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050  4 QA911 .K85 2012 
082 0  532.50151 
100 1  Kuksin, Sergei 
245 10 Mathematics of Two-Dimensional Turbulence 
264  1 Cambridge :|bCambridge University Press,|c2012 
264  4 |c©2012 
300    1 online resource (338 pages) 
336    text|btxt|2rdacontent 
337    computer|bc|2rdamedia 
338    online resource|bcr|2rdacarrier 
490 1  Cambridge Tracts in Mathematics ;|vv.194 
505 0  Cover -- Mathematics of Two-Dimensional Turbulence -- 
       Title -- Copyright -- Dedication -- Contents -- Preface --
       Equations and forces -- What is in this book? -- Other 
       equations -- Readers of this book -- Acknowledgements -- 1
       Preliminaries -- 1.1 Function spaces -- 1.1.1 Functions of
       the space variables -- Lebesgue spaces -- Sobolev spaces -
       - 1.1.2 Functions of space and time variables -- 1.2 Basic
       facts from measure theory -- 1.2.1 s-algebras and measures
       -- 1.2.2 Convergence of integrals -- 1.2.3 Metrics on the 
       space of probabilities and convergence of measures -- 
       Total variation distance -- Dual-Lipschitz distance -- 
       Kantorovich distance -- 1.2.4 Couplings and maximal 
       couplings of probability measures -- 1.2.5 Kantorovich 
       functionals -- 1.3 Markov processes and random dynamical 
       systems -- 1.3.1 Markov processes -- 1.3.2 Random 
       dynamical systems -- 1.3.3 Markov RDS -- 1.3.4 Invariant 
       and stationary measures -- Notes and comments -- 2 Two-
       dimensional Navier-Stokes equations -- 2.1 Cauchy problem 
       for the deterministic system -- 2.1.1 Equations and 
       boundary conditions -- 2.1.2 Leray decomposition -- 2.1.3 
       Properties of some multilinear maps -- 2.1.4 Reduction to 
       an abstract evolution equation -- 2.1.5 Existence and 
       uniqueness of solution -- 2.1.6 Regularity of solutions --
       2.1.7 Navier-Stokes process -- 2.1.8 Foias-Prodi estimates
       -- 2.1.9 Some hydrodynamical terminology -- 2.2 Stochastic
       Navier-Stokes equations -- 2.3 Navier-Stokes equations 
       perturbed by a random kick force -- 2.3.1 Existence and 
       uniqueness of solution -- 2.3.2 Markov chain and RDS -- 
       2.3.3 Additional results higher Sobolev norms and time 
       averages -- 2.4 Navier-Stokes equations perturbed by 
       spatially regular white noise -- 2.4.1 Existence and 
       uniqueness of solution, and Markov process -- 2.4.2 
       Additional results: energy balance, higher Sobolev norms, 
       and time averages 
505 8  2.4.3 Universality of white-noise forces -- 2.4.4 RDS 
       associated with Navier-Stokes equations -- 2.5 Existence 
       of a stationary distribution -- 2.5.1 The Bogolyubov-
       Krylov argument -- 2.5.2 Application to Navier-Stokes 
       equations -- 2.6 Appendix: some technical proofs -- Notes 
       and comments -- 3 Uniqueness of stationary measure -- 
       Contraction of the space of measures -- Coupling argument 
       -- 3.1 Three results on uniqueness and mixing -- 3.1.1 
       Decay of a Kantorovich functional -- 3.1.2 Coupling method
       : uniqueness and mixing -- 3.1.3 Coupling method: 
       exponential mixing -- 3.2 Dissipative RDS with bounded 
       kicks -- 3.2.1 Main result -- 3.2.2 Coupling -- 3.2.3 
       Proof of Theorem -- 3.2.4 Application to Navier-Stokes 
       equations -- 3.3 Navier-Stokes system perturbed by white 
       noise -- 3.3.1 Main result and scheme of its proof -- 
       3.3.2 Recurrence: proof of Proposition -- 3.3.3 Stability:
       proof of Proposition -- 3.4 Navier-Stokes system with 
       unbounded kicks -- 3.4.1 Formulation of the result -- 
       3.4.2 Proof of Theorem -- 3.5 Further results and 
       generalisations -- 3.5.1 The Flandoli-Maslowski theorem --
       3.5.2 Exponential mixing for the Navier-Stokes system with
       white noise -- 3.5.3 Convergence for functionals on higher
       Sobolev spaces -- 3.5.4 Mixing for Navier-Stokes equations
       perturbed by a compound Poisson process -- 3.5.5 
       Description of some results on uniqueness and mixing for 
       other PDEs -- 3.5.6 An alternative proof of mixing for 
