Troup Mechanics
Troup Mechanics
Troup Mechanics
Introduction
Stating that "a symplectic camel cannot go through the eye of a needle" (V.I. Arnold, [1]).
532 S.B. Kuksin
2
equation we study, could have it. Such a measure was first constructed by L.
F riedlander [ F ] for the <p4 equation
w = uxx u3, u u(t, x), xeT 1
lately invariant G ibbs measures were obtained for some other x one dimensional
equations with coercive hamiltonians by P. Z hidkov, H . M cK ean K . Vaninski and
J. Bourgain (see [Bo] and references therein).
In this paper we construct for ham iltonian P D E 's the second invariant
symplectic capacity. F or each open subset & of the phase space Z of x dependent
vector functions we define its capacity c(G) in such a way that the following
properties are fulfilled:
1) translational inaance:
c() = c( + ) for eZ;
2) monotonocity:
c(1)^c(2) if G1=>G2;
3) 2 homogeneity:
c() = 2c()
4) nontrivialty:
0 < c(G) < oc if G 0 is bounded.
Our definition of the capacity c(G) is based on finite dimensional approxima
tions of G. Thus the capacity c inherits the very im portan t normalization of the
finite dimensional symplectic capacity:
5) c(Br) = c({rk)) = r\
where B is an r ball in Z and ^ is the cylinder based on an r disc in the plane
spanned by the vectors ^, ^ from a fixed D arboux basis f, }, ... in Z .
(Possibly the ball and the cylinder are n ot centered at zero.)
We prove that the capacity c is preserved by the flow maps S t of
a (nonautonom ous) hamiltonian equation, provided that
S t = linear operator + compact sm ooth operator, (1)
where the linear operator is the direct sum of rotation s in the planes spanned by
j~ and j (j = 1,2, ... ) and compactness of the nonlinear term "agrees with the
basis {f }" (these two assumptions hold trivially if the linear operator is the
identity map).
The assumption (1) does not hold for general ham iltonian P D E , but still it
holds for some im portan t equations. F or example, for
Nonlinear string equation
u = uxx + p{u;t,x\ u = u ( , x) , xe T 15 (2)
2
The G ibbs measure depends on the hamiltonian. Therefore it can be used to study autonomous
equations only.
Infinite Dimensional Symplectic Capacities 533
Schrdinger equation
where w* is the convolution with a fixed real function and G is a real valued
smooth function.
U sually there is an arbitrariness in choosing a phase space for a hamiltonian
P D E . In contrast, to construct the invariant capacity one should take for a phase
space of the equation the H ubert space distinguished by the property that some
H ubert basis of the space is in the same time a D arboux basis for the corresponding
symplectic structure. F or a nonlinear P D E such a distinguished phase space Z is
essentially unique. In particular, for Eq. (2) the phase space is
Z = H il2{) xH ll2{T \ formed by the vector functions
Squeezing Theorem. / /
and {ps = Us + ws~} are the complex F ourier coefficients of the solution w(, x).
The theorem implies that if \ pk {t)\ < r for all solutions with u(09 )e BR, then r ^ R.
Another immediate consequence of the theorem is that a bounded solution of
the equation cannot be "uniformly asymptotically stable" as t > oo, because for any
p neighbourhood Bp of the initial point diameter of the set S t(Bp) cannot tend to
zero.
The theorem is also connected with the following im portan t question: does the
energy of nonlinear conservative oscillations spread to higher frequencies? A possible
mathematical reformulation of this physical question is the following.
P rovide the phase space of x dependent vector functions with some H ubert
norm || || ^ . D enote the corresponding H ubert space as Z and its H ubert basis as
534 S.B. Kuksin
{f }. Take a ball BR = {|| z || ^ < K} in the space of initial data and consider the
problem: is it true that for fixed k "part of energy leaves A th m ode":
S T (BR) + \ Pk +
2
* < P} (7)
for some T > 0, p < R
If Z = Z , then (7) is the squeezing, which is impossible due to the theorem.
