Randomly Forced Nonlinear Pdes and Statistical Hydrodynamics in 2 Space Dimensions

2 The deterministic 2D Navier-Stokes Equation 10 2.1 Leray decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Properties of the nonlinearity B . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 The existence and uniqueness theorem . . . . . . . . . . . . . . . . . . . . 15 2.4 Improving the smoothness of solutions . . . . . . . . . . . . . . . . . . . . 19 2.5 The NS semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Singular forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 Some hydrodynamical terminology . . . . . . . . . . . . . . . . . . . . . . 26

[1]  R. Temam.  Navier-Stokes Equations , 1977 .

[2]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[3]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[4]  Jonathan C. Mattingly,et al.  Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing , 2004, math/0406087.

[5]  C. Doering,et al.  Applied analysis of the Navier-Stokes equations: Index , 1995 .

[6]  V. Arnold,et al.  Topological methods in hydrodynamics , 1998 .

[7]  Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise , 2000, nlin/0009028.

[8]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[9]  E Weinan,et al.  Invariant measures for Burgers equation with stochastic forcing , 2000, math/0005306.

[10]  R. M. Dudley,et al.  Real Analysis and Probability , 1989 .

[11]  Peter Constantin,et al.  Navier-Stokes equations and area of interfaces , 1990 .

[12]  Benedetta Ferrario,et al.  Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure , 1999 .

[13]  Franco Flandoli,et al.  Ergodicity of the 2-D Navier-Stokes equation under random perturbations , 1995 .

[14]  A. Shirikyan.  Analyticity of solutions for randomly forced two-dimensional Navier-Stokes equations , 2002 .

[15]  Armen Shirikyan,et al.  A Coupling Approach¶to Randomly Forced Nonlinear PDE's. I , 2001 .

[16]  Martin Hairer,et al.  Exponential mixing properties of stochastic PDEs through asymptotic coupling , 2001, math/0109115.

[17]  G. Batchelor,et al.  The theory of homogeneous turbulence , 1954 .

[18]  Meinhard E. Mayer,et al.  Navier-Stokes Equations and Turbulence , 2008 .

[19]  David Ruelle,et al.  Statistical mechanics of a one-dimensional lattice gas , 1968 .

[20]  I. Gyöngy,et al.  On the splitting-up method and stochastic partial differential equations , 2003 .

[21]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[22]  I. Chavel.  Eigenvalues in Riemannian geometry , 1984 .

[23]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[24]  Franco Flandoli,et al.  Dissipativity and invariant measures for stochastic Navier-Stokes equations , 1994 .

[25]  G. Gallavotti,et al.  Foundations of Fluid Dynamics , 2002 .

[26]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

[27]  Y. Kifer.  Ergodic theory of random transformations , 1986 .

[28]  L. Arnold.  Random Dynamical Systems , 2003 .

[29]  R. Durrett.  Probability: Theory and Examples , 1993 .

[30]  R. Bowen.  Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[31]  Jonathan C. Mattingly.  Ergodicity of 2D Navier–Stokes Equations with¶Random Forcing and Large Viscosity , 1999 .