November  2016, 21(9): 2927-2947. doi: 10.3934/dcdsb.2016080

Statistical properties of stochastic 2D Navier-Stokes equations from linear models

1. 

University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY 82071

2. 

Università di Pavia, Dipartimento di Matematica, via Ferrata 5, 27100 Pavia, Italy

Received  January 2016 Revised  March 2016 Published  October 2016

A new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence has been proposed and tested through numerical simulations. This is achieved by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem.
    In this paper, we investigate this conjecture for the 2D Navier-Stokes equations driven by an additive noise. In order to check this conjecture, we analyze the coupled system Navier-Stokes/linear advection system in the unknowns $(u,w)$. We introduce a parameter $\lambda$ which gives a system $(u^\lambda,w^\lambda)$; this system is studied for any $\lambda$ proving its well posedness and the uniqueness of its invariant measure $\mu^\lambda$.
    The key point is that for any $\lambda \neq 0$ the fields $u^\lambda$ and $w^\lambda$ have the same scaling exponents, by assuming universality of the scaling exponents to the force. In order to prove the same for the original fields $u$ and $w$, we investigate the limit as $\lambda \to 0$, proving that $\mu^\lambda$ weakly converges to $\mu^0$, where $\mu^0$ is the only invariant measure for the joint system for $(u,w)$ when $\lambda=0$.
Citation: Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080
References:
[1]

L. Angheluta, R. Benzi, L. Biferale, I. Procaccia and T. Toschi, Anomalous scaling exponents in nonlinear models of turbulence, Phys. Rev. Lett., 97 (2006), 160601. doi: 10.1103/PhysRevLett.97.160601.

[2]

I. Arad, L. Biferale, A. Celani, I. Procaccia and M. Vergassola, Statistical conservation laws in turbulent transport, Phys. Rev. Lett., 87 (2001), 164502. doi: 10.1103/PhysRevLett.87.164502.

[3]

R. Benzi, B. Levant, I. Procaccia and E. S. Titi, Statistical properties of nonlinear shell models of turbulence from linear advection model: rigorous results, Nonlinearity, 20 (2007), 1431-1441. doi: 10.1088/0951-7715/20/6/006.

[4]

H. Bessaih and B. Ferrario, Inviscid limit of stochastic damped 2D Navier-Stokes equations, Nonlinearity, 27 (2014), 1-15. doi: 10.1088/0951-7715/27/1/1.

[5]

H. Bessaih, F. Flandoli and E. S. Titi, Stochastic attractors for shell phenomenological models of turbulence, J. Stat. Phys., 140 (2010), 688-717. doi: 10.1007/s10955-010-0010-0.

[6]

P. L. Chow, Stationary solutions of two-dimensional Navier-Stokes equations with random perturbation, Nonlinear stochastic PDEs, (Minneapolis, MN, 1994), IMA Vol. Math. Appl., Springer, 77 (1996), 237-245. doi: 10.1007/978-1-4613-8468-7_13.

[7]

Y. Cohen, T. Gilbert and I. Procaccia, Statistically preserved structures in shell models of passive scalar advection, Phys. Rev. E., 65 (2002), 026314. doi: 10.1103/PhysRevE.65.026314.

[8]

G. Da Prato, S. Kwapień and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics, 23 (1987), 1-23. doi: 10.1080/17442508708833480.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[10]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.

[11]

G. Falkovich, K. Gawędzki and M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod. Phys., 73 (2001), 913-975. doi: 10.1103/RevModPhys.73.913.

[12]

B. Ferrario, Ergodic results for stochastic Navier-Stokes equation, Stochastics Stochastics Rep., 60 (1997), 271-288. doi: 10.1080/17442509708834110.

[13]

B. Ferrario, Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure, Ann. Mat. Pura Appl., 177 (1999), 331-347. doi: 10.1007/BF02505916.

[14]

B. Ferrario, Uniqueness result for the 2D Navier-Stokes equation with additive noise, Stoch. Stoch. Rep., 75 (2003), 435-442. doi: 10.1080/10451120310001644485.

[15]

F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 403-423. doi: 10.1007/BF01194988.

[16]

F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.

[17]

F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys., 172 (1995), 119-141. doi: 10.1007/BF02104513.

[18]

U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.

[19]

K. Gawędzki and A. Kupiainen, Anomalous Scaling of the Passive Scalar, Phys. Rev. Lett., 75 (1995), 3834-3837.

[20]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math., 164 (2006), 993-1032. doi: 10.4007/annals.2006.164.993.