To explain why we treat (7) as the spread of energy, suppose that the phase
space of the system we consider is the space of pairs of x dependent functions with
n n
x e 1" W/ (2 ) . Fix d ^ 0 and choose the tilde basis equal to {* (x)\ se },
where
<5> = 1 + |s| and {s{x)} is the usual trigonometric basis of L 2 (T "). F or a vec
tor function denote by Zf the coefficients of its decomposition in the basis
{(s(x), 0), (0, s(x))} (i.e., its Fourier coefficients) and denote by E s = zs+ 2 + zs~ 2 the
th
energy of the s mode. Below we draw idealised pictures of the energy distribution
for the points from BR (at left) and from p with k = 1 (at right):
1 2 1 2
So if p < R and the m ap S sent the left set inside the right one, then a part of the
energy indeed left the first mode. F or more information on the energy transition
subject we refer the reader, say, to [ P ] .
F or Eqs. (2) (5) above the distinguished phase spaces Z have the bases of the
form (8) with d = \ , i, 1,0 correspondingly. The question if the "squeezing" (7) is
possible can be also stated with the distinguished phase spaces Z replaced by some
smoother spaces. The most interesting seems the case of energetic spaces. F or Eqs.
(2) (5) these are respectively Sobolev spaces H 1, H\ H 2, H 1 (having the bases of
the form (8) with d = 1, 1, 2,1). In such a case we do not know if (7) is possible or
not for general Eqs. (2) (5). But for Eq. (2) with a time dependent analytic function
p(u) of the form
where m > 0, the answer again is "n o " if p in (7) is sufficiently small. The reason is
that for Eq. (2), (9) has in rich supply small amplitude time quasiperiodic solutions.
See [BK ] for the case a = 0, b = const 0 and [ K ] for the case m = m(x), where
Infinite Dimensional Symplectic Capacities 535
3
m( ) is a "typical function," even in x. In particular, (2) has time periodic solutions
which lie in BR near the plane IR IR ^ 4
We provide the linear space IR 2" = IR x IR with the usual symplectic structure
dp dq; for a smooth function f{p,q) we denote by Vf the corresponding hamil
tonian vectorfield.
G iven m > 0 and an open domain c IR 2" we call a function fe C($)
m admissible if
i) 0 ^ / ^ m ;
ii) / vanishes in a nonempty subdomain of \
iii) / = m in a neighborhood of d(P.
Following [H Z 1, H Z 2] we define the capacity c2n((9) of the domain (9 as
c2n() inflm^l for each m > m% and each m admissible/ in the vectorfield
Vf has a nontrivial periodic solution of period ^ 1}.
Clearly c2n is a monotonic symplectic invariant:
c2n((9i)Sc2n((92) if ^lC= d ?2 (1.1)
and
c2n((9) = c 2(0(?)) (1.2)
2
if : + IR " is a symplectomorphism.
D enote by , the ball
The following equalities gives the main property of the capacity c2n:
c2n(Br) = c2n(r) = r 2 (1.3)
(see [H Z 1, H Z 2] for a proof).
An immediate consequence of (1.1)(1.3) is the famous squeezing theorem of M.
G romov [G ]: the ball Br can be symplectically embedded into the cylinder R only
if R ^ r.
Take any rc vector r = (r 1 ? . . . , rn\ 0 < r ; ^ oo , where some r, is finite. Then
3
More exactly, [BK, K ] provide (2), (9) with time quasiperiodic, x periodic solutions which are
even or odd in x since theorems of these works are applicable to even periodic and odd periodic
boundary conditions (which are equivalent to Dirichlet and N eumann boundary conditions on
the half period).
4
These solutions form symplectic ribs which prevent the symplectic camel from the first footnote
to go through the eye of a needle (cf. Sect. 1 in [A]).
