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2 The Navier–Stokes and Euler Equations – Fluid and Gas Dynamics
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where Dg denotes the material derivative of the scalar function g:
Dt
Dg
= g t + u.grad g .
Dt

Thus, for two-dimensional flows, the vorticity gets convected by the velocity
field, is diffused with diffusion coefficient ν and externally produced/destroyed
by the curl of the external force. For three dimensional flows an additional
term appears in the vorticity formulation of the Navier–Stokes equations, which
corresponds to vorticity distortion.
The Navier–Stokes and Euler equations had tremendous impact on applied
9
mathematics in the 20th century, e.g. they have given rise to Prandtl’s boundary
layer theory which is at the origin of modern singular perturbation theory.
Nevertheless the analytical understanding of the Navier–Stokes equations is still
somewhat limited: In three space dimensions, with smooth, decaying (in the far
field) initial datum and force field, a global-in-time weak solution is known to
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exist (Leray solution ), however it is not known whether this weak solution is
unique and the existence/uniqueness of global-in-time smooth solutions is also
unknown for three-dimensional flows with arbitrarily large smooth initial data
and forcing fields, decaying in the far field. In fact, this is precisely the content
of a Clay Institute Millennium Problem 11 with an award of USD 1000000!!
A very deep theorem (see [2]) proves that possible singularity sets of weak
solutions of the three-dimensional Navier–Stokes equations are ‘small’ (e.g.
they cannot contain a space-time curve) but it has not been shown that they are
empty …
We remark that the theory of two dimensional incompressible flows is much
simpler, in fact smooth global 2 − d solutions exist for arbitrarily large smooth
data in the viscid and inviscid case (see [6]).
Whyisitsoimportant to know whethertime-globalsmoothsolutions of the
incompressible Navier–Stokes system exist for all smooth data? If smoothness
breaks down in finite time then – close to break-down time – the velocity field u
of the fluid becomes unbounded. Obviously, we conceive flows of viscous real
fluids as smooth with a locally finite velocity field, so breakdown of smoothness
in finitetimewould be highly counterintuitive.Hereour natural conception of
theworld surrounding usisatstake!
The theory of mathematical hydrology is a direct important consequence
of the Navier–Stokes or, resp., Euler equations. The flow of rivers in general –
and in particular in waterfalls like the famous ones of the Rio Iguassu on the
Argentinian-Brazilian border, of the Oranje river in the South African Augra-
bies National Park and others shown in the Figs. 2.1–2.6, are often modeled
by the so called Saint–Venant system, named after the French civil engineer

9 http://www.fluidmech.net/msc/prandtl.htm
10
http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Leray.html
11
http://www.claymath.org/millennium/Navier–Stokes_Equations/