🚀 go-pugleaf

In article <1992Jul21.211921.17976@galois.mit.edu> jbaez@nevanlinna.mit.edu (John C. Baez) writes:
>In article <1992Jul20.153122.29180@murdoch.acc.Virginia.EDU> crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass) writes:
>>In article <1076@kepler1.rentec.com> andrew@rentec.com (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