Mathematics of two-dimensional turbulence /:
"This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier-Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t, x) that physici...
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| Hauptverfasser: | , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [England] ; New York :
Cambridge University Press,
2012.
|
| Schriftenreihe: | Cambridge tracts in mathematics ;
194. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | "This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier-Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t, x) that physicists assume in their work. They rigorously prove that u(t, x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t, x) - proving, in particular, that observables f(u(t, .)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces"-- "This book deals with basic problems and questions, interesting for physicists and engineers working in the theory of turbulence. Accordingly Chapters 3-5 (which form the main part of this book) end with sections, where we explain the physical relevance of the obtained results. These sections also provide brief summaries of the corresponding chapters. In Chapters 3 and 4, our main goal is to justify, for the 2D case, the statistical properties of fluid's velocity"-- |
| Beschreibung: | 1 online resource |
| Bibliographie: | Includes bibliographical references (pages 307-318) and index. |
| ISBN: | 9781139569194 1139569198 9781139137119 1139137115 9781139571005 1139571001 9781139572750 113957275X 1139888986 9781139888981 1139579576 9781139579575 1139573527 9781139573528 1139570099 9781139570091 |
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| 100 | 1 | |a Kuksin, Sergej B., |d 1955- |e author. |1 https://id.oclc.org/worldcat/entity/E39PBJvRQy7rrXtWBtrpHPbTpP |0 http://id.loc.gov/authorities/names/n93089555 | |
| 245 | 1 | 0 | |a Mathematics of two-dimensional turbulence / |c Sergei Kuksin, Armen Shirikyan. |
| 260 | |a Cambridge [England] ; |a New York : |b Cambridge University Press, |c 2012. | ||
| 300 | |a 1 online resource | ||
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| 490 | 1 | |a Cambridge tracts in mathematics ; |v 194 | |
| 520 | |a "This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier-Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t, x) that physicists assume in their work. They rigorously prove that u(t, x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t, x) - proving, in particular, that observables f(u(t, .)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces"-- |c Provided by publisher | ||
| 520 | |a "This book deals with basic problems and questions, interesting for physicists and engineers working in the theory of turbulence. Accordingly Chapters 3-5 (which form the main part of this book) end with sections, where we explain the physical relevance of the obtained results. These sections also provide brief summaries of the corresponding chapters. In Chapters 3 and 4, our main goal is to justify, for the 2D case, the statistical properties of fluid's velocity"-- |c Provided by publisher | ||
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a 1. Preliminaries -- 2. Two-dimensional Navier-Stokes equations -- 3. Uniqueness of stationary measure and mixing -- 4. Ergodicity and limiting theorems -- 5. Inviscid limit -- 6. Miscellanies -- 7. Appendix -- 8. Solutions to some exercises. | |
| 504 | |a Includes bibliographical references (pages 307-318) and index. | ||
| 546 | |a English. | ||
| 650 | 0 | |a Hydrodynamics |x Statistical methods. | |
| 650 | 0 | |a Turbulence |x Mathematics. | |
| 650 | 4 | |a Hydrodynamics |x Statistical methods. | |
| 650 | 4 | |a Turbulence |x Mathematics. | |
| 650 | 6 | |a Hydrodynamique |x Méthodes statistiques. | |
| 650 | 6 | |a Turbulence |x Mathématiques. | |
| 650 | 7 | |a MATHEMATICS |x Probability & Statistics |x General. |2 bisacsh | |
| 650 | 7 | |a SCIENCE |x Mechanics |x Fluids. |2 bisacsh | |
| 650 | 7 | |a Hidrodinámica |2 embne | |
| 650 | 7 | |a Hydrodynamics |x Statistical methods |2 fast | |
| 650 | 7 | |a Turbulence |x Mathematics |2 fast | |
| 700 | 1 | |a Shirikyan, Armen, |e author. | |
| 758 | |i has work: |a Mathematics of two-dimensional turbulence (Text) |1 https://id.oclc.org/worldcat/entity/E39PCGKJ6VFjq9yF36F9K9WyMP |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Kuksin, Sergej B., 1955- |t Mathematics of two-dimensional turbulence. |d Cambridge, [England] ; New York : Cambridge University Press, 2012 |z 9781107022829 |w (DLC) 2012024345 |w (OCoLC)793221740 |
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| 880 | 8 | |6 505-00/(S |a 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. | |
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| author | Kuksin, Sergej B., 1955- Shirikyan, Armen |
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| author_facet | Kuksin, Sergej B., 1955- Shirikyan, Armen |
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| contents | 1. Preliminaries -- 2. Two-dimensional Navier-Stokes equations -- 3. Uniqueness of stationary measure and mixing -- 4. Ergodicity and limiting theorems -- 5. Inviscid limit -- 6. Miscellanies -- 7. Appendix -- 8. Solutions to some exercises. |
