Alphabeta MathLocal smooth periodic Euler and Navier–Stokes
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hereditary and Productive Behaviour of the Separation Axioms
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Orthonormal Bases, Parseval and Fourier Series
- Outer Measure and the Caratheodory Extension Theorem
- Periodic fluid Fourier Sobolev spaces
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Sound-speed variables for polytropic Euler
Definition
Smooth bounded-wall and exterior solution domains
Theorem
Global smooth periodic linear acoustics
Definition
Flat periodic and infinite channel domains
Theorem
Global compatible no-slip shear evolution
Lemma
Continuation of bounded polynomial ODE solutions
Theorem
Local smooth periodic Euler and Navier–Stokes
Theorem
Local smooth positive-density polytropic Euler
Sources
- François Golse, The Boltzmann Equation and Its Hydrodynamic Limits, §7.4
- David Tong, Fluid Mechanics, Chapter 2, §§3.2.1–3.2.3
- David Tong, Fluid Mechanics, Chapter 3, §4.4.4
- François Golse, The Boltzmann Equation and Its Hydrodynamic Limits
- Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, §2.6, Lemma 2.14 and Corollary 2.15
- Brian F. Morton, Local-in-Time Well-Posedness of the Incompressible Euler and Navier–Stokes Equations, §§7.1–7.3
- Emil Wiedemann, Lecture Notes: Navier–Stokes Equations, §4.3
- François Golse, The Boltzmann Equation and Its Hydrodynamic Limits, §7.4, Theorems 7.8–7.9
- T. Sideris, Formation of Singularities in Three-Dimensional Compressible Fluids, §1