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Preface
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 |
  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 $\lambda=0$.
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).
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).
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.
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.
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.
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, 77 (1996), 237.
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).
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.
doi: 10.1080/17442508708833480. |
[9] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).
doi: 10.1017/CBO9780511666223. |
[10] |
G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems,, London Mathematical Society Lecture Note Series, (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.
doi: 10.1103/RevModPhys.73.913. |
[12] |
B. Ferrario, Ergodic results for stochastic Navier-Stokes equation,, Stochastics Stochastics Rep., 60 (1997), 271.
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.
doi: 10.1007/BF02505916. |
[14] |
B. Ferrario, Uniqueness result for the 2D Navier-Stokes equation with additive noise,, Stoch. Stoch. Rep., 75 (2003), 435.
doi: 10.1080/10451120310001644485. |
[15] |
F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations,, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 403.
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.
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.
doi: 10.1007/BF02104513. |
[18] |
U. Frisch, Turbulence,, Cambridge University Press, (1995).
|
[19] |
K. Gawędzki and A. Kupiainen, Anomalous Scaling of the Passive Scalar,, Phys. Rev. Lett., 75 (1995), 3834. Google Scholar |
[20] |
M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing,, Ann. of Math., 164 (2006), 993.
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.
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.
doi: 10.1080/17442509308833854. |
[24] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979).
|
[25] |
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1983).
|
[26] |
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics,, Mathematics and its Applications, (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, (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).
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).
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.
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.
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.
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, 77 (1996), 237.
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).
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.
doi: 10.1080/17442508708833480. |
[9] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).
doi: 10.1017/CBO9780511666223. |
[10] |
G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems,, London Mathematical Society Lecture Note Series, (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.
doi: 10.1103/RevModPhys.73.913. |
[12] |
B. Ferrario, Ergodic results for stochastic Navier-Stokes equation,, Stochastics Stochastics Rep., 60 (1997), 271.
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.
doi: 10.1007/BF02505916. |
[14] |
B. Ferrario, Uniqueness result for the 2D Navier-Stokes equation with additive noise,, Stoch. Stoch. Rep., 75 (2003), 435.
doi: 10.1080/10451120310001644485. |
[15] |
F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations,, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 403.
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.
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.
doi: 10.1007/BF02104513. |
[18] |
U. Frisch, Turbulence,, Cambridge University Press, (1995).
|
[19] |
K. Gawędzki and A. Kupiainen, Anomalous Scaling of the Passive Scalar,, Phys. Rev. Lett., 75 (1995), 3834. Google Scholar |
[20] |
M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing,, Ann. of Math., 164 (2006), 993.
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.
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.
doi: 10.1080/17442509308833854. |
[24] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979).
|
[25] |
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1983).
|
[26] |
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics,, Mathematics and its Applications, (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, (1995).
|
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