Preface ....................................................... vii
Acknowledgement ................................................ xi
1. Physical background .......................................... 1
1.1. Kinematics, description of motion ....................... 1
1.2. Balance laws ............................................ 3
1.3. Constitutive equations .................................. 6
1.4. Barotropic flows ....................................... 13
1.5. The Navier-Stokes system ............................... 15
1.6. Bibliographical notes .................................. 17
2. Mathematical preliminaries .................................. 20
2.1. Function spaces ........................................ 20
2.2. Weak convergence ....................................... 28
2.3. Vector functions of one real variable .................. 37
2.4. Bibliographical notes .................................. 39
3. A priori estimates .......................................... 40
3.1. Estimates based on the maximum principle ............... 42
3.2. Total mass conservation ................................ 44
3.3. Energy estimates ....................................... 44
3.4. Viscous dissipation .................................... 46
3.5. A priori estimates — summary ........................... 51
3.6. Bibliographical notes .................................. 52
4. Variational solutions ....................................... 54
4.1. The equation of continuity ............................. 54
4.2. Momentum equation ...................................... 66
4.3. Thermal energy equation ................................ 74
4.4. Bibliographical notes .................................. 83
5. Pressure and temperature estimates .......................... 86
5.1. Local pressure estimates ............................... 86
5.2. Temperature estimates .................................. 94
5.3. Bibliographical notes .................................. 99
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