theorem 10.96 Rankine–Hugoniot condition

open in the book · parts/02-mathematical-methods/08-pdes.tex:2515 · p. 367

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 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 definition 6.2 Open set ¶ ↺
      - depends_on equation 3.51 eq:set-indexed ¶
- 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 axiom 7.1 Completeness of $\R$ ¶ ↺
  - 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 axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - 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 theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
  - 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 10.99 Entropy condition ¶
- 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.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 ¶
depends_on example 10.97 Burgers shock ¶
depends_on example 10.98 Weak solutions are not unique ¶
depends_on proposition 10.102 Jump form of the entropy inequality ¶ ↺

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:2523
`depends_on`	→	Green	declared	parts/02-mathematical-methods/08-pdes.tex:2523
`depends_on`	←	Entropy condition	declared	parts/02-mathematical-methods/08-pdes.tex:2609
`depends_on`	←	Burgers shock	declared	parts/02-mathematical-methods/08-pdes.tex:2571
`depends_on`	←	Weak solutions are not unique	declared	parts/02-mathematical-methods/08-pdes.tex:2597
`depends_on`	←	Jump form of the entropy inequality	declared	parts/02-mathematical-methods/08-pdes.tex:2668
`proves`	←	ch:08-pdes@proof-40	declared	parts/02-mathematical-methods/08-pdes.tex:2526