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I. D. Chueshov
Title:
Introduction to the Theory
of InfiniteDimensional
Dissipative Systems
«A CTA » 200
Author:
ISBN:
966–7021
966
7021–64
64–5
I. D. Chueshov
I ntroduction to the Theory
of Infinite-Dimensional
D issipative
S ystems
Universitylecturesincontemporarymathematics
This book provides an exhaustive introduction to the scope
of main ideas and methods of the
theory of infinite-dimensional dissipative dynamical systems which
has been rapidly developing in recent years. In the examples
systems generated by nonlinear
partial differential equations
arising in the different problems
of modern mechanics of continua
are considered. The main goal
of the book is to help the reader
to master the basic strategies used
in the study of infinite-dimensional
dissipative systems and to qualify
him/her for an independent scientific research in the given branch.
Experts in nonlinear dynamics will
find many fundamental facts in the
convenient and practical form
in this book.
The core of the book is composed of the courses given by the
author at the Department
of Mechanics and Mathematics
at Kharkov University during
a number of years. This book contains a large number of exercises
which make the main text more
complete. It is sufficient to know
the fundamentals of functional
analysis and ordinary differential
equations to read the book.
Translated by
You can O R D E R this book
while visiting the website
of «ACTA» Scientific Publishing House
http://www.acta.com.ua
www.acta.com.ua/en/
Constantin I. Chueshov
from the Russian edition («ACTA», 1999)
Translation edited by
Maryna B. Khorolska
Chapter
3
Inertial Manifolds
Contents
....§1
Basic Equation and Concept of Inertial Manifold . . . . . . . . . 149
....§2
Integral Equation for Determination of Inertial Manifold . . 155
....§3
Existence and Properties of Inertial Manifolds . . . . . . . . . . 161
....§4
Continuous Dependence of Inertial Manifold
on Problem Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
....§5
Examples and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
....§6
Approximate Inertial Manifolds
for Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . 182
....§7
Inertial Manifold for Second Order in Time Equations . . . . 189
....§8
Approximate Inertial Manifolds for Second Order
in Time Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
....§9
Idea of Nonlinear Galerkin Method . . . . . . . . . . . . . . . . . . . . 209
....
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
If an infinite-dimensional dynamical system possesses a global attractor of finite
dimension (see the definitions in Chapter 1), then there is, at least theoretically,
a possibility to reduce the study of its asymptotic regimes to the investigation of properties of a finite-dimensional system. However, as the structure of attractor cannot
be described in details for the most interesting cases, the constructive investigation
of this finite-dimensional system cannot be carried out. In this respect some ideas
related to the method of integral manifolds and to the reduction principle are very
useful. They have led to appearance and intensive use of the concept of inertial manifold of an infinite-dimensional dynamical system (see [1]–[8] and the references
therein). This manifold is a finite-dimensional invariant surface, it contains a global
attractor and attracts trajectories exponentially fast. Moreover, there is a possibility
to reduce the study of limit regimes of the original infinite-dimensional system
to solving of a similar problem for a class of ordinary differential equations.
In this chapter we present one of the approaches to the construction of inertial
manifolds (IM) for an evolutionary equation of the type:
d-----u
+ A u = B (u, t) ,
dt
u t = 0 = u0 .
(0.1)
Here u ( t ) is a function of the real variable t with the values in a separable Hilbert
space H . We pay the main attention to the case when A is a positive linear operator
with discrete spectrum and B ( u , t ) is a nonlinear mapping of H subordinated to A
in some sense. The approach used here for the construction of inertial manifolds is
based on a variant of the Lyapunov-Perron method presented in the paper [2]. Other
approaches can be found in [1], [3]–[7], [9], and [10]. However, it should be noted
that all the methods for the construction of IM known at present time require a quite
strong condition on the spectrum of the operator A : the difference l N + 1 - l N
of two neighbouring eigenvalues of the operator A should grow sufficiently fast
as N ® ¥ .
§1
Basic Equation and Concept
of Inertial Manifold
In a separable Hilbert space H we consider a Cauchy problem of the type
du
------ + A u = B ( u , t ) ,
dt
t > s,
u t = s = u0 ,
s ÎR,
(1.1)
where A is a positive operator with discrete spectrum (for the definition see Section
1 of Chapter 2) and B ( . , . ) is a nonlinear continuous mapping from D ( Aq ) ´ R
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Inertial Manifolds
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into H, 0 £ q < 1 , possessing the properties
3
B ( u , t ) £ M ( 1 + Aq u )
(1.2)
B ( u1 , t ) - B ( u2 , t ) £ M Aq ( u1 - u2 )
(1.3)
and
for all u , u1 , and u2 from the domain .q = D ( Aq ) of the operator Aq . Here M is
a positive constant independent of t and . is the norm in the space H . Further it
is assumed that { e k } is the orthonormal basis in H consisting of the eigenfunctions
of the operator A :
A ek = lk ek ,
0 < l1 £ l2 £ ¼ ,
lim l k = ¥ .
k®¥
Theorem 2.3 of Chapter 2 implies that for any initial condition u0 Î . q problem (1.1) has a unique mild (in .q ) solution u ( t ) on every half-interval [ s , s + T ) ,
i.e. there exists a unique function u ( t ) Î C ( s , s + T ; .q ) which satisfies the integral equation
t
u (t) = e
-( t - s ) A
u0 +
òe
-( t - t ) A
B ( u ( t ) , t ) dt
(1.4)
s
for all t Î [ s , s + T ) . This solution possesses the property (see (2.6) in Chapter 2)
Ab ( u ( t + s ) - u ( t ) ) £ C s q - b ,
0 £ b £ q
for 0 < s < 1 and t > s . Moreover, for any pair of mild solutions u1 ( t ) and u2 ( t ) to
problem (1.1) the following inequalities hold (see (2.2.15)):
Aq u ( t ) £ a1 e
a2 ( t - s )
Aq u ( s ) ,
t ³ s
(1.5)
and (cf. (2.2.18))
ì -l
(t - s)
a ( t - s) ü
+ M ( 1 + k ) a1 l-N1++1q e 2
QN Aq u ( t ) £ í e N + 1
ý Aq u ( s ) , (1.6)
î
þ
where u ( t ) = u1 ( t ) - u2 ( t ) , a1 and a2 are positive numbers depending on q , l1 ,
and M only. Hereinafter QN = I - PN , where PN is the orthoprojector onto the first
N eigenvectors of the operator A . Moreover, we use the notation
¥
k = qq
òx
-q -x
e
dx for q > 0
and
k = 0 for q = 0 .
(1.7)
0
Further we will also use the following so-called dichotomy estimates proved
in Lemma 1.1 of Chapter 2:
q
Aq e -t A PN £ lN
e
-t A
e
QN £ e
lN t
-lN + 1 t
,
,
t ÎR;
t ³ 0;
(1.8)
Basic Equation and Concept of Inertial Manifold
q
Aq e -t A Q N £ [ ( q ¤ t ) q + l N
+ 1] e
-lN + 1 t
,
t > 0,
q > 0.
The inertial manifold (IM) of problem (1.1) is a collection of surfaces
{ Mt , t Î R } in H of the form
Mt = { p + F ( p , t ) : p Î PN H , F ( p , t ) Î ( 1 - PN ) .q } ,
where F ( p , t ) is a mapping from PN H ´ R into ( 1 - PN ) .q satisfying the Lipschitz
condition
Aq( F ( p1 , t ) - F ( p2 , t ) ) £ C Aq ( p1 - p 2 )
(1.9)
with the constant C independent of pj and t . We also require the fulfillment of the
invariance condition (if u 0 Î Ms , then the solution u ( t ) to problem (1.1) possesses the property u ( t ) Î Mt , t ³ s ) and the condition of the uniform exponential
attraction of bounded sets: there exists g > 0 such that for any bounded set B Ì H
there exist numbers C B and tB > s such that
ì
sup í dist . ( u ( t , u0 ) ,
q
î
ü
Mt ) : u0 Î B ý £ CB e
-g ( t - tB )
þ
for all t ³ tB . Here u ( t , u0 ) is a mild solution to problem (1.1).
From the point of view of applications the existence of an inertial manifold (IM)
means that a regular separation of fast (in the subspace ( I - PN ) H ) and slow (in the
subspace PN H ) motions is possible. Moreover, the subspace of slow motions turns
out to be finite-dimensional. It should be noted in advance that such separation is
not unique. However, if the global attractor exists, then every IM contains it.
When constructing IM we usually use the methods developed in the theory
of integral manifolds for central and central-unstable cases (see [11], [12]).
If the inertial manifold exists, then it continuously depends on t , i.e.
lim Aq ( F ( p , s ) - F ( p , t ) ) = 0
t®s
for any p Î PN H and s Î R . Indeed, let u ( t ) be the solution to problem (1.1) with
u0 = p + F ( p , s ) , p Î PN H . Then u ( t ) Î Mt for t ³ s and hence
u ( t ) = PN u ( t ) + F ( PN u ( t ) , t ) .
Therefore,
F( p , t ) - F ( p , s ) = [ F ( p , t ) - F ( PN u ( t ) , t ) ] +
+ [ u ( t ) - u0 ] + [ p - PN u ( t ) ] .
Consequently, Lipschitz condition (1.9) leads to the estimate
Aq ( F( p , s ) - F ( p , t ) ) £ C Aq ( u ( t ) - u 0 ) .
Since u ( t ) Î C ( s , + ¥ , D ( Aq ) ) , this estimate gives us the required continuity property of F( p , t ) .
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Inertial Manifolds
E x e r c i s e 1.1
Prove that the estimate
Ab ( F ( p , t + s ) - F ( p , t ) ) £ Cb ( p , N ) s q - b
holds for F ( p , t ) when 0 £ s £ 1 , 0 £ b £ q , t Î R .
The notion of the inertial manifold is closely related to the notion of the inertial
form . If we rewrite the solution u ( t ) in the form u ( t ) = p ( t ) + q ( t ) , where
p ( t ) = PN u ( t ) , q ( t ) = Q N u ( t ) , and Q N = I - PN , then equation (1.1) can be rewritten as a system of two equations
ì
ï
ï
ï
í
ï
ï
ï
î
d
----- p ( t ) + A p ( t ) = PN B ( p ( t ) + q ( t ) ) ,
dt
d
----- q ( t ) + A q ( t ) = Q N B ( p ( t ) + q ( t ) ) ,
dt
p t = s = p0 º PN u0 ,
q t = s = q0 º QN u0 .
By virtue of the invariance property of IM the condition ( p0 , q0 ) Î M s implies that
( p ( t ) , q ( t ) ) Î Mt , i.e. the equality q0 = F ( p0 , s ) implies that q ( t ) = F ( p ( t ) , t ) .
Therefore, if we know the function F ( p , t ) that gives IM, then the solution u ( t )
lying in Mt can be found in two stages: at first we solve the problem
d
----- p ( t ) + A p ( t ) = PN B ( p ( t ) + F ( p ( t ) , t ) ) ,
dt
p t = s = p0 ,
(1.10)
and then we take u ( t ) = p ( t ) + F ( p ( t ) , t ) . Thus, the qualitative behaviour of solutions u ( t ) lying in IM is completely determined by the properties of differential
equation (1.10) in the finite-dimensional space PN H . Equation (1.10) is said to be
the inertial form (IF) of problem (1.1). In the autonomous case ( B ( u , t ) º B ( u ) )
one can use the attraction property for IM and the reduction principle (see Theorem
7.4 of Chapter 1) in order to state that the finite-dimensional IF completely determines the asymptotic behaviour of the dynamical system generated by problem (1.1).
E x e r c i s e 1.2 Let F ( p , t ) give the inertial manifold for problem (1.1).
Show that IF (1.10) is uniquely solvable on the whole real axis, i.e.
there exists a unique function p ( t ) Î C ( - ¥ , ¥ ; PN H ) such that
equation (1.10) holds.
E x e r c i s e 1.3 Let p ( t ) be a solution to IF (1.10) defined for all t Î R . Prove
that u ( t ) = p ( t ) + F ( p ( t ) , t ) is a mild solution to problem (1.1) de= p0 + F ( p0 , t ) .
fined on the whole time axis and such that u
t=s
Use the results of Exercises 1.2 and 1.3 to show that if IM
{ Mt } exists, then it is strictly invariant, i.e. for any u Î Mt and
s < t there exists u0 Î Mt such that u = u ( t ) is a solution to problem (1.1).
E x e r c i s e 1.4
Basic Equation and Concept of Inertial Manifold
In the sections to follow the construction of IM is based on a version of the LyapunovPerron method presented in the paper by Chow-Lu [2]. This method is based on the
following simple fact.
Lemma 1.1.
Let f ( t ) be a continuous function on R with the values in H such that
QN f ( t )
£ C,
t ÎR.
