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d!T>Qe_`91K/U!fhg>ij9kNPl
Non-Linear Stability Analysis of
Higher Order Dissipative Partial Differential Equations
m
m
J.-P. Eckmann1 2 and C.E. Wayne3 4
1
Dépt. de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland
2
Section de Mathématiques, Université de Genève, CH-1211 Genève 4, Switzerland
3
Dept. of Mathematics, Boston University, 111 Cummington St., Boston, MA 02215, USA
4
Center for BioDynamics, Boston University, 111 Cummington St., Boston, MA 02215, USA
nop/qOrts*u6q6v
We extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential
equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One
important example of this type of equation which we analyze in some detail is the Cahn-Hilliard equation. We analyze
the marginally stable solutions of this equation in some detail. A second context in which such equations arise is in the
Ginzburg-Landau equation, or other pattern forming equations, near a codimension-two bifurcation.
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
‰BŠW‹ŒŽ‘“’•”–—˜ŽŒš™jŒ’œ›™žŸŒŽ B¡¢‘›”¤£¥›
2
In this paper, we extend the methods developed in [W1], [W2], [EWW], to study the asymptotic
behavior of marginally stable non-linear PDE’s. These are PDE’s such as
¦a§¥¨ª©¬«®­F¯°¥±³²a´F¨~µ·¶¹¸¥­Z¨´Bº
¨»©¼¨­)½&ºO¾/´ , with ½À¿ÂÁÄà , and where « is a polynomial. In the papers cited above, we
where
«Å­ÇÆa´W©œ¯ÈÆ . In this paper,
have treated essentially parabolic problems, i.e., the case«Åwhere
­)Æ´É© ¯“Æ , where «Å­P¯°˜± ² ´ has
we extend the problem to non-parabolic cases such as
2
4
continuous spectrum all the way up to 0. We deal in particular with the stability analysis of
the Cahn-Hilliard equation [CH] in an infinite domain. Where appropriate, we indicate how to
formulate the assumptions for more general differential operators and non-linearities.
The Cahn-Hilliard equation models the dynamics of a material with the following 3 properties:
i) The material prefers one of two concentrations that can coexist at a given temperature.
ii) The material prefers to be spatially uniform.
iii) The total mass is conserved.
¶ ­Z¨´…©Ë­ ¯$¨ ´ Ì
Ê
¦D§¥¨Í©¬ÎÐÏO¯…㨳µÑ¶¹¸¥­Z¨´/Òº
¦D§¥¨ª© ¯…Î ¨®¯ 4γ¨~µ 4Î~¨ Ó
The first point above means that we should consider a potential with 2 minima with equal
2 2
critical values, and for concreteness, we will choose
1
. The Cahn-Hilliard
equation is then
11
or, expanding,
2
¨­)½&ºO¾O´jÔͨ
3
­ Ó´
­ 1 Ó 2´
We will be interested specifically in the non-linear stability of the spatially uniform states,
0.
It is obvious that constants are solutions of (1.2), for any 0 . Furthermore, it is easy to
3 1 2 , and linearly unstable for
check that these solutions are (locally) linearly stable for 0
3 1 2 . We concentrate our analysis on the remaining case, namely 0
3 1 2 . In this
0
case, linearizing about 0 3 1 2 leads to the linear equation
Õ ¨ ÕÙ × Ø
¨
¨ © × Ø
­P¯Þͺ
Õ ¨ ÕÖ × Ø
¦ §¥Ü ©Ý¯…Î Ü º
2
«Å­ÇÆa´©ß¯ÈÆ
¾jà Þ
¨ ©ÛÚ × Ø
­ 1 Ó 3´
4
which has spectrum in
0] and corresponds to the case
. For this linearized
and the purpose of this
problem, bounded initial data lead to solutions which tend to 0 as
paper is to study under which conditions the addition of the nonlinear terms does not change the
stability of the solutions. This is difficult for two reasons: First, as we have said, the spectrum
of the linearized problem extends all the way to 0, and second, the nonlinearity does not have a
sign.
Considering the Ginzburg-Landau equations (on ),
â
Á
¦ § ¨Í©á¦ ² ¨~µ$¨Å¯Â¨ Õ ¨ Õ º
2
2
­ 1 Ó 4´
In our example, the curvatures of the two minima are equal. This does not seem to be necessary for our proofs.
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
3
we can construct another example of a similar nature. It is provided by those time-independent
solutions of (1.4) which are exactly on the borderline between being Eckhaus stable and Eckhaus
unstable. These solutions are
2
1
¨ãJ­)½´ä©œåæ ãO²ç ¯Ðè º
èW© éaê
with
1 3, cf. [EG]. We believe that the asymptotic behavior of solutions for this problem
is of the same nature as that of the Cahn-Hilliard equations. Here, we describe a program which
we believe would lead to a proof. The first part of the analysis of this problem would follow
rather closely that given in [EWW] for the Swift-Hohenberg equation. Letting
for the
critical value
1 3, and writing
, the equation for is
èW© éaê
¨¢©Í¨ Ì µ Ü
Ü
¦ §¥Ü ©á¦ ²ìÜ µ Ü ¯ 2Ü Õ ¨ Ì Õ Í
¯ Ü í ­Z¨ Ì ´ µªî­ Ü ´ Ó
2
¨Ì
2
2
2
¨Ìë
© ¨ã
­ 1 Ó 5´
It has a linear part which is like a Schrödinger operator in a periodic potential (the inhomogeneity
). This can be handled by going to Floquet variables, namely setting
ã
Ü ò is &ó é è -periodic in ½ :
where
Ü ­Ç½ºO¾/´³©ðï ã dñ å æ5ò ² Ü ò ­)½&ºO¾/´jº
×
Ü ò ­)½&ºO¾O´³© ô å æ õ Oã ² Ü ò m õ )­ ¾/´ Ó
õ•ö÷
2
Üò
¯•î­ ñ ´
The linear part of (1.5) leaves the subspaces spanned by the
invariant, and has discrete
4
spectrum in each such subspace. The spectrum is in
0 and the largest eigenvalue is
when equals its critical value
1 3 (which is the case we discuss here). In this sense,
the problem of the marginal Eckhaus instability resembles the problem of the Cahn-Hilliard
equation. At this point, the discussion of the problem follows the techniques we developed
in [EWW]. We would like to rescale as we will do below for the Cahn-Hilliard equation and
its generalizations, but the problem will be more complicated because the Brillouin zone is
[
]. We then have to check that the non-linearity is “irrelevant” in the
restricted to
terminology developed below. Again, as in [EWW], we believe that this will not be quite the
case, but the saving grace will be that the projection of the potentially non-irrelevant modes
4
onto the eigenstates corresponding to the
term vanish to some higher degrees because
of translation invariance of the original problem, cf. [EWW, Section 4], and [S].
In fact, as T. Gallay pointed out to us after a first version of this paper was completed, one
can probably avoid the use of Floquet variables in this example by defining a new dependent
2
variable through
1
1
. Then the linearized equation for
has constant coefficients and one does not need to introduce Floquet variables to study its
spectrum. As in the previous argument, the spectrum of this linearized operator behaves like
4
when
is near zero, (and
1 3), and hence we expect that the analysis which
follows would allow one to study the long-time behavior of the full nonlinear equation.
