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Navier-Stokes Equations

Table of Contents

1. Overview

When we model fluid flow with viscosity (friction among the fluid parcels), we end up with the Navier-Stokes equation. It's notoriously difficult, and if anyone can find a suitable solution to it…they'll win a million dollars.

Be warned: when mathematicians discuss the Navier-Stokes equations, they typically mean the incompressible version.

2. References

  • O.A. Ladyzhenskaya,
    The Mathematical Theory of Viscous Incompressible Flow.
    Published 1963. NB: Ladyzhenskaya is an amazing mathematician, she first proved the finite difference method works for numerically computing Navier-Stokes.
  • Pierre Gilles Lemarie-Rieusset,
    The Navier-Stokes problem in the 21st century.
    CRC Press, 2011.
  • Roger Temam,
    Navier-Stokes Equations: Theory and Numerical Analysis.
    AMS Press, 2001.

2.1. Recommended

The following have been recommended to me (from Tai-Peng Tsai's Lectures on Navier-Stokes Equations), though I have yet to look at further:

  • Gregory Seregin,
    Lecture Notes On Regularity Theory For The Navier-Stokes Equations.
    World Scientific, 2014.
  • Peter Constantin and Ciprian Foias,
    Navier-Stokes equations.
    U. Chicago Press, 1988
  • Pierre-Louis Lions,
    Mathematical topics in fluid mechanics. Vol. 1.
    Oxford Press, 1996.
  • Hermann Sohr,
    The Navier-Stokes equations: An elementary functional analytic approach.
    Birkhauser, 2001.
  • Pierre Gilles Lemarie-Rieusset,
    Recent developments in the Navier-Stokes problem.
    Chapman & Hall, 2002.
  • Hajer Bahouri, Jean-Yves Chemin, and Raphael Danchin,
    Fourier analysis and nonlinear partial differential equations.
    Springer, 2011.
  • James C. Robinson, Jose L. Rodrigo, and Witold Sadowski,
    The three-dimensional Navier-Stokes equations.
    Cambridge University Press, 2016.

Books recommended to me from random people on Math stackexchange or mathoverflow:

  • C. Foias, O. Manley, R. Rosa, R. Temam,
    Navier-Stokes Equations and Turbulence.
    CUP, 2008.
  • Roger Temam,
    Navier-Stokes Equations and Nonlinear Functional Analysis.
    SIAM, 1987.

2.2. Steady State Solutions

  • Giovanni Galdi,
    An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems.
    Springer, 2nd ed., 2011.

Last Updated 2022-04-07 Thu 08:43.