Then for the mild solution u ( t ) (on the whole axis) to equation
d
---- u + A u = f ( t )
(1.11)
dt
to be bounded in the subspace QN .q it is necessary and sufficient that
t
t
u (t) =
e -( t - s ) A p
+
ò
e -( t - t ) A PN
f ( t ) dt +
òe
-( t - t ) A Q
N
f ( t ) dt (1.12)
-¥
s
for t Î R , where p is an element from PN H and s is an arbitrary real
number.
We note that the solution to problem (1.11) on the whole axis is a function u ( t ) Î
Î C ( R , H ) satisfying the equation
t
u ( t ) = e -( t - s ) A u(s) +
òe
-( t - t ) A
f ( t ) dt
s
for any s Î R .
Proof.
It is easy to prove (do it yourself) that equation (1.12) gives a mild solution
to (1.11) with the required property of boundedness. Vice versa, let u ( t ) be
a solution to equation (1.11) such that QN u ( t ) q is bounded. Then the function q ( t ) = QN u ( t ) is a bounded solution to equation
d
---- q ( t ) + A q ( t ) = Q N f ( t ) .
dt
Consequently, Lemma 2.1.2 implies that
t
q (t) =
òe
-( t - t ) A
QN f ( t ) dt .
-¥
Therefore, in order to prove (1.12) it is sufficient to use the constant variation
formula for a solution to the finite-dimensional equation
dp
------ + Ap = PN f ( t ) ,
dt
Thus, Lemma 1.1 is proved.
p ( t ) = PN u ( t ) .
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Lemma 1.1 enables us to obtain an equation to determine the function F ( p , t ) .
Indeed, let us assume that B ( u , t ) is bounded and there exists Mt with the function F ( p , t ) possessing the property Aq F ( p , t ) £ C for all p Î PN H and t Î R .
Then the solution to problem (1.1) lying in Mt has the form
u (t) = p (t) + F(p (t) , t) .
3
It is bounded in the subspace Q N H and therefore it satisfies the equation of the
form
u ( t ) = e -( t - s ) A p +
t
t
+
òe
-( t - t ) A
PN B ( u ( t ) , t ) dt +
òe
-( t - t ) A
Q N B ( u ( t ) , t ) dt ,
( t Î R ) . (1.13)
-¥
s
Moreover,
s
F ( p , s ) = QN u ( s ) =
òe
-( s - t ) A
QN B ( u ( t ) , t ) dt .
(1.14)
-¥
Actually it is this fact that forms the core of the Lyapunov-Perron method. It is
proved below that under some conditions (i) integral equation (1.13) is uniquely
solvable for any p Î PN H and (ii) the function F ( p , s ) defined by equality (1.14)
gives IM.
In the construction of IM with the help of the Lyapunov-Perron method an important role is also played by the results given in the following exercises.
E x e r c i s e 1.5 Assume that sup { e -g ( s - t ) f (t) : t < s } < ¥ , where g is any
number from the interval ( lN , lN + 1 ) and s Î R . Let u ( t ) be a mild
solution (on the whole axis) to equation (1.11). Show that u ( t ) possesses the property
sup { e -g ( s - t ) Aq u ( t ) } < ¥
t< s
if and only if equation (1.12) holds for t < s .
Hint: consider the new unknown function
w( t ) = eg (t - s) u( t)
instead of u ( t ) .
Assume that f ( t ) is a continuous function on the semiaxis
[ s , + ¥ ) with the values in H such that for some g from the interval
( lN , l N + 1 ) the equation
E x e r c i s e 1.6
sup { e -g ( s - t ) f ( t ) : t Î [ s , + ¥ ) } < ¥
holds. Prove that for a mild solution u ( t ) to equation (1.11) on the
semiaxis [ s , + ¥ ) to possess the property
Integral Equation for Determination of Inertial Manifold
sup { e -g ( s - t ) Aq u ( t ) : t Î [ s , + ¥ ) } < ¥
it is necessary and sufficient that
t
u ( t ) = e -( t - s ) A q +
òe
-( t - t ) A
Q N f ( t ) dt -
s
+¥
-
òe
-( t - t ) A
PN f ( t ) dt ,
(1.15)
t
where t ³ s and q is an element of Q N D ( Aq) . Hint: see the hint to
Exercise 1.5.
§2
Integral Equation for Determination
of Inertial Manifold
In this section we study the solvability and the properties of solutions to a class of integral equations which contains equation (1.13) as a limit case. Broader treatment of
the equation of the type (1.13) is useful in connection with some problems of the approximation theory for IM.
For s Î R and 0 < L £ ¥ we define the space C s º C g , q ( s - L , s ) as the set
of continuous functions v ( t ) on the segment [ s - L , s ] with the values in D ( Aq )
and such that
vs º
sup
t Î[ s - L , s]
{ e -g ( s - t ) Aq u ( t ) } < ¥ .
Here g is a positive number. In this space we consider the integral equation
v (t) =
Bps, L [ v ] ( t ) ,
s -L £ t £ s,
(2.1)
where
s
Bps, L [ v ] ( t )
=
e -( t - s ) A p
-
òe
- ( t - t ) A PB ( v ( t ) ,
t ) dt +
t
t
+
òe
- ( t - t ) A QB ( v ( t ) ,
t ) dt .
s -L
Hereinafter the index N of the projectors PN and Q N is omitted, i.e. P is the orthoprojector onto Lin { e 1 , ¼ , e N } and Q = 1 - P . It should be noted that the most significant case for the construction of IM is when L = ¥ .
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Lemma 2.1.
3
or 0 < L £ ¥ and
Let at least one of two conditions be fulfilled:
0< L< ¥
and
q
qq
M æ ------------- L 1 - q + L l N + 1ö £ q < 1
è1 - q
ø
q
q
M
-------lN + 1 - lN ³ 2
q ( ( 1 + k ) lN + 1 + lN ) ,
0 < q < 1,
(2.2)
(2.3)
where k is defined by equation (1.7). Then for any fixed s Î R there
exists a unique function vs ( t ; p ) Î Cs satisfying equation (2.1) for all
t Î [ s - L , s ] , where g is an arbitrary number from the segment [ l N ,
q
in the case of (2.3).
l N + 1 ] in the case of (2.2) and g = l N + ( 2 M ¤ q ) l N
Moreover,
v ( . ; p1 ) - v ( . ; p1 ) s £ ( 1 - q ) -1 Aq ( p1 - p2 )
(2.4)
vs s £ ( 1 - q ) -1 { D 1 + Aq p } ,
(2.5)
- 1+ q
-1+q
.
D1 = M ( 1 + k ) lN
+ 1 + M lN
(2.6)
and
where
Proof.
Let us apply the fixed point method to equation (2.1). Using (1.8) it is easy
to check (similar estimates are given in Chapter 2) that
Aq ( Bps , L ( v1) ( t ) - Bps , L ( v2 ) ( t ) )
1
2
£
s
£ e
lN ( s - t )
Aq ( p
- p2 ) +
1
òl
q lN ( t - t )
M
Ne
v 1 ( t ) - v 2 ( t ) q dt +
t
t
+
ò
-lN + 1 ( t - t)
q -ö q
q
æ ---------+ lN
M v1 ( t ) - v2 ( t ) q dt £
+1 e
è t - tø
s -L
£ e
lN ( s - t )
Aq ( p1 - p 2 ) + ( q1 ( s , t ) + q2 ( s , t ) ) e g ( s - t ) v1 - v2 s ,
where
t
q1 ( s , t ) = M
ò
s -L
-( lN + 1 - g ) ( t - t )
q -ö q
q
æ ---------+ lN
dt
+1 e
è t - tø
(2.7)
Integral Equation for Determination of Inertial Manifold
and
s
q2 ( s , t ) = M
òl
q
N
e
( lN - g ) ( t - t )
dt .
(2.8)
t
Therefore, if the estimate
q1 ( s , t ) + q2 ( s , t ) £ q ,
s -L £ t £ s
(2.9)
holds, then
Bps , L [ v1 ] - Bps , L [ v2 ] s £ Aq ( p1 - p2 ) + q v1 - v2 s .
1
2
(2.10)
Let us estimate the values q1 ( s , t ) and q2 ( s , t ) . Assume that (2.2) is fulfilled.
Then it is evident that
t
q1 ( s , t ) £
M qq
ò (t - t)
- q dt
q
+ M lN
+ 1 (t - s + L) =
s -L
qq
q
= M ------------ ( t - s + L ) 1 - q + M l N
+ 1 (t - s + L)
1- q
and
q
q
( s - t ) £ M lN
q2 ( s , t ) £ M lN
+ 1 (s - t)
for lN £ g £ l N + 1 . Therefore,
qq
q
ö
q1 ( s , t ) + q2 ( s , t ) £ M æ ----------- ( t - s + L ) 1 - q + l N
+ 1 Lø .
è1- q
Consequently, equation (2.2) implies (2.9). Now let the spectral condition (2.3)
be fulfilled. Then
t
q1 ( s , t ) £
ò
-¥
q
M lN
- g) (t - t)
M q q -( l
+1
------------------- e N + 1
dt + --------------------lN + 1 - g
(t - t) q
for all g < lN + 1 . We change the variable in integration x = ( lN + 1 - g ) ( t - t )
and find that
q
M lN
Mk
+1
-,
q1 ( s , t ) £ ------------------------------------- + --------------------lN + 1 - g
( lN + 1 - g ) 1 - q
where the constant k is defined by (1.7). It is also evident that
q
M lN
q2 ( s , t ) £ --------------g - lN
q
provided that g > lN . Equation (2.3) implies that g = lN + ( 2 M ¤ q ) l N
lies in
the interval ( lN , l N + 1 ) . If we choose the parameter g in such way, then we get
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Inertial Manifolds
q
M ( 1 + k ) lN
q
+1
- + --- .
q1 ( s , t ) + q2 ( s , t ) £ -------------------------------------------------2
q
2
M
l N + 1 - l N - -------q lN
Hence, equation (2.3) implies (2.9). Therefore, estimate (2.10) is valid, provided that the hypotheses of the lemma hold. Moreover, similar reasoning enables
us to show that
Bps, L [ v ] s £ D1 + Aq p + q v s ,
(2.11)
where D1 is defined by formula (2.6). In particular, estimates (2.10) and (2.11)
mean that when s , L , and p are fixed, the operator Bps, L maps C s into itself
and is contractive. Therefore, there exists a unique fixed point vs ( t , p ) . Evidently it possesses properties (2.4) and (2.5). Lemma 2.1 is proved.
Lemma 2.1 enables us to define a collection of manifolds { MLs } by the formula
MsL = { p + F L ( p , s ) : p Î P H } ,
where
s
FL(p ,
s) =
òe
-( s - t) A
Q B ( v ( t ) , t ) dt º Q v ( s ; p ) .
(2.12)
s -L
Here v ( t ) = v ( t ; p ) is the solution to integral equation (2.1). Some properties of
the manifolds { MLs } and the function F L(p , s) are given in the following assertion.
Theorem 2.1.
Assume that at leas
least one of two conditions (2.2) and (2.3) is satisfied.
Then the mapping F L ( . , s ) from P H into Q H possesses the properties
a)
Aq F L ( p , s )
£ D 2 + q ( 1 - q ) -1 { D1 + Aq p }
(2.13)
for any p Î PH , hereinafter D1 is defined by formula (2.6) and
-1 + q
D2 = M ( 1 + k ) lN
+1 ;
b) the manifold
(2.14)
MsL is a Lipschitzian surface and
q
Aq F L ( p1 , s ) - F L ( p2 , s ) £ ------------ Aq ( p1 - p 2 )
1 -q
(2.15)
for all p1 , p2 Î P H and s Î R ;
c) if u ( t ) º u ( t , s ; p + F sL ( p ) ) is the solution to problem (1.1) with the
initial data u0 = p + F L ( p , s ) , p Î PH , then Qu ( t ) = F L ( Pu ( t ) , t )
for L = ¥ . In case of L < ¥ the inequality
Aq ( Qu ( t ) - FL ( P u ( t ) , t ) ) £
£ D 2 ( 1 - q ) -1 e -g L + q ( 1 - q ) -2 e -g ( t - s ) { D 1 + Aq p }
(2.16)
Integral Equation for Determination of Inertial Manifold
holds for all s £ t £ s + L , where g is an arbitrary number from the
q
segment [ l N , l N + 1 ] if (2.2) is fulfilled and g = l N + ( 2 M ¤ q ) l N
when
(2.3) is fulfilled;
d) if B ( u , t ) º B ( u ) does not depend on t , then F L ( p , s ) º F L ( p ) , i.e.
F L ( p , t ) is independent of t .
Proof.
Equations (2.12) and (1.8) imply that
s
Aq F L ( p ,
s
£ M
ò
s)
£ M
-lN + 1 ( s - t )
q öq
q
æ ----------- + lN
( 1 + Aq v ( t ) ) dt £
+1 e
è s - tø
ò
s -L
-lN + 1 ( s - t )
q -ö q
q
æ ----------+ lN
dt + q1 ( s , s ) v s .
