example 10.98 Weak solutions are not unique

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 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 definition 6.5 Open cover ¶
    • 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 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 theorem 7.25 Heine–Cantor: uniform continuity ¶
        • 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 definition 7.26 Derivative of a function at a point ¶
      • 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 ¶

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