generic hydrodynamic instability via contact homology

1
generic hydrodynamic instability via contact
homology

• robert ghrist
• department of mathematics
• university of illinois
• urbana-champaign, usa

ams fall sectional 2004
joint work with john etnyre mathematics u penn
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the setup
consider steady, inviscid velocity fields on R3
/ periodic
classical interesting examples abc flows
arnold, henon, childriss,
x A cos z B sin y y B cos x C sin z z
C cos y A sin x
force-free fields
these fields have several interesting features
chaotic dynamics (observed numerically)
curl eigenfields ( vorticity
?velocity )
the abc fields are merely the principal
eigenfields of curl
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everyone who knows...
that water is wet knows that generic 3-d euler
flows are hydrodynamically unstable...
...but this statement is problematic
the space of all steady 3-d euler flows is not
well-understood
theorem arnold any real-analytic steady
euler field in 3-d is either totally
integrable, or else it is a curl eigenfield
it follows that it is very difficult to assign a
parameter with which to discuss genericity...
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idea add a parameter space
vary the geometry of the flow domain (the
riemannian structure)
theorem etnyre-g for generic Cr riemannian
structure on R3/periodic (r2), all
curl-eigenfields (except ?0) are unstable
(linear, L2 norm).
by generic we mean residual (countable
intersection of open, dense sets --- all spaces
baire hence generic gt dense
the proof uses three disparate ingredients
an instability criterion friedlander-vishik
transversality theory for the curl operator cf.
uhlenbeck
contact homology eliashberg-givental-hofer,
bourgeois
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the instability criterion
as proved by friedlander-vishik, utilizing
perspectives from lifshitz-hameiri, bayly,
arnold, and others...
the instability criterion any steady euler
flow possessing a nondegenerate fixed point or
periodic orbit with exponential stretching forces
the field to be linearly unstable in energy
norm.
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transversality and curl
proposition for generic geometry, the
eigendecomposition of the curl operator is
simple. (??0)
spectrum is discrete and eigenspaces are 1-d
proposition for generic geometry, all fixed
points are nondegenerate. (??0)
incompressibility gt instability criterion
satisfied
proposition if no fixed points, then for generic
geometry, all periodic orbits are
nondegenerate. (??0)
each is either hyperbolic (good) or elliptic (bad)
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what if...
there are no periodic orbits?
all the periodic orbits are elliptic?
theorem e-g any nondegenerate curl-eigenfield
(??0) without fixed points must possess a
hyperbolic-type periodic orbit
this is a highly nontrivial result, relying on
very deep work of Hofer, Eliashberg, Bourgeois,
et al.
proof uses a new homology theory coming from
contact topology
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contact-curl connection
a contact structure is a plane field which is
nowhere integrable.
frobenius theorem ? contact structures are
orthogonal to curl eigenfields
the topological idea classify contact structures
via the topological features of the associated
reeb fields
(reeb fields beltrami fields without the notion
of a metric)
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morse homology contact homology
given M manifold
given ? contact structure on M
choose function fM?R
choose 1-form a for ?
examine gradient field X -grad(f)
examine reeb field X for a
count fixed points of X
count periodic orbits of X
lemma generically nondegenerate
lemma generically nondegenerate
grading morse index
grading conley-zehnder index
chain groups Cn(M,f)
chain algebra A(?) over Z2
boundary map ?Cn(M,f)?Cn-1(M,f)
boundary map counts punctured surfaces
counts number of heteroclinic curves from one
fixed point to others of one lower index
in M x R which are pseudoholomorphic
and asymptote to periodic orbits of X
thm ?20 and homology does not depend on
choices
thm ?20 and homology does not depend on
choices
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and you do what with this?
classify the contact structures on R3/Z3
(Giroux, Kanda)
compute the contact homology
(techniques of Bourgeois)
do tricks...
prove there is always a saddle-type orbit
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whats bad about this proof...
it says nothing about the euclidean case
...and some people care about this (degenerate)
metric...
this is not a measure-theoretic genericity
the instability criterion holds in greater
generality...
its only linear instability
and its very high-frequency at that
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whats worth doing
investigate the complementary class of integrable
fields
do they really exist? (generically)
what properties of the euler equations are
geometrically stable?
steady solutions? singularities?
examine other parameter spaces
shape of boundary, forcing terms, etc.