+1 e
è s - tø
s -L
By virtue of (2.9) we have that q1 ( s , s ) < q . Therefore, when we change the variable in integration x = lN + 1 ( s - t ) with the help of equation (2.5) we obtain (2.13).
Similarly, using (2.4) and (1.8) one can prove property (2.15).
Let us prove assertion (c). We fix t0 Î [ s , s + L ] and assume that w ( t ) is a
function on the segment [ s , s + L] such that w ( t ) = u ( t ) for t Î [ s , t0 ] and w ( t ) =
= vs ( t ) for t Î [ s - L , s ] . Here vs ( t ) is the solution to integral equation (2.1). Using
equations (1.4) and (2.1) we obtain that
t
w (t) =
e -( t - s ) A ( p
+ FL(p ,
s) ) +
= e
p+
-( t - t ) A
B ( w ( t ) , t ) dt =
s
t
-( t - s ) A
òe
òe
-( t - t ) A
t
PB ( w ( t ) , t ) dt +
òe
-( t - t ) A
QB ( w ( t ) , t ) dt
(2.17)
s -L
s
for s £ t £ t0 . Evidently, equation (2.17) also remains true for t Î [ s - L , s ] . Equation (1.4) gives us that
s
p=e
-( s - t0 ) A
p ( t0 ) +
òe
-( s - t ) A
PB ( w ( t ) , t ) dt .
t0
Therefore, the substitution in (2.17) gives us that
w( t) =
t ,L
Bp0( t ) [ w ] ( t ) + bL ( t0 , s ; t )
0
(2.18)
for all t Î [ t0 - L , t0 ] , where p ( t ) = P u ( t ) and
t0 - L
bL ( t0 , s ; t ) =
òe
s -L
-( t - t ) A
Q B ( v s ( t ) , t ) dt .
(2.19)
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In particular, if L = ¥ equation (2.18) turns into equation (2.1) with s = t0 and
p = p ( t0 ) . Therefore, equation (2.12) implies the invariance property Q u ( t0 ) =
= F ¥ ( P u ( t0 ) , t0 ) . Let us estimate the value (2.19). If we reason in the same way as
in the proof of Lemma 2.1, then we obtain that
3
Aq bL ( t0 , s ; t )
£ e
-( t - t0 + L ) lN + 1 ì *
í q1 ( s , t0 - L )
î
+ q 1 ( s , t0 - L ) e
g ( s - t0 + L )
ü
vs s ý ,
þ
where q1 ( s , t ) is defined by formula (2.7) and
t
q1* ( s ,
t) = M
q öq
q
æ æ ---------ö -lN + 1 ( t - t ) dt .
- + lN
+ 1ø e
è è t - tø
ò
(2.20)
s -L
Therefore, simple calculations give us that
Aq bL ( t0 , s ; t )
£ e
-( t - t0 + L ) lN + 1
ì
ü
g ( s - t0 + L )
q vs s ý ,
í D2 + e
î
þ
(2.21)
where D 2 is defined by formula (2.14). Let vt ( t ) be the solution to integral equa0
tion (2.1) for s = t0 and p = P u ( t0 ) . Then using (2.12), (2.18), and (2.1) we find
that
Q u ( t0 ) - F L ( P u ( t0 ) , t0 ) = Q ( w ( t0 ) - vt ( t0 ) ) .
0
(2.22)
However, for all t Î [ t0 - L , t0 ] we have that
w ( t ) - vt ( t ) =
0
t ,L
t ,L
Bp0( t ) [ w ] ( t ) - Bp0( t ) [ vt ] ( t ) + bL ( t0 , s ; t ) .
0
0
0
Therefore, the contractibility property of the operator
( 1 - q ) w - vt
0 t0
£
sup
t Î [ t0 - L ,
t ,L
Bp0
gives us that
ì -g ( t0 - t ) q
ü
A bL ( t0 , s ; t ) ý .
íe
t 0 ]î
þ
Hence, it follows from (2.21) and (2.22) that
Aq ( Q u ( t0 ) ) - F L ( P u ( t 0 ) , t0 ) £
£ w - vt
0 t0
Aq ( w ( t0 ) - vt ( t0 ) )
0
£
ì
ü
-g ( t0 - s )
£ ( 1 - q ) -1 í e -g L D2 + q e
vs s ý .
î
þ
This and equation (2.5) imply (2.16). Therefore, assertion (c) is proved.
In order to prove assertion (d) it should be kept in mind that if * ( u , t ) º
º * ( u ) , then the structure of the operator Bps, L enables us to state that
Bps, L [ v ] ( t - h ) = Bps + h , L [ vh ] ( t )
for s + h - L £ t £ s + h , where vh ( t ) = v ( t - h ) . Therefore, if v ( t ) Î Cg , q ( s - L , s )
is a solution to integral equation (2.1), then the function
vh ( t ) º v ( t - h ) Î Cg , q ( s + h - L , s + h )
Existence and Properties of Inertial Manifolds
is its solution when s + h is written instead of s . Consequently, equation (2.12)
gives us that
FL ( p , s + h ) = Q vh ( s + h ) = Q v ( s ) = FL ( p , s ) .
Thus, Theorem 2.1 is proved.
E x e r c i s e 2.1 Show that if B ( u , t ) £ M , then inequalities (2.13) and
(2.16) can be replaced by the relations
Aq F L ( p , s ) £ D2 ,
Aq ( Q u ( t ) - F L ( Pu ( ( t ) , t ) ) )
£ D2 ( 1 - q ) -1 e -g L ,
(2.23)
(2.24)
where D 2 is defined by formula (2.14).
§ 3
Existence and Properties
of Inertial Manifolds
In particular, assertion (c) of Theorem 2.1 shows that if the spectral gap condition
q
q
M
-------lN + 1 - lN ³ 2
q ( ( 1 + k ) lN + 1 + lN ) ,
0 < q < 1,
(3.1)
is fulfilled, then the collection of surfaces
Ms = { p + F ( p , s ) : p Î P H } ,
s ÎR,
(3.2)
is invariant, i.e.
U ( t , s ) Ms Ì
F¥ ( p ,
Mt ,
-¥ < s £ t < ¥ .
(3.3)
s ) is defined by formula (2.12) for L = ¥ and U ( t , s ) is
Here F ( p , s ) =
the evolutionary operator corresponding to problem (1.1). It is defined by the formula U ( t , s ) u0 = u ( t ) , where u ( t ) is a mild solution to problem (1.1).
In this section we show that collection (3.2) possesses the property of exponential uniform attraction. Hence, { Mt } is an inertial manifold for problem (1.1). Moreover, Theorem 3.1 below states that { Mt } is an exponentially asymptotically
˜
complete IM, i.e. for any solution u ( t ) = U ( t , s ) u 0 there exists a solution u ( t ) =
˜
˜
= U (t , s) u
0 lying in the manifold ( u ( t ) Î M t , t ³ s ) such that
˜
h > 0, t > s.
Aq ( u ( t ) - u ( t ) ) £ C e -h ( t - s ) ,
˜
In this case the solution u ( t ) is said to be an induced trajectory for u ( t ) on the
manifold Mt . In particular, the existence of induced trajectories means that the solution to original infinite-dimensional problem (1.1) can be naturally associated with
the solution to the system of ordinary differential equations (1.10).
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Theorem 3.1.
Assume that spectral gap condition (3.1) is valid for some q < 2 - 2 .
Then the manifold { Ms , s Î R } given by formula (3.2) is inertial for problem (1.1).. Moreover, for any solution u ( t ) = U ( t , s ) u 0 there exists an induced trajectory u* ( t ) = U ( t , s ) u*0 such that u* ( t ) Î Mt for t ³ s and
Aq ( u ( t ) - u* ( t ) )
2 (1 - q)
- e -g ( t - s ) Aq ( Qu 0 - F ( P0 u , s ) ) ,
£ ----------------------------(2 - q) 2 - 2
(3.4)
M q
-------where g = l N + 2
q l N and t ³ s .
Proof.
Obviously it is sufficient just to prove the existence of an induced trajectory
*
u ( t ) Î Mt possessing property (3.4). Let u ( t ) be a mild solution to problem (1.1),
u ( t ) = U ( t , s ) u 0 . We construct the induced trajectory u* ( t ) = U ( t , s ) u*0 for u ( t )
in the form u* ( t ) = u ( t ) + w ( t ) , where w ( t ) lies in the space C s+ º C s , g ( s , + ¥ ,
D ( Aq) ) of continuous functions on the semiaxis [ s , + ¥ ) such that
w s , + º sup { e g ( t - s ) Aq w ( t ) } < ¥ ,
t ³ s
(3.5)
q
where g = lN + ( 2 M ¤ q ) lN
. We introduce the notation
F (w, t) = B (u (t) + w, t) - B (u (t) )
(3.6)
and consider the integral equation (cf. (1.15))
t
w (t) =
ò
Bs+ [ w ] ( t ) º e -( t - s ) A q ( w ) + e -( t - t ) A Q F ( w ( t ) , t ) dt s
+¥
-
òe
-( t - t ) A
P F ( w ( t ) , t ) dt ,
t Î [s, + ¥ ) ,
(3.7)
t
in the space Cs+ . Here the value q ( w ) Î Q D ( Aq) is chosen from the condition
u* ( s ) = u ( s ) + w ( s ) Î Ms ,
i.e. such that
Q u0 + Q w ( s ) = F ( Pu 0 + P w ( s ) , s ) .
Therefore, by virtue of (3.7) we have
+¥
æ
ç
q ( w ) = - Q u0 + F P u0 e -( s - t ) A P F ( w ( t ) , t ) dt ,
ç
è
s
ò
ö
s÷ .
÷
ø
(3.8)
Thus, in order to prove inequality (3.4) it is sufficient to prove the solvability of integral equation (3.7) in the space Cs+ and to obtain the estimate of the solution. The
preparatory steps for doing this are hidden in the following exercises.
Existence and Properties of Inertial Manifolds
E x e r c i s e 3.1
Assume that F ( w , t ) has the form (3.6). Show that for any
w ( t ) , w ( t ) Î Cs+ = C s , g ( s , + ¥ ; D ( Aq) )
and for t ³ s the following inequalities hold:
E x e r c i s e 3.2
F ( w ( t ) , t ) £ e -g ( t - s ) M w s , + ,
(3.9)
F ( w ( t ) , t ) - F ( w ( t ) , t ) £ e -g ( t - s ) M w - w s , + .
(3.10)
Using (1.8) prove that the equations
q
æ +¥
ö
lN
ç
Aq e -( t - t ) A P e -g ( t - s ) dt÷ £ ---------------- × e -g ( t - s ) , (3.11)
ç
÷
g - lN
è t
ø
ò
æ
ç
ç
è
ö
Aq e -( t - t ) A Q e -g ( t - s ) dt÷ £
÷
ø
s
q
q
k ( lN + 1 - g ) + lN + 1
£ ------------------------------------------------------- × e -g ( t - s )
lN + 1 - g
t
ò
(3.12)
hold for l N < g < l N + 1 and t ³ s . Here k is defined by formula
(1.7).
Lemma 3.1.
Assume that spectral gap condition (3.1) holds with q < 2 - 2 . Then
Bs+ is a continuous contractive mapping of the space Cs+ into itself.
The unique fixed point w of this mapping satisfies the estimate
2 (1 - q)
- Aq ( Q u0 - F ( P u0 , s ) ) .
w s , + £ ---------------------------(2 - q)2 - 2
(3.13)
Proof.
If we use (3.7), then we find that
Aq Bs+ [ w ] ( t )
£ e
-( t - s ) lN + 1
Aq q ( w ) +
+¥
t
+
ò
Aq e -( t - t ) A Q
F ( w ( t ) , t ) dt +
s
ò
Aq e -( t - t ) A P F ( w ( t ) , t ) dt
t
for t > s . Therefore, (3.9), (3.11), and (3.12) give us that
Aq B+s [ w ] ( t )
£ e
-( t - s ) lN + 1
Aq q ( w ) +
q
q
( 1 + k ) lN
ì lN
+1 ü
- + -------------------------------- ý M e -g ( t - s ) w s ,
+ í --------------lN + 1 - g þ
î g - lN
+
.
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q
Since g = l N + ( 2 M ¤ q ) l N
, spectral gap condition (3.1) implies that
Aq B+s [ w ] ( t )
£ e
-( t - s ) lN + 1
Aq q ( w ) + q e -g ( t - s ) w s , + .
