From: crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass)
Subject: Re: Chaos
Organization: University of Virginia
Date: Sun, 26 Jul 1992 00:09:17 GMT

In article <1116@kepler1.rentec.com> andrew@rentec.com (Andrew Mullhaupt) writes:
>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.

     That is what I thought.  Earlier stuff, I seem to remember,
     relied on postulating that smooth IC's remained smooth as well.

     The problem I have is that I do not expect global regularity.
     For two dimensions, I can easily see that it might be true (and
     in fact it is).  It is the fact that vorticity can 'pop' out
     into the third dimension that allows fluid turbulence.  I have
     always expected (at least since third grade) that this would
     cause problems in 3D proofs of this sort.

>>>     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.

     Actually, this is not true.  It has been a number of years
     since the two dimensional results were worked out (> 5).  
     Since then, I have seen basically only variations on a theme.

     As I said, one could have expected such results to work
     in two-dimensions since there is no turbulence.
     
>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.

     With symmetric helicity, you have taken away a very important degree 
     of freedom.  I'll become a believer when Temam comes up with a 
     3D proof of global regularity with Dirichlet or Neumann BC's.

     Until then, I'm not holding my breath.  I actually hope that
     the equations are more interesting than that.

                                   dale bass
--
C. R. Bass                                           crb7q@virginia.edu
Department of Mechanical and Aerospace Engineering
University of Virginia
Charlottesville, Virginia                            (804) 924-7926
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