This book gives an account of recent achievements in the mathematical theory of two-dimensional turbulence, described by the 2D Navier-Stokes equation, perturbed by a random force. The main results presented here were obtained during the last five to ten years and, up to now, have been available only in papers in the primary literature. Their summary and synthesis here, beginning with some preliminaries on partial differential equations and stochastics, make this book a self-contained account that will appeal to readers with a general background in analysis. After laying the groundwork, the author goes on to recent results on ergodicity of random dynamical systems, which the randomly forced Navier-Stokes equation defines in the function space of divergence-free vector fields, including a Central Limit Theorem. The physical meaning of these results is discussed as well as their relations with the theory of attractors. Next, the author studies the behaviour of solutions when the viscosity goes to zero. In the final section these dynamical methods are used to derive the so-called balance relations--the infinitely many algebraical relations satisfied by the solutions. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Sergei B. Kuksin
Series: Zurich Lectures in Advanced Mathematics
Publisher: European Mathematical Society 6
Year: 2006
Language: English
Pages: 103
Contents......Page 5
0 Introduction......Page 7
1.1 Function spaces for functions of x......Page 11
1.2 Functions of t and x......Page 13
2.1 Leray decomposition......Page 15
2.2 Properties of the nonlinearity B......Page 18
2.3 The existence and uniqueness theorem......Page 20
2.4 Improving the smoothness of solutions......Page 24
2.5 The NS semigroup......Page 28
2.6 Singular forces......Page 29
2.7 Some hydrodynamical terminology......Page 32
3.1 Ingredients for the constructions......Page 34
3.2 The kicked NSE......Page 35
3.3 Stationary measures......Page 37
3.4 More estimates......Page 38
4.1 White in time forces......Page 40
4.2 The white-forced 2D NSE......Page 41
4.3 Estimates for solutions......Page 43
4.4 Stationary measures......Page 46
4.5 High-frequency random kicks......Page 47
5.1 Weak convergence of measures and Lipschitz-dual distance......Page 49
5.2 Variational distance......Page 50
5.3 Coupling......Page 51
5.4 Kantorovich functionals......Page 52
6.1 The main lemma......Page 53
6.2 Weak solution of (6.1)......Page 55
6.3 The theorem......Page 56
6.4 Corollaries from the theorem......Page 60
6.5 3D NSE with small random kicks......Page 61
6.6 Stationary measures and random attractors......Page 62
6.7 Appendix: Summary of the proof of Theorem 6.4......Page 63
7.1 The main theorem......Page 66
7.2 Stationary measures for equation, perturbed by high frequency kicks......Page 68
8 Ergodicity and the strong law of large numbers......Page 70
9 The martingale approximation and CLT......Page 73
10.1 White-forces, proportional to the square-root of the viscosity......Page 76
10.2 One negative result......Page 81
10.3 Other scalings......Page 83
10.4 Discussion......Page 84
10.5 Kicked equations......Page 85
11.1 The balance relations......Page 87
11.2 The co-area form of the balance relations......Page 90
12 Comments......Page 93
Bibliography......Page 98
Index......Page 103
