proposition A.163 The viscous problem produces the entropy inequality

Rests on

• depends_on definition 10.101 Entropy pair; entropy inequality ¶
  • depends_on definition 10.99 Entropy condition ¶
    • 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 proposition 10.15 Method of characteristics, first order ¶
      • 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 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 remark 7.128 What the derivations below take as given ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • proves proof ch:05-real-analysis@proof-77 ¶
      • proves proof ch:08-pdes@proof-40 ¶
  • depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
• depends_on equation 10.81 eq:pde-conservation-law ¶
• depends_on theorem 10.80 Parabolic maximum principle ¶
  • depends_on definition 10.79 Parabolic boundary ¶
    • depends_on definition 10.6 Type of a second-order operator ¶
      • depends_on definition 10.4 The second-order operator ¶
        • depends_on definition 10.3 Principal part and principal symbol ¶ ↺
        • depends_on proposition 7.105 Clairaut–Schwarz ¶
      • depends_on theorem 5.116 Sylvester's law of inertia ¶
        • depends_on definition 5.110 Bilinear map ¶
        • depends_on definition 5.111 Symmetric and antisymmetric forms ¶
        • depends_on equation 5.139 eq:lin-bilinear-congruence ¶
        • proves proof ch:03-linear-algebra-representations@proof-53 ¶
    • depends_on definition 6.3 Closed 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 6.2 Open set ¶
        • depends_on definition 6.1 Topological space ¶
  • depends_on theorem 10.75 Weak maximum principle ¶
    • depends_on definition 10.66 Harmonic function ¶
      • 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 equation 10.24 eq:pde-poisson-equation ¶
    • depends_on theorem 6.11 Heine–Borel on $\R$ ¶
      • depends_on axiom 7.1 Completeness of $\R$ ¶
      • depends_on definition 6.9 Compact set ¶
        • depends_on definition 6.5 Open cover ¶
      • proves proof ch:04-topology@proof-2 ¶
    • proves proof ch:08-pdes@proof-30 ¶
  • proves proof ch:08-pdes@proof-33 ¶
• proves proof app:A-long-proofs@proof-98 ¶

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