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- From: jbaez@riesz.mit.edu (John C. Baez)
- Subject: Re: Chaos
- Message-ID: <1992Jul23.151434.5110@galois.mit.edu>
- Sender: news@galois.mit.edu
- Nntp-Posting-Host: riesz
- Organization: MIT Department of Mathematics, Cambridge, MA
- References: <1992Jul20.153122.29180@murdoch.acc.Virginia.EDU> <1992Jul21.211921.17976@galois.mit.edu> <1992Jul22.211155.9389@murdoch.acc.Virginia.EDU>
- Date: Thu, 23 Jul 92 15:14:34 GMT
- Lines: 70
-
- 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. :-)
-