è
øÂù
èÄ© é ê
ñ ¿ ¯èDºtè
¯•î­ ñ ´
¨ú­)½&ºO¾/´Ž© ç ¯Àè åcæ Oã ² ­ µ Ü ­)½&ºO¾/´O´
¯•î­ ñ ´
ÕSñÕ
èÅ© éaê
Ü ­)½&ºO¾/´
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
4
We place our examples in the following more general setting. Consider equations of the
¦¨ © F­ ¯ ´Fûü γû¨ýµþÅ­Z¨ºkÿ*¦² ¨>´Bº
¦
¾
1
¯ 1, and ½¹¿ÛÁ à , ¾
satisfy Õ Õjù 2
form
1
¨
¨
þ
­ 1 Ó 6´
where the multi-indices 1. Furthermore,
is a
polynomial in and its derivatives. We wish to study the asymptotics of the solution of (1.6)
as
. First, one introduces scaling variables by defining
¾Bà Þ
¨ú­)½&ºO¾/´ä© ¾ Ã 1 û
Ü Ï ¾ ½ û
º log ¾/Ò Ó
­ 1 Ó 7´
Ø
Ø Æ ©Ë½ é ¾ Ø û
and © log ¾ , the initial value problem (1.6) with
Introducing new
variables
¾j© 1 is transformed to the non-autonomous problem
initial data at
¦ Ü © ­F¯ ´Fûü γû Ü µ 1 Æ *± Ü µ Ü µÑå ìþÅ­`å Ü ºkÿ*å ¦ Ü >´Bº ­ ´
¦
1
1Ó 8
× × 2
2
© 0. The analysis of this equation involves two steps:
with initial data at 2
1
1
2
2
2
1
2
2
2
i) An analysis of the linear operator
ii) A determination of which non-linear terms are relevant.
é ­ ´PƱ
plays an important rôle in the analysis of this linear
As we will see, the term 1 2
operator as it allows us to push the continuous spectrum of the operator more and more into the
stable region by working in Sobolev spaces with higher and higher polynomial weights. These
weights force the functions to decrease more and more rapidly near
. Taking Fourier
transforms on both sides of (1.8) we obtain:
Õ½ Õ © Þ
¦!Ü © ¯ ­#"$%"´Pû&Ü ¯ 1 "'*±)(Ü µÑå ìþ ­`å Ü ºkÿ*å ­P¯°*"´!Ü >´hº ­ ´
¦
1Ó 9
Ì × × 2
þ
þ
where Ì is the polynomial , written in terms of convolution products, (see the discussion of
2
2
2
2
the non-linearities below).
We will discuss the form of the non-linear terms below, and consider first the linear operator
+
© ¯ ­#"'%"´ û ¯
A straightforward calculation shows that
,.-
+
has the countable set of eigenvalues
© ¯0/ º
2
± ( Ó
1 "
2
/
© 0 º 1 º 2 º ÓcÓcÓ º
with eigenfunctions (written in multi-index notation),
Õ Õ ©
and 1
/ .
­2"´©3" å ×
(4 ( º
­ 1 Ó 10´
­ 1 Ó 11´
­ 1 Ó 12´
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
+
If we consider
5
as acting on the Sobolev spaces
© 8!Ü
Õ “Õù?> º Õ @ÈÕìù?ACB º
+
there
a constant ø õ Ö 0 such that will have continuous spectrumÞ in the half-plane
¯
, exists
ø õ in addition to the eigenvalues above. Since ø õ is increasing to with A , we can
Re Ù
mõ
576
: 9
¦ Ü
": :(<
;
9
L2
Ù Þ º
for all =
force the continuous spectrum arbitrarily far into the left half-plane by choosing A appropriately,
and the dominant behavior of the linear operator will be dictated by the eigenvalues with the
largest real part.
Á$DFEHGJILKNM
+
In order to switch back and forth from the Fourier transform representation of
to the un-Fourier transformed representation of this operator with ease, we also consider the
Sobolev spaces
© 8Ü
Õ ÕùP> º Õ @ÈÕìùQARBÍÓ
5 6
5 6
Note that Fourier transformation is an isomorphism from m õ to m õ .
+
Note that is not sectorial, and therefore we know of no way to bound the semi-group
mõ
5O6
: 9
¦² ½:; Ü
9
L2
Ù Þ º
for all +
generated by by spectral information alone. However, in Appendix A, we develop an integral
representation of the semi-group and we then show that it satisfies the estimates needed for the
invariant manifold theorem.
We next discuss which terms in the non-linearity are “relevant.” Consider a monomial
¦² ¨ ò º
©
-
­ 1 Ó 13´
U T X ZY\[^] _ [
-WV
S
0
where the are distinct multi-indices. After changing variables as in (1.7), and taking Fourier
transforms in this becomes
½
©
­ 2 µ ´ Ja
S
µ Ò T Ï Õ
¯
ô
Õ
ñ Zed
exp `
exp bc
-WV
2
2
Ï ­F¯“°*"´ ZY ] ÜDÒ Ì ò g hh g Ï ­F¯“*° "´ ZYji] ÜDÒ Ì ò i Ó
0
f
­ 1 Ó 14´
0
0
Here, g denotes the convolution product. If we combine the powers of in the exponential, we
see that if
2
µ
Ù ô-kV T Ï Õ Õ µ Ò ñ - º
­ 1 Ó 15´
0
then the coefficient of this term will go to zero exponentially fast in , and hence it will be
irrelevant from the point of view of the long time behavior of the solutions.
l Dnm!oqprLptsoquvM
A monomial like (1.14) is called irrelevant if it satisfies the inequality (1.15). It
is called critical if the l.h.s. of (1.15) is equal to the r.h.s, and relevant in the remaining case.
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
6
These definitions are suggested by the following which is our first main result:
w x Dns:ILDFE
y
z:M{z:M
all terms in the non-linearity in (1.6)
For
any
solution
¨­)½&ºO¾O´ of (1.6) withAssume
­
¯
´
µ
5 6 are irrelevant.
õ
é 2 and
sufficiently
small
initial
conditions
in
(with
•
>
Ö
2
1
m
­
µ
µ
µ
´
A Ö 2> ñ 1 ), there is a constant |Ì , depending on the initial conditions, such that
for every } Ö 0,
½ ´ƒ ©
{§ lim
~€ ¾  :× ‚ ƒƒ ¨­)½&ºO¾/´&¯ ¾ à | Ì û v„ Ì ­ ¾
0Ó
Ø hØ û ƒƒ
ƒ
ƒ
ƒ
1
2
2
Here,
„
1
2
­Ì )Æa´³© ­ 1´ à ï dà "¤å æ (4 å ×
2ó Ø
2
L
(4 ( ­ 1 Ó 16´
Ó
Á$DFEHGJILKNM This theorem is a special case of a more detailed analysis which will be given below.
That
us to compute, in principle, the form of the solutions of (1.6) up to
î­)¾ × ò analysis
´ , for anywillñ allow
Ö 0. We note that if one only wanted the first order asymptotics of the
solution, one could also use the renormalization group analysis of [BKL]. ¨Â©
µ † , the
Q
We †now apply Theorem 1.1 to the Cahn-Hilliard equation. Writing
3× Ø
1 2
function
is seen to satisfy
¦† ©Ý¯…Î †$µ ÎÉ­‡† ´µ Î ­{† ´
¦
¾
Ó
4ê 3
4
2
­`¦a² ¤´
ÎÉ­‡† ´
2
­ 1 Ó 17´
3
2
Upon expanding
we obtain two types of terms—those of the form
Lˆ‰† 2
. In both cases,
the form
ô ­ Õ= - Õ µ ´ ñ - ©
~­Z¦ ²Lˆ‰†•´ and those of
†
2
µ 2Ó
© 2 in this example, these terms will be irrelevant if 4 µ Ù 2 µ 2, that is in dimensions
Since Î ­‡† ´ is irrelevant for Ö 1. Thus, as a corollary to Theorem 1.1 we
Ö 2. Also, the term
2
3
get immediately
Š s:ILs‹{‹tGŒIŽz:MM
µ ­ é ´
¨¢Ô × Ø
¨­)½ºO¾/´³©
Á$DFEHGJILKNM
mõ
Ö
3,µ with
initial
­
µ
´
2 ) to the
Ö
Solutions of the Cahn-Hilliard equation in dimension 5 6
2 , and A
conditions sufficiently close (in
, with >
3
2 >
1 2
constant solution
3
behave asymptotically as
µ ¾ |Ã Ì „ Ì Ï ¾ ½ Ò µƒî Ï ¾ 1à ü 1
Ò Ó
­ 1 Ó 18´
Ø
Ø
Ø :× ‚
3Ø
© 1 º 2. The case © 2 is of
We will examine below what happens in the cases 1
1 2
4
1 4
1
4
particular interest because its non-linearity is critical in the renormalization group terminology.
