GLOBAL ANALYSIS OF WAVELET
METHODS FOR EULER’S EQUATION
Wayne Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
Tel (65) 874-2749
RIGID BODIES
Euler’s equation

A  (A)  
for their inertial motion
g : R  SO(3)
angular velocity in the body
  v  g g v
inertia operator (from mass distribution)
1
A
Theoria et ad motus corporum solidorum seu rigodorum
ex primiis nostrae cognitionis principiis stbilita onmes
motus qui inhuiusmodi corpora cadere possunt
accommodata, Memoirs de l'Acad'emie des Sciences
Berlin, 1765.
IDEAL FLUIDS
u n  0
u  u u  grad p,   u  0
Euler’s equation
g : R  SDiff (D)
1
u  g g pressure p
n of domain D
for their inertial motion
velocity in space
outward normal
Commentationes mechanicae ad theoriam corporum
fluidorum pertinentes, M'emoirs de l'Acad'emie des
Sciences Berlin, 1765.
GEODESICS
Moreau observed that these classical equations
describe geodesics, on the Lie groups that
parameterize their configurations, with respect to
the left, right invariant Riemannian metric
determined by the inertia operator (determined from
kinetic energy) on the associated Lie algebra
Une method de cinematique fonctionnelle en
hydrodynamique, C. R. Acad. Sci. Paris 249(1959),
2156-2158
EULER’S EQUATION ON LIE GROUPS
Arnold derived Euler’s equation
*
Au   adu Au
that describe geodesics on Lie groups with respect to
left, right
  1,  1 invariant Riemannian metrics
Mathematical Methods of Classical Mechanics,
Springer, New York, 1978
GLOBAL ANALYSIS
based on this geometric formulation provides
a powerful tool for studying fluid dynamics
Arnold used it to explain sensitivity to initial
conditions in terms of curvature
Ebin, Marsden, and Shkoller used it to derive
existence, uniqueness and regularity results for both
Euler’s and Navier-Stokes equations
These ideas are fundamental for the study of a large
class of nonlinear partial differential equations and
have developed into the extensive field of
topological hydrodynamics
REPRESENTATIONS

G
dual
G Lie group G Lie algebra

exp : G  G   ,   : G  G  R
For g  G, u, v  G,  G define the adjoint
1
and coadjoint representations Ad g u  g u g

 Ad g , v    , Ad g v 
d
ad u v  Adexp(  tu) Adexp( tu) v
dt


 ad u , v    , ad u v , u, v  G,  G
WEAK FORMULATION
The inertia operator A : G  G is self-adjoint
and positive definite and defines a bilinear form
(u, v)   Au , v , u, v  G
Then
iff
*
Au   adu Au
(u , v)  (ad u v, u ), u, v  G
in this case the energy is constant
1
1
E   Au , u   (u, u )
2
2
LAGRANGIAN FORMULATION
A trajectory
g:R G
is a geodesic
if and only if the associated momentum satisfies

Au  Ad g  Au (0)
where for a left, right invariant Riemannian metric
1
u  g g , g g
1
is the angular velocity in the body, in space
The momentum lies within a coadjoint orbit which
has a sympletic structure and thus even dimension
CARTAN-KILLING OPERATOR
Define the Cartan-Killing operator

B : G G
 Bu, v   trace ad u ad v , u, v  G
B is self-adjoint and satisfies
 B ad w u, v    B u, ad w v   0
B is nonsingular iff G is semisimple (Cartan)
B is positive semidefinite iff G is compact (Weyl)
VORTICITY EQUATION
g : R  G ,   1,1
1
1
and associated u  g g
 , g g : R  G
Then c : R  G satisfies the vorticity equation
Consider a trajectory
iff
c   adu c
c  Ad g  c(0)
in this case the enstrophy is constant
1
   Bc, c 
2
VORTICITY FORMULATION
Define the Greens operator
1
L  A B : G G
c : R  G satisfies the vorticity equation
then u  Lc satisfies Euler’s equation
If B is nonsingular then the converse holds
If c0  G then u 0  Lc 0 is a stationary point
If
iff
B ad u 0 c0  0
CANONICAL FOURIER BASIS
There exist a basis
e1,, e1 for G
1 , ,  d  R , | i
1
2
|
and
 freq ei
such that
Le i  i ei and  Bei , ei   sgn( i )
If c 
i ei then
d
1

2

i 1
i
2
sgn( i ) i ,
d
1
E
2

i 1
2
| i |  i
CANONICAL REPRESENTATION
k
Cij , i, j, k  1,, d
d
Structure Constants
defined by
ad ei e j   C ek
k 1
k
ij
yield vorticity equation
 k  

i, j
k
 i  j (i   j )Cij
SPARSER REPRESENTATIONS ???
Symmetric forms (each k) can be diagonalized
Can they be simultaneously sparsified?
Existence of higher order invariants suggests so
For any representation
 : G  GL (n )
p
p  trace ((c) ), p  1,, n
are constant; furthermore, Ado’s theorem ensures the
existence of faithful finite dimensional representations
IDEAL FLUID FLOW IN
G  SDiff (R )
2
R
2
G  Div ( R )
1
2
u  g g ,   1,  Au , v  
ad u v  [u, v]

uv
Poisson bracket (commutator)
The weak form of Euler’s equation
(u , v)  ([ u , v], u )
provides the Faedo-Galerkin approximation method
STREAM FUNCTION
 0 1
u  Jf
J

  1 0
iso
1
2
J : H (R ) 
 G


Then c    u  f
c   adf c
(orthogonal coordinates ad f c  f x c y  f y c x )
1
1
The Green’s operator L  A B   
1
has convolution kernel G 2 ( x )   ln | x |
R

c
is constant along particles in the flow, therefore
the moments

c , m 1
m
are invariant
T
IDEAL FLUID FLOW IN
Identified with ideal flows in
R
2
2
that are
periodic with respect to the subgroup
2
2Z
with average value zero, for the spectral basis
ip

x
E p (x)  e
,
p  Z \ {0}
2
of the complexified stream function Lie algebra
ad E p E q  (p  q) E p  q
 BE p , Eq    p,q
FAIRLIER, FLETCHER AND ZACHOS
defined the map
n : G su (n)
for odd
n
p1p 2 / 2 p1 p 2
n
n (E(p1 ,p 2 ) ) 

F H
4i
e
1
0
F  
0
0

0
0


0

  
0
... 
0
 
4i / n

0 
 
0 
n 1 


0
0
0
H  
0
1
 0
1
0
0
1
0
...  1 
 0
   
0
  0

ZEITLIN
used the approximation
 ad n (E p ) n (E q )
 n (Ep )n (Eq )  n (Eq )n (Ep )
2

n
 sin
(p  q) E p  q
2
n
 (p  q) E p  q  ad E p E q
2
to approximate flow on T by flows on SU( n )
m
limit trace (n (c) ) 
n 

m
c , m 1
WAVELET BASES
Neither the canonical Fourier basis nor the
canonical sparse matrix basis provides a sparse
representation of Euler’s equation on SU(n)
Wavelet vorticity bases provide nearly sparse
representations for Euler’s equations because
(i) Green’s operator is Calderon-Zygmund
(ii) Poisson bracket is exponentially localized
Wavelet bases provide simple approximations
for invariant moments and energy
We are using wavelet bases to study Okubo-Weiss
criteria for two-dimensional turbulence