       kick force models -- 3.6 Appendix: some technical proofs -
       - 3.6.1 Proof of Lemma -- 3.6.2 Recurrence for Navier-
       Stokes equations with unbounded kicks -- 3.6.3 Exponential
       squeezing for Navier-Stokes equations with unbounded kicks
       -- 3.7 Relevance of the results for physics -- Notes and 
       comments -- 4 Ergodicity and limiting theorems -- 4.1 
       Ergodic theorems -- 4.1.1 Strong law of large numbers -- 
       4.1.2 Law of the iterated logarithm 
505 8  4.1.3 Central limit theorem -- 4.2 Random attractors and 
       stationary distributions -- 4.2.1 Random point attractors 
       -- 4.2.2 The Ledrappier-Le Jan-Crauel theorem -- 4.2.3 
       Ergodic RDS and minimal attractors -- 4.2.4 Application to
       the Navier-Stokes system -- 4.3 Dependence of stationary 
       measure on the random force -- 4.3.1 Regular dependence on
       parameters -- 4.3.2 Universality of white-noise 
       perturbations -- 4.4 Relevance of the results for physics 
       -- Notes and comments -- 5 Inviscid limit -- 5.1 Balance 
       relations -- 5.1.1 Energy and enstrophy -- 5.1.2 Balance 
       relations -- 5.1.3 Pointwise exponential estimates -- 5.2 
       Limiting measures -- 5.2.1 Existence of accumulation 
       points -- 5.2.2 Estimates for the densities of the energy 
       and enstrophy -- 5.2.3 Further properties of the limiting 
       measures -- 5.2.4 Other scalings -- 5.2.5 Kicked Navier-
       Stokes system -- 5.2.6 Inviscid limit for the complex 
       Ginzburg-Landau equation -- 5.3 Relevance of the results 
       for physics -- Notes and comments -- 6 Miscellanies -- 6.1
       3D Navier-Stokes system in thin domains -- 6.1.1 
       Preliminaries on the Cauchy problem -- 6.1.2 Large-time 
       asymptotics of solutions -- 6.1.3 The limit ε → 0 -- 
       Relevance of the results for physics -- 6.2 Ergodicity and
       Markov selection -- 6.2.1 Finite-dimensional stochastic 
       differential equations -- 6.2.2 The Da Prato-Debussche-
       Odasso theorem -- 6.2.3 The Flandoli-Romito theorem -- 6.3
       Navier-Stokes equations with very degenerate noise -- 
       6.3.1 2D Navier-Stokes equations: controllability and 
       mixing properties -- 6.3.2 3D Navier-Stokes equations with
       degenerate noise -- Appendix -- A.1 Monotone class theorem
       -- A.2 Standard measurable spaces -- A.3 Projection 
       theorem -- A.4 Gaussian random variables -- A.5 Weak 
       convergence of random measures -- A.6 The Gelfand triple 
       and Yosida approximation -- A.7 Itô formula in Hilbert 
       spaces 
505 8  A.8 Local time for continuous Itô processes -- A.9 
       Krylov's estimate -- A.10 Girsanov's theorem -- A.11 
       Martingales submartingales, and supermartingales -- Doob-
       Kolmogorov inequality -- Doob's moment inequality -- 
       Exponential supermartingale -- Doob's optional sampling 
       theorem -- Convergence theorem -- A.12 Limit theorems for 
       discrete-time martingales -- A.13 Martingale approximation
       for Markov processes -- A.14 Generalised Poincaré 
       inequality -- A.15 Functions in Sobolev spaces with a 
       discrete essential range -- Solutions to selected 
       exercises -- Notation and conventions -- Abstract spaces 
       and functions -- Functional spaces -- Measures and 
       applications -- References -- Index 
520    Presents recent progress in two-dimensional mathematical 
       hydrodynamics, including rigorous results on turbulence in
       space-periodic fluid flows 
588    Description based on publisher supplied metadata and other
       sources 
590    Electronic reproduction. Ann Arbor, Michigan : ProQuest 
       Ebook Central, 2020. Available via World Wide Web. Access 
       may be limited to ProQuest Ebook Central affiliated 
       libraries 
650  0 Hydrodynamics -- Statistical methods.;Turbulence -- 
       Mathematics 
655  4 Electronic books 
700 1  Shirikyan, Armen 
776 08 |iPrint version:|aKuksin, Sergei|tMathematics of Two-
       Dimensional Turbulence|dCambridge : Cambridge University 
       Press,c2012|z9781107022829 
830  0 Cambridge Tracts in Mathematics 
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