0. Introduction ............................................... vii
1. Function spaces .............................................. 1
1.1. Function spaces for functions of x ..................... 1
1.2. Functions of t and x ................................... 3
2. The deterministic 2D Navier-Stokes Equation .................. 5
2.1. Leray decomposition .................................... 5
2.2. Properties of the nonlinearity B ....................... 8
2.3. The existence and uniqueness theorem .................. 10
2.4. Improving the smoothness of solutions ................. 14
2.5. The NS semigroup ...................................... 18
2.6. Singular forces ....................................... 19
2.7. Some hydrodynamical terminology ....................... 22
3. Random kick-forces .......................................... 24
3.1. Ingredients for the constructions ..................... 24
3.2. The kicked NSE ........................................ 25
3.3. Stationary measures ................................... 27
3.4. More estimates ........................................ 28
4. White-forced equations ...................................... 30
4.1. White in time forces .................................. 30
4.2. The white-forced 2D NSE ............................... 31
4.3. Estimates for solutions ............................... 33
4.4. Stationary measures ................................... 36
4.5. High-frequency random kicks ........................... 37
5. Preliminaries from measure theory ........................... 39
5.1. Weak convergence of measures and Lipschitz-dual
distance .............................................. 39
5.2. Variational distance .................................. 40
5.3. Coupling .............................................. 41
5.4. Kantorovich functionals ............................... 42
6. Uniqueness of a stationary measure: kick-forces ............. 43
6.1. The main lemma ........................................ 43
6.2. Weak solution of (6.1) ................................ 45
6.3. The theorem ........................................... 46
6.4. Corollaries from the theorem .......................... 50
6.5. 3D NSE with small random kicks ........................ 51
6.6. Stationary measures and random attractors ............. 52
6.7. Appendix: Summary of the proof of Theorem 6.4 ......... 53
7. Uniqueness of a stationary measure: white-forces ............ 56
7.1. The main theorem ...................................... 56
7.2. Stationary measures for equation, perturbed
by high frequency kicks ............................... 58
8. Ergodicity and the strong law of large numbers .............. 60
9. The martingale approximation and CLT ........................ 63
10.The Eulerian limit .......................................... 66
10.1. White-forces, proportional to the square-root
of the viscosity ...................................... 66
10.2. One negative result ................................... 71
10.3. Other scalings ........................................ 73
10.4. Discussion ............................................ 74
10.5. Kicked equations ...................................... 75
11.Balance relations for the white-forced NSE .................. 77
11.1. The balance relations ................................. 77
11.2. The co-area form of the balance relations ............. 80
12.Comments .................................................... 83
Bibliography ................................................... 88
Index .......................................................... 93
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