(3.14)
Similarly with the help of (3.10)–(3.12) we have that
Aq ( B+s [ w ] ( t ) - B+s [ w ] ( t ) )
£ e
-( t - s) lN + 1
for any w , w
£
Aq ( q ( w ) - q ( w ) ) + q e -g ( t - s ) w - w s , +
(3.15)
Î C s+ . From equations (3.8), (3.9), and (2.15) we obtain that
+¥
Aq ( q ( w )
+ Q u0 - F ( Pu0 , s ) )
qM
£ -----------1 -q
ò
Aq e -( s - t ) A P e -g ( t - s ) dt × w s , + .
s
Therefore, (3.11) implies that
Aq q ( w )
£
q2
Aq ( Q u0 - F ( P u0 , s ) ) + -------------------- w s , + .
2 (1 - q)
Similarly we have that
q2
Aq ( q ( w ) - q ( w ) ) £ -------------------- w - w s , + .
2 (1 - q)
It follows from (3.14)–(3.17) that
B+s [ w ] s ,
+
(3.17)
q 2 -q
£ Aq ( Q u 0 - F ( Pu 0 , s ) ) + --- × ------------ w s ,
2 1 -q
Bs+ [ w ] - Bs+ [ w ] s ,
+
q 2 -q
£ --- × ------------ w - w s ,
2 1 -q
+
+,
(3.18)
.
Therefore, if q < 2 - 2 , then the operator Bs+ is continuous and contractive in
C s+ . Estimate (3.13) of its fixed point follows from (3.18). Lemma 3.1 is proved.
In order to complete the proof of Theorem 3.1 we must prove that the function
u* ( t ) = u ( t ) + w ( t )
is a mild solution to problem (1.1) lying in { Mt , t ³ s } (here w ( t ) is a solution
to integral equation (3.7) ). We can do that by using the result of Exercise 1.2, the invariance of the collection { Mt } , and the fact that equality (3.8) is equivalent to the
equation u* ( s ) Î Ms . Theorem 3.1 is completely proved.
E x e r c i s e 3.3 Show that if the hypotheses of Theorem 3.1 hold, then the induced trajectory u* ( t ) is uniquely defined in the following sense: if
there exists a trajectory u** ( t ) such that u** ( t ) Î Mt for t ³ s and
Aq ( u ( t ) - u** ( t ) ) £ C e -g ( t - s )
M lq
-------u** ( t ) º u* ( t ) for t ³ s .
with g ³ l N + 2
q N , then
Existence and Properties of Inertial Manifolds
The construction presented in the proof of Theorem 3.1 shows that in order to build
the induced trajectory for a solution u ( t ) with the exponential order of decrease g
given, it is necessary to have the information on the behaviour of the solution u ( t )
for all values t ³ s . In this connection the following simple fact on the exponential
closeness of the solution u ( t ) to its projection P u ( t ) + F ( P u ( t ) , t ) onto the manifold appears to be useful sometimes.
E x e r c i s e 3.4 Show that if the hypotheses of Theorem 3.1 hold, then the estimate
Aq ( Q u ( t ) - F ( P u ( t ) , t ) ) £
2
- e -g ( t - s ) Aq ( Pu 0 - F ( P u0 , t ) )
£ ----------------------------(2 - q)2 - 2
is valid for any solution u ( t ) to problem (1.1). Here g = l N +
q
+ ( 2 M ¤ q ) lN
and t ³ s (Hint: add the value F (( Pu* ( t ) , t ) *
- Qu ( t )) = 0 to the expression under the norm sign in the left-hand
side. Here u* ( t ) is the induced trajectory for u ( t ) ).
It is evident that the inertial manifold { Mt } consists of the solutions u ( t ) to problem
(1.1) which are defined for all real t (see Exercises 1.3 and 1.4). These solutions can
be characterized as follows.
Theorem 3.2.
Assume that spectral gap condition (3.1) holds with q < 2 - 2 and
{ Mt } is the inertial manifold for problem (1.1) constructed in Theorem 3.1..
Then for a solution u ( t ) to problem (1.1) defined for all t Î R to lie in the
inertial manifold ( u ( t ) Î Mt ) , it is necessary and sufficient that
u s º sup { e -g ( s - t ) Aq u ( t ) : -¥ < t £ s } < ¥
(3.19)
M q
-------for each s Î R , where g = l N + 2
q lN .
Proof.
If u ( t ) Î Mt , then u ( t ) = Pu ( t ) + F ( P u ( t ) , t ) . Therefore, equation (2.13)
implies that
Aq u ( t )
q D1
1
£ D 2 + -----------+ ------------ Aq Pu ( t ) .
1 -q 1 -q
The function p ( t ) = Pu ( t ) satisfies the equation
t
p (t) =
e -( t - s ) A p ( s )
+
òe
-( t - t) A
s
for all real t and s . Therefore, we have that
PB ( u ( t ) , t ) dt
(3.20)
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s
Aq p ( t )
£ e
( s - t ) lN
Aq p ( s )
+
q
M lN
òe
( t - t ) lN
( 1 + Aq u ( t ) ) dt
t
for t £ s . With the help of (3.20) we find that
3
Aq p ( t )
£ C (s, N, q) e
( s - t ) lN
M lqN
+ -----------1 -q
s
òe
( t - t) lN
Aq p ( t ) dt
t
for t £ s , where
q D1 ö 1
------------- .
C ( s , N , q ) = Aq p ( s ) + M æ 1 + D 2 + -----------è
1 - qø l1 - q
N
Hence, the inequality
M lqN
j(t) £ C ( s , N , q ) + -----------1- q
s
òj(t) dt
t
Aq p ( t )
( t - s) lN
holds for the function j(t) =
and t £ s . If we introduce the funce
s
tion y(t) = ò j (t) dt , then the last inequlity can be rewritten in the form
t
M lqN
y'(t) + ------------- y ( t ) ³ -C ( s , N , q ) ,
1- q
t £ s,
or
q
ì M lN ü
d
- tý
---y ( t ) exp í -----------dt
î 1- q þ
q
ì M lN ü
³ -C ( s , N , q ) exp í ------------tý ,
î 1- q þ
t £ s.
After the integration over the segment [ t , s ] and a simple transformation it is easy
to obtain the estimate
M lqNö
ì
ü
£ C ( s , N , q ) exp í æ lN + ------------ (s - t) ý .
è
ø
1- q
î
þ
Obviously for q < 2 - 2 we have that
Aq p ( t )
(3.21)
M lqN
M q
-------- < g = lN + 2
lN + -----------q lN .
1- q
Therefore, equations (3.21) and (3.20) imply (3.19).
Vice versa, we assume that equation (3.19) holds for the solution u ( t ) . Then
B (u (t) )
£ e g (s - t) M (1 + u s) ,
t £ s.
(3.22)
It is evident that q ( t ) = e -g ( s - t ) Q u ( t ) is a bounded (on ( - ¥ , s ] ) solution to the
equation
dq
------ + ( A - g ) q = F ( t ) ,
dt
Existence and Properties of Inertial Manifolds
where F ( t ) = exp { - g ( s - t ) } Q B ( u ( t ) ) . By virtue of (3.22) the function F ( t ) is
bounded in Q H . It is also clear that Ag = A - g is a positive operator with discrete
spectrum in Q H . Therefore, Lemma 1.1 is applicable. It gives
t
Q u (t) =
òe
- ( t - t ) A Q B ( u ( t ) ) dt .
-¥
Using the equation for P u ( t ) it is now easy to find that
u (t) =
Bps, ¥ [ u ] ( t ) ,
t £ s,
where p = P u ( s ) and Bps, ¥ [ u ] is the integral operator similar to the one in (2.1).
Hence, we have that Q u ( s ) = F ( P u ( s ) , s ) accoring to definition (2.12) of the
function F ( p , s ) = F ¥ ( p , s ) . Thus, Theorem 3.2 is proved.
The following assertion shows that IM Ms º Ms¥ can be approximated by the manifolds { MsL } , L < ¥ , with the exponential accuracy (see (2.12)).
Theorem 3.3.
Assume that spectral gap condition (3.1) is fulfilled with q < 1 . We also
assume that the function F L ( p , s ) is defined by equality (2.12) for
0 < L £ ¥ . Then the estimate
L
L
Aq ( F 1 ( p , s ) - F 2 ( p , s ) )
£ D2 ( 1 - q ) -1 e
- g N Lmin
£
1+ q
-d L
+ ---------------------2- { D 1 + Aq p } e N min ,
2 (1 - q)
(3.23)
is valid with L min = min ( L1 , L2 ) , 0 < L 1 , L2 £ ¥ ; the constants D 1 and D 2
are defined by equations (2.6) and (2.14);;
M q
-------gN = lN + 2
q lN ,
2 M (1 - q)
d N = ------------------------- lqN .
q (1 + q)
Proof.
Let 0 < L 1 < L2 < ¥ . Definition (2.12) implies that
L
L
F 1 ( p , s ) - F 2 ( p , s ) = Q ( v1 ( s ) - v2 ( s ) ) ,
(3.24)
where vj ( t ) is a solution to integral equation (2.1) with L = Lj , j = 1 , 2 . The operator Bps, L2 acting in C g , q ( s - L2 , s ) (see (2.1)) can be represented in the form
s , L2
[ v] (t)
Bp
s , L1
[ v] (t)
= Bp
+ b (v ; t, s) ,
t Î [ s - L1 , s ] ,
where
s - L1
b (v ; t, s) =
ò
s - L2
e -( t - t ) A Q B ( v ( t ) , t ) dt
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and v ( t ) is an arbitrary element in C g , q ( s - L2 , s ) . Therefore, if vj ( t ) is a solution
to problem (2.1) with L = L j , then
3
v1 ( t ) - v2 ( t ) =
s , L1
Bp
s , L1
[ v1 ] - Bp
[ v2 ] - b ( v2 ; t , s )
(3.25)
for all s - L1 £ t £ s . Let us estimate the value b ( v2 ; t , s ) . As before it is easy to
verify that
Aq b ( v2 ; t , s )
£ e
-( t - s + L1 ) lN + 1
{ r1 ( s - L1 ) + r2 ( s - L1 ) v2
s, *
}
for all t Î [ s - L1 , s ] , where
t
r1 ( t ) = M
ò
Aq e -( t - t ) A Q dt ,
-¥
t
r2 ( t ) = M
ò
Aq e -( t - t ) A Q e
g (s - t)
*
dt ,
-¥
1+ q
M q
------and the norm v2 s , is defined using the constants q * = ----------- and g* = l N + 2
q*- lN
2
*
by the formula
ì -g ( s - t ) q
ü
v2 s , = sup í e *
A v2 ( t ) : t Î [ s - L2 , s ] ý .
*
î
þ
Evidently, spectral gap condition (3.1) implies the same equation with the parameter
q * instead of q . Therefore, simple calculations based on (1.8) give us that
r1 ( t ) £ D 2
and
r2 £ e
-g ( s - t )
*
q*
----- ,
2
where D2 is defined by formula (2.14). Using Lemma 2.1 under condition (2.3) with
q * instead of q we obtain that
v2 s ,
*
£ ( 1 - q * ) -1 { D 1 + Aq p } ,
where D1 is given by formula (2.6). Therefore, finaly we have that
Aq b ( v2 ; t , s )
£ e
-( t - s + L1 ) l N + 1
ì
ü
q*
g L
- e * 1 ( D1 + Aq p ) ý
í D2 + --------------------*)
(
2
1
q
î
þ
for all t Î [ s - L1 , s ] . Consequently,
ì
ü
sup í e g ( t - s ) Aq b ( v2 ; t , s ) : t Î [ s - L1 , s ] ý £
î
þ
£ e
- g L1
ì
ü
q*
g L
- e * 1 ( D 1 + Aq p ) ý .
í D2 + --------------------*)
(
2
1
q
î
þ
Existence and Properties of Inertial Manifolds
Therefore, since
gives us that
s , L1
Bp
is a contractive operator in C g , q ( s - L1 , s ) , equation (3.25)
( 1 - q ) v1 - v2 C
£ e
-g L1
g , q ( s - L1 , s )
£
1 + q -( g - g* ) L1 æ D + Aq p ö
--- × ----------- e
D2 + 1
è 1
ø .
2 1- q
--- ( 1 + q ) . Hence, estimate (3.23) follows from
Here we also use the equality q * = 1
2
(3.24). Theorem 3.3 is proved.
E x e r c i s e 3.5 Show that in the case when B ( u , t ) £ M equation (3.23)
can be replaced by the inequality
Aq ( F
L1
(p, s) - F
L2
( p , s ) ) £ D2 ( 1 - q ) -1 e
- g N Lmin
.
E x e r c i s e 3.6 Assume that the hypotheses of Theorem 3.1 hold. Then the
estimate
Aq ( Q u ( t ) - F L ( P u ( t ) , t ) ) £
q
-g ( t - s)
- a lN L
£ C æ 1 + Aq u0 ö e N
+ CR e
è
ø
holds for t ³ t and for any solution u ( t ) to problem (1.1) possess*
ing the dissipativity property: Aq u ( t ) £ R for t ³ t* ³ s and for
2 M l q and the constant a > 0
some R and t . Here gN = l N + -------q N
*
does not depend on N .