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
‘ ŠW‹“
Œ ’ ™ —˜™jŒŽ ž ™jŒ¤—P¡4h£P’¤›
7
+
Note that spectral subspaces corresponding to eigenvalues of
are automatically invariant
manifolds for the semi-flow defined by the linear part of (1.9). The aim of this section is to
demonstrate that the full non-linear problem has similar invariant manifolds in a neighborhood
of the origin. This then shows that the conceptual understanding of what is happening can be
+
gained purely from a knowledge of , (and the scaling behavior of the non-linearity).
+
We begin with a proposition concerning the linear semi-group generated by .
«ò
” I•sŒ–—s:uhprLp{s:o˜- M‡zŒM
Let
denote the projection onto
576 the spectral subspace associated with the
k
V
eigenvalues ™ 2›š 0 , and let œ
1
(in
). If A
2 >
1 , then there
+
0 such that the semi-group generated by satisfies
exists 
hŽž
6
6

1
21
exp
0 1
2
1
9Wœ
œ
9
9 9
2
2
” I•sŸsŒ ¡M
­ µ µ ñµ ´
ò ©Ê­ À¯ « ò ´
m õ ðÖ
ò Ö
åò ò Ü õ ù ¾ ã ò û Ï ¯ ñ µ Ò Ü ã õ ºtè © º º ÓcÓcÓ º ¯ Ó ­ Ó ´
m
Ø
× m
The proof,
© 1.which is presented in Appendix A, is modeled on the proof in [EWW] which
treats the case Given such estimates on the linear evolution, the construction of invariant manifolds
is
«
straightforward. Denote by ¢ the coordinates on the (finite-dimensional) range of ò , and let
£ © œ ò Ü . Finally let ¤ ©ëå × hØ û ©ë¾ × Ø û . Then, applying the projection operators « ò and
œ ò to (1.9), it can be written as the system of equations
© ¦ ò ¢ µ „ ­ ¤ º ¢ º £ ´jº
¥ §
¢
µ ¨­ ¤ º ¢ º £ ´úº
­ 2 Ó 2´
£¥ © œ ò + £ R
¥ ©Ý¯
¤
û¤ º
j× û ò
2
1
2
1
2
¥
where “ ” denotes differentiation w.r.t. . We next need a bound on the non-linearity:
•Ö 2 ¯ 1 µ é 2, and assume
µ
ô
T ­ µ ´ ù -WV Õ Õ ñ Ó
2
5 6
ûü m õ norm bounded by
Then the non-linear term (1.14) has
×
( U
(
T Ü 6ò [ ©
Ü 6 º
)¤ -kV 9 9 m õ
¤ 9 9W« m õ
"Ä©­¬ -kV ­ µ ´ - ¯ƒ­ µ ´
©
­
¬ -kV ©
k
º
ÿ
Ï
- ºkÿ ¯ Ò
Õ= Õ ñ 2 , ®
 ñ
with
.
T
T ñ , and 
©ëå × Ø û
, Eq.(1.14)
” I•sŸsŒ ¡M
Taking the inverse Fourier transform of (1.14), and substituting ¤
becomes
( U X
T ¦ Y\[^] Ü _ ò [
¤ -WV
Ó
© D.EªEHGPMM
Assume
Ü ¿
5 6
mõ
with >
0
2
1
0
0
0
2
0
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
8
 to each factor, and observing
The result then follows from the Sobolev embedding theorem
that the choice of > guarantees that each factor is in fact in L . Note that the lemma has the
immediate corollary (because is a polynomial):
þ
Š s:ILs‹{‹tGŒIŽ­M±°ŸM
Á
Under the hypotheses
of 6 Lemma 2.2, for every ²
5 6
5
(1.9) is a ³Ÿ´ function from f
to
2
1 .
mõ
ûü m õ
×
This corollary in turn implies that the terms in (1.14) and (2.2) are all ³´
1, the non-linear term in
functions. This, in
conjunction with the estimates on the linear semi-group is sufficient to establish the following
wyx Dns:ILDFE
¯ µ é 2 and AáÖ 2 ­ > µ µ ñ µ 1´ . Suppose further
Ö
Mjµ¶M
Suppose that >
2
1
that all terms in the nonlinearity satisfy
µ
T ­ µ ´ ô
­ 2 Ó 3´
ù -WV Õ Õ ñ Ó
2
üŸ function · ­ ¤ º ¢ ´ , with ÛÖ 0, defined in some neighborhood of the
Then there exists a ³
Á f Á
j¸¹ , such that the manifold £ © · ­ ¤ º ¢ ´ is left invariant by the semiorigin in
flow of (2.2). Furthermore, any solution of (2.2) which remains near the origin ¹for all times
î Ï å º» Ò .
approaches a solution of (2.2)—restricted to the invariant manifold—at a rate
0
1
dim range
1
2
” I•sŸsŒ ¡M
The existence of the invariant manifold, given the assumptions on the linear semi-group
and the non-linearity, seems, to our knowledge, not to be explicitly spelled out in the literature.
The formulation which comes closest to our needs is the one given in [H], where the assumptions
on the non-linearity are those we have in our case, but the semi-group is supposed to be analytic.
However, Henry’s construction of the invariant manifold only uses certain bounds on the decay
of the semi-group, and not the stronger assumption of analyticity. Those bounds are true in our
case, by Proposition 2.1.
To be more precise about exactly how one constructs the Winvariant
manifold, note that we
are looking for a function · ¤ ¢ , whose graph ¤ ¢ · ¤ ¢
is invariant with respect to the
semiflow defined by (2.2). A standard
calculation then shows that · should satisfy

­ º ´
­ º ´
­ º ´ä© »¯ ï ×
· ¤ ¢
ÿD­ º ´6º ­ º ´
­ òå×
œ
0
žŸ
œ
ò ´¼¨ Ï 1 ­ ¤ º ¢ ´6º ·½ 1 ­ ¤ º ¢ ´ Ò º
where 1 ¤ ¢ is the flow defined by
¥
1
¤
¤
2
¥ ¾¦
¢
¢ „ ¤ ¢ · ¤ ¢
À¿
kÁ
To prove that · exists, one finds a fixed point of the map · 1
· 1
by

žŸ ¼¨ 1
œ
¤ ¢ ·Â½ 1 ¤ ¢
· 1 ¤ ¢
œ
©š¯
º
© ò µ Ï º º ­ º ´Ò Ó
­ º ´ à Ï þÅ­ º ´6º ­ · º 1 ´ Ò
­ 2 Ó 4´
­ 2 Ó 5´
defined
Åþ ­ º k´ ­ º ´~©š¯ ï × ­ ò å × ò ´ Ï ­ º ´6º
­ º ´Ò º
­ 2 Ó 6´
§
§
§
§
Á®­ º 1 ´F1§ ­ º ´~©
å û º åŽÃ ¹ ¢ µ ï .Ä Lå à ¹Ž × T „ Ï 1 ­ ¤ º ¢ ´6º ·Â½ 1 ­ ¤ º ¢ ´ Ò ašÓ
·
¤ ¢
` × Ø ¤
0
2
0
T
T
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
9
«ò
Á Áò Ã
mõ
From this point we follow closely the argument of Gallay [G, pp.257–258]. Let
the
ÈÇ be
¹ , and
dimension of the range of , and define Å:Æ to be a neighborhood of the origin in
5 6
let Å T be a neighborhood of the origin in the range of œ
(equipped with the
norm). We
1
assume that · and are elements of the metric spaces
ò
­ º ´©
Õ
­ º ´¯ ­ º ´ õ ù·ø ­ Õ ¯ Õ µ ¯ ´ ¤º
m º ´© ­ º ´
©
ÿ
Á
à
­
­ 2 Ó 7´
ü§ ­ º ´j© ­ º ´ Õ
® ;
¾hº § §
¾
¾@¿ÁΕº
§ ­ º ´¯ ­ º ´ …ù å ­ Õ ¯ Õ µ ¯ ´
ÁÄÃ ¹ for ¢ , and is a positive constant, smaller
where­ we use
the ordinary Euclidean norm in
´
þ Á
than ñé 2 . We now apply the contraction mapping theorem to prove that and in (2.7)
have fixed points. The only difference with the estimates of [G] are that in the present case the
§ derivatives and we must take advantage of the smoothing properties of
nonlinear terms “loose”
ž
å
the semigroup œ ò
œ ò to recover them.
