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2 The Navier–Stokes and Euler Equations – Fluid and Gas Dynamics
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Adh´ emar Jean Claude Barr´ e de Saint–Venant .The main issueistoincor-
porate thefreeboundaryrepresentingthe height-over-bottom h = h(x, t)of
the water (measured vertically from the bottom of the river). Let Z = Z(x)
be the height of the bottom of the river measured vertically from a con-
stant 0-level below the bottom (thus describing the river bottom topogra-
phy), which in the most simple setting is assumed to have a small variation.
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Note that here the space variable x in R or R denotes the horizontal di-
rection(s) und u = u(x, t) the horizontal velocity component(s), the vertical
velocity component is assumed to vanish. The dependence on the vertical co-
ordinate enters only through the free boundary h. Then, under certain assump-
tions, most notably incompressibility, vanishing viscosity, small variation of the
riverbottomtopography andsmall waterheight h, the Saint–Venant system
reads:
∂h
+div (hu) = 0
∂t
∂(hu) g 2
+div (hu ⊗ u)+grad h + gh grad Z = 0
∂t 2
Here g denotes the gravity constant. Note that h + Z is the local level of the
water surface, measured vertically again from the constant 0-level below the
bottom of the river. For analytical and numerical work on (even more general)
Saint–Venant systems we refer to the paper [4]. Spectacular simulations of the
breaking of a dam and of river flooding using Saint–Venant systems can be
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found in Benoit Perthame’s webpage .
Many gas flows cannot generically be considered to be incompressible, par-
ticularly at sufficiently large velocities. Then the incompressibility constraint
div u = 0 on the velocity field has to be dropped and the compressible Euler or
Navier–Stokes systems, depending on whether the viscosity is small or not, have
to be used to model the flow.
Herewestatethesesystemsunderthesimplifyingassumptionofanisentropic
flow, i.e. the pressure p is a given function of the (nonconstant!) gas density:
p = p(ρ), where p is, say, an increasing differentiable function of ρ. Under this
constitutive assumption the compressible Navier–Stokes equations read:
ρ t +div (ρu) = 0
(ρu) t +div (ρu ⊗ u)+grad p(ρ) = νΔu +(λ + ν) grad(div u)+ ρf .

Here λ is the so called shear viscosity and ν + λ is non-negative.
For a comprehensive review of modern results on the compressible Navier–
Stokes equations we refer to the text [5].
For the compressible Euler equations, obtained by setting λ = 0and ν = 0,
globally smooth solutions do not exist in general. Consider the one-dimensional
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http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Saint–Venant.html
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http://www.dma.ens.fr/users/perthame/