Therefore, if the hypotheses of Theorem 3.1 hold, then a bounded solution to problem (1.1) gets into the exponentially small (with respect to lqN and L ) vicinity of
the manifold { MsL : - ¥ < s < ¥ } at an exponential velocity.
According to (2.12) in order to build an approximation { MsL } of the inertial
manifold { Ms } we should solve integral equation (2.1) for L large enough. This
equation has the same structure both for L < ¥ and for L = ¥ . Therefore, it is impossible to use the surfaces { MsL } directly for the effective approximation of { Ms } .
However, by virtue of contractiveness of the operator Bps, ¥ in the space Cs– =
= Cg , q ( -¥ , s ) , its fixed point vs ( t ) which determines Ms can be found with the
help of iterations. This fact enables us to construct the collection { Mn , s } of approximations for { Ms } as follows. Let v0 = v0 , s ( t ; p ) be an element of C s– . We take
vn º vn , s ( t , p ) =
Bps,
¥
[ vn - 1 ] ( t ) ,
n = 1, 2, ¼ ,
and define the surfaces { Mn , s } by the formula
Mn , s = { p + Fn ( p , s ) : p Î P H } ,
where Fn ( p , s ) = Q vn , s ( p , s ) ,
n = 1, 2, ¼
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E x e r c i s e 3.7
Let v0 º p and let B ( u , t ) º B ( u ) . Show that
F0 ( p , s ) º 0 and F1 ( p , s ) = A -1 Q B ( p ) .
E x e r c i s e 3.8
that
Assume that spectral gap condition (3.1) is fulfilled. Show
Aq ( Fn ( p , s ) - F ( p , s ) )
£ q n æ v0 s + ( 1 - q ) -1 D 1 + Aq p ö ,
è
ø
where D1 is defined by formula (2.6) and F ( p , s ) is the function
that determines the inertial manifold.
E x e r c i s e 3.9 Prove the assertion for Fn ( p , s ) similar to the one in Exercise 3.5.
Theorems represented above can also be used in the case when the original system is
dissipative and estimates (1.2) and (1.3) are not assumed to be uniform with respect
to u Î D ( Aq) . The dissipativity property enables us to restrict ourselves to the consideration of the trajectories lying in a vicinity of the absorbing set when we study
the asymptotic behaviour of solutions to problem (0.1). In this case it is convenient
to modify the original problem. Assume that the mapping B ( u , t ) is continuous with
respect to its arguments and possesses the properties
B ( u , t ) £ Cr ,
B ( u1 , t ) - B ( u2 , t )
£ C r Aq ( u1 - u 2 )
(3.26)
for any r > 0 and for all u , u1 , and u 2 lying in the ball Br = { v : Aq v £ r } .
Let c ( s ) be an infinitely differentiable function on R+ = [ 0 , ¥ ) such that
c (s) = 1 ,
0 £ s £ 1;
0 £ c (s) £ 1 ,
c (s) = 0 , s ³ 2 ;
c ¢ (s) £ 2 ,
We define the mapping BR ( u , t ) by assuming that
BR ( u , t ) = c ( R -1 Aq u ) B ( u , t ) ,
s Î R+ .
u Î D ( Aq ) .
(3.27)
E x e r c i s e 3.10 Show that the mapping BR ( u , t ) possesses the properties
Aq B R ( u , t ) £ M ,
BR ( u1 , t ) - BR ( u2 , t ) £ M Aq ( u1 - u2 ) ,
(3.28)
where M = C2 R ( 1 + 2 ¤ R ) and C r is a constant from (3.26).
Let us now assume that B ( u , t ) satisfies condition (3.26) and the problem
du
------ + Au = B ( u , t ) ,
dt
u t = 0 = u0 ,
(3.29)
has a unique mild solution on any segment [ s , s + T ] and possesses the following
dissipativity property: there exists R0 > 0 such that for any R > 0 the relation
Continuous Dependence of Inertial Manifold on Problem Parameters
Aq u ( t , s ; u 0 ) £ R0
for all
t - s ³ t0 ( R )
(3.30)
holds, provided that Aq u0 £ R . Here u ( t , s ; u0 ) is the solution to problem (3.29).
E x e r c i s e 3.11 Show that the asymptotic behaviour of solutions to problem
(3.29) completely coincides with the asymptotic behaviour of solutions to the problem
du
------ + A u = B 2 R ( u , t ) ,
dt
0
u t = s = u0 ,
(3.31)
where B 2 R is defined by formula (3.27) and R 0 is the constant
0
from equation (3.30).
E x e r c i s e 3.12 Assume that for a solution to problem (3.29) the invariance
property of the absorbing ball is fulfilled: if Aq u0 £ R0 , then
Aq u ( t , s ; u0 ) £ R for all t £ s . Let Mt be the invariant manifold
q
0
of problem (3.31). Then the set MR
t = M t Ç { u : A u £ R 0 } is inR0
variant for problem (3.29): if u0 Î Ms , then u ( t , s ; u 0 ) Î MsR0 ,
t ³ s.
Thus, if the appropriate spectral gap condition for problem (3.29) is fulfilled, then
there exists a finite-dimensional surface which is a locally invariant exponentially attracting set.
In conclusion of this section we note that the version of the Lyapunov-Perron method represented here can also be used for the construction (see [13]) of inertial
manifolds for retarded semilinear parabolic equations similar to the ones considered
in Section 8 of Chapter 2. In this case both the smallness of retardation and the fulfilment of the spectral gap condition of the form (3.1) are required.
§ 4
Continuous Dependence of Inertial
Manifold on Problem Parameters
Let us consider the Cauchy problem
d-----u
+ A u = B* (u , t ) ,
dt
u t = s = u0 ,
s ÎR
(4.1)
in the space H together with problem (1.1). Assume that B * ( u , t ) is a nonlinear
mapping from D ( Aq ) ´ R into H possessing properties (1.2) and (1.3) with the
same constant M as in problem (1.1). If spectral gap condition (3.1) is fulfilled, then
problem (4.1) (as well as (1.1)) possesses an invariant manifold
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Ms* = { p + F*( p , s ) : p Î P H } ,
s ÎR.
(4.2)
The aim of this section is to obtain an estimate for the distance between the
manifolds M s and Ms* . The main result is the following assertion.
Theorem 4.1.
Assume that conditions (1.2),, (1.3),, and (3.1) are fulfilled both for
problems (1.1) and (4.1).. We also assume that
B ( v , t ) - B * ( v , t ) £ r1 + r2 Aq v
(4.3)
for all v Î D ( Aq ) and t Î R , where r1 and r2 are positive numbers. Then
the equation
sup Aq ( F ( p , s ) - F*( p , s ) )
s ÎR
r1 + r2
- + C2 ( q , q , M ) r2 Aq p
£ C1 ( q , q ) -----------------l1N- q
is valid for the functions F ( p , s ) and F*( p , s ) which give the invariant
manifolds for problems (1.1) and (4.1) respectively. Here the numbers
C1 ( q , q ) and C 2 ( q , q , M ) do not depend on N and rj .
Proof.
Equation (2.12) with L = ¥ implies that
s
Aq ( F ( p , s ) - F*( p , s ) )
£
ò
Aq e -( s - t ) A Q B ( v ( t ) , t ) - B * ( v *( t ) , t ) dt ,
-¥
where v ( t ) and v*( t ) are solutions to the integral equations of the type (2.1) corresponding to problems (1.1) and (4.1) respectively. Equations (1.3) and (4.3) give
us that
B ( v ( t ) , t ) - B *( v*( t ) , t )
£ M Aq ( v ( t ) - v*( t ) ) + æ r1 + r2 Aq v ( t ) ö £
è
ø
£ e g ( s - t ) ( M v - v * s + r2 v s ) + r1
(4.4)
for t £ s , where
ì
ü
w s = ess sup í e -g ( s - t ) AQ w ( t ) ý
t£ s
î
þ
(4.5)
M q
-------and g = lN + 2
q l N as before. Hence, after simple calculations as in Section 2 we
find that
r
1+k
q
Aq ( F ( p , s ) - F*( p , s ) ) £ --- æ v - v* s + -----2- v sö + r1 -------------- .
è
ø
2
M
l1N-+q1
(4.6)
Let us estimate the value v - v * s . Since v and v * are fixed points of the corresponding operator Bps, ¥ , we have that
Continuous Dependence of Inertial Manifold on Problem Parameters
s
Aq ( v ( t )
ò
£
òAe
q -( t - t ) A P
B ( v (t ) , t ) - B *( v* ( t ) , t ) dt +
t
s
+
- v* ( t ) )
Aq e -( t - t ) A Q B ( v ( t ) , t ) - B *( v* ( t ) , t ) dt .
-¥
Therefore, by using spectral gap condition (3.1) and estimate (4.4) as above it is
easy to find that
qr
1 + k-ö
æ 1 - + -------------v - v * s £ q v - v* s + ---------2- × v s + r1 ç ----------.
1- q
1-q ÷
M
l
l
è N
N + 1ø
Consequently,
r1 2 + k
r
q
- × ------------- .
v - v * s £ ----------- × -----2- × v s + ---------1 - q l1 - q
1- q M
N
Therefore, equation (4.6) implies that
Aq ( F ( p , s ) - F*( p , s ) )
r2
q
2-q 2 + k
£ ----------- × -------× v s + r1 × --------------- × ------------ .
1- q 2 M
2 - 2 q l1 - q
N
Hence, estimate (2.5) gives us the inequality
Aq ( F ( p , s ) - F*( p , s ) )
r1 + r2 2 + k
r
q
- × ------------- + ---------------------2- × -----2- Aq p .
£ -----------------2
1
q
(1 - q) l
2 (1- q) M
N
This implies the assertion of Theorem 4.1.
Let us now consider the Galerkin approximations um ( t ) of problem (1.1). We remind (see Chapter 2) that the Galerkin approximation of the order m is defined as a
function u m ( t ) with the values in Pm H , this function being a solution to the problem
d um
----------- + A um = Pm B ( um ) ,
0
dt
um
t=s
= u0 m .
(4.7)
Here Pm is the orthoprojector onto the span of elements { e1 , ¼ , em } in H .
E x e r c i s e 4.1 Assume that spectral gap condition (3.1) holds and m ³ N + 1 .
Show that problem (4.7) possesses an invariant manifold of the form
Ms( m ) = { p + F ( m ) ( p , s ) : p Î P H }
in Pm H , where the function F ( m ) ( p , s ) : P H ® ( Pm - P ) H is defined by equation similar to (2.12).
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The following assertion holds.
3
Theorem 4.2.
Assume that spectral gap condition (3.1) holds. Let F ( p , s ) and
F ( m ) ( p , s ) be the functions defined by the formulae of the type (2.12) and
let these functions give invariant manifolds for problems (1.1) and (4.7) for
m ³ N + 1 respectively. Then the estimate
Aq ( F( p ,
s)
- F(m) ( p ,
s) )
ì
ü
D1 + Aq p ï
C (q, M, q) ï
£ ------------------------------ í 1 + --------------------------- ý
lN + 1 ï
l1m-+q1
ï
1 - --------------î
þ
l
(4.8)
m+1
is valid, where the constant D1 is defined by formula (2.6)..
Proof.
It is evident that
( F ( p , s ) - F( m) ( p , s ) ) = Q [ v ( s ; p ) - v( m) ( s ; p ) ] ,
(4.9)
where v ( t , p ) and v ( m ) ( t , p ) are solutions to the integral equations
v (t) =
Bps,
¥
[ v] (t) ,
-¥ < t £ s ,
and
v ( m ) ( t ) = Pm Bps, ¥ [ v ( m ) ] ( t ) ,
Here
-¥ < t £ s .
Bps, ¥ is defined as in (2.1). Since
v ( t ) - v ( m ) ( t ) = ( I - Pm ) v ( t ) + Pm [ Bps, ¥ [ v ] ( t ) - Bps, ¥ [ v ( m ) ] ( t ) ] ,
we have
Aq ( v ( t ) - v ( m ) ( t ) ) = Aq ( 1 - Pm ) v ( t ) + Aq [ Bps, ¥ [ v ] ( t ) - Bps, ¥ [ v ( m ) ] ( t ) ] .
The contractiveness property of the operator
Bps,
¥
leads to the equation
Aq ( v ( t ) - v ( m ) ( t ) ) = Aq ( 1 - Pm ) v ( t ) + q × v - v ( m ) s e g ( s - t ) .
In particular, this implies that
v - v ( m ) s º sup e -g ( s - t ) Aq ( v ( t ) - v ( m ) ( t ) ) £ ( 1 - q ) -1 ( 1 - Pm ) v s .
t< s
Hence, with the help of (4.9) we find that
Aq ( F ( p , s ) - F ( m ) ( p , s ) )
£ ( 1 - q ) -1 ( 1 - Pm ) v s .
