definition 10.2 Linear, semilinear, quasilinear

open in the book · parts/02-mathematical-methods/08-pdes.tex:75 · p. 339

ground object -- no derivation owed

Rests on

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 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - 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 definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
- depends_on definition 6.2 Open set ¶
  - depends_on definition 6.1 Topological space ¶
    - depends_on definition 3.31 Empty set ¶
    - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶

Supports

depends_on definition A.103 First-order quasilinear system with zero data ¶
- depends_on lemma A.104 The recursion ¶
  - depends_on lemma A.109 Domination ¶
    - depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶
  - depends_on lemma A.105 Formal reduction of a normal problem ¶
- depends_on proposition A.110 Explicit solution of the majorant problem ¶
  - depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶ ↺
depends_on definition 10.93 Scalar conservation law ¶
- 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 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 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 example 10.97 Burgers shock ¶ ↺
- depends_on proposition 10.94 Gradient catastrophe ¶
depends_on definition 10.3 Principal part and principal symbol ¶
- depends_on definition 10.10 Characteristic surface ¶
  - depends_on proposition 10.13 Canonical form in two variables ¶
    - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
    - depends_on theorem 10.55 d'Alembert's formula ¶
      - depends_on corollary 10.56 Domain of dependence and influence ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
  - depends_on proposition 10.11 What a characteristic surface is ¶
    - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
      - depends_on lemma A.105 Formal reduction of a normal problem ¶ ↺
    - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
- depends_on definition 10.4 The second-order operator ¶
  - depends_on definition 10.10 Characteristic surface ¶ ↺
  - depends_on definition 10.42 Green's function ¶
    - depends_on definition 10.50 Retarded and advanced Green's functions ¶
    - depends_on example 10.49 The method of images ¶
    - depends_on proposition 28.24 Impulse response of the damped oscillator ¶
    - depends_on proposition 10.47 Bilinear expansion ¶
    - depends_on proposition 10.43 Representation of the solution ¶
    - depends_on theorem 10.45 Symmetry of the Green's function ¶
  - depends_on definition 10.6 Type of a second-order operator ¶
    - depends_on definition 10.79 Parabolic boundary ¶
      - depends_on theorem 10.80 Parabolic maximum principle ¶
    - depends_on example 10.9 ex:pde-three-types ¶
    - depends_on proposition 10.25 The backward heat problem is ill posed ¶
    - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶ ↺
    - depends_on theorem 10.8 The type is a coordinate invariant ¶
      - depends_on lemma A.90 Affine normalization ¶
      - depends_on proposition 10.13 Canonical form in two variables ¶ ↺
      - depends_on proposition 10.11 What a characteristic surface is ¶ ↺
- depends_on definition 10.83 Weak solution ¶
  - 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.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on lemma A.452 Young and Hölder ¶
    - depends_on definition 10.87 Weak form of an elliptic problem ¶ ↺
  - depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
  - depends_on proposition 10.84 Consistency ¶
depends_on lemma 10.28 The separation constant ¶
- depends_on definition A.118 Simple separation of the Helmholtz equation ¶
  - depends_on theorem A.120 Stäckel and Robertson conditions ¶
    - depends_on example A.122 Cartesian ¶
    - depends_on example A.123 Circular cylindrical ¶
    - depends_on example A.124 Spherical ¶
    - depends_on proposition A.128 The ellipsoidal system separates ¶
      - depends_on proposition A.132 The count ¶
- depends_on example 10.31 Cartesian separation ¶
  - depends_on example A.122 Cartesian ¶ ↺
- depends_on example 10.32 Cylindrical separation ¶
  - depends_on example A.123 Circular cylindrical ¶ ↺
- depends_on example 10.33 Spherical separation ¶
  - depends_on example A.124 Spherical ¶ ↺
- depends_on proposition 23.9 Separation of the time ¶
  - depends_on proposition 23.10 The characteristic function is the abbreviated action ¶
    - depends_on remark 23.11 Whose principle, and where it fails ¶
  - depends_on remark 23.12 Two functions, two roles ¶
- depends_on proposition 10.29 Time separation reduces the canonical equations to Helmholtz ¶
  - depends_on example 10.38 The vibrating string ¶
  - depends_on phenomenon 28.53 Chladni's figures ¶
  - depends_on theorem 10.36 Solution of the heat problem on an interval ¶
- depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
  - depends_on remark 23.19 The same conditions govern the Helmholtz equation ¶
depends_on proposition 28.44 Superposition ¶
depends_on proposition 10.15 Method of characteristics, first order ¶
- depends_on definition 10.93 Scalar conservation law ¶ ↺
- depends_on proposition 10.16 Cauchy's characteristic strips ¶
- depends_on proposition 10.94 Gradient catastrophe ¶ ↺
- depends_on remark 31.90 Why a compression wave must steepen ¶
- depends_on remark 23.3 What Part II owes this chapter ¶
depends_on proposition 10.43 Representation of the solution ¶ ↺
depends_on theorem 10.22 Cauchy–Kovalevskaya ¶ ↺
depends_on theorem 10.52 Duhamel's principle ¶
- depends_on corollary 10.53 Duhamel for the wave equation ¶

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`	→	Partial differential equation; order	declared	parts/02-mathematical-methods/08-pdes.tex:90
`depends_on`	←	First-order quasilinear system with zero data	declared	appendices/A-long-proofs.tex:6135
`depends_on`	←	Scalar conservation law	declared	parts/02-mathematical-methods/08-pdes.tex:2460
`depends_on`	←	Principal part and principal symbol	declared	parts/02-mathematical-methods/08-pdes.tex:111
`depends_on`	←	The separation constant	declared	parts/02-mathematical-methods/08-pdes.tex:810
`depends_on`	←	Superposition	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1533
`depends_on`	←	Method of characteristics, first order	declared	parts/02-mathematical-methods/08-pdes.tex:435
`depends_on`	←	Representation of the solution	declared	parts/02-mathematical-methods/08-pdes.tex:1171
`depends_on`	←	Cauchy–Kovalevskaya	declared	parts/02-mathematical-methods/08-pdes.tex:627
`depends_on`	←	Duhamel's principle	declared	parts/02-mathematical-methods/08-pdes.tex:1397