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Contents
Contents
Preface
Introduction
1.1 Conservation and constitutive laws . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Constitutive relations for the Cauchy stress tensor . . . . 7
1.1.2 Constitutive relations for the heat flux vector . . .. . . . . . 11
1.2 The (p−q) coupled fluid-energy systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 The (p−q) coupled fluid-energy system. The stationary case . . . . . . . . 12
1.2.2 The (p−q) coupled fluid-energy system. The non-stationary case . . . . . 14
1.3 Nonstandard boundary conditions . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Slip friction boundary conditions . . . . . .. . . . . . . . . . . . . 16
1.3.2 Convective/radiative heat-transfer boundary conditions . 18
Preliminaries of functional analysis
2.1 Sobolev spaces . . . . . . . . . . . . . . . . . . . .. .. . .. 21
2.1.1 Notations and preliminaries. . . . . . . . . . 21
2.1.2 Trace operator. . . . . . . . . . . . . . . . . . . . 25
2.2 Green formula. . . . . . . . . . . . . . . . . .. . . . . . . . . 28
2.3 Bochner spaces. . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Theorems of existence. . . . . . . .. . . . . . . . . . . 31
2.5 Duality theory on convex analysis. . . . .. . . . . . 37
2.6 Fixed points for multivalued functions. . . . . . . . 41
2.7 Additional mathematical tools. . . . . . . . .. . . . . . 42
2.7.1 Friederichs mollifiers. . . . . . . . . . . . . . . 43
2.7.2 Nemytskii operators. . . . . . . . . . . . . . . 44
Stationary heat flows
3.1 Functional space framework and main result . . . . 46
3.2 Primal problem (P)w,ξ . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Dual problem (P∗)w,ξ . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Energy problem (Q)Ψ,ξ . . . . . . . . . . . . . . . . . . . . .. . 57
3.5 Existence of weak solutions . . . . . . . . . . . . . . . . . . 67
Non-stationary heat flows
4.1 Functional space framework and main results . . . .72
4.2 Primal problem (P)ξ . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Dual problem (P∗)ξ . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Energy problem (Q)Ψ,ξ . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Existence of weak solutions . . . . . . . . . . .. . . . . . . 98
Local boundary conditions
5.1 Functional space framework . . . . . . . . . . . . . . . . 102
5.2 The stationary case . . . . . . . . . . . . . .. . . . . . . . . . 103
5.2.1 Auxiliary existence results . . . . . . . . . . . . 106
5.2.2 Proof of Theorem 5.2.1 . . .. . . . . . . . . . . . 116
5.2.3 Proof of Theorem 5.2.2 . . . .. . . . . . . . . . . 118
5.3 The non-stationary case . . . . . . . . . . . . . . . . .. . . . 119
5.3.1 Auxiliary existence results . . . . . . . . . . . . . 121
5.3.2 Existence of weak solutions . . . . . . .. . . . . 129
Non-standard boundary conditions
6.1 Weak formulation and main result . . . . . . .. . . . . . . 134
6.2 Existence of weak auxiliary solutions . . . . .. . . . . . 136
6.3 Existence of weak solutions . . . . . . . . . . . . . . . . . . 140
Conclusions
Appendix: Physical background
Bibliography
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Corrigendum
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