£ Aq ( v ( s ) - v ( m ) ( s ) ) £ v - v ( m ) s £ (4.10)
Continuous Dependence of Inertial Manifold on Problem Parameters
Let us estimate the value ( 1 - Pm ) v s . It is clear that
t
( 1 - Pm ) v ( t ) =
òe
- ( t - t ) A ( 1 - P ) B ( v ( t ) ) dt .
m
-¥
Therefore, Lemma 2.1.1 (see also (1.8)) gives us that
t
Aq ( 1 - Pm ) v ( t )
t
+M
ò
£ M
ò
-( t - t ) lm + 1
q - ö q + lq
æ ---------dt +
m +1 e
è t - tø
-¥
-lm + 1 ( t - t ) ( t - t ) g
q - ö q + lq
æ ---------dt v s × e g ( s - t ) ,
e
m +1 e
è t - tø
-¥
where
M q
-------g º lN + 2
q lN < lN + 1 < lm + 1
as above. Simple calculations analogous to the ones in Lemma 2.1 imply that
Aq ( 1 - Pm ) v ( t )
q
M ( 1 + k ) M ( 1 + k ) lm + 1 g ( s - t )
- e
£ ----------------------- + -------------------------------------vs,
lm + 1 - g
l1m-+q1
where the constant k has the form (1.7). Consequently, using (2.5) we obtain
lN + 1 ö -1
M (1 + k)
( 1 - Pm ) v s £ ----------------------- æ 1 + æ 1 - -------------v sö £
è
è
ø
l m + 1ø
l1m-+q1
lN + 1 ö -1
M (1 + k)
£ ---------------------- æ 1 + æ 1 - -------------( 1 - q ) -1 ( D 1 + Aq p )ö .
è
è
ø
ø
1-q
l
lm + 1
m +1
This and (4.10) imply estimate (4.8). Theorem 4.2 is proved.
E x e r c i s e 4.2 In addition assume that the hypotheses of Theorem 4.2 hold
and B ( u , t ) £ M . Show that in this case estimate (4.8) has the form
Aq ( F ( p , s ) - F ( m ) ( p , s ) )
-1 + q
£ C ( q ; M , q ) lm
+1 .
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E x a m p l e 5.1
3
Let us consider the nonlinear heat equation
§ 5
ì
ï
ï
í
ï
ï
î
Examples and Discussion
¶2 u
¶u
------ = n ---------2- + f ( x , u , t ) ,
¶t
¶x
u x=0 = u x=l = 0 ,
0 < x < l, t > 0 ,
( 5.1 )
( 5.2 )
u t = 0 = u0 ( x ) .
( 5.3 )
Assume that n is a positive parameter and f ( x , u , t ) is a continuous function
of its variables which possesses the properties
M
f ( x , 0 , t ) £ ------ .
l
f ( x , u1 , t ) - f ( x , u2 , t ) £ M u1 - u2 ,
Problem (5.1)–(5.3) generates a dynamical system in L 2 ( 0 , l ) (see Section 3
of Chapter 2). Therewith
d2
A = -n --------2- ,
dx
D ( A ) = H 01 ( 0 , L ) Ç H 2 ( 0 , l ) ,
where H s ( 0 , l ) is the Sobolev space of the order s . The mapping B ( . , t ) given
by the formula u ( x ) ® f ( x , u ( x ) , t ) satisfies conditions (1.2) and (1.3) with
q = 0 . In this case spectral gap condition (2.3) has the form
p 2M
-------n ----( ( N + 1 ) 2 - N2) ³ 4
q .
l2
Thus, problem (5.1)–(5.3) possesses an inertial manifold of the dimension N ,
provided that
l 2M -------------- + 2
N > -1
2 n q p2
(5.4)
for some q < 2 - 2 .
E x e r c i s e 5.1 Find the conditions under which the inertial manifold of problem (5.1)–(5.3) is one-dimensional. What is the structure of the corresponding inertial form?
E x e r c i s e 5.2 Consider problem (5.1) and (5.3) with the Neumann boundary conditions:
¶-----u
¶x
x=0
¶u
= -----¶x
x=l
=0
(5.5)
Show that problem (5.1), (5.3), and (5.5) has an inertial manifold
of the dimension N + 1 , provided condition (5.4) holds for some
Examples and Discussion
N ³ 0 . (Hint: A = - n ( d 2 ¤ dx 2 ) + e with condition (5.5), B ( u , t ) =
= - e u + f ( x , u , t ) , where e > 0 is small enough).
E x e r c i s e 5.3 Find the conditions on the parameters of problem (5.1), (5.3),
and (5.5) under which there exists a one-dimensional inertial manifold. Show that if f ( x , u , t ) º f ( u , t ) , then the corresponding inertial form is of the type
p· ( t ) = f ( p ( t ) , t ) ,
p t = 0 = p0 .
E x a m p l e 5.2
Consider the problem
ì
ï
í
ï
î
¶2 u
¶-----¶u
u= n ---------2- + f æ x , u , ------ , tö ,
è
¶t
¶x ø
¶x
u x=0 = u x=l = 0,
0 < x < l, t > 0 ,
(5.6)
u t = 0 = u0 ( x ) .
Here n > 0 and f ( x , u , x , t ) is a continuous function of its variables such that
f ( x , u1 , x , t ) - f ( x , u2 , x , t ) £ L1 u1 - u2 + L2 x1 - x2
(5.7)
for all x Î ( 0 , l ) , t ³ 0 and
l
ò [ f (x , 0 , 0 , t) ] dx £ L
2
2
3,
0
where Lj are nonnegative numbers. As in Example 5.1 we assume that
d2
A = -n --------2- ,
dx
D ( A ) = H01 ( 0 , l ) Ç H 2 ( 0 , l ) ,
¶u
B ( u , t ) = f æ x , u , ------ , tö .
è
¶x ø
It is evident that
B ( u1 , t ) - B ( u2 , t )
Here
¶u
¶u
£ L1 u1 - u2 + L 2 --------1- - --------2- .
¶x
¶x
. is the norm in L2 ( 0 , l ) . By using the obvious inequality
¶u
-----¶x
2
p 2
³ æ ---ö u
è lø
2
,
u Î H01 ( 0 , l ) ,
we find that
1 - æ L --l- + L ö A1 / 2 ( u - u ) .
B ( u1 , t ) - B ( u 2 , t ) £ -----2ø
1
2
nè 1p
Hence, conditions (1.2) and (1.3) are fulfilled with
--- ,
q=1
2
ì
ü
1 l
M = max í L3 ; ------- æ --- L 1 + L2ö ý .
è
ø
p
n
î
þ
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Therewith spectral gap condition (2.3) acquires the form
p
2M
n --- ( 2 N + 1 ) ³ -------q (2 N + 1 + k (N + 1) ) ,
l
where
x
3
1
k = ------2
òx
-1 / 2 -x
e
dx =
p
--- .
2
0
Thus, the equation
1+
pq n
p N + 1£ ------------------ × ----------------2 2N + 1
2lM
or
pq n
p
p
1 - £ --------------2 + --- + --- × ----------------2
2 2N + 1
lM
must be valid for some 0 < q < 2 - 2 . We can ensure the fulfilment of this condition only in the case when
p q0 n
p
2 + --- < -----------------,
lM
2
q0 = 2 - 2 ,
i.e. if
ì
2 -1
1 - æ --l- L + L ö ü < 2 p ------n- × ------------------------.
M º max í L3 ; -----2ø ý
èp 1
l
n
2 2+ p
î
þ
(5.9)
Thus, in order to apply the above-presented theorems to the construction of the
inertial manifold for problem (5.6) one should pose some additional conditions
(see (5.7) and (5.9)) on the nonlinear term f ( x , u , ¶u ¤ ¶x , t ) or require that the
diffusion coefficient n be large enough.
E x e r c i s e 5.4 Assume that f ( x , u , x , t ) = e f ( x , u , x , t ) in (5.6), where
the function f possesses properties (5.7) and (5.8) with arbitrary
Lj ³ 0 . Show that problem (5.6) has an inertial manifold for any
0 < e < e0 , where
ì
ü
n
1 l2 -1
e0 = 2 p ------- × ------------------------- × max í L3 ; --- æ -----L1 + L2ö ý
è
ø
n p
l 2 2+ p
î
þ
-1
.
Characterize the dependence of the dimension of inertial manifold
on e .
E x e r c i s e 5.5 Study the question on the existence of an inertial manifold for
problem (5.6) in which the Dirichlet boundary condition is replaced
by the Neumann boundary condition (5.5).
Examples and Discussion
It should be noted that
ln = Cd n 2 / d ( 1 + o ( 1 ) ) ,
n ® ¥,
d = dim W ,
where l n are the eigenvalues of the linear part of the equation of the type
¶-----u
= n Du + f ( x , u , Ñ u , t ) ,
x ÎW, t > 0,
¶t
in a multidimensional bounded domain W . Therefore, we can not expect that Theorem 3.1 is directly applicable in this case. In this connection we point out the paper
[3] in which the existence of IM for the nonlinear heat equation is proved in a bounded domain W Ì R d ( d £ 3 ) that satisfies the so-called “principle of spatial averaging” (the class of these domains contains two- and three-dimensional cubes).
It is evident that the most severe constraint that essentially restricts an application of Theorem 3.1 is spectral gap condition (3.1). In some cases it is possible to
weaken or modify it a little. In this connection we mention papers [6] and [7]
in which spectral gap condition (3.1) is given with the parameters q = 2 and k = 0
for 0 £ q < 1 . Besides it is not necessary to assume that the spectrum of the operator A is discrete. It is sufficient just to require that the selfadjoint operator A possess a gap in the positive part of the spectrum such that for its edges the spectral
condition holds. We can also assume the operator A to be sectorial rather than selfadjoint (for example, see [6]).
Unfortunately, we cannot get rid of the spectral conditions in the construction
of the inertial manifold. One of the approaches to overcome this difficulty runs as
follows: let us consider the regularization of problem (0.1) of the form
d-----u+ A u + e Am u = B ( u , t ) ,
u t = 0 = u0 .
(5.10)
dt
˜
Here e > 0 and the number m > 0 is chosen such that the operator A = A + e A m
possesses spectral gap condition (3.1). Therewith IM for problem (5.10) should be
naturally called an approximate IM for system (0.1). Other approaches to the construction of the approximate IM are presented below.
It should also be noted that in spite of the arising difficulties the number of equations
of mathematical physics for which it is possible to prove the existence of IM is large
enough. Among these equations we can name the Cahn-Hillard equations in the domain W = ( 0 , L) d , d = dim W £ 2 , the Ginzburg-Landau equations ( W = ( 0 , L) d ,
d £ 2 ), the Kuramoto-Sivashinsky equation, some equations of the theory of oscillations ( d = 1 ), a number of reaction-diffusion equations, the Swift-Hohenberg equation, and a non-local version of the Burgers equation. The corresponding references
and an extended list of equations can be found in survey [8].
In conclusion of this section we give one more interesting application of the
theorem on the existence of an inertial manifold.
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E x a m p l e 5.3
Let us consider the system of reaction-diffusion equations
¶-----u
= n Du + f ( u , Ñ u ) ,
¶t
¶-----u
¶n
¶W
= 0,
(5.11)
in a bounded domain W Ì R d . Here u = ( u1 , ¼ , u m ) and the function f ( u , x )
satisfies the global Lipschitz condition:
f ( u , x ) - f ( v , h ) £ L { u - v 2 + x - h 2 }1 / 2 ,
(5.12)
where u , v Î R m , x , h Î R m d , and L > 0 . We also assume that f ( 0 , 0 ) £ L .
Problem (5.11) can be rewritten in the form (0.1) in the space H = [ L 2 ( W ) ] m
if we suppose
A u = - n Du + u ,
B ( u ) = u + f ( u , Ñu ) .
It is clear that the operator A is positive in its natural domain and it has a discrete spectrum. Equation (5.12) implies that the relation
B (u) - B (v)
ì
£ Lí u -v
î
2
ü1 / 2
+ Ñ( u - v ) 2 ý + u - v
þ
æ
ì
1 - üö ì u - v
£ ç 1 + L maxí 1 ; -----ý÷ í
n þø î
è
î
2
ü
+ n Ñ( u - v ) 2 ý
þ
£
1/2
is valid for B(u) . Thus,
B (u) - B (v)
£ M A1 / 2 ( u - v ) ,
where
ì
1- ü .
M = 1 + L maxí 1 ; -----ý
nþ
î
Therefore, problem (5.11) generates an evolutionary semigroup St (see Chapter 2) in the space D ( A1 / 2 ) . An important property of St is the following: the
subspace L which consists of constant vectors is invariant with respect to this
semigroup. The dimension of this subspace is equal to m . The action of the
semigroup in this subspace is generated by a system of ordinary differential
equations
du
------- = f ( u , 0 ) ,
dt
u (t) Î L .
