From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Chaos
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 23 Jul 92 15:14:34 GMT

In article <1992Jul22.211155.9389@murdoch.acc.Virginia.EDU> crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass) 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:

>>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....
 
Whoops - when I see NS I think "nonlinear Schrodinger," not
Navier-Stokes.

Anyway, there follows a terse and lucid answer:

>      First, an attractor is a subset of a phase space that all orbits
>      in a that begin in a 'basin of attraction' tend to.  Among other
>      things it is a bounded invariant subset of the particular 
>      space (Hilbert space H for NS) that we are interested in.  From
>      proposition 13.1 in Constantin and Foias's book, we take
>      S(t) u_0 = u(t) as the solutions to the two-dimensional forced
>      Navier-Stokes equations where S is a map from H to H. We have
>      for various reasons and among other things
>
>            "(v)  There exists 
>
>                    B^V_\rho = {u: ||u|| <= \rho} \subset V
>                 
>                  which is an absorbing set, i.e. for every u_0 \in H
>                  there exists t_0(|u_0|) such that, for t >= t(|u_0|),
>                  S(t) u_0 \in B^V_\rho. "
>
>      Which comes from various energy estimates of the NS equations
>      in this setting.

Let me see if I have the right gut feeling.  The NS equations are not
conservative and I guess without forcing the energy would always
approach zero as t -> infinity.  The forcing can pump in energy and so
the work is to show that for more energetic states the rate of
dissipation of energy is higher so there is some \rho such that
eventually every solution has energy <= \rho.  (Let me know if this is
seriously wrong.) 

>      This basically allows them to define a set X such that
>
>              X = \bigcap S(t) B^V_\rho 
>      where we get
>
>             "(i)     X is compact in H
>              (ii)    S(t) X = X for all t>=0
>              (iii)   If we have Z bounded in H satisfying S(t) Z = Z
>                      for all t >= 0, then Z \subset X
>              (iv)    For every u_0 \in H
>                      lim[t->\infty] dist(S(t) u_0,X) = 0
>              (v)     X is connected."

Let's see.  (ii) is trivial, (iv) is believable, I guess (v) is easy
(an intersection of nested connected sets is connected), (iii) seems
easy (first of all Z must be in the ball B^V_\rho, but since S(t)Z = Z,
Z is in S(t) B^V\rho for all all t>0).  That leaves (i), which I find to
be the interesting part... it's the compactness that could make one hope
this attractor is vaguely finite-dimensional... and I wonder what
creates the compactness.

Of course for an equation like the heat equation on a compact manifold,
time evolution maps the unit ball of Hilbert space into a compact set.
So maybe the forced NS equation in 2d acts rather similarly - due to the
viscosity?  

So my impression is that this result uses fairly standard techniques in
PDE, and is not a mind-boggling, revolutionary sort of result.  Still, I
like it.  I'm not trying to understand turbulence, of course.  :-)
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