Mathematical tools for the study of the incompressible Navier-Stokes equations and related models

F. Boyer, P. Fabrie

# Erratum and complements

• The definition of the sound speed in the fluid on top of page 35 should be $$\frac{1}{c} = \sqrt{ \left(\frac{\partial \rho}{\partial p}\right)_T }$$
• Some precisions are needed concerning Theorem VI.1.6 and Lemma VI.1.7 : Details are given in this document
• In System (VI.38) the inflow part of the boundary $\Gamma_v^-(t)$ actually does not depend on $t$ since $v=v_b$ on the boundary and $v_b$ does not depend on time. However, all the analysis proposed here can be adapted to a time-dependent boundary data $v_b$, provided we assume suitable regularity properties.
• At the end of the statement of Theorem VI.2.1, in the case $\inf \rho_0>0$, it is not true that $v$ belongs to the space $H$ since its normal component does not vanish on the boundary. The correct formulation is the following $$v-v_b\in L^\infty(]0,T[,H)\cap N_2^{\frac{1}{4}}(]0,T[,H).$$

### Chapter I : The equations of fluid mechanics

1. Continuous description of a fluid
2. The transport theorem
3. Conservation equations
4. Fundamental laws: Newtonian fluids and thermodynamics laws
5. Summary of the equations
6. Incompressible models
7. Some exact steady solutions

### Chapter II : Analysis tools

1. Main notation
2. Fundamental results from functional analysis
3. Basic compactness results
4. Functions of one real variable
5. Spaces of Banach-valued functions
6. Some results in spectral analysis of unbounded operators

### Chapter III : Sobolev spaces

1. Domains
2. Sobolev spaces on Lipschitz domains
3. Calculus near the boundary of domains
4. The Laplace problem

### Chapter IV : Steady Stokes equations

1. Necas inequality
2. Characterisation of gradient fields. De Rham's theorem
3. The divergence operator and related spaces
4. The curl operator and related spaces
5. The Stokes problem
6. Regularity of the Stokes problem
7. The Stokes problem with stress boundary conditions
8. The interface Stokes problem
9. The Stokes problem with vorticity boundary conditions

### Chapter V : Navier-Stokes equations for homogeneous fluids

1. Leray's Theorem
2. Strong solutions
3. The steady Navier-Stokes equations

### Chapter VI : Non-homogeneous fluids

1. Weak solutions of the transport equation
2. The nonhomogeneous incompressible Navier-Stokes equations

### Chapter VII : Boundary conditions modeling

1. Outflow boundary conditions
2. Dirichlet boundary conditions through a penalty method