(5.13)
E x e r c i s e 5.6 Assume that equation (5.12) holds for x = h = 0 . Show that
equation (5.13) is uniquely solvable on the whole time axis for any
initial condition and the equation
ì -L ( s - t )
ü
sup í e
u (s) ý < ¥
þ
holds for any s Î R .
t £ sî
(5.14)
Examples and Discussion
The subspace L consists of the eigenvectors of the operator A corresponding to the
eigenvalue l 1 = 1 . The next eigenvalue has the form l2 = n m1 + 1 , where m 1 is
the first nonzero eigenvalue of the Laplace operator with the Neumann boundary
condition on ¶ W . Therefore, spectral gap equation (3.1) can be rewritten in the
form
æ
ì
1 - üö æ æ 1 + p
--q- ç 1 + L maxí 1 ; -----n m1 ³ 2
--- öø n m 1 + 1 + 1öø
ý÷
2
n þø è è
è
î
(5.15)
for N = m and q = 1 ¤ 2 , where 0 < q < 2 - 2 . It is clear that there exists n0 > 0
such that equation (5.15) holds for all n ³ n0 . Therefore, we can apply Theorem 3.1
to find that if n is large enough, then there exists IM of the type
ì
ü
M = í p + F ( p) : p Î L , F : L ® H
Lý .
î
þ
The invariance of the subspace L and estimate (5.14) enable us to use Theorem 3.2
and to state that L Ì M . This easily implies that F ( p) º 0 , i.e. M = L . Thus,
Theorem 3.1 gives us that for any solution u ( t ) to problem (5.11) there exists a so˜
lution u ( t ) to the system of ordinary differential equations (5.13) such that
˜
u ( t ) - u ( t ) 1 £ C e -g t ,
t ³ 0,
where the constant g > 0 does not depend on u ( t ) and
of the first order.
E x e r c i s e 5.7
.
1
is the Sobolev norm
Consider the problem
¶2u
¶u
------ = n ---------2 + f ( x , u ) , 0 < x < p ;
¶t
¶x
u x = 0 = u x = p = 0 , (5.16)
where the function f ( x , u ) has the form
f ( x , u ) = g1 ( u1 , u2 ) sin x + g2 ( u 1 , u2 ) sin 2 x .
Here
p
ò
2
u j = --- u ( x ) sin j x dx ,
p
j = 1, 2 ,
0
and gj ( u1 , u 2 ) are continuous functions such that
gj ( u 1 , u2 ) - gj ( v1 , v2 ) £
£ Lj æ u1 - v1 2 + u2 - v2 2ö
è
ø
Show that if
1/2
;
gj ( 0 , 0 ) = 0 .
2
- p ( L12 + L 22 ) ,
n > -----------------------5 ( 2 - 1)
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then the dynamical system generated by problem (5.16) has the
two-dimensional (flat) inertial manifold
M = { p1 sin x + p2 sin 2 x : p1 , p2 Î R }
and the corresponding inertial form is:
p· 1 + n p 1 = g1 ( p1 , p 2 ) ,
p· 2 + 4 n p 2 = g2 ( p1 , p2 ) .
E x e r c i s e 5.8 Study the question on the existence of an inertial manifold
for the Hopf model of turbulence appearance (see Section 7 of Chapter 2).
§ 6 Approximate Inertial Manifolds
for Semilinear Parabolic Equations
Even in the cases when the existence of IM can be proved, the question concerning
the effective use of the inertial form
¶t p + A p = PB ( p + F( p , t ) , t )
(6.1)
is not simple. The fact is that it is not practically possible to find a more or less explicit solution to the integral equation for F ( p , t ) even in the finite-dimensional
case. In this connection we face the problem of approximate or asymptotic construction of an invariant (inertial) manifold. Various aspects of this problem related to finite-dimensional systems are presented in the book by Ya. Baris and O. Lykova [14].
For infinite-dimensional systems the problem of construction of an approximate IM can be interpreted as a problem of reduction, i.e. as a problem of constructive description of finite-dimensional projectors P and functions F ( . , t ) : P H ®
® ( 1 - P ) H such that an equation of form (6.1) “inherits” (of course, this needs
to be specified) all the peculiarities of the long-time behaviour of the original system
(0.1). It is clear that the manifolds arising in this case have to be close in some sense
to the real IM (in fact, the dynamics on IM reproduces all the essential features of
the qualitative behaviour of the original system). Under such a formulation a problem of construction of IM acquires secondary importance, so one can directly construct a sequence of approximate IMs. Usually (see the references in survey [8]) the
problem of the construction of an approximate IM can be formulated as follows: find
a surface of the form
Mt = { p + F ( p , t ) : p Î P H } ,
(6.2)
which attracts all the trajectories of the system in its small vicinity. The character of
closeness is determined by the parameter l-N1+ 1 related to the decomposition
Approximate Inertial Manifolds for Semilinear Parabolic Equations
ì
ï
í
ï
î
dp
------ + A p = P B ( p + q , t ) ,
dt
(6.3)
dq
------ + A q = ( 1 - P ) B ( p + q , t ) .
dt
( 0)
We obtain the trivial approximate IM Mt if we put F ( p , t ) = F0 ( p , t ) º 0 in (6.2).
In this case M(t 0 ) is a finite-dimensional subspace in 0 whereas inertial form (6.1)
turns into the standard Galerkin approximation of problem (0.1) corresponding to
this subspace. One can find the simplest non-trivial approximation Mt( 1 ) using formula (6.2) and assuming that
F ( p , t ) = F1 ( p , t ) º A -1 ( 1 - P ) B ( p , t ) .
(6.4)
The consideration of system (0.1) on Mt( 1 ) leads to the second equation of equations (6.3) being replaced by the equality A q = ( 1 - P ) B ( p , t ) . The results of the
computer simulation (see the references in survey [8]) show that the use of just the
first approximation to IM has a number of advantages in comparison with the traditional Galerkin method (some peculiarities of the qualitative behaviour of the system
can be observed for a smaller number of modes).
There exist several methods of the construction of an approximate IM. We present
the approach based on Lemma 2.1 which enables us to construct an approximate IM
of the exponential order, i.e. the surfaces in the phase space H such that their exponentially small (with respect to the parameter lN +1 ) vicinities uniformly attract all
the trajectories of the system. For the first time this approach was used in paper [15]
for a class of stochastic equations in the Hilbert space. Here we give its deterministic
version.
Let us consider the integral equation (see(2.1))
v (t) =
Bps, L [ v ] ( t ) ,
s -L £ t £ s
and assume that L = r l-Nq+ 1 , where the parameter r possesses the property
qq - l -( 1 - q ) r 1 - q + rö < 1 .
q º M æ ----------è1 - q 1
ø
(6.5)
In this case equations (2.2) hold. Hence, Lemma 2.1 enables us to construct a collec-q
tion of manifolds { M sL } for L = r l N
+ 1 with the help of the formula
MsL = { p + F L( p , s ) : p Î P H } ,
(6.6)
where
s
FL( p , s ) =
òe
s-L
-( s - t ) A
Q B ( v ( t ) , t ) dt º Q v ( s , p ) .
Here v ( t ) = v ( t , p ) is a solution to integral equation (2.1) and L = r l-Nq+ 1 .
(6.7)
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E x e r c i s e 6.1 Show that both the function F L(p , s) and the surface
do not depend on s in the autonomous case ( B(u , t) º B(t) ) .
MsL
The following assertion is valid.
Theorem 6.1.
There exist positive numbers r1 = r1 ( M , q , l 1 ) and L = L ( M , q , l 1 )
such that if
l1N-+q1 ³ L r -1 ,
-q
L = r lN + 1 ,
0 < r £ r1 ,
(6.8)
then the mappings F L ( . , s ) : P H ® Q H defined by equation (6.7) possess
the property
Aq ( Q u ( t ) - F L ( P u ( t ) , t ) )
£
ì s0 q
ü
ì r 1- q ü
(1)
(2)
£ CR exp í - ----r- l N + 1 ( t - t* ) ý + C R exp í - --2- l N + 1 ý
î
þ
î
þ
(6.9)
for all t ³ t* + L ¤ 2 . Here s 0 > 0 is an absolute constant and u ( t ) is a mild
solution to problem (1.1) such that
Aq u ( t ) £ R
t Î [ t* , + ¥ ) .
If B ( u , t ) £ M , then estimate (6.9) can be rewritten as follows:
for
(6.10)
Aq ( Q u ( t ) - F L ( Pu ( t ) , t ) ) £
ì s0 q
ü
1- q
£ CR exp í - ----r- lN + 1 ( t - t* ) ý + D 2 exp { -r l N + 1 } ,
î
þ
(6.11)
where D 2 is defined by equality (2.14)..
Proof.
Let
ˆ ( t ) = U ( t , s ; Pu ( s ) + F L ( Pu ( s ) , s ) ) ,
u
t* £ s £ t ,
where U ( t , s ; v ) is a mild solution to problem (1.1) with the initial condition
v Î D ( Aq ) at the moment s . Therewith u ( t ) = U ( t , 0 ; u0 ) . It is evident that
ˆ (t) ) +
Q u ( t ) - FL( P u ( t) , t) = Q ( u ( t ) - u
ˆ (t ) , t) + FL( P u
ˆ (t) , t) - FL(P u (t ) , t) .
ˆ ( t) - FL(P u
+ Qu
(6.12)
Let us estimate each term in this decomposition. Equation (1.6) implies that
ˆ (t))
Aq Q ( u ( t ) - u
£ a N ( t - s ) Aq ( Q u ( s ) - F L ( P u ( s ) , s ) ) ,
(6.13)
Approximate Inertial Manifolds for Semilinear Parabolic Equations
where
aN ( t ) = e
- l N + 1t
+ M ( 1 + k ) a1 l-N1++1q e
a2 t
.
Using (2.16) we find that
ˆ ( s) - FL( P u
ˆ (s) , s) ) £ b (L , t - s)
Aq ( Q u
N
(6.14)
where
bN ( L , t ) = D2 ( 1 - q ) -1 e -g L + q ( 1 - q ) -2 ( R + D 1 ) e - g t ,
moreover, the second term in bN ( L , t ) can be omitted if B ( u , t ) £ M (see Exercise 2.1). At last equations (2.15) and (1.5) imply that
ˆ (t) , t) ) - FL(P u (t) , t)
Aq ( F L ( P u
£
q
a (t - s) q
£ a1 ----------- e 2
A ( Qu ( s ) - F L ( Pu ( s ) , s ) ) .
1- q
(6.15)
Thus, equations (6.12)–(6.15) give us the inequality
d ( t ) £ a N ( t - s ) d ( s ) + bN ( L , t - s )
(6.16)
for t ³ s ³ t , where
*
d ( t ) = Aq ( Q u ( t ) - F L ( Pu ( t ) , t ) )
and
a t
q
aN ( t ) = aN ( t ) + a1 ---------- e 2 =
1- q
= e
-lN + 1 t
-1 + q
-1
+ a1 M ( 1 + k ) lN
e
+ 1 + q (1 - q)
a2 t
.
It follows from (6.16) that under the condition s + L ¤ 2 £ t £ s + L the equation
d ( t ) £ aN , L d ( s ) + bN ( L , L ¤ 2 )
(6.17)
holds with
L
- l N + 1 --2
aN , L = e
q
a L
+ a1 M ( 1 + k ) l-N1++1q + ----------- e 2 .
1- q
It is clear that aN , L £ 1 ¤ 2 if
lN + 1 L > 4 ln 2 ,
l-N1++1q > 16 a1 M ( 1 + k )
and
a2 L < ln 2 ,
q £ ( 1 + 16 a1 ) -1 .
(6.18)
r l-Nq+ 1
Let r1 = r1 ( M , q , l 1 ) be such that equation (6.18) holds for L =
and for
the parameter q of the form (6.5) with 0 < r £ r1 . Then equation (6.8) with
L = 4 ( 1 + 4 a1 M ( 1 + k ) r1 ) implies that aN , L £ 1 ¤ 2 . Let tn = t* + ( 1 ¤ 2 ) n L . Then
it follows from (6.17) that
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--- d ( tn ) + bN ( L , L ¤ 2 ) ,
d ( tn + 1 ) £ 1
2
n = 0, 1, 2, ¼
After iterations we find that
L
d ( tn ) £ 2 -n d ( t0 ) + 2 bN æè L , ---öø ,
2
n = 0, 1, 2, ¼
(6.19)
Equation (6.17) also gives us that
---ö ,
--- d ( tn ) + bN æ L , L
d (t) £ 1
è 2ø
2
--- £ t £ tn + L .
tn + L
2
Therefore, it follows from (6.19) that
ì
ü
---ö
d ( t ) £ 2 exp í - --2- ( t - t* ) ln 2 ý d ( t* ) + 2 bN æ L , L
è 2ø
L
î
þ
for all t ³ t + L ¤ 2 . This implies (6.9) and (6.11) if we take g = l N + 1 in the equa*
tion for bN ( L , L ¤ 2 ) . Thus, Theorem 6.1 is proved.
