definition 10.99 Entropy condition

open in the book · parts/02-mathematical-methods/08-pdes.tex:2600 · p. 367

ground object -- no derivation owed

Rests on

depends_on definition 10.95 Weak solution of a conservation law ¶
- depends_on definition 10.93 Scalar conservation law ¶
  - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
    - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.2 Open set ¶
  - depends_on proposition 10.15 Method of characteristics, first order ¶
    - depends_on definition 10.2 Linear, semilinear, quasilinear ¶ ↺
    - depends_on proposition 7.31 Chain rule ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-15 ¶
    - proves proof ch:08-pdes@proof-5 ¶
- depends_on definition 10.83 Weak solution ¶
  - depends_on definition 17.38 Schwartz space and tempered distributions ¶
  - depends_on definition 10.3 Principal part and principal symbol ¶
    - depends_on definition 10.2 Linear, semilinear, quasilinear ¶ ↺
    - depends_on equation 10.2 eq:pde-multi-index ¶
depends_on theorem 10.96 Rankine–Hugoniot condition ¶
- depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
- depends_on theorem 7.131 Green ¶
  - depends_on definition 7.127 Simple regions ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    - depends_on definition 7.41 Antiderivative ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-24 ¶
    - proves proof ch:05-real-analysis@proof-25 ¶
  - proves proof ch:05-real-analysis@proof-77 ¶
- proves proof ch:08-pdes@proof-40 ¶

Supports

depends_on definition 10.101 Entropy pair; entropy inequality ¶
- depends_on definition A.150 Kruzhkov entropy pair ¶
  - depends_on lemma A.157 Two elementary identities ¶
    - depends_on proposition A.159 The doubled inequality ¶
      - depends_on proposition A.160 The cancellation ¶
    - depends_on proposition A.162 The limit $\delta\to0$ ¶
  - depends_on proposition A.156 The Kruzhkov inequality ¶
    - depends_on proposition A.159 The doubled inequality ¶ ↺
  - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - depends_on corollary A.152 Uniqueness ¶
- depends_on proposition A.156 The Kruzhkov inequality ¶ ↺
- depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
- depends_on proposition 10.102 Jump form of the entropy inequality ¶
- depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
- depends_on theorem 10.103 Uniqueness in the entropy class ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Weak solution of a conservation law	declared	parts/02-mathematical-methods/08-pdes.tex:2609
`depends_on`	→	Rankine–Hugoniot condition	declared	parts/02-mathematical-methods/08-pdes.tex:2609
`depends_on`	←	Entropy pair; entropy inequality	declared	parts/02-mathematical-methods/08-pdes.tex:2644