þ®­ º ´ 5É , paying particular
We first show, following the estimate of [G, p.258] that · 1 is in
©Ýÿ ·
5É
:Å Æ
à
Å T L· 0 0 0 ;
6
Ê
Ë W
9· ¤ ¢
· ¤ ¢ 9
¤ ¤
9k¢ ¢9
f Å Æ
1 :
Å Æ 1 0¤ ¢
¤ ¢ ;
1 00
0 0 for all 1 is continuous in ;
Ë
1 ¤ ¢ 9 ?Ì Ž ¤ ʤ
91 ¤ ¢
9k¢ ¢Í9 for all
× ­ û º ü º m õ ´
attention to the way¨ in which our case differs from Gallay’s. From Corollary
2.3 we see that
Ç
57
6
Å T to
the nonlinear term in (2.2) is ¨ a Lipshitz function as a map from Å Æ
and
2
1
if >Ï is the Lipshitz constant of , one can make >ÐÏ arbitrarily small by restricting ¤ ¢ £ to a
sufficiently small neighborhood of the origin. We can estimate
þÅ­ · º 1  ´k­ ¤ º ¢ ´¯Àþ®­ · º 1 ´k­¤ º¢ ´ 9 6 m õ
ù ï ×  y9 ­ œ ò å × žŸ œ ò ´ Ï ¨ Ï ­ 1 ­ ¤ º ¢ ´6º ·›½ 1 ­ ¤ º ¢ ´ Ò ¯¾¨ Ï 1 ­¤ º¢ ´6º ·›½ 1 ­¤ º¢ ´ ÒÒ
ù  ï × ýÕ úÕ<× • º å × ¹ Ò Ò
Ñ
f ƒ ¨ Ï 1 ­ ¤ º ¢ ´1º ·›½ 1 ­ ¤ º ¢ ´ Ò ¯¾¨ Ï 1 ­h¤ º¢ ´6º ·½ 1 ­¤ º¢ ´ ƒ 6
ƒ
ƒ × û × mõ
ùѝ ï × ýÕ úÕ<× • º å × ¹ Ò Ò
f > Ï Ï 9 1 ­ ¤ º ¢ ´&¯ 1 ­ ¤ º ¢ ´ 9 µ 9·)½ 1 ­ ¤ º ¢ ´¯ ·Ó½ 1 ­¤ º¢ ´ 9 6 õ Ò

m
¹
ù  ï × ýÕ úÕ × • º å × Ò Ò > Ï ­ 1 µ ø ´/åŽ Ò Ò ­ Õ ¤ ¯Ê¤&Õ µ 9k¢ ¯Ô¢Í9 ´
Ñ
­ º ´
¯ ¤&Õ µ 9k¢ ¯Ô¢¶9 ´ Ó
 ñ
ù ¯ƒ­ ñ µ 1´ é ­ 2 ´ > Ï ­ 1 µ ø ´k­ Õ ¤ Ô
þ®­ º ´ 5É provided
Thus, we find that · 1 is in
­ º ´
 ñ
¯ ƒ­ ñ µ 1´ é ­ 2 ´ > Ï ­ 1 µ ø ´ Ù·ø»Ó
9
0
2
1
2
9
6
mõ
1
2
0
2
2
1
2
1
1
2
0
2
1
2
1
2
0
­ 2 Ó 8´
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
10
­ º º ´
which we can insure by taking ¤ ¢ £ in a sufficiently small neighborhood of the origin (since
as we noted above, this results in > Ï becoming small).
Note that the only difference between this estimate and the corresponding estimate of [G,
p.258] is that we used the smoothing property of the semigroup that comes from Proposition 2.1,
1
® ;
while Gallay did not assume
Á that his semigroup was smoothing. The proof that ® ·
and the proofs that and are contractions follow
¨
žŸ as in [G, pp.258–259], with the one change
œ
œ
to
overcome
the
loss
of
derivatives
in
.
that we must use the smoothing property
of
Á
Once we have shown that and are contractions, the fixed point · gives the invariant
manifold
whose existence is asserted in TheoremŸ2.4, though this shows only that · is Lipshitz,
not ³ 1 as claimed. To prove that · is in fact ³ 1 , one follows the proof
žŸ of [G, Lemma 2.10],
again with the sole change that when one estimates the factors of œ
œ which occur in the
Á
mappings Õ and Õ , one must use the smoothing of the semigroup.
Once one knows that the manifold exists, it is also easy to show that any solution which
remains near the origin must approach a solution on the invariant manifold (see, e.g. [C]). Note
that even though our non-linearity is quite smooth, we Ÿcannot hope, in general, to obtain an
invariant manifold whose smoothness
is greater than ³ 1 , since this smoothness is related to
¦
+
the gap between the spectrum of , and that of œ
œ , (see, e.g. [LW]).
þ
ü
@Š
­ º ´¿
òå
þ
ò
ü
þ
ò
¤£F—˜–™—PhŒ›
òå
ò
ò
ò
ü
Ö ÓטØÂØ
Here, we show how the existence of the invariant manifold implies Theorem 1.1 and related
results. To prove Theorem 1.1, we assume that all terms in the non-linearity are irrelevant.
This means that (2.3) holds. Suppose further that
1 and that >
2
1 2 and
2 >
1 . These hypotheses guarantee that Theorem 2.4 applies and hence any
A
2 º» 2
solution near the origin must approach a solution on the invariant manifold, at a rate
576
in
.
The equations on the invariant manifold can be written as a system of ordinary differential
equations:
Ù
ŽÚ
¥
ñ ©
¯ µ é
î Ïå
ÀÖ
Ö ­ µ µ ñµ ´
Ò
mõ
© 1 Ì Õ „ Ï ¢ º · ­ ¤ º ¢ ´6º ¤ Ò º
¢
© 1 º ÓcÓcÓ º º
­ 3 Ó 1´
- µ Ù 1 - Ï º ­ º ´6º ÒLÚº
¥ - © ¯
/
¢ m
¢ m
Ì
Õ
û
„ ¢ · ¤ ¢ ¤
m
¥ © ¯
¤
û¤ º
­ ´
,
, ©ð¯
where 1 Ì and 1 Ì m are the projections onto the spectral subspace of
1 é 2 ,
and
,
respectively. Note that has a -dimensional spectral subspace.
