From: crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass)
Subject: Re: Chaos
Organization: University of Virginia
Date: Wed, 22 Jul 1992 21:11:55 GMT

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 dimension is about 500 (as one
>>>expects from various other estimates). Renormalization is not necessary
>>>for an attractor, although the Orszag-Yahot theory is essentially a
>>>renormalization theory. The Foias stuff can be found in the book by
>>>Foias and Constantine, but I think the attractor theorems came after
>>>his book with Roger Temam, although my memory may be playing tricks.
>
>>     Is this the 'universal attractor' for 2-D NS with periodic and
>>     dirichlet BC's?  
>
>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"?

      I hate to be the one to respond, since others may
      view such results in a more positive light.  But since no one
      else has piped in ...

      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.

      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."

      It is this set X that is called the universal attractor.

      What they proved is that the forced NS equations in two dimensions with
      the associated periodic boundary conditions in this case (since
      H is defined as an L^2 space of periodic divergence-free functions
      (Physica D 30:284 (1988))) have such a set X.  I think that
      there may be some results of a similar nature using Dirichelet
      BC's, but I cannot put my finger on a paper (or even remember the
      relevant arguments since they would have to be different
      than the ones presented in the paper referenced above). Over the
      years, I have also seen published allusions by Foias or Temam to 
      such results in 3D, however I do not believe that there are currently
      any such results in 3D.

      As far as various other nonlinearities go, I do not know.
      The nonlinearity they consider obove is, of course, quadratic.

      Editorials at a minimum...

                              dale bass


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