## Ergodicity of the finite dimensional approximation of the 3d navier–stokes equations forced by a degenerate (2002)

Citations: | 11 - 1 self |

### BibTeX

@TECHREPORT{Romito02ergodicityof,

author = {Marco Romito},

title = {Ergodicity of the finite dimensional approximation of the 3d navier–stokes equations forced by a degenerate},

institution = {},

year = {2002}

}

### OpenURL

### Abstract

Abstract. We prove ergodicity of the finite dimensional approximations of the three dimensional Navier-Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential. 1.

### Citations

68 |
Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes
- MEYN, TWEEDIE
- 1993
(Show Context)
Citation Context ...e irreducibility property implies also that the support of the invariant measure is the whole state space. The next theorem shows, by means of general techniques developed in Meyn and Tweedie [11], [=-=12]-=-, that thesnite approximation of Navier-Stokes equations has good dissipation properties, strong enough to ensure the exponential mixing of the dynamics given by the Markov process. In order to state ... |

50 |
On the support of diffusion processes with applications to the strong maximum principle
- Stroock, Varadhan
- 1972
(Show Context)
Citation Context ...s of the generated Lie algebra span U. A Markov semigroup is irreducible if it gives positive mass to any open set for each initial condition and each time. It is well known (see Stroock and Varadhan =-=[15]-=-) that irreducibility is true if the control problem (see equations (6.1)) associated to problem (2.1) is controllable. The last statement follows from Theorem 6.5. Finally, the irreducibility propert... |

44 |
Stability of markovian processes II: continuous-time processes and sampled chains
- Meyn, Tweedie
- 1993
(Show Context)
Citation Context ...ly, the irreducibility property implies also that the support of the invariant measure is the whole state space. The next theorem shows, by means of general techniques developed in Meyn and Tweedie [=-=11]-=-, [12], that thesnite approximation of Navier-Stokes equations has good dissipation properties, strong enough to ensure the exponential mixing of the dynamics given by the Markov process. In order to ... |

40 | Hairer: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise
- Eckmann, M
- 2001
(Show Context)
Citation Context ...eory for Markov chains and Markov processes collected and developed by Meyn and Tweedie (one can see their book [10]). Among many others we quote the papers by E and Mattingly [2], Eckmann and Hairer =-=[3-=-], Hairer [6], Rey-Bellet and Thomas [13], and some of the references therein. The paper is organised as follows. In thesrst section the main denitions are given, together with the statements of the m... |

39 |
Ergodicity for infinite-dimensional systems
- Prato, Zabczyk
- 1996
(Show Context)
Citation Context ...the existence of the invariant measure. The method is classical and based on the Krylov-Bogoliubov method (see for example Theorem 3.1.1 of Da Prato and Zabczyk 3∑ j=1 q s k,ij ∂ ∂s j k ,4 M. ROMITO =-=[1]-=-). The compactness follows by the following argument. Let ‖u‖2 = ∑ k∈˜ K |uk| 2 , then by Itô formula (using also the first property of Lemma 7.1), d‖u(t)‖ 2 = ∑( 2uk ⋅ Fk(u) + Tr(q T k ⋅ qk) ) dt + 2... |

33 | Exponential convergence to non-equilibrium stationary states in classical statistical mechanics
- Rey-Bellet, Thomas
(Show Context)
Citation Context ...sses collected and developed by Meyn and Tweedie (one can see their book [10]). Among many others we quote the papers by E and Mattingly [2], Eckmann and Hairer [3], Hairer [6], Rey-Bellet and Thomas =-=[13-=-], and some of the references therein. The paper is organised as follows. In thesrst section the main denitions are given, together with the statements of the main results and an outline of their proo... |

30 |
Markov Chains and Stochastic Stability. Communications and Control Engineering Series
- Meyn, Tweedie
- 1993
(Show Context)
Citation Context ... have used here, such us the hypoellipticity for degenerate diusions, or the general theory for Markov chains and Markov processes collected and developed by Meyn and Tweedie (one can see their book [=-=10]-=-). Among many others we quote the papers by E and Mattingly [2], Eckmann and Hairer [3], Hairer [6], Rey-Bellet and Thomas [13], and some of the references therein. The paper is organised as follows. ... |

23 |
Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation
- Weinan, Mattingly
(Show Context)
Citation Context ...ity is proved for thesnite dimensional approximations of the three-dimensional Navier-Stokes equations, driven by a random force. The same problem has been solved in two dimensions by E and Mattingly =-=[2]-=-. Such result can have a qualitative interest for the statistical behaviour of an incompressiblesuid. Indeed, if the Kolmogorov theory of turbulence is taken into account, one can believe that the cas... |

13 |
Geometric control theory, Cambridge
- Jurdjevic
- 1997
(Show Context)
Citation Context ...at the constant vectorselds of LS(F) span R n . Then A F (x; t) = R n for each x 2 R n and t > 0. Proof. From the previous theorem, AF (x; t) = R n . Moreover, by Theorem 13, Chapter 3 of Jurdjevic [=-=-=-8] (see also the remarks after Theorem 11, Chapter 5 of [8]) it follows that also A F (x; t) = R n . In the following, we will need the following two lemmata. Thesrst lemma permits the enlargement o... |