0
0
1
2
1
2
1
0
1
1
1
0
1
1
The important observation to make at this point is that since the non-linearity is assumed
to be irrelevant, there exist constants  0 and  1 such that
ÛÙ 1
Û
"R
for some
behave as
(
(
ÛÙ 1
º
­
º
1
´
º
º
º
­
º
6
´
º
Ï
hÚ ÛÛ
Ò
Ï
ŽÚ ÛÛ
Ò
Û
Ì Õ „ ¢ · ¤ ¢ ¤ ùܝ ¤
Ì Õ „ ¢ · ¤ ¢ ¤ ùܝ ¤ º
­ ´…©Ÿå × Ø û
¤ ­ 0´ , this implies immediately that solutions of (3.1)
1. Since ¤ ­ ´ä© | Ì µªî­`å × Ø û
F´hº
¢ - ­ ´ä© î­`å Ø û
P´
¢ m Ó
×
0
0
1
1
2
2
0
2
1
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
11
(
"Ý
(
Furthermore, because the nonlinear terms are proportional to ¤ , for some
1, the
function · whose graph gives the invariant manifold will also be proportional to ¤ . This means
that as 0 (i.e., ¤
0), the center manifold becomes “flat” —that is, it coincides with the
,
,
eigendirections
to the eigenvalues 0 and 1 . The eigenfunction with eigenvalue
(4 ( corresponding
+
5 6
, or taking inverse Fourier transform, „ , cf. (1.16). Thus, in
solutions
0 of is (
4
(
of (1.9) behave as
à
à
å×
mõ
Ì
µªî­Zå × Ø û
´ Ó
¨­)½&ºO¾O´ and using the Sobolev lemma

Reverting from scaling
variables to the unscaled variables
5 6
to estimate the L norm in terms of the
we
m õ norm, we obtain Theorem 1.1. Since
observed above that the non-linearity in the Cahn-Hilliard equation is irrelevant when¨Ä Ô 3, we
immediately see in this case that (1.18) holds for initial conditions which are close to
3× Ø ,
Ü ­#"º ´³© |Ì å ×
2
1 2
which yields Corollary 1.2.
Š Û–— —P–™j£³–™ ›
Þ ÓßHà
©
We now consider the Cahn-Hilliard equation in dimension 2, which is the critical case
in terms of the renormalization group terminology [BKL]. This means that in some non-linear
terms the inequality (1.15) becomes an equality.
In the Cahn-Hilliard equation, when 2 (and 2), we see that the quadratic term is
critical, and the cubic term is irrelevant. Note that Theorem 2.4 still implies the existence of an
,
invariant manifold tangent at the origin to the eigenspace of 0 . This means that when written
in the form of (2.2), the non-linearity can be written as the sum of 2 pieces—one quadratic in
¢ and £ which is independent of ¤ (and hence critical) and a cubic piece in ¢ and £ which is
linear in ¤ (and hence irrelevant). This implies that the Eqs.(3.1), when reduced to the invariant
manifold, take the form
©
Õ „ Ï ¢ º · ­ ¤ º ¢ ´6º ¤ ÒŽÚ µáÙ 1 Ì Õ „ Ï ¢ º · ­ ¤ º ¢ ´1º ¤ ÒhÚ º
Ì
© 1 º 2 º ­ 4 Ó 1´
¥ - © ¯
- µáÙ 1 - Ï º ­ º ´6º ÒŽÚ µáÙ 1 - Ï º ­ º ´6º ÒLÚ º
/
¢ m
¢ m
Õ
¢
·
¤
¢
¤
Õ
¢
·
¤
¢
¤
Ì
Ì
„
„
m
m
¥ © ¯
¤
¤Ó
Î ­‡† ´&µ 4Î ­{† ´ ,
ê
We now exploit the form of the non-linear
term
in
(1.17),
namely
4
3
Ô 1. Thus if we integrate by parts, we find that
plus the fact that the eigenfunction 1 Ì
Ù1
Ì Õ „ Ï ¢ º · ­ ¤ º ¢ ´1º ¤ hÒ Ú µ Ù 1 Ì Õ „ Ï ¢ º · ­ ¤ º ¢ ´6º ¤ ŽÒ Ú © 0 º
­)¾/´B© ¢ ­ 0´ . This means that the invariant manifold contains
¥ Ô
so that in (4.1), ¢
0 and thus ¢
a curve of fixed points,
that any solution near the origin approaches one of these fixed
and
î
`
­
å
´
Ø
points with a rate
× . Note further that since the quadratic term in the nonlinearity
¢
1
¥
0
©
©
Ù1
2
3
0
1
4 1
1
4
0
2
3
1
1
2
3
0
2
3
0
0
0
0
0
4
is independent of ¤ in this case, the center manifold will not be “flat” as it was in the case
of an irrelevant nonlinearity. Thus, while to lowest order, the fixed points in the invariant
manifold will be proportional to the eigenvector with eigenvalue zero, there will be higher order
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
12
corrections which can be computed perturbatively by computing the terms in the Taylor series
for the invariant manifold.
4
Since from (4.1) we also see that ¢ 1
, we find upon reverting to the unscaled
variables the second main result:
wyx Dns:ILDFE⵶M{z:M
For 5
´
m © î ­`å × Ø
©
¨¢Ô × Ø
¨­)½ºO¾/´³© 1 µ ¾ 1 „ Ì Ï ¾ ½ Ò µƒî Ï ¾ 1 Ò º
Ø
Ø
Ø :× ‚
3Ø
where „ Ì is one of the fixed points on the invariant manifold. We denote it by „ Ì to indicate
+
that to lowest order it is equal to the function „ Ì which is the eigenfunction of the operator
close in
2, if the initial conditions of the Cahn-Hilliard equation are sufficiently
3 1 2 , then the solution behaves asymptotically as
5 18 to the stationary state
m
1 2
1 2
1 4
3 4
with eigenvalue zero, but it will have higher order corrections coming from the curvature of the
invariant manifold.
Á$DFEHGJILKNM
Note that this result implies that in contrast to the situation in 3, the long-time
asymptotics are no longer given be the solution of the linearized equation—nonlinear effects
enter even at lowest order.
Š Û‘£˜Ÿ’ ™jŒŽ·–™ ›
ã ÓßHà
©
Here, we consider the case of 1 where one term of the non-linearity is relevant. This
necessitates a change of strategy, because the quadratic term is proportional to ¤ 1 and hence
the non-linear terms in (2.2) are not smooth enough to apply the invariant manifold theorem. In
order to circumvent this difficulty, we choose a scaling
different from (1.7). Consider again the
P†
3 1 2
. In 1, we get
Cahn-Hilliard equation, (1.17), with
×
¨¢© × Ø µ
©
¦† ©Ý¯ ¦ $† µ
¦ φ Ò µ ¦ φ Ò
­ 5 Ó 1´
¦!¾
¦!½
4 ê 3 ¦!½
4 ¦!½
Ó
†~­)½&ºO¾/´ ©Í¾ Ø ¶ ­Ç½ ¾ Ø º
¾/´
¶ satisfies
Now let
log . Then
é
×
¦¶ © ¯¦ ¶ µ 1 ƛ¦ ¶ µ 1 ¶ µ Ø ¦ Ï ¶ Ò µÑå Ø ¦ Ï ¶ Ò
­ 5 Ó 2´
¦
Ó
3
×
4
2
¯…¦ µ Æì¦ µ , which in Fourier
Proceeding as in the other cases, we define the linear operator
variables becomes
¯ " ¯ "!¦ ( µ º
+ © N
º ÓcÓcÓ . Thus, unlike the operator + , we have one
- ©
× , / © 0 º 1©Û
so that it has eigenvalues ä
å × Ø , and let ¢ and ¢ denote the amplitudes
eigenvalue lying in the right half-plane. Let ¤
of the eigenvectors with eigenvalues ä and ä . Then (5.2) takes the form
µ „ ­ ¢ º ¢ º ¤ º ¢Få ´hº
¥ ©
¢
¢
­ º º º ´hº
¥ ©
¢
­ 5 Ó 3´
„ ¢ ¢ ¤ ¢Få
º
¥ ©Ý¯
¤
¤
µ „ å ­¢ º¢ º¤ º¢ å ´ Ó
+
¥ ©
¢ å
œ
¢ å
4
2
2
2
4
1 2
3
2
2
1 4
1 2 2
4
2
2 2
4
4
1
4
1
1
4
1
2
1
4
1
4
8
0
0
0
1
4 0
1
1
1
0
0
0
1
1
1
8
1
0
1
3
1
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
©
13
¶
Here, œ is the projection onto the complement of the eigenspaces corresponding to ä 0 and
, and „ 0 , „ 1 , and „ å are the projections of the non-linearity onto the various
ä 1 , ¢Zå
œ
subspaces.
