152 19 G2 for Euler
The Euler equations are formally reversible: Changing the sign of time t and
the velocity u, obviously leave the equations unchanged. In particular, if ˆu is
a solution to the Euler equations with initial velocity u
0
at time t = 0 and
final value ˆu(
ˆ
t ) at time t =
ˆ
t, then the function ˆv(t) = (−u(
ˆ
t − t), p(
ˆ
t − t))
satisfies Euler’s equations for t ∈ (0,
ˆ
t ] with initial data v(0) = −u(
ˆ
t ) and
final velocity v(
ˆ
t ) = −u(0). Thus, reversing the velocities u(
ˆ
t ) and letting
time pass backwards, would bring back the velocities to −u(0) from −u(
ˆ
t ).
The basic energy estimate (16.2) with ν = 0 states that the total kinetic
energy
1
2
ku(t)k
2
of a pointwise solution ˆu of (19.1) with k·k the [L
2
(Ω)]
3
-norm,
stays constant if f = 0:
1
2
ku(
ˆ
t )k
2
=
1
2
ku(0)k
2
, (19.2)
which follows by multiplication of the momentum equation by u and integra-
tion. Thus it appears that a system governed by the Euler equations allows
the design of a perpetuum mobile by (somehow) reversing the velocity at t = 0
and t =
ˆ
t, corresponding to a system bouncing back and forth for ever. We
conclude that if the Euler equations admit a pointwise solution with point-
wise zero residual, then that solution would be reversible and represent a
perpetuum mobile of the first kind, which is a machine running forever with-
out consuming any energy. On the other hand, NS equations would not allow
such a design because kinetic energy would be lost to the ν-term in the energy
balance (16.2), and turned into heat.
We further recall ’dAlembert’s Mystery stating that a bluff body, sub-
ject to inviscid flow described by the Euler equations, has zero drag. Both a
perpetuum mobile and a bluff body with zero drag are at variance with obser-
vations, and something seems to be seriously wrong with the Euler equations.
But what could it be, since they after all just express Newton’s 2nd Law and
incompressibility? Prandtl blamed the assumption of inviscid flow with ν = 0,
but we have instead pointed to the instability of any exact solutions of the Eu-
ler equations developing into turbulent approximate solutions. In particular,
any conclusion made from an assumption of existence of an exact pointwise
solution may be completely wrong, including the energy conservation (19.2)
and d’Alembert’s computations of zero drag.
We have claimed that non-existence of pointwise solutions of the Euler
equations follows from the observation that solutions to the NS equations in
general are turbulent if ν is small, and that it is unthinkable that these tur-
bulent solutions could converge to a pointwise solution of the Euler equations
as ν tends to zero. The reason is that as we let ν tend to zero, the correspond-
ing NS solutions develop ever finer scales of turbulence which is incompatible
with convergence to a pointwise solution of the Euler equations. If the NS
solutions had stayed laminar as ν tends to zero, pointwise convergence would
have been possible, but NS solutions invariably become turbulent if ν is small,
and thus convergence simply cannot take place. We thus have clear evidence
that in general pointwise solutions of the Euler equations are non-existent.