Table of contents

CONTENTS Introduction § 1. Classical statistical physics on a two-dimensional lattice and quantum mechanics on a chain § 2. Connection with the inverse problem method § 3. The six-vertex model § 4. Generating vectors and permutation relations § 5. The general Bethe Ansatz § 6. Integral equations Conclusion Appendix 1 Appendix 2 References

CONTENTS Introduction Chapter I. G-convergence of operators § 1. G-convergence of abstract operators § 2. G-convergence of elliptic operators of higher order. The N-condition. Auxiliary propositions § 3. Basic properties of G-convergence of elliptic operators § 4. Some examples § 5. Further results on the G-convergence of elliptic operators Chapter II. The averaging of elliptic operators § 1. Auxiliary results. Statements of the main theorems on averaging § 2. Analysis of the equations for the functions N_γ § 3. Proofs of the main theorems on averaging of elliptic operators Chapter III. Some applications § 1. The asymptotic behaviour at infinity of solutions of second-order elliptic equations § 2. The asymptotic behaviour of fundamental solutions of second-order parabolic equations. Stabilization References

CONTENTS Introduction Chapter I. Space-time statistical solutions of the Navier-Stokes system in a bounded domain § 1. The Navier-Stokes system. Function spaces § 2. Space-time statistical solutions § 3. Galerkin approximations of a statistical solution § 4. Weak compactness of the statistical solutions of the Galerkin equations § 5. P is a statistical solution. Properties of P § 6. A uniqueness theorem for statistical solutions Chapter II. Statistical solutions of the stochastic Navier-Stokes system in a bounded domain § 1. The stochastic Navier-Stokes system. Function spaces § 2. Statistical solutions of the stochastic Navier-Stokes system § 3. Galerkin approximations of a statistical solution § 4. The construction of a statistical solution Chapter III. The Hopf equation and the direct Kolmogorov equation § 1. The Hopf equation § 2. The direct Kolmogorov equation Chapter IV. Stationary solutions of the stochastic Navier-Stokes system § 1. Stationary statistical solutions of a stochastic system § 2. The construction of a stationary statistical solution § 3. On the mean speed of dissipation of energy Chapter V. Moments of statistical solutions of the Navier-Stokes system § 1. Moments of a spatial statistical solution. The chain of moment equations § 2. Moment theory in the case of small Reynolds numbers Chapter VI. Translation-homogeneous statistical solutions § 1. Homogeneous measures. Mean density of energy. Examples § 2. Function spaces § 3. Homogeneous statistical solutions of the Navier-Stokes system § 4. On individual solutions of the Cauchy problem with infinite energy § 5. Periodic finite-dimensional approximations of the stochastic Cauchy problem § 6. An estimate of the time derivative § 7. Passage to the limit § 8. On a question of Kolmogorov References