+
1
Since the spectrum of œ 1 œ lies in the half-plane Re ä
4 , we can construct an invariant
manifold for (5.3) which is the graph of a function · å ¢ 0 ¢ 1 ¤ , and every solution of (5.3)
4
.
which remains in a neighborhood of the origin will approach this manifold at a rate
What is more, the equations on the invariant manifold are extremely simple in this case, since
the projections onto the “0” and “1” components correspond to integrating with respect to the
functions 1 and , respectively. Applying these projections to the non-linearity and integrating
once, resp. twice by parts, we see that these projections of the non-linear terms vanish. Thus,
the equations on the invariant manifold of (5.3) are simply
­ »º ù º ´
î­`å × Ø ´
½
¢
©
¥
0
©
¢
1
4 0
©
º
¢
¥
1
©
0
º
¤
¥
© ¯ ¤ Ó
1
8
­ ´
Note that these equations again imply that there is a line of fixed points in the invariant manifold
corresponding to ¢ 0 0 and ¢ 1 ¢ 1 0 . Just as in the two dimensional case, these fixed points
+
will be tangent at the origin to the eigenvector of 1 with eigenvalue zero, and higher order
corrections to this first approximation can be computed perturbatively from the equation for the
invariant manifold. Thus, as long as the solution of (5.3) remains in a neighborhood of the
origin, it will be of the form
4
¢0
¢00
¢
1
¤
¢ å
­ ´©¬å Ø ­ ´hº
­ ´© ¢ ­ 0´hº
­ ´©¬å × Ø ¤ ­ 0´Bº
­ ´© î­`å × Ø ´Bº
­ 5 Ó 4´
1
8
4
and we see that the solution either leaves the neighborhood of the origin, or it approaches one
of the fixed points on the invariant manifold. Note that the solutions that remain near the origin
must have ¢ 0 0. Thus:
©
wyx Dns:ILDFE
æM{zŒM
Suppose
of the Cahn-Hilliard
equation
† that the initial condition
is of the form
6
5
1 2 á†

3
norm for some >
4 and A
15. Assume
0
0 with† 0 small in the

d 0
0. Then the solution is of the form
furthermore that ç
¨ © × Ø µ
where „
” I•sŸsŒ ¡M
Ì4Ì
×
½ ­)½´B©
¨­)½ºO¾/´³©
mõ
µ ¾ 1 „ Ì4Ì Ï ¾ ½ Ò µªî Ï ¾ 1 Ò º
Ø
Ø
Ø :× ‚
3Ø
1
1 2
1 2
1 4
3 4
is one of the fixed points on the invariant manifold.
The proof is an obvious modification of the one of Theorem 1.1, taking into account
the special form of the eigenfunctions corresponding to the eigenvalues ä 0 and ä 1 .
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
טØÂØ Œ’•‰— è ê
Š é Ž”¤Œ’¤› hŒ Zà £P—FŒ™  ›žŸ—^ë•ìB‘Ž”ÂØ
14
In this appendix, we sketch the proof of Proposition 2.1. The proof is quite similar to the
estimates on the linear semi-group in Appendix B of [EWW], (which was given for the case of
1) so we concentrate only on
a one-dimensional Laplacian, or in the present notation the points where the present argument differs from the one in [EWW].
We begin with the representation
© ©
Ï åŽŽž ÜÒ ­)½´© å ¯ à ï dà £ ¨ ­ £ º ´ ÜúÏ å )­ ½³µ £ ´ Ò º
2ó
­ A Ó 1´
­ º ´ä© ï dà ñ å æ5ò 4=í exp Ï ¯ ­ ñ ñ F´ û­ 1 ¯ å × a´ Ò Ó
­ A Ó 2´
2
2
where
¨ £
As in [EWW], the action of the semi-group is analyzed by considering separately the behavior
of the part far from the origin and that close
¨ to the origin. The new difficulty here is that we
do not have an explicit representation of as in the case 1. However, the technique of
estimating the long-time behavior will remain essentially the same. Let î¶ï be a smooth cutoff
?ð and is equal to 1 for
function which vanishes for
4ð 3. We start by studying the
region far from the origin. The analog of Proposition B.2 of [EWW] is
Õ ½ ÕÙ
” I•sŒ–—s:uhprLp{s:oòñêM‡zŒM
such that for all
Ü ¿
Õ ½ ÕÖ
57For
6 every > 0 and every A
one has
mõ
ƒ6
ƒ
 > A
‰ô
öõ
2
2
2
é
Ö
0, there exist a ó
­ º ´ å Ã1ü 6 X å õ Ø µ å × ÷ ï
ù
×
­
´
Ï
Ò
ƒ
ƒ mõ
è © 0 º 1 º cÓ ÓcÓ º 2 ¯ 1. Here, ô ­ ´j© 1 ¯ å × .
for
å Ü
hŽž
Ħ
ƒî ï
©
2
Žø Y 2 Lº 1%] _
0 and a 
Ü
9 9
6
­> ºA ´ Ù Þ
­ A Ó 3´
×
ã mõ º
­ º ´
¨
The crucial step in proving this estimate is to derive the asymptotics of £ for large
£ . This will replace the explicit (Gaussian) estimates for the 1, 1 case analyzed in
[EWW]. This estimate is provided by the following
” I•sŒ–—s:uhprLp{s:oÈñêMtM
The kernel
In fact, one has the estimate
Á$DFEHGJILKNM
©
­ º ´
ÕÕ
¨ £
decays faster than any inverse power of £ for £ large.
Õ ¨­ £ º ´ Õjùܝ ô ­ ´ × exp Ï ¯ ó ­ Õ £ Õ û é ô ­ ´O´ •º Ò º
© ó ­ º ´ Ö 0.
© 1, we recover the explicit bound on the Green’s function:
If 2
2
for some ó
©
ô
í
å
Wà ¡Ø ù• L
m
­ ´ Ã Ø û
×:÷
Ó

2
2
2
1
2
1
­ A Ó 4´
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
” I•sŸsŒ ¡M
15
We need to estimate the quantity
Ϭ
û m à © ï dà ñ å × ù• L
ò[Ò åæ¬
ò[²[ Ó
­ A Ó 5´
½©ð­)½ º º º ´
By rotational
symmetry,
it© suffices
½
½
á
©
#"ºtèJ´ expression
­
" ¿ƒÁ for è ¿ Á 0à ÓcÓcÓ 0
ô ­ ´ to£ û bound the preceding
with
0. Setting
2
, with
, and
× , and ñ
× , this
ú
2
1
^[ û
‰[ û
2
1
1
1
1
1
1
means that we must bound
©ðï
Ã × è exp Ï ¯ ô ­ ´k­#" µèÂcèì´ û ¯ 2° " ô ­ ´ £ û × Ò Ó
"¢© £ ¾
èW© £ Ä , then we have
If we rescale the variables as
, and
Ï ¯ ô ­ ´ £ û Ï ­)¾ µ Ä Ä ´ û µ 2° ¾ ÒÒ Ó
ü © £ Ã ï ¾ Ã
d d × Ä exp
Á DFEHGJILKNM Note that the polynomial ­)¾ µ Ä Ä ´Fû~¯ 2° ¾ is independent of £ .
$
ü
"
1
d d
2
1
2
2
1
2
2
ü
We will bound
by taking advantage of the fact that the integrand is an entire function
and translate the contour of integration so that it passes through at least one critical point of the
exponent. These critical points occur at Ä
0 and the roots of 2 1
—that is, at the
Lý 4 1 points
exp 2 2 1 ,
0 1 2
2
2.
