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2. The Navier–Stokes and Euler Equations –
Fluid and Gas Dynamics

Fluid and gas dynamics have a decisive impact on our daily lives. There are the
fine droplets of water which sprinkle down in our morning shower, the waves
which we face swimming or surfing in the ocean, the river which adapts to the
topographybyformingawaterfall,theturbulentaircurrentswhichoftendisturb
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our transatlantic flight in a jet plane, the tsunami which can wreck an entire
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region of our world, the athmospheric flows creating tornados and hurricanes ,
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the live-giving flow of blood in our arteries and veins …All theseflowshave
a great complexity from the geometrical, (bio)physical and (bio)mechanical
viewpoints and their mathematical modeling is a highly challenging task.
Clearly, the dynamics of fluids and gases is governed by the interaction
of their atoms/molecules, which theoretically can be modeled microscopically,
i.e. by individual particle dynamics, relying on a grand Hamiltonian function
depending on 3N space coordinates and 3N momentum coordinates, where N
is the number of particles in the fluid/gas. Note that the Newtonian ensemble
trajectories live in 6N dimensional phase space! For most practical purposes this
is prohibitive and it is essential to carry out the thermodynamic Boltzmann–
Grad limit, which – under certain hypothesis on the particle interactions – gives
the Boltzmann equation of gas dynamics (see Chapter 1 on kinetic equations)
for the evolution of the effective mass density function in 6-dimensional phase
space.
Under the assumption of a small particle mean free path (i.e. in the colli-
sion dominated regime) a further approximation is possible, leading to time-
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dependent macroscopic equations in position space R ,referredtoasNavier–
Stokes and Euler systems. These systems of nonlinear partial differential equa-
tions are absolutely central in the modeling of fluid and gas flows.
For more (precise) information on this modeling hierarchy we refer to [3].
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The Navier–Stokes system was written down in the 19th century. It is named
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after the French engineer and physicist Claude–Luis Navier and the Irish math-
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ematician and physicist George Gabriel Stokes .

1 http://www.tsunami.org/
2 http://www.spc.noaa.gov/faq/tornado/
3 http://www.nhc.noaa.gov/
4 http://iacs.epfl.ch/cmcs/NewResearch/vascular.php3
5 http://www.navier–stokes.net/
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http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Navier.html
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http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Stokes.html