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

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F. Boyer, P. Fabrie
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====== 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 : {{:publications:erratum_bf_dec_2012.pdf|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

==== Appendix A : Classic differential operators ====
==== Appendix B : Thermodynamics supplement ====
==== References ====
==== Index ====