Mathematical tools for the study of the Navier-Stokes equations

F. Boyer and P. Fabrie

Résumé et objectifs du livre

Table des matières

• Preface
• Chapter I : The equations of fluid mechanics
  1. Lagrangian and Eulerian coordinates
  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. Main results
  2. The transport equation
  3. The approximate problem for the nonhomogeneous Navier-Stokes equations
  4. Existence of a weak solution
• Chapter VII : Boundary conditions modeling
  1. Outflow boundary conditions
  2. Dirichlet boundary conditions by penalty
• Appendix A : Classic differential operators
• Appendix B : Thermodynamics supplement
• References