definition 10.95 Weak solution of a conservation law

open in the book · parts/02-mathematical-methods/08-pdes.tex:2501 · p. 366

ground object -- no derivation owed

Rests on

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 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- 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 definition 7.9 Real function ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-11 ¶
    - 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 ¶

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 definition 10.101 Entropy pair; entropy inequality ¶ ↺
depends_on example 10.98 Weak solutions are not unique ¶
depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
depends_on theorem 10.103 Uniqueness in the entropy class ¶ ↺
depends_on theorem 10.96 Rankine–Hugoniot condition ¶
- depends_on definition 10.99 Entropy condition ¶ ↺
- 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`	→	Scalar conservation law	declared	parts/02-mathematical-methods/08-pdes.tex:2512
`depends_on`	→	Weak solution	declared	parts/02-mathematical-methods/08-pdes.tex:2512
`depends_on`	←	Entropy 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
`depends_on`	←	Weak solutions are not unique	declared	parts/02-mathematical-methods/08-pdes.tex:2597
`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
`depends_on`	←	Rankine–Hugoniot condition	declared	parts/02-mathematical-methods/08-pdes.tex:2523