ü
Inserting this expression into the exponent of the integrand of , we see that the value of
the polynomial at the critical points is
¾ò ©
©
¾ û × ©š¯°
­)° û ò ü ´ ñ © º º º cÓ ÓcÓ º ¯
×
¯ ô ­ ´ £ û ` exp ­)° 2ó­ ­ 4 ñ ¯ µ ´1´ ´¯ 2 ° exp ­Z° ó ­ ­ 4ñ µ¯ 1´´ ´ a
2 2
1
2 2
1
µ
´
­
© ­ 2 ¯ 1´ ô ­ ´ £ û exp ­)° ó ­ 4 ñ ¯ 2 ´ ´ Ó
2 2
1
© 0, then the real part of the critical value is
In particular, if we take ñ
­ 2 ¯ 1´ ô ­ ´ £ û cos Ï óé 2 µ ó&é ­ 4 ¯ 2´ ÒÎþ ¯ ô ­ ´ £ û ó º
2
ƒ
Á
µ¾
when is large (and is negative for all Ö 0). Integrating over the region
2
2
2
2
0 and observing
that there is only one critical point on this line, we get, using standard techniques of stationary
phase:
í
ü
with 
leads to
Ö
­ ´ × Ã Øh û ¥å × ÿ ùL L
º
à Þ , when £ à Þ . Reverting to the original variables, this
­ ´ × Ã Ø û å × ² Žø Y •º ] ¡Ø ù• L
ø Y •º ] º
þ
ô
û à ó&é 2 as ú
û mà þ ô
û ©  û é ­ 2 ´ Ø û × . This completes the proof of Proposition A.2.
0 and 
2
where Ì
2
2
1
2
1
2
2
1
1
2
1
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
16
­)½´
We now consider the action of the semi-group on functions localized inside a ball of radius
denote the eigenfunction (written
1 2
0 for all
(note that 1 0
0
ð . A key observation here is the following lemma. Let 1 0
+
1
in position space) of with eigenvalue 0 and let
­)½´B©
½ ).
© D.EªEHGPñêM±°M
© ×
5
The operator+
has the same eigenvalues as .
Á$DFEHGJILKNM
+
1
+ ­)½´ Ø
­)½´ Ö
is self-adjoint on (a dense domain in) L2
­ZÁ Ã ´
and
5
The domains of , and have to be chosen carefully here. The correct choices
have been explained in detail in [EWW, p.197].
” I•sŸsŒ ¡M
The proof is a straightforward calculation. We note further that if 1
are the eigen+
5
11
functions of then the eigenfunctions of are .
#"
If we take the inverse Fourier transform of the eigenfunctions 1
of (1.12), we see that
­)½´B© Ï ×
Õ 1 )­ ½´ Õ þ ¢Õ ½ Õ Ò
ƒÖ
Õ½ Õ
Ò ­)½´
­ ´
­P¯ ŽÕ ½ Õ • º ´hº
Ò exp ó
2
2
1
for some ó
0, using the same sort of estimates as those used to bound the kernel
semi-group. Thus, for
sufficiently large, we get
Õ )­ ½´ Õ þ ¢Õ ½ Õ Ò
Ò exp
5
¨
of the
­F¯ óŽÕ ½ Õ • º ´ Ó
2
1
2
2
1
The usefulness of introducing the operator
is that it is sectorial, since it is self-adjoint
and bounded below. Therefore, the associated semi-group can be estimated from spectral
0 denotes the projection onto the spectral subspace
information alone. In particular, if
0 is defined by
spanned by the eigenfunctions with eigenvalues 0, 21 , 2 2 , , 2 , and œ
0 , then we have a bound on the operator norm of œ 0 œ 0 œ 0 1
«ò
ò © À¯ « ò
«ò
ò å
Ž
9Wœ 0
œ
¯ û × û ÓcÓcÓ × û ò
ò å
ò 9ùܝ ò å × ò ü Ø û Ó
0
1
ò
ò
2
+
­ A Ó 6´
We can use this information to bound the semi-group associated with . Note that if we denote
+
5
by
and œ the projection associated with the spectral subspaces of (as we did for ), then
we have the identity:
ò
å Žž œ ò Ü ©áå Žž œ ò ­ 1 ¯
Ü
´ Ü µå Žž œ ò î ï Ü Ó
î ï
Since î ï is localized away from the origin, it can be studied with the help of Proposition A.1,
so we focus on the other term. There we get
9
å Žž œ ò ­ 1 ¯
î ï
´Ü 96 õ ©
m ©
× å Žž × œ ò × ­ 1 ¯ î ï ´ Ü 9 6 m õ
ƒ Ï¥å œ ò ÒÏ × ­ 1 ¯ î ï ´ Ü Ò ƒ 6 m õ
ƒ
ƒ
µ
ù  exp ÏO¯ ñ 2 1 Ò ƒƒ × ­ 1 ¯ î ï ´ Ü ƒƒ 6 m õ Ó
9
1
1
0
1
1
1
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
Õ × )­ ½´ ìÕ ù˜ exp ­ ó Õ ½ Õ L º ´ and that ­ 1 ¯
´ Ü ƒ 6 õ ù  exp Ï ó ­ 4ð é 3´ • º Ò 9 Ü 9 6 õ º
m
ƒ m
Using now the bounds on 1 0 , we see that
when
4ð 3, we get
Õ ½ ÕÖ
é
ƒ
× ­1¯
ò ­1¯
´Ü
ƒ
so that finally
å
ƒ Ž ž œ
ƒ
ƒ
î ï
1
î ï
ƒ6
ƒ
ƒ
2
1
2
1
î ï
m õ ù
Ï ­ ´
exp ó 4ð é 3
2
• 2 º
1
µ 1Ò Ü 6
O
Ï
¯
ñ
exp 9 9 mõ Ó
2
Ò
1
Ü ¿
Under the hypotheses of Proposition A.1, there exist constants 
5O6
0, such that for all
, one has
Ö
0 and ó
å
ò ­1¯
ƒ hŽž œ
ƒ
ƒ
î ï
´Ü
ƒ6
ƒ
ƒ
0
2
2
Thus we have proven:
” I•sŒ–—s:uhprLp{s:oPñêM*µÍM
Ü ´\­)½´j©
17
mõ
m õ ù
Ï ­ ´
exp ó 4ð é 3
2
• 2 º
1
­> º A ´ Ö
µ 1Ò Ü 6
¯
Ï
ñ
9 9 mõ Ó
exp 2
Ò
We now return to the:
” I•sŸsŒ sŒ ” I•sŒ–—s:uhprLp{s:o3M{z:M
Žž 6 As in [EWW] it is only necessary to consider the term with
highest derivative in 9kî ï
9 . All other terms are easier to estimate. Also, as in that paper,
6
6
6
we use the fact that
Žž
h 2
Žž
D
D
A7
6
where D is a shorthand notation for a product of derivatives w.r.t. the of total degree > . Thus,
2
6
6
¨ £
Žž
2
£
£
D
D
d
A8
å Ü mõ
Ïå
First consider the
9kî
†
èW©
å ©áå Ø û å
º
­ Ó´
½
Ü Ò ­)½´ä© å ­ Ã Ø ´ Ãû ï Ã ­ º ´ Ï Ü Ò ­`å Ø û ­)½~µ O´ ´ Ó
2ó
­ Ó´
0 case of (A.3). Then
åï Žž Ü 9 6 õ ù hå ­ Ã Ø ´ Ãû
ï dà £ Õ ¨­ £ º ´ Õ ƒ † õ î ï Ï D6 ÜÒ ­`å Ø û
­ Ó µ £ ´O´ ƒ º ­ A Ó 9´
m
ƒ
ƒ
2ó
­ µ»½€F½´ Ø . Note that the conclusions of Lemma
is the operator of multiplication by 1
2
2
2
1 2
where
¨
B.4 of [EWW] do not depend on the exact form of and so it also holds in the present situation
and we have
© D.EªEHGCñ MæM
ƒƒ
Á$DFEHGJILKNM
we work.