536 S.B. Kuksin
n
with some 1 ^ j = * The ellipsoid
B{r) = {(p,q)\ ri2(Pj + qf) < 1}
2
contains the ball Br an d is contained in the cylinder {pf0 + qf0 <r }. So we get
from (1.3) (and (1.1), (1.2)) that
c2n(B(r)) = r2 . (1.4)
Let Z be a H ubert space with the scalar product < , ) an d a H ubert basis
{<Pf\ j7t 1}. F o r e N we denote by Z " the linear envelope of the vectors
{f 11 ^ 7 ^ n) an d denote by 77" the n atural projector
n Z *Z n .
We also denote Z n = Z Q Z ", use the decomposition of Z ,
Z = Z nZ n, (2.1)
and write accordingly z e Z a s
z ^ z ", *, ) , z"eZ z e Z n .
We define the skewsymmetric linear operator J,
J Z^Z, f ^ + <^/ ,
and supply Z with a symplectic structure by means of the 2 form
= (Jdz,dz} . 5
We take a selfadjoint (possibly unbounded) linear operator A such that
f jf Vj (2.2)
(the operator is selfadjoint in Z with the n atural domain of definition) and consider
a hamiltonian
Remark. The assumption (2.2) concerning Eq. (2.4) is not very restrictive, see Part
6 for concrete examples. Besides, this assumption can be achieved if the "un
bounded linear part" JU of Eq. (2.4) has a discrete spectrum which is semisimple
and imaginary we can put the operator JAU to the normal form (2.2) with some
D arboux basis {f } and introduce a H ubert structure in Z with {f} as a H ubert
basis. The "bounded linear part" JAb can be added to the nonlinear term JVh, so it
may contain Jordan cells and hyperbolic eigenvalues.
We say that a continuous curve z() e C(T 1, T 2;Z) is a solution of (2.4) in Z if
for 7\ ^ t ^ 2 ,
Remark. Define Z A as a H ubert space with the norm \ \ z\ \ A = <(1 + A2)~ 1 z,z) z.
Then Z cz Z A and the linear m ap
Z+ <Z <Z .
and impose the following compactness assumption: for some triad Z + < Z < Z _ as
above the function h can be extended to a C 2 smooth function on Z_ x R. Then
the gradient map Vht:Z Z can be extended to maps
Proo/ . It is sufficient to prove the statement for replaced by CBR with arbitrary
R > 0. In Appendix 1 below we prove (in a traditional way) that the maps S tlt tl(u)
for u e nBR, 0 ^ tl912 S T, exist if | x t2\ ^ (, T ) and are dififeomorphisms.
We can replace (R, T) by ' = (R f,T) ^ (R, T) and write S t as
and we have
|| () || ^ Ci (, ^( )) || 17(0) ||, Ot^T. (2.11)
Lemma 2. Under the assumption of Lemma 1 the flow maps S t are symplectic:
S? = .
a n
Proof We should check that for any two vectors ^10,^20 d 0 ^ t rg 7\
S *[; 1o,^2o] = const .
That is, we should check that
l(t)\ = (Jvi(t) 9v2(t)} = const ,
where Vj(t) is the solution of (2.10) with Vj(0) = vjoj = 1,2.
Infinite Dimensional Symplectic Capacities 539
D enote
lN (t) = (JvMN v2(}.
We have
^lN (t) = (J1,N 2) + (JvuN 2y
at
= < J2(Av + dVh,(z)v),N v2y
+ (JV,N J(Av2 + dVht(z)v2)}
N
= (dVh,(z)v1, v2) + {vi,N dVh,(z)v2y .
Thus,
due to Lebesgue theorem, because by (2.11) the function under the integral is
bounded by some (t, iV) independent constant and pointwise tends to zero.
We shall study Eq. (2.4) for w(0) = u0 e a Z and 0 g t ^ T, where () ^ T.
In addition we suppose that the maps S t: (9 > Z are uniformly bounded for
O^t ST . That is, for each R > 0 there exists R' such that
S t(nBR) c:BR, VO ^ ^ T . (2.12)
To study the solutions with w0 e nBR we can replace / ^(M) by
ht(u) = g(\\u\\i)ht(u), (2 13)
2
where the function geC^(t) is such that g(r) = 1 for |r | ^ . R ' . D enote by
j : Z _ ^ Z + the duality isomorphism:
Then
and the^function h meets (2.8) (2.8"). The equation with the transformed hamil
tonian h has the same solutions for M(0) G nBR and 0 ^ < T.
