definition 10.83 Weak solution

open in the book · parts/02-mathematical-methods/08-pdes.tex:2241 · p. 364

ground object -- no derivation owed

Rests on

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 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 equation 10.2 eq:pde-multi-index ¶

Supports

depends_on definition 10.87 Weak form of an elliptic problem ¶
- depends_on corollary 10.89 Existence for the elliptic problem ¶
- depends_on proposition 10.91 Dirichlet principle ¶
- depends_on theorem 10.88 Lax–Milgram ¶
  - depends_on corollary 10.89 Existence for the elliptic problem ¶ ↺
  - depends_on proposition 10.91 Dirichlet principle ¶ ↺
depends_on definition 10.85 Sobolev space ¶
- depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
  - depends_on definition A.462 The weighted space and the energy space ¶
    - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
      - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
  - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
    - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
      - depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
  - depends_on lemma A.452 Young and Hölder ¶
    - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on theorem A.490 Equicontinuity of the normalised class ¶
    - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶ ↺
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶ ↺
- depends_on definition 10.87 Weak form of an elliptic problem ¶ ↺
depends_on definition 10.95 Weak solution of a conservation law ¶
- 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.156 The Kruzhkov inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - 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 ¶ ↺
depends_on proposition 10.84 Consistency ¶

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`	→	Schwartz space and tempered distributions	declared	parts/02-mathematical-methods/08-pdes.tex:2256
`depends_on`	→	Principal part and principal symbol	declared	parts/02-mathematical-methods/08-pdes.tex:2256
`depends_on`	←	Weak form of an elliptic problem	declared	parts/02-mathematical-methods/08-pdes.tex:2329
`depends_on`	←	Sobolev space	declared	parts/02-mathematical-methods/08-pdes.tex:2295
`depends_on`	←	Weak solution of a conservation law	declared	parts/02-mathematical-methods/08-pdes.tex:2512
`depends_on`	←	Consistency	declared	parts/02-mathematical-methods/08-pdes.tex:2262