†
One has the bounds
´hî ï
Ü Z­ å Ø û ­ Ó µ £ ´/´ ƒ ù
ƒ
2
2
2


å × 9 Ü 9 m õ º
­ 1 µ Õ £ Õ ´ ´n9 Ü 9 m õ º
2
0
2
2
0
Õ Õaù
Õ ÕaÖ
if £
if £
7ð
7ð
é 8,
é 8.
Note that the proof in [EWW, p.199] is also unaffected by the dimension in which
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
18
Now use Lemma A.5 to bound the integral in (A.9) by writing it as an integral over
7 ð 8 and an integral over £
7ð 8. The integral over £
7ð 8 is bounded with the
aid of Lemma A.5 as
Õ £ ÕDù
é

Õ ÕÖ
å Ã Ø û
ï
­ 2ó ´ Ã
2
í
Ò Ò
7 ï
é
Õ Õaù
é
¨­ £ º ´ å õ Ø Ü 6
­ ´/å ¯ å ×
Õ
Õ ×
9 9 m õ ùܝ Ø
2
2 2
8
­ A Ó 10´
Ü 6 mõ º
¨­ º ´
d £ Õ £ Õù˝ , with 
9 9
where the last step used the estimates of Theorem 4.1 to show that ç
independent of .
To estimate the integral in the outer region, we use the second part of Lemma A.5 and then
bound it by
åhà Øh û
ï
­ 2ó ´ Ã
¨­ £ º ´ ­ µ £ ´ õ Ø Ü 6
­ º ´Få Ã Ø û exp Ï ¯ óð • º Ò 9 Ü 9 6 õ º
Õ
Õ 1 Õ Õ
9 9 m õ ù A
Ò Ò ï Ø
­ A Ó 11m ´
for some ó Ö 0, where, again, we have used the estimates of decay in Proposition A.2 both to
Ï ¯ • º Ò as well as to bound the integral over £ . Combining (A.1),
extract the factor of exp óŸð èW
© 0 case of (A.3).
(A.10), and (A.11), we get the
è
èý© 1. We can
We next indicate how to treat the Ö 0 cases of (A.3). Consider the case

2
í
2
7
2
2
2
2
1
8
2
2
1
rewrite (A.8) by integrating by parts once w.r.t. one component of £ , for example £ 1 . Then,
Ï å Žž D6 ÜDÒ ­)½´ä© å à hØ ­ û
å ´× Ã Ø û ï dà £ Ï Dí ¨ Ò ­ £ º ´ Ï D6 × ÜDÒ ­`å Øh û
­)½~µ £ O´ ´ Ó ­ A Ó 12´
2ó
2
2
1
2
1
Differentiating (A.2) w.r.t. £
1
gives
­ Dí ¨ Ò ­ £ º ´ä© ï dà ñ ° ñ exp ­Z° è £ ´ exp Ï ­ ñ ñ ´Fû­ 1 ¯ å × a´ Ò Ó
­ A Ó 13´
" - © ô ­ ´ Øh û
To estimate (A.13), first replace ñ by
ñ , where, as before ô ­ ´W© 1 ¯ å × .
Then,
­ Dí ¨ Ò ­ £ º ´³© ô ­ ´ ð û ô ­ ´ 1 û ï d¯Ã "—" exp Ï ­#" "´Fû Ò exp Ï °*"“ £ é ô ­ ´ Ø û
Ò Ó ­ A Ó 14´
hØ Ø 1
1
1
2
1
1
2
1
2
2
1
We estimate this integral" by using the method of stationary phase as in the proof of Proposition
A.2. The extra factor of 1 does not cause any trouble as it is easily offset by the exponentially
decaying terms.
One now uses the Schwarz inequality to rewrite
9kî ï
åhŽž D6 Ü 9 6 m õ ù å ­ Ã Ø ´ Ãû
ï dà £
2ó
­ º ´
2
è©
Û í ¨
Û
ÛÛ D £ ÛÛ ƒ †
ƒƒ
1
õ
î ï
Ï D6 × ÜÒ ­ZåŽ Ø ­ Ó µ £ ´O´ ƒƒ ³º ­ A Ó 15´
ƒ
1
2
2
and then proceeds as in the case when
0, breaking the integral over £ into the same two pieces
as before. These two pieces are then estimated with the aid of Lemma A.5. Note that while the
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
19
ô ­ ´ Ã û factor × Ø of (A.14) will be absorbed when one integrates w.r.t £ , the remaining factor
è © 2 º 3 º ÓcÓcÓ 2 ¯ 1
ô ­ ´ Ø û of ×
will remain in the final bound of (A.3). The bounds for
2
1
2
follow in a similar fashion.
To complete the proof of Proposition 2.1, first rewrite
åhŽž œ ò ©¬åŽŽž Ø œ ò åhŽž Ø ©¬åŽŽž Ø œ ò î¶ï hå Žž Ø µåhŽž Ø œ ò ­ 1 ¯ î¶ï ´/åhŽž Ø Ó ­ A Ó 16´
å Žž Ø œ ò on a function localized
The second of these terms involves an estimate of the action of
near the origin, so by Proposition A.4, we get a bound
µ 1 Ü 6 ã ­ ´
å9 hŽž Ø œ ò ­ 1 ¯ î¶ï ´/åhŽž Ø Ü 9 6 õ Ü
Ï
ã
4ð Ò û Ø û ¯
ñ
a 9 9
×
m ù  exp ` ó 3
2
× m õ Ó A Ó 17
2
2
2
2
2
2
2
2
2
2
2
1
1
2
We use Proposition A.1 to bound the first term of (A.16):
­ º ´å
­ ´
 > A
õ
ô
Y 2 ]
2
6
üÃW
X å ×
µÑå ×÷ ï
4
2
As a preliminary step, we note that if we first choose A
å
2 6
üÃW
X å ×
4
µÑå ×÷ ï
2
Žø Y 2 •º 1] _
then for sufficiently small ä (roughly speaking ä
and (A.18) imply
9
åhŽž œ ò 9ù ô ­ ´  ã û
å × hØ Á$DFEHGJILKNM
Ü
Žø Y 2 L º 1] _
9 9
6
and ð such that
ù  å × { ò ü Ø û
t
º
Ï 1 µ¹­ 4é 3´ û Ø û × Ò ×
1
1
2
­ A Ó 18´
ã mõ Ó
×
2
2
2
1
1
), the Eqs.(A.17)
ã ò © ô ­ ´  ã û
å × Ò ¹ Ò 9 Ü 9 6 ã ò Ó
Ø × m
× m
­ µ µ ä ­ ñ µ 1´O´ , which is where the
Note that this requires that we choose Aυ 2 >
2
Ü
¹ 1
6
2
9 9
1
2
restriction on A in Proposition 2.1 (and hence Theorem 1.1 ) arises.
This shows that the projection of the semi-group onto the complement of the eigenspace
,
spanned by the first eigenvalues decays with a rate proportional to the eigenvalue
1 . We
,
1 }
can sharpen the decay rate so that we obtain a rate like exp
1 by the techniques
of [EWW], (see Eq. B.14 and following) and this completes the proof of Proposition 2.1
ñ
ÏO¯ ­ ¯ ´ Õ ò ü Õ Ò
òü
– ŒÑ£˜’Âìhž Œ ›
× It is a pleasure to thank Thierry Gallay for helpful comments about this paper and for pointing
out an error in a previous version of the manuscript. The research of CEW was supported in
part by the NSF Grant DMS-9803164. This work was also supported by the Fonds National
Suisse. JPE also thanks Center for BioDynamics and the Department of Mathematics at Boston
University for their hospitality.
wx&yz{4|~}}5x€ƒ‚…„†x‡}ˆ&z
¡4‘&Œ–›
[BKL]
[EG]
[C]
[CH]
[EWW]
[G]
[H]
[LW]
[S]
[W1]
[W2]
20
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