Mathematics of Open Fluid Systems

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
3.3.5 Weak Solutions
3.4 Relative Energy Revisited
3.5 Weak–Strong Uniqueness
3.5.1 Estimates Based on the Momentum Balance
3.5.2 Estimating Pressure
3.5.3 Boundary Integrals
3.5.4 Weak–Strong Uniqueness, Conditional Result
3.6 Ballistic Energy
3.7 Concluding Remarks
4 Constitutive Theory and Weak–Strong Uniqueness Revisited
4.1 Constitutive Relations
4.1.1 Equation of State
4.1.2 Transport Coefficients
4.2 Weak–Strong Uniqueness Revisited
4.2.1 Essential vs. Residual Component
4.2.2 Temperature Gradient
4.2.3 Velocity Gradient
4.2.4 Weak–Strong Uniqueness, Unconditional Result
4.3 Hard Sphere model
4.4 Concluding Remarks
5 Existence Theory, Basic Approximation Scheme
5.1 Approximation Scheme
5.1.1 Regularization of the Constitutive Relations
5.1.2 Domain Approximation
5.1.3 Regularity of the Boundary Data
5.1.4 Galerkin Approximation
5.1.5 Approximate Equation of Continuity
5.1.6 Approximate Momentum Balance
5.1.7 Internal Energy Balance
5.2 Solvability of the Basic Level Approximate Problem
5.2.1 Linear Parabolic Problem
5.2.2 Quasilinear Parabolic Problem
5.2.3 Approximate Equation of Continuity
5.2.4 Approximate Internal Energy Balance
5.3 Existence of Basic Approximate Solutions
6 Vanishing Galerkin Limit and Domain Approximation
6.1 Weak Formulation, Entropy, and Total Energy Balance
6.1.1 Equation of Continuity
6.1.2 Kinetic Energy Balance
6.1.3 Total Energy Balance
6.1.4 Entropy Inequality
6.2 Uniform Bounds
6.3 Convergence n →∞
6.3.1 Approximate Equation of Continuity
6.3.2 Pointwise Convergence of the Temperature and the Limit in the Approximate Entropy Inequality
6.3.3 Momentum Equation
6.3.3.1 Total Energy Balance
6.4 Limit System
6.4.1 Density Renormalization
7 Vanishing Artificial Diffusion Limit
7.1 Uniform Bounds
7.1.1 Pressure Estimates
7.2 Convergence →0
7.2.1 Approximate Equation of Continuity
7.2.2 Pointwise Convergence of the Temperature
7.2.3 Pointwise Convergence of the Density
7.2.3.1 Lions Identity
7.2.3.2 Oscillation Defect
7.3 Limit System
8 Vanishing Artificial Pressure Limit
8.1 Vanishing Artificial Pressure—Real Gas EOS
8.1.1 Energy Estimates
8.1.2 Pressure Estimates
8.1.3 Positivity of the Absolute Temperature
8.1.4 Convergence δ→0
8.1.5 Limit in the Momentum Equation
8.1.6 Limit in the Energy Balance
8.1.7 Pointwise Convergence of the Temperature
8.1.8 Pointwise Convergence of the Density
8.2 Vanishing Artificial Pressure—Hard Sphere EOS
8.2.1 Energy Estimates
8.2.2 Pressure Estimates
8.2.3 Convergence δ→0
9 Existence Theory: Main Results
9.1 Physical Domain and Boundary/Initial Data
9.1.1 Boundary Data
9.1.2 Initial Data
9.1.3 Global Existence: Real Gas EOS
9.1.4 Global Existence—Hard Sphere Gas
9.2 Concluding Remarks
Part III Qualitative Properties
10 Long Time Behaviour
10.1 Asymptotic Compactness
10.1.1 Statement of the Result
10.1.2 Time Evolution of the Density Oscillation Defect
10.1.3 Strong Convergence of the Density
10.2 Dissipativity, Bounded Absorbing Sets
10.2.1 Preliminary Discussion
10.2.2 Bounded Absorbing Sets: Hard Sphere Gas
10.2.2.1 Extension of the Boundary Velocity
10.2.2.2 Uniform Bounds
10.3 ω-Limit Sets Generated by Bounded Trajectories
10.3.1 Trajectory Space
10.3.2 ω-Limit Sets
11 Statistical Solutions, Ergodic Hypothesis, andTurbulence
11.1 Bounded Invariant Sets
11.2 Krylov–Bogolyubov Approach
11.3 Application of Birkhoff–Khinchin Theorem
11.4 Concluding Remarks
12 Systems with Prescribed Boundary Temperature
12.1 Dirichlet Boundary Conditions for the Temperature
12.2 Weak Formulation
12.3 Relative Energy and Weak–Strong Uniqueness
12.3.1 Relative Energy
12.3.2 Relative Energy Inequality
12.3.3 Weak–Strong Uniqueness
12.4 Existence of Weak Solutions
12.4.1 Spatially Homogeneous Boundary Temperature
12.4.2 General Boundary Conditions
12.4.3 Conclusion
12.5 Concluding Remarks
Appendix
A.1 Skorokhod Spaces
A.1.1 The Space D([0,T]; R)
A.1.2 Extension to Unbounded Intervals
A.1.3 Extension to Infinite-Dimensional Spaces
B.1 Equation of Continuity, Renormalization
B.1.1 Renormalization
B.1.2 Weak Sequential Stability
B.1.3 Extension Outside Γ0in
B.1.4 Extension Outside Γ0out
B.1.5 Extension of the Class of Test Functions
References
Index