Table des matières
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 $$v-v_b\in L^\infty(]0,T[,H)\cap N_2^{\frac{1}{4}}(]0,T[,H).$$
Table of Contents
Preface
Chapter I : The equations of fluid mechanics
- Continuous description of a fluid
- The transport theorem
- Conservation equations
- Fundamental laws: Newtonian fluids and thermodynamics laws
- Summary of the equations
- Incompressible models
- Some exact steady solutions
Chapter II : Analysis tools
- Main notation
- Fundamental results from functional analysis
- Basic compactness results
- Functions of one real variable
- Spaces of Banach-valued functions
- Some results in spectral analysis of unbounded operators
Chapter III : Sobolev spaces
- Domains
- Sobolev spaces on Lipschitz domains
- Calculus near the boundary of domains
- The Laplace problem
Chapter IV : Steady Stokes equations
- Necas inequality
- Characterisation of gradient fields. De Rham's theorem
- The divergence operator and related spaces
- The curl operator and related spaces
- The Stokes problem
- Regularity of the Stokes problem
- The Stokes problem with stress boundary conditions
- The interface Stokes problem
- The Stokes problem with vorticity boundary conditions
Chapter V : Navier-Stokes equations for homogeneous fluids
- Leray's Theorem
- Strong solutions
- The steady Navier-Stokes equations
Chapter VI : Non-homogeneous fluids
- Weak solutions of the transport equation
- The nonhomogeneous incompressible Navier-Stokes equations
Chapter VII : Boundary conditions modeling
- Outflow boundary conditions
- Dirichlet boundary conditions through a penalty method