NetNews Usenet Archive 1992 #16/NN_1992_16.iso/spool/sci/physics/11652

Internet Message Format | 1992-07-25 | 2.4 KB

Xref: sparky sci.physics:11652 sci.math:9549
Path: sparky!uunet!psinntp!kepler1!andrew
From: [e-mail] (Andrew Mullhaupt)
Newsgroups: sci.physics,sci.math
Subject: Re: Chaos
Message-ID: <[e-mail]>
Date: 24 Jul 92 21:40:55 GMT
References: <[e-mail]> <[e-mail]> <[e-mail]>
Followup-To: sci.physics
Organization: Renaissance Technologies Corp., Setauket, NY.
Lines: 36

In article <[e-mail]> [e-mail] (John C. Baez) writes:
>In article <[e-mail]> [e-mail] (Cameron Randale Bass) writes:
>>In article <[e-mail]> [e-mail] (Andrew Mullhaupt) writes:
>>>OK but there is still a universal attractor for the N-S equations as
>>>rigorously proved by Foias et. al.

The hard core theory seems to be well known in 2 space dimensions. I hadn't
remembered this, but a quick look at the literature shows that over the past
ten years, the 2-D case is pretty well worked out. The 3-D case is not done,
but you can get pretty close - to get the 3-D case you need to know that the
do not develop singularites. (Global regularity). In the 3-D case, you can
do stuff, such as Ghidaglia and Temam "Lower bound for the dimension of the
attractor for the Navier Stokes equations in space dimension 3" in the
collection: _Mechanics, analysis and geometry: 200 years after Lagrange_,
M. Francaviglia, ed. North-Holland 1991.

>>     Is this the 'universal attractor' for 2-D NS with periodic and
>>     dirichlet BC's?

OK, so the rigorous stuff is still at 2-D, but the hope that there
won't be an attractor in the 3-D case looks a bit thin, and getting
thinner as time goes by. Note that there are more different 2-D problems
which have been worked out. I think that helically symmetric 3-D N-S solutions
are an invariant subspace to which this theory applies, too.

>Could someone elaborate a bit more on this idea and what precisely was
>proved?  I find it rather odd.  I seem to recall that the 2-D nonlinear
>Schrodinger equation with cubic nonlinearity is completely integrable
>(let's take periodic boundary conditions, say).  What sort of
>nonlinearities does Foias consider, and what is a "universal attractor"?

Real nonlinearities - like the Navier Stokes equations. You can also
do stuff with complex Ginzburg-Landau, etc.

Later,
Andrew Mullhaupt