Mathematics of open fluid systems
Intro -- Preface -- Acknowledgements -- Contents -- Preliminaries, Notation -- 0.1 Vectors, Tensors, Sets -- 0.2 Relations -- 0.3 Differential Operators -- 0.4 Special Functions -- 0.5 State Variables in Fluid Mechanics -- 0.6 Geometry of Spatial Domains -- 0.6.1 Weakly Lipschitz Domains -- 0.6.2 Di...
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| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
Cham
Birkhäuser
2022
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| Schriftenreihe: | Nec̆as Center Series
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| Online Zugang: | zbMATH |
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Tag hinzufügen | Zusammenfassung: | Intro -- Preface -- Acknowledgements -- Contents -- Preliminaries, Notation -- 0.1 Vectors, Tensors, Sets -- 0.2 Relations -- 0.3 Differential Operators -- 0.4 Special Functions -- 0.5 State Variables in Fluid Mechanics -- 0.6 Geometry of Spatial Domains -- 0.6.1 Weakly Lipschitz Domains -- 0.6.2 Distance Function, Nearest Point -- 0.7 Function Spaces -- 0.7.1 Spaces of Continuous Functions -- 0.7.2 Spaces of Integrable Vector-Valued Functions -- 0.7.3 Sobolev Spaces -- 0.7.4 Measures -- 0.8 Inverse of Div-Operator -- 0.9 By Parts Integration for Vector-Valued Functions -- 0.10 Compensated Compactness -- 0.10.1 Div-Curl Lemma -- 0.10.2 Commutators Involving Riesz Operator -- 0.10.3 Commutator Lemma -- 0.11 Measures on Infinite-Dimensional Spaces -- 0.12 Miscellaneous Results Used in the Text -- 0.12.1 Korn-Poincaré Inequality -- 0.12.2 Lions-Aubin Compactness Argument -- Part I Modelling -- 1 Mathematical Models of Fluids in ContinuumMechanics -- 1.1 Conservation/Balance Laws -- 1.1.1 Balance Laws of Continuum Fluid Dynamics -- 1.1.2 Constitutive Relations -- 1.1.2.1 Thermodynamics -- 1.1.2.2 Transport Coefficients -- 1.2 Navier-Stokes-Fourier System -- 1.3 Thermodynamic Stability -- 1.4 Concluding Remarks -- 2 Open vs. Closed Systems -- 2.1 Closed and Isolated Systems -- 2.2 Open Systems -- 2.3 Global Form of Conservation Laws -- 2.3.1 Total Mass Balance -- 2.3.2 Total Energy Balance -- 2.4 Concluding Remarks -- Part II Analysis -- 3 Generalized Solutions -- 3.1 Relative Energy -- 3.1.1 Relative Energy as Bregman Distance -- 3.2 Energy Balance Equations -- 3.2.1 Kinetic Energy Balance -- 3.2.2 Internal and Total Energy Balance -- 3.2.2.1 Barotropic Fluids -- 3.3 Weak Formulation -- 3.3.1 Equation of Continuity -- 3.3.2 Momentum Equation -- 3.3.3 Entropy Balance -- 3.3.4 Total Energy Balance -- 3.3.4.1 Energy Balance for the Barotropic System. The goal of this monograph is to develop a mathematical theory of open fluid systems in the framework of continuum thermodynamics. Part I discusses the difference between open and closed fluid systems and introduces the Navier-Stokes-Fourier system as the mathematical model of a fluid in motion that will be used throughout the text. A class of generalized solutions to the Navier-Stokes-Fourier system is considered in Part II in order to show existence of global-in-time solutions for any finite energy initial data, as well as to establish the weak-strong uniqueness principle. Finally, Part III addresses questions of asymptotic compactness and global boundedness of trajectories and briefly considers the statistical theory of turbulence and the validity of the ergodic hypothesis. |
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| Beschreibung: | Literaturverzeichnis: Seite 277-282 |
| Beschreibung: | xxvii, 284 Seiten Illustrationen, Diagramme |
| ISBN: | 9783030947927 978-3-030-94792-7 |