[21]

M. Hairer and J. C. Mattingly, A theory of hypo-ellipticity and unique ergodicity for semi-linear stochastic PDEs, Electron. J. Probab., 16 (2011), 658-738. doi: 10.1214/EJP.v16-875.

[22]

S. B. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge Tracts in Mathematics, 2012. doi: 10.1017/CBO9781139137119.

[23]

B. Maslowski, On probability distributions of solutions of semilinear stochastic evolution equations, Stochastics Stochastics Rep., 45 (1993), 17-44. doi: 10.1080/17442509308833854.

[24]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[25]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 41. SIAM, Philadelphia, PA, 1983.

[26]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Mathematics and its Applications, Springer, 1988. doi: 10.1007/978-94-009-1423-0.

[27]

K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

show all references

References:
[1]

L. Angheluta, R. Benzi, L. Biferale, I. Procaccia and T. Toschi, Anomalous scaling exponents in nonlinear models of turbulence, Phys. Rev. Lett., 97 (2006), 160601. doi: 10.1103/PhysRevLett.97.160601.

[2]

I. Arad, L. Biferale, A. Celani, I. Procaccia and M. Vergassola, Statistical conservation laws in turbulent transport, Phys. Rev. Lett., 87 (2001), 164502. doi: 10.1103/PhysRevLett.87.164502.

[3]

R. Benzi, B. Levant, I. Procaccia and E. S. Titi, Statistical properties of nonlinear shell models of turbulence from linear advection model: rigorous results, Nonlinearity, 20 (2007), 1431-1441. doi: 10.1088/0951-7715/20/6/006.

[4]

H. Bessaih and B. Ferrario, Inviscid limit of stochastic damped 2D Navier-Stokes equations, Nonlinearity, 27 (2014), 1-15. doi: 10.1088/0951-7715/27/1/1.

[5]

H. Bessaih, F. Flandoli and E. S. Titi, Stochastic attractors for shell phenomenological models of turbulence, J. Stat. Phys., 140 (2010), 688-717. doi: 10.1007/s10955-010-0010-0.

[6]

P. L. Chow, Stationary solutions of two-dimensional Navier-Stokes equations with random perturbation, Nonlinear stochastic PDEs, (Minneapolis, MN, 1994), IMA Vol. Math. Appl., Springer, 77 (1996), 237-245. doi: 10.1007/978-1-4613-8468-7_13.

[7]

Y. Cohen, T. Gilbert and I. Procaccia, Statistically preserved structures in shell models of passive scalar advection, Phys. Rev. E., 65 (2002), 026314. doi: 10.1103/PhysRevE.65.026314.

[8]

G. Da Prato, S. Kwapień and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics, 23 (1987), 1-23. doi: 10.1080/17442508708833480.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[10]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.

[11]

G. Falkovich, K. Gawędzki and M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod. Phys., 73 (2001), 913-975. doi: 10.1103/RevModPhys.73.913.

[12]

B. Ferrario, Ergodic results for stochastic Navier-Stokes equation, Stochastics Stochastics Rep., 60 (1997), 271-288. doi: 10.1080/17442509708834110.

[13]

B. Ferrario, Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure, Ann. Mat. Pura Appl., 177 (1999), 331-347. doi: 10.1007/BF02505916.

[14]

B. Ferrario, Uniqueness result for the 2D Navier-Stokes equation with additive noise, Stoch. Stoch. Rep., 75 (2003), 435-442. doi: 10.1080/10451120310001644485.

[15]

F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 403-423. doi: 10.1007/BF01194988.

[16]

F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.

[17]

F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys., 172 (1995), 119-141. doi: 10.1007/BF02104513.

[18]

U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.

[19]

K. Gawędzki and A. Kupiainen, Anomalous Scaling of the Passive Scalar, Phys. Rev. Lett., 75 (1995), 3834-3837.

[20]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math., 164 (2006), 993-1032. doi: 10.4007/annals.2006.164.993.

[21]

M. Hairer and J. C. Mattingly, A theory of hypo-ellipticity and unique ergodicity for semi-linear stochastic PDEs, Electron. J. Probab., 16 (2011), 658-738. doi: 10.1214/EJP.v16-875.

[22]

S. B. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge Tracts in Mathematics, 2012. doi: 10.1017/CBO9781139137119.

[23]

B. Maslowski, On probability distributions of solutions of semilinear stochastic evolution equations, Stochastics Stochastics Rep., 45 (1993), 17-44. doi: 10.1080/17442509308833854.

[24]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[25]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 41. SIAM, Philadelphia, PA, 1983.

[26]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Mathematics and its Applications, Springer, 1988. doi: 10.1007/978-94-009-1423-0.

[27]

K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

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