Chapter 1 Introduction 1
1.1 The main models 1
1.2 Notations and some preliminary lemmas 10
Chapter 2 The Navier-Stokes Equations with Damping 13
2.1 Introduction 13
2.2 Existence of weak solutions 15
2.3 Existence and uniqueness of strong solutions 21
Chapter 3 Decay of Navier-Stokes Equations with Damping 29
3.1 Introduction 29
3.2 A priori estimates on upper bound ofdecay 32
3.3 A priori estimates on lower bound ofdecay 38
3.4 The decay ofthe weak solutions 42
Chapter 4 Stokes Approximation of Non-homogeneous Navier-Stokes Equations 48
4.1 Introduction 48
4.2 Existence of weak solutions 51
4.3 Existence and uniqueness of strong solutions 60
Chapter 5 Large Solutions to Non-homogeneous Navier-Stokes Equations 67
5.1 Introduction 67
5.2 The global existence ofsolutions 69
5.3 The global stability ofsolutions 74
Chapter 6 Some Remarks on Planar Boussinesq Equations 89
6.1 Introduction and the main results 89
6.2 The case ofsmooth initial data 91
References 103