12 |
Some applications of stochastic calculus to partial differential equations. Eleventh Saint Flour probability summer school—1981 (Saint Flour
- Stroock
- 1981
(Show Context)
Citation Context ...the dynamics (2.1) is strongly Feller and irreducible. A Markov semigroup P t is strongly Feller if P t ' is bounded continuous in time and space when ' is bounded measurable. By a theorem of Stroock =-=[14-=-], the transition semigroup is strongly Feller if the Hormander condition holds: the Lie algebra generated by the vectorselds in (2.3), evaluated at each point, is the state space U . Since N is a det... |

7 |
Basic Algebra I, Second Edition
- Jacobson
- 1985
(Show Context)
Citation Context ...ebraic system of generators for the group (Z 3 ; +) and N KN , then N is a determining set of indices for the ultraviolet cut-o N . 12 M. ROMITO For the sake of completeness, we give (see Jacobson [7=-=-=-], Theorem 3.8 and Theorem 3.9) a necessary and sucient condition for a set of indices N to be a system of generators of the whole group Z 3 . Theorem 5.1. A set N Z 3 is a system of generators of Z ... |

5 |
Irreducibility of the 3-D stochastic Navier-Stokes equation
- Flandoli
- 1997
(Show Context)
Citation Context ...visited with positive probability at each time. For this aim we study the associated control problem (see Section 6). Irreducibility for the innite dimensional equations wassrstly proved by Flandoli [=-=4-=-], under the assumption that the noise acts on all modes. 2000 Mathematics Subject Classication. Primary 76D05; Secondary 35Q30, 76M35, 76F55. Key words and phrases. Navier-Stokes equations, invariant... |

5 |
Polynomial control systems
- JURDJEVIC, KUPKA
- 1985
(Show Context)
Citation Context ... thesnite dimensional approximations of Navier-Stokes equations. Thesrst part contains some generalities on polynomial control systems. The approach and the results are taken from Jurdjevic and Kupka =-=[9]-=-. In the second part we adapt the proof of a theorem (again of Jurdjevic and Kupka [9]) ERGODICITY OF THE FINITE DIMENSIONAL... 13 to our case. The original theorem applied to polynomials of odd degre... |

5 |
On the support of diusion processes with applications to the strong maximum principle
- Stroock, Varadhan
- 1972
(Show Context)
Citation Context ... of the generated Lie algebra span U . A Markov semigroup is irreducible if it gives positive mass to any open set for each initial condition and each time. It is well known (see Stroock and Varadhan =-=[15]-=-) that irreducibility is true if the control problem (see equations (6.1)) associated to problem (2.1) is controllable. The last statement follows from Theorem 6.5. Finally, the irreducibility propert... |

3 |
Foundations of fluid dynamics. Translated from the Italian. Texts and Monographs in Physics
- Gallavotti
- 2002
(Show Context)
Citation Context ...project the equations in the space of divergence-free vector fields, so that the pressure disappears. We obtain the following infinite system of stochastic differential equations (see also Gallavotti =-=[5]-=-, Chapter 2, where the author gives also a6 M. ROMITO interpretation of the physics of the fluid in terms of the Fourier coordinates) duk = [ −ν|k| 2 uk −i ∑ (k ⋅ uh) ( ul − uk ⋅ k = 0 h,l∈Z 3 h+l=k ... |

1 |
Ergodicity for in systems
- Prato, Zabczyk
- 1996
(Show Context)
Citation Context ...et. Proof. First, we prove the existence of the invariant measure. The method is classical and based on the Krylov-Bogoliubov method (see for example Theorem 3.1.1 of Da Prato and Zabczyk 4 M. ROMITO =-=[-=-1]). The compactness follows by the following argument. Let kuk 2 = P k2 e K ju k j 2 , then by It^o formula (using also thesrst property of Lemma 7.1), dku(t)k 2 = X k2 e K 2u k F k (u) + Tr(q T k ... |

1 |
Foundations of dynamics, translated from the Italian. Texts and Monographs in Physics
- Gallavotti
- 2002
(Show Context)
Citation Context ... we project the equations in the space of divergence-free vectorselds, so that the pressure disappears. We obtain the following innite system of stochastic dierential equations (see also Gallavotti [5], Chapter 2, where the author gives also a 6 M. ROMITO interpretation of the physics of thesuid in terms of the Fourier coordinates) du k = jkj 2 u k i X h;l2Z 3 h+l=k (k u h ) u l k u l jkj 2 ... |

1 |
Exponential mixing for a stochastic partial dierential equation driven by degenerate noise, Nonlinearity 15
- Hairer
(Show Context)
Citation Context ...kov chains and Markov processes collected and developed by Meyn and Tweedie (one can see their book [10]). Among many others we quote the papers by E and Mattingly [2], Eckmann and Hairer [3], Hairer =-=[6-=-], Rey-Bellet and Thomas [13], and some of the references therein. The paper is organised as follows. In thesrst section the main denitions are given, together with the statements of the main results ... |

1 |
Exponential mixing for a stochastic partial differential equation driven by degenerate noise, Nonlinearity 15
- Hairer
(Show Context)
Citation Context ...kov chains and Markov processes collected and developed by Meyn and Tweedie (one can see their book [10]). Among many others we quote the papers by E and Mattingly [2], Eckmann and Hairer [3], Hairer =-=[6]-=-, Rey-Bellet and Thomas [13], and some of the references therein. The paper is organised as follows. In the first section the main definitions are given, together with the statements of the main resul... |