definition 10.101 Entropy pair; entropy inequality

open in the book · parts/02-mathematical-methods/08-pdes.tex:2628 · p. 368

ground object -- no derivation owed

Rests on

depends_on definition 10.99 Entropy condition ¶
- 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 proposition 10.15 Method of characteristics, first order ¶
      - depends_on definition 10.2 Linear, semilinear, quasilinear ¶ ↺
      - depends_on proposition 7.31 Chain rule ¶
      - 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 remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - 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 theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-77 ¶
  - proves proof ch:08-pdes@proof-40 ¶
depends_on definition 10.95 Weak solution of a conservation law ¶ ↺

Supports

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.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`	→	Entropy condition	declared	parts/02-mathematical-methods/08-pdes.tex:2644
`depends_on`	→	Weak solution of a conservation law	declared	parts/02-mathematical-methods/08-pdes.tex:2644
`depends_on`	←	Kruzhkov entropy pair	declared	appendices/A-long-proofs.tex:8093
`depends_on`	←	The Kruzhkov inequality	declared	appendices/A-long-proofs.tex:8241
`depends_on`	←	The viscous problem produces the entropy inequality	declared	appendices/A-long-proofs.tex:8677
`depends_on`	←	Jump form of the entropy inequality	declared	parts/02-mathematical-methods/08-pdes.tex:2668
`depends_on`	←	$L^{1}$ contraction on a cone	declared	appendices/A-long-proofs.tex:8114
`depends_on`	←	Uniqueness in the entropy class	declared	parts/02-mathematical-methods/08-pdes.tex:2720