🚀 go-pugleaf

In article <1992Jul20.194122.4545@murdoch.acc.Virginia.EDU> crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass) writes:

>     Work on existence and uniqueness and 'attractors' in 2-D with
>     usually special forcings and/or boundary conditions is hardly
>     of earth-shattering import.  Much more interesting from Foias
>     is his work on the Hopf functional formulation of turbulence
>     This is, of course, unrelated to 'chaos'.  Most people do not know
>     that the Navier-Stokes equations have been 'solved' (formally
>     by functional methods J. Rat. Mech. Anal. 1:87 (1952) and all
>     that followed from that work).  This could be very interesting
>     if we were better at functional integration.  I suggest that work in 
>     this area (of course, there are people working in this area)
>     is probably more potentially earthshaking, and there are applications
>     in other research backwaters (e.g. QM).

Hmm... I suppose I should just read the reference, but I wonder in what
sense the Navier-Stokes equation can be said to be "solved".  There is
no good global existence theorem for solutions, for one thing.  It's
true that the Navier-Stokes equation, like most decent PDE's, can be
written as a ODE with values in an infinite-dimensional vector space

df/dt = G(f)

and then "solved" by an integral equation

f(t) = f(0) + int_0^t G(f(s)) ds .

While this technique is really useful for studying PDE it would be
misleading to say to ordinary folks that the Navier-Stokes equation had
been solved if this was all there was to it.  Of course, you say it was
"solved" with quotes.   If it's possible to clarify without too much
work that'd be nice.

By the way, it's very enjoyable to read you writing about this stuff - as
opposed to the SSC - I guess I like physics more than politics.