Alphabeta MathDistributional systems and entropy flux pairs
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1 · Prerequisites
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- 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
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Distributions Test Functions and Differentiation
- 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
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- 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
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Polynomial Rings, the Division Algorithm and Roots
- 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
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Lebesgue Integral and the Convergence Theorems
- The Maximal Function and Lebesgue Differentiation
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Lemma
Locally integrable functions define distributions
Definition
Distributional systems and entropy flux pairs
Definition
Planar shocks and piecewise smooth traces
Theorem
Vector Rankine–Hugoniot condition
Proposition
Entropy jump residual
Definition
Strict hyperbolicity and Lax shock pattern
Theorem
Strictly convex scalar Riemann construction
Proposition
Entropy inequality under an assumed viscous limit
Sources
- Razvan Gelca, Functional Analysis, lecture notes, Chapter 7, Section 7.3
- Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd ed., Chapter III §3.2 and Chapter IV §§4.3–4.5
- Lawrence C. Evans, Entropy and Partial Differential Equations, Chapter V §§2–3
- Piotr Gwiazda, Martin Michálek, and Agnieszka Świerczewska-Gwiazda, A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation, §1
- Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd ed., Chapter III §3.1 and Chapter IV §4.3
- Lawrence C. Evans, Entropy and Partial Differential Equations, Chapter V §3
- Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd ed., Chapter III §3.2 and Chapter IV §4.5
- Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd ed., Chapter III §3.1 and Chapter VIII §8.3
- Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd ed., Chapters IV §§4.3–4.5 and Chapter VIII §8.3
- Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd ed., Chapter IV §4.6
- Lawrence C. Evans, Entropy and Partial Differential Equations, Chapter V §2