan, S.S.: Stochastic Analysis of Tidal Dynamics Equation. Infinite Dimensional Stochastic Analysis, pp. 90–113, QP–PQ: Quantum Probab. White Noise Anal., 22, World Science, Hackensack (2008)
Manna U., Menaldi J.L., Sritharan S.S.: Stochastic 2-D Navier–Stokes equation with artificial compressibility. Commun. Stoch. Anal. 1(1), 123–139 (2007)
Menaldi J.L., Sritharan S.S.: Stochastic 2-D Navier–Stokes Equation. Appl. Math. Optim. 46, 31–53 (2002)
Metivier M.: Stochastic Partial Differential Equations in Infinite Dimensional Spaces. Quaderni, Scuola Normale Superiore, Pisa (1988)
Ohkitani K., Yamada M.: Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence. Prog. Theor. Phys. 89, 329–341 (1989)
Pardoux, E.: Equations aux derivées partielles stochastiques non linéaires monotones. Etude de solutions fortes de type Itô; Thése Doct. Sci. Math. Univ. Paris Sud (1975)
Pardoux E.: Sur des equations aux dérivées partielles stochastiques monotones. C. R. Acad. Sci. 275(2), A101–A103 (1972)
Quastel J., Yau H.-T.: Lattice gases, large deviations, and the incompressible Navier–Stokes equations. Ann. Math. 2nd Ser. 148(1), 51–108 (1998)
Sowers R.: Large deviations for a reaction diffusion equation with non-Gaussian perturbations. Ann. Probab. 20, 504–537 (1992)
Sritharan S.S., Sundar P.: Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise. Stoc. Process. Appl. 116, 1636–1659 (2006)
Stroock D.: An Introduction to the Theory of Large Deviations. Springer, Universitext (1984)
Temam R.: Navier–Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1984)
Varadhan, S.R.S.: Large deviations and its Applications, 46, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia (1984)
Vishik M.J., Fursikov A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer, Boston (1980)