We say that the hamiltonian / is admissible if it satisfies (2.2), (2.8) (2.8") for
some triad Z + < Z < Z _ it is admissible for 0 ^ ^ T and w0 e C? cz Z if also
T S {) and (2.12) holds for each R.
Definition 2. A symplectomorphism :Z ~=> > Z is called elementary if it is the
S T mapfor some hamiltonian equation which is admissible for 0 : t ^ T and u0 e .
Each elementary symplectomorphism = S t admits the representation (2.9),
where the m ap S t is compact by the assumption (2.8). Each compact m ap can be
approximated by finite dimensional maps. Below we need a symplectic version of
this statement:
Lemma 3. For each elementary , each > 0 and R < oc there exists N such that
(u) = eTJA(I + )(I + )( U) (2.14)
540 S.B. Kuksin
F o r a n o p e n n o n e m p t y d o m a i n c Z a n d ^ l w e d e n o t e
contain a quadratic in z part. Thus the linear operator A is fixed modulo compact
linear operators only. Equations (2.4) with fixed "un boun ded part" of the operator
A forms a natural class of equations which can be studied with the same capacity c.
Compact perturbations of the operator A imply perturbations of the basis
{f }. The corresponding stability result for the capacity c can be stated as follows:
1
Proposition 1. The capacity c() of a domain will not change if the basis {r } is
replaced by another Hilbert basis Darboux {r }, which is quadratically close to the
initial one,
Proof F or m < c() there exists an m admissible function/ which is not fast. That
is, one can find a sequence {nj} such that the vectorfield Vfn has no fast solution.
F or large n^ the fun ction / ^ is m admissible in Hj. Thus, 'c2n(n) : m and the
statement follows.
Proposition 4. For any domain a Z and e Z we have
c{) = c( + ) .
Proof Let us denote = + . It is sufficient to check that c() ^ c(\ because
= so then also c() S c{). We decompose as = n + no (see (2.1);
n0 will be fixed later) and denote x = + n\ Clearly, c(G) = c() (see also
Lemma 5 below) and = (9X + no.
We take any m admissible function/ in with m > c{) and wish to check that
this function is fast. As || n \ \ > 0 as n oc , then
dist(d&,d)^\ \ n\ \ *O as n +oo.
542 S.B. Kuksin
Clearly,
B(r) c D{r) . (3.5)
Theorem 1. For each sequence r as in (3.4),
c(B(r)) = c(D(r)) = r2 .
Proof of the theorem is given in P art 4 below.
Corollary. For r > 0
c(Br) = r2 .
F rom this statement we get nontriviality of the capacity c:
Theorem 2. For each nonempty bounded open domain we have
0 < c() < oo .
Proof. We can find p, R > 0 an d e Z such that
Bp + cz(9czBR.
Thus by P roposition 4 an d the corollary we have p2 ^ c() ^ ^R2.
N ow we turn to invariance of the capacity. We start with a trivial observation.
Lemma 5. / / a map F :Z Z has the form
F (z", z n ) = (F "( Z "), z ), (z",zn)eZ"xZ n, (3.6)
n n
where F is a smooth symplectomorphism ofZ , then c(F{)) = c()for each domain
.
Proof The m ap F an d its inverse are Lipschitz uniformly in bounded subsets. So
F * sends m admissible functions in F() to similar ones in &. F or N ^ n it
transforms the vectorfield Vfs,f = f F, to VfN . So the classes of admissible and fast
functions are preserved by F, an d the result follows.
Infinite Dimensional Symplectic Capacities 543
A much more essential property of the capacity c is its invariance with respect
to elementary symplectomorphisms.
Theorem 3. For any elementary symplectomorphism and any domain we have
= c(())>
provided that the map ~ : (# ) > is bounded (it sends bounded sets to bounded
sets).
