In the spring of 1999, I taught (at CARNEGIEMELLON University) a
graduate course entitled Partial Di?erential Equations Models in
Oceanography, and I wrote lecture notes which I distributed to the
students; these notes were then made available on the Internet, and they
were distributed to the participants of a Summer School held in Lisbon,
Portugal, in July 1999. After a few years, I feel it will be useful to
make the text available to a larger audience by publishing a revised
version. To an uninformed observer, it may seem that there is more
interest in the Navier-Stokes equation nowadays, but many who claim to
be interested show such a lack of knowledge about continuum mechanics
that one may wonder about such a super?cial attraction. Could one of the
Clay Millennium Prizes
bethereasonbehindthisrenewedinterest?Readingthetextoftheconjectures to
be solved for winning that particular prize leaves the impression that
the subject was not chosen by people interested in continuum mechanics,
as the selected questions have almost no physical content. Invariance by
translation or scaling is mentioned, but why is invariance by rotations
not pointed out 1 andwhyisGalileaninvariance omitted,
asitistheessentialfactwhichmakes 1 Velocities involved for ordinary
?uids being much smaller than the velocity of light c, no relativistic
corrections are necessary and Galilean invariance should then be used,
but one should be aware that once the mathematical equation has been
written it is not automatic that its solutions will only use velocities
bounded by c.
