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Author Kuksin, Sergei
Title Mathematics of Two-Dimensional Turbulence
Imprint Cambridge : Cambridge University Press, 2012
©2012
book jacket
Descript 1 online resource (338 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Cambridge Tracts in Mathematics ; v.194
Cambridge Tracts in Mathematics
Note 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
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
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
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
Presents recent progress in two-dimensional mathematical hydrodynamics, including rigorous results on turbulence in space-periodic fluid flows
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Kuksin, Sergei Mathematics of Two-Dimensional Turbulence Cambridge : Cambridge University Press,c2012 9781107022829
Subject Hydrodynamics -- Statistical methods.;Turbulence -- Mathematics
Electronic books
Alt Author Shirikyan, Armen
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