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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.</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">Askews and Holts Library Services</subfield><subfield code="b">ASKH</subfield><subfield code="n">AH34205805</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">Askews and Holts Library Services</subfield><subfield code="b">ASKH</subfield><subfield code="n">AH37563276</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">Askews and Holts Library Services</subfield><subfield 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| id | ZDB-4-EBA-ocn812481755 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:24:59Z |
| institution | BVB |
| isbn | 9781139569194 1139569198 9781139137119 1139137115 9781139571005 1139571001 9781139572750 113957275X 1139888986 9781139888981 1139579576 9781139579575 1139573527 9781139573528 1139570099 9781139570091 |
| language | English |
| oclc_num | 812481755 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource |
| psigel | ZDB-4-EBA |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | Cambridge tracts in mathematics ; |
| series2 | Cambridge tracts in mathematics ; |
| spelling | Kuksin, Sergej B., 1955- author. https://id.oclc.org/worldcat/entity/E39PBJvRQy7rrXtWBtrpHPbTpP http://id.loc.gov/authorities/names/n93089555 Mathematics of two-dimensional turbulence / Sergei Kuksin, Armen Shirikyan. Cambridge [England] ; New York : Cambridge University Press, 2012. 1 online resource text txt rdacontent computer c rdamedia online resource cr rdacarrier data file Cambridge tracts in mathematics ; 194 "This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier-Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t, x) that physicists assume in their work. They rigorously prove that u(t, x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t, x) - proving, in particular, that observables f(u(t, .)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces"-- Provided by publisher "This book deals with basic problems and questions, interesting for physicists and engineers working in the theory of turbulence. Accordingly Chapters 3-5 (which form the main part of this book) end with sections, where we explain the physical relevance of the obtained results. These sections also provide brief summaries of the corresponding chapters. In Chapters 3 and 4, our main goal is to justify, for the 2D case, the statistical properties of fluid's velocity"-- Provided by publisher Print version record. 1. Preliminaries -- 2. Two-dimensional Navier-Stokes equations -- 3. Uniqueness of stationary measure and mixing -- 4. Ergodicity and limiting theorems -- 5. Inviscid limit -- 6. Miscellanies -- 7. Appendix -- 8. Solutions to some exercises. Includes bibliographical references (pages 307-318) and index. English. Hydrodynamics Statistical methods. Turbulence Mathematics. Hydrodynamique Méthodes statistiques. Turbulence Mathématiques. MATHEMATICS Probability & Statistics General. bisacsh SCIENCE Mechanics Fluids. bisacsh Hidrodinámica embne Hydrodynamics Statistical methods fast Turbulence Mathematics fast Shirikyan, Armen, author. has work: Mathematics of two-dimensional turbulence (Text) https://id.oclc.org/worldcat/entity/E39PCGKJ6VFjq9yF36F9K9WyMP https://id.oclc.org/worldcat/ontology/hasWork Print version: Kuksin, Sergej B., 1955- Mathematics of two-dimensional turbulence. Cambridge, [England] ; New York : Cambridge University Press, 2012 9781107022829 (DLC) 2012024345 (OCoLC)793221740 Cambridge tracts in mathematics ; 194. http://id.loc.gov/authorities/names/n42005726 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=480318 Volltext 505-00/(S 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. |
| spellingShingle | Kuksin, Sergej B., 1955- Shirikyan, Armen Mathematics of two-dimensional turbulence / Cambridge tracts in mathematics ; 1. Preliminaries -- 2. Two-dimensional Navier-Stokes equations -- 3. Uniqueness of stationary measure and mixing -- 4. Ergodicity and limiting theorems -- 5. Inviscid limit -- 6. Miscellanies -- 7. Appendix -- 8. Solutions to some exercises. Hydrodynamics Statistical methods. Turbulence Mathematics. Hydrodynamique Méthodes statistiques. Turbulence Mathématiques. MATHEMATICS Probability & Statistics General. bisacsh SCIENCE Mechanics Fluids. bisacsh Hidrodinámica embne Hydrodynamics Statistical methods fast Turbulence Mathematics fast |
| title | Mathematics of two-dimensional turbulence / |
| title_auth | Mathematics of two-dimensional turbulence / |
| title_exact_search | Mathematics of two-dimensional turbulence / |
| title_full | Mathematics of two-dimensional turbulence / Sergei Kuksin, Armen Shirikyan. |
| title_fullStr | Mathematics of two-dimensional turbulence / Sergei Kuksin, Armen Shirikyan. |
| title_full_unstemmed | Mathematics of two-dimensional turbulence / Sergei Kuksin, Armen Shirikyan. |
| title_short | Mathematics of two-dimensional turbulence / |
| title_sort | mathematics of two dimensional turbulence |
| topic | Hydrodynamics Statistical methods. Turbulence Mathematics. Hydrodynamique Méthodes statistiques. Turbulence Mathématiques. MATHEMATICS Probability & Statistics General. bisacsh SCIENCE Mechanics Fluids. bisacsh Hidrodinámica embne Hydrodynamics Statistical methods fast Turbulence Mathematics fast |
| topic_facet | Hydrodynamics Statistical methods. Turbulence Mathematics. Hydrodynamique Méthodes statistiques. Turbulence Mathématiques. MATHEMATICS Probability & Statistics General. SCIENCE Mechanics Fluids. Hidrodinámica Hydrodynamics Statistical methods Turbulence Mathematics |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=480318 |
| work_keys_str_mv | AT kuksinsergejb mathematicsoftwodimensionalturbulence AT shirikyanarmen mathematicsoftwodimensionalturbulence |