Luc TARTAR Compensated Compactness with more ... - ICMS

I was puzzled that they considered a not so realistic EOS (equation
of state), since incompressibility implies an infinite speed
of sound, without saying what to do for a realistic situation,
of a (compressible) fluid occupying a domain Ω ⊂ R 3 having a
boundary (while compact manifolds have no boundary)!
Thirty years after I was again puzzled that for writing the
questions for the million dollar Clay Prize on Navier–Stokes
equation, Charles FEFFERMAN could not find (in Princeton
or elsewhere) a knowledgeable person (mathematician or not)
who could tell him what the equation is about: he should have
mentioned the basic conservation laws of Continuum Mechanics
and explained why he did not use the conservation of energy;
among the groups of invariance of the equation he should have
mentioned the invariance by rotation (since one considers an
isotropic fluid) and Galilean invariance; in selecting cases without
boundary (R 3 or a flat torus) he should have mentioned
that realistic domains have a boundary, and because bounding
the vorticity is the main difficulty for proving global existence
of smooth solutions, he should have mentioned the importance
of the boundary, since it seems that vorticity is created there.
In 1970, I first read about covariant derivatives, used for writing
Navier–Stokes equation on a Riemannian manifold, but
later I wondered why geometers think it is a good way to
write such equations: is their framework useful for more general
EOS For example, realistic fluids have their viscosity depending
upon temperature, maybe also upon pressure.
Geometers (and later harmonic analysts) seemed to think that
they were doing a useful job in writing equations of Continuum
Mechanics in their language, but has it led to a natural way of
obtaining a priori estimates for flows of realistic fluids
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