Proof of the theorem is given in P art 5.
Remarks. 1) The last assumption of the theorem holds trivially if the set is
bounded.
2) If the m ap " 1 is not bounded, we can only state that c() rg c(()).
3) We need the compactness assumptions Vh e C 1 ( I R x Z _ , Z + ) and (2.8")
only to get the decomposition (2.14) which is essentially equivalent to compactness
of the operator S t in (2.9). So what we really need to prove, that the maps S t preserve
the capacity, is their smoothness jointly with the decomposition (1) from the
introduction.
Corollary (the Squeezing Theorem). If an elementary symplectomorphism sends
a ball
{zeZ\ \ \ z z\ \ <r}
to a cylinder
4. Proof of Theorem 1
varies from zero to . We can find p > 0, yx < and a smooth function/ () such
that 0 f'(t) <, 1, 0 Sf^ m and
544 S.B. Kuksin
Define
2
F(z)= f(^\ \ u\ \
The function F is smooth in Z, it vanishes near zero and equals m outside the ball
B and in the set {u\ ||u || > r^fy~Jy}. Thus, F = m in some neighborhood of x <9#r
in Z, and so it is m admissible.
We wish to show that the function F is n ot fast. We introduce in Z " the
action angle variables / , ,
1
2 2
Ij = (pj h qf), ^ j= arctan (g/ pj) ,
So
We start with
Lemma 6. / / :(9 > Q a Z is an elementary symplectomorphism such that the
inverse map '1 is bounded then the symplectomorphism ~ 1 :Q > also is elemen
tary.
Proof. If is the S flow for some Eq. (2.4) satisfying (2.2), (2.8) (2.8"), then ~
is the .S IOW for the equation with the h am ilt o n ian /
l
f= (Az,zy h{z,T t),
Proof of Theorem 3. We denote Q = {&) and take any m admissible fun ction / in
Q, where m > c(). If we can prove that the function/ is fast, then c(Q) ^ c() and
the result follows, because the suplectomorphism " 1 also is elementary by
Lemma 6 and so c{) ^ c(Q) as well.
We apply Lemma 3 with N so large that < jd(f) and denote by u 2 the
intermediate domains which arise from the decomposition (2.13):
We also denote
TJ
f2= fe \ ei.
As the m ap eTJ is an isometry, then
and the function / i s fast. So for each n P 1 the vectorfield Vjn has in ?" a fast
solution. By Lemma 4 this solution lies in Supp / which equals Su pp/ 2 by (3.3).
Therefore the solution is also a fast trajectory of Vf2t, the function f2 is fast as well
as the function / and the result follows.
L = L 2<>L l9
where (JV) > 0 as N > 00 . F or the m ap L we have got an analog of the decompo
sition (2.14). So we can complete the proof by repeating the arguments we used
above to prove Theorem 3.
546 S.B. Kuksin
6. Examples
where G is a real valued smooth function and u * is the convolution with a fixed
real valued function e if^ "). D enote the second term in (6.1) as ht(u). Then
where positive constants Ck are bounded for bounded t and nonnegative M fc's are
^ independent. We denote by B the operator B = ( d2/ dx2 + 1) 1 / 2 and write the
equation in the form
= Bv,
v = Bu + B 1f(u;t,x). (6.5)
Define the phase space Z of Eq. (6.5) as Z = H 112 xH i/ 2, where
x
) is the Sobolev space with the scalar product
2
< WI , M 2 ) = j Bu(x)u2(x)dx/ 2 .
o
F or the symplectic H ubert basis { ^ |j 6 Z } of Z we take
where
12 sin jx,
' cosj'x, j ^ 0.
6
We remark that this statement is trivial if G =f(\ U\ 2; t,x) and w = 0, p = q = 0 in the
definitions of J?p and / 7 because in such a case the flow maps preserve the norm in Z.
7
We write the nonlinear term in the form u f(u;t,x) for convenience; possibly
548 S.B. Kuksin
J (u, v) = ( v, u) .
Vht = (B 1f{u,{x);t,x),O).
which in its turn results from (6.4) because the space H 1/ 2 is embedded in each
space L p,p < oo .
The relations (2.8'), (2.8") both follow from the continuity of the m ap (6.7).
D
Thus Eq. (6.5) in the form (2.4) is admissible in the space Z and its flow maps
S t preserve the capacity c() of a domain a Z if () ^ . So the "squeezing" (6)
(see Introduction) is impossible for the nonlinear wave equation in the phase space
H 1/ 2 x H 1/ 2 till the blow up time.
We remark that solutions of (6.3) never blow up in particular, if M o = 0 (i.e., if
the function/ is bounded and meets (6.4)). This readily results from representing the
equation in the form (6.5).
Infinite Dimensional Symplectic Capacities 549
A( M , I > ) = f fif(u*,)/ (2) ,
T"
and one again can rewrite the equation in the form (2.4) with Z =
H 1/ 2(J") x H 1 / 2 ( l ) , choose Z _ L 2 x L 2 , Z + H 1 x H 1 and see that the equa
tion is admissible.
N ow Eq. (6.9) (in the form (6.5)) is a hamiltonian equation (2.4) with
By (2.8) the m ap
[ 0, T ] x 2R + Z , (,3) ,V( 3;) (Al)
{t )JA
is bounded an d Lipschitz; the m ap e ~ J is a linear isometry of Z . So F defines
a contraction of the ball B if | 2 t1 | ^ , where is sufficiently small. Therefore
the m ap F has the only stationary point ( ) which defines the unique solution z(t)
of (2.6),
We skip a traditional proof of the fact that the maps S tlt:BR > B2R are
sm ooth. 8 They are diffeomorphisms because the maps S tttl are their inverse.
Proof. We can modify the hamiltonian h as in (2.13). After this modification the
m ap is S flow for some Eq. (2.4), where
ht(u) O for ||u||_ > R
with some R > R'.
We shall use the following statement which will be proven later.
Lemma. For | | ^ T, any R and w, v e BR(Z) we have
\ \ Vht(u) Vht()\ \ SC(R)\ \ u \ \ , (A2)
and
\ \ Vht(u) N Vht(u)\ \ SN), (A3)
where Si(N) > 0 as N oo .
N ow we set hN = htN . Then
and for 0 ^ t ^ T solutions (t) with v(0) in BR(Z) do not leave some ball BR{Z). So
the flow S? is well defined for 0 S t ^ T and sends bounded sets to bounded sets.
By Lemma 1
TJA
S? = N = e (I + N ), (A5)
TJA
where (/ f N ) is a smooth symplectomorphism (as well as S% and e ) and
^v has the form (2.16). Besides
1 N TJA
(/ + ^ ) " = S e
is a smooth bounded m ap of Z .
N ext we estimate the difference N . We denote w() = u(t) (), where
N
u(t) and t () = (y , %)() are solutions of (2.4) and (A4) with w(0) = v(0) = u.
F or w( we get the equation
w JAw = J(Vht(u) Vht(v) + Vht(v) VhN (v)) = (t\ w(0) = 0.
By (A2), (A3) ||zl()|| ^ C ||() v(\ \ + 1(N). So
t t
|| w() || ^ j || zl () || d ^ C j || w() || <i
o o
N ow G ronwals lemma implies that
Ct
\ \ w(\ \ ^C1(N)e .
As
N eTJA( N )
(A6)
for M in BR(Z).
We can write as
= eTJA(I + $N +
( N ))(I + N y\ l + N )
1
e ( / + ( N )o(/ + ^ ) ) ^ + N ) .
To prove (A7) we suppose that the convergence does n ot hold. Then we can find
a sequence {u {n) } a K such that || (/ n)u{n) || ^ > 0. As K is precompact, then
u{n) ^ueZ. For rij > 1 we have
||( / 7 7 "' ) ||< / 2 , ||n( nj) u\ \ < / 2 .
j
So ||((J " )u(rj) || < and we get a contradiction which proves (A7).
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