In particular, it should be noted that relations (6.9) and (6.11) also mean that a solution to problem (0.1) possessing the property (6.10) reaches the layer of the thickness e N = c 1 exp { - c2 l1N-+q1 } adjacent to the surface { MLt } given by equation (6.6)
for t large enough. Moreover, it is clear that if problem (0.1) is autonomous
( B(u , t) º B(u) ) and if it possesses a global attractor, then the attractor lies in this
layer. In the autonomous case M L does not depend on t (see Exercise 6.1). These
observations give us some information about the position of the attractor in the
phase space. Sometimes they enable us to establish the so-called localization theorems for the global attractor.
E x e r c i s e 6.2
that
Let B ( u , t ) £ M . Use equations (1.4) and (1.8) to show
Aq u ( t )
£ e
-l1 ( t - s )
Aq u0 + R 0 ,
where R0 = M ( 1 + k ) l-11 + q .
In particular, the result of this exercise means that assumption (6.10) holds for any
R > R0 and for t* large enough under the condition B ( u , t ) £ M . In the general
case equation (6.10) is a variant of the dissipativity property.
E x e r c i s e 6.3 Let v 0 = v 0L, s ( t , s ) be a function from Cg , q ( s - L , s ) .
Assume that
vnL º vnL , s ( t , s ) =
Bps, L [ vn - 1 ] ( t ) ,
n = 1, 2, ¼
and
F nL ( p , s ) = Q v nL , s ( s , p ) ,
n = 0, 1, 2, ¼
Show that the assertions of Theorem 6.1 remain true for the function
F nL ( p , s ) if we add the term
Approximate Inertial Manifolds for Semilinear Parabolic Equations
q n ( v0 s + c0 ( D1 + R ) )
to the right-hand sides of equations (6.9) and (6.11). Here q is defined by equality (6.5) and v0 s is the norm of the function v0 in the
space C g , q ( s - L , s ) .
Therefore, the function F nL ( p , s ) generates a collection of approximate inertial
manifolds of the exponential (with respect to l N + 1 ) order for n large enough.
E x a m p l e 6.1
Let us consider the nonlinear heat equation in a bounded domain W Ì R d :
ì
ï
í
ï
î
¶u
------ = Du + f ( u , Ñu ) , x Î W , t > s ,
¶t
u ¶ W = 0 , u t = s = u0 ( x ) .
(6.20)
Assume that the function f ( w , x ) possesses the properties
f ( u 1 , x 1 ) - f ( u 2 , x 2 ) £ C1 ( u1 - u2 + x 1 - x 2 ) ,
f ( u , x ) £ C2 .
We use Theorem 6.1 and the asymptotic formula
lN ~ c0 N 2 / d ,
N ® ¥,
for the eigenvalues of the operator - D in W Ì R d to obtain that in the Sobolev
space H10 ( W ) for any N there exists a finite-dimensional Lipschitzian surface
MN of the dimension N such that
dist
H01 ( W )
( u ( t ) , MN ) £ C1 exp { -s 1 N 1 / d ( t - t* ) } + C2 exp { -s 2 N 1 / d }
for t ³ t and for any mild (in H01 ( W ) ) solution u ( t ) to problem (6.20). Here t
*
*
is large enough, Cj and sj are positive constants.
E x e r c i s e 6.4 Consider the abstract form of the two-dimensional system of
the Navier-Stokes equations
du
------ + n Au + b ( u , u ) = f ( t ) ,
u t = 0 = u0
(6.21)
dt
(see Example 3.5 and Exercises 4.10 and 4.11 of Chapter 2). Assume
that A1 / 2 f ( t ) < C for t ³ 0 . Use the dissipativity property for
(6.21) and the formula
l
c 0 k £ -----k- £ c 1 k
l1
for the eigenvalues of the operator A to show that there exists a collection of functions { F ( p , t ) : t ³ 1 } from PD ( A ) into ( 1 - P ) D ( A )
possessing the properties
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a)
A F ( p , t ) £ c1 N -1 / 2 ;
A ( F ( p1 , t ) - F ( p2 , t ) ) £ c2 A ( p1 - p2 )
for any p , p1 , p2 Î P D ( A) ;
b)
for any solution u ( t ) Î D ( A ) to problem (6.21) there
exists t * > 1 such that
3
AQ ( u ( t ) - F ( Pu ( t ) , t ) )
£
£ c 3 exp { -s 0 N 1 / 2 ( t - t* ) } + c 4 exp { -s 1 N 1 / 2 } .
Here P is the orthoprojector onto the first N eigenelements of the
operator A .
E x e r c i s e 6.5 Use Theorem 6.1 to construct approximate inertial manifolds
for (a) the nonlocal Burgers equation, (b) the Cahn-Hilliard equation, and (c) the system of reaction-diffusion equations (see Sections 3 and 4 of Chapter 2).
In conclusion of the section we note (see [8], [9]) that in the autonomous case the approximate IM can also be built using the equation
á F¢( p ) ; - Ap + P B ( p + F ( p ) )ñ + A F ( p ) = Q B ( p + F ( p ) ) .
(6.22)
Here p Î PH , Q = I - P , F¢ ( p ) is the Frechét derivative and á F¢( p ) , wñ is its
value at the point p on the element w . At least formally, equation (6.22) can be obtained if we substitute the pair { p ( t ) ; F( p ( t ) ) } into equation (6.3). The second of
-1
equations (6.3) implicitly contains a small parameter lN
+ 1 . Therefore, using (6.22)
we can suggest an iteration process of calculation of the sequence { Fm } giving the
approximate IM:
A Fk ( p ) = Q B ( p + Fn ( k ) ( p ) ) +
1
+ á Fn¢
2 (k)
( p ) ; Ap - P B ( p + Fn
3 (k)
( p ) )ñ ,
k ³ 1,
(6.23)
where the integers ni ( k ) are such that
0 £ ni ( k ) £ k - 1 ,
lim n i ( k ) = ¥ ,
k®¥
i = 1, 2, 3 .
One should also choose the zeroth approximation and concretely define the form of
the values ni ( k ) (for example, we can take F0 ( v ) º 0 and n i ( k ) = k - 1 , i = 1 , 2 ,
3 ). When constructing a sequence of approximate IMs one has to solve only a linear
stationary problem on each step. From the point of view of concrete calculations this
gives certain advantages in comparison with the construction used in Theorem 6.1.
However, these manifolds have the power order of approximation only (for detailed
discussion of this construction and for proofs see [9]).
E x e r c i s e 6.6 Prove that the mapping F1 ( v ) has the form (6.4) under the
condition F0 ( v ) º 0 . Write down the equation for F2 ( v ) when
n1( 2 ) = 1 , n2 ( 2 ) = n 3 ( 2 ) = 0 .
Inertial Manifold for Second Order in Time Equations
§7
Inertial Manifold for Second Order
in Time Equations
The approach to the construction of IM given in Sections 2–4 is essentially based on
the fact that the system has form (0.1) with a selfadjoint positive operator A . However, there exists a wide class of problems which cannot be reduced to this form.
From the point of view of applications the important representatives of this class are
second order in time systems arising in the theory of nonlinear oscillations:
ì d2 u
u
- + 2 e d-----+ A u = B (u, t) , t > s , e > 0 ,
ï --------2
d
t
d
t
ï
(7.1)
í
du
ïu
= u0 , ------= u1 .
ï t=s
dt t = s
î
In this section we study the existence of IM for problem (7.1). We assume that
A is a selfadjoint positive operator with discrete spectrum ( m k and ek are the corresponding eigenvalues and eigenelements) and the mapping B ( u , t ) possesses the
properties of the type (1.2) and (1.3) for 0 £ q £ 1 ¤ 2 , i.e. B ( u , t ) is a continuous
mapping from D ( Aq ) ´ R into H such that
B ( 0 , t ) £ M0 ,
B ( u1 , t ) - B ( u2 , t )
£ M1 Aq ( u1 - u2 ) ,
(7.2)
where 0 £ q £ 1 ¤ 2 and u 1 , u2 Î D ()q ) º .q .
The simplest example of a system of the form (7.1) is the following nonlinear
wave equation with dissipation:
ì
ï
ï
ï
í
ï
ï
ï
î
¶u ¶ 2 u
¶u
¶2u
--------+ 2 e ------ - ---------2 + f æ x , t ; u , ------ö = 0 ,
2
è
¶t ¶x
¶x ø
¶t
0 < x < L, t > s ,
u x=0 = u x=L = 0 ,
¶u
= u1( x ) .
u t = s = u 0 ( x ) , -----¶t t = s
(7.3)
Let 0 = D ( A1 / 2 ) ´ H . It is clear that 0 is a separable Hilbert space with the
inner product
( U , V ) = ( Au0 , v0 ) + ( u 1 , v1 ) ,
(7.4)
where U = ( u 0 ; u 1 ) and V = ( v0 ; v1 ) are elements of 0 . In the space 0 problem (7.1) can be rewritten as a system of the first order:
d
----- U ( t ) + A U ( t ) =
dt
B (U(t) , t) , t > s ;
U t = s = U0 .
(7.5)
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Here
3
¶u ( t )
U ( t ) = æ u ( t ) ; -------------ö ,
è
¶t ø
The linear operator
U0 = ( u 0 ; u 1 ) Î 0 .
A and the mapping B( U, t ) are defined by the equations:
A U = ( -u1 ; A u0 + 2 e u1 ) ,
D ( A ) = D ( A ) ´ D ( A1 / 2 ) ,
(7.6)
B ( U, t ) = ( 0 ; B ( u0 , t ) ) , U = ( u0 ; u1 ) .
E x e r c i s e 7.1 Prove that the eigenvalues and eigenvectors of the operator
A have the form:
l n± = e ± e 2 - m n ,
fn± = ( e n ; - ln± e n ) ,
n = 1 , 2 , ¼ , (7.7)
where mn and e n are the eigenvalues and eigenvectors of A .
E x e r c i s e 7.2 Display graphically the spectrum of the operator A on the
complex plane.
These exercises show that although problem (7.1) can be represented in the form
(7.5) which is formally identical to (0.1) we cannot use Theorem 3.1 here. Nevertheless, after a small modification the reasoning of Sections 2–4 enables us to prove the
existence of IM for problem (7.1). Such a modification based on an idea from [16] is
given below.
First of all we prove the solvability of problem (7.1). Let us first consider the linear problem
ì
ï
ï
í
ï
ï
î
d2u
du
--------+ 2 e ------- + Au = h ( t ) , t > s ,
2
dt
dt
du
= u1 .
u t = s = u0 , ------dt t = s
(7.8)
These equations can also be rewritten in the form (cf. (7.5))
d---U t = s = U0 ,
U( t ) + AU(t ) = H(t ) ,
(7.9)
dt
where U ( t ) = ( u ( t ) ; u· ( t ) ) and H ( t ) = ( 0 ; h ( t ) ) . We define a mild solution to
problem (7.8) (or (7.9)) on the segment [ s , s + T ] as a function u ( t ) from the class
Ls , T º C ( s , s + T ; .1 ¤ 2 ) Ç C 1 ( s , s + T ; H ) Ç C 2 ( s , s + T ; .-1 ¤ 2 )
which satisfies equations (7.8). Here .q = D ( Aq ) as before (see Chapter 2). One
can prove the existence and uniqueness of mild solutions to (7.8) using the Galerkin
method, for example. The approximate Galerkin solution of the order m is
defined as a function
Inertial Manifold for Second Order in Time Equations
m
um ( t ) =
å g (t) e
k
k
k=1
satisfying the equations
ì ( u·· ( t ) , e ) + 2 e ( u· ( t ) , e ) + ( Au ( t ) , e ) = ( h ( t ) , e ) ,
m
j
j
m
j
j
ï m
í
ï ( u ( s ) , e ) = ( u , e ) , ( u· m ( s ) , e ) = ( u , e )
0
1
j
j
j
j
î m
t >s ,
(7.10)
for j = 1 , 2 , ¼ , m . Moreover, we assume that gj ( t ) Î C 1 ( s , s + T ) and g· j ( t ) is
absolutely continuous. Hereinafter we use the notation v· ( t ) = dv ¤ d t . Evidently
equations (7.10) can be rewritten in the form
ì u·· ( t ) + 2 e u· ( t ) + A u ( t ) = p h ( t ) ,
m
m
m
ï m
í
= pm u0 , u· m
= pm u 1 ,
ï um
t=s
t=s
î
(7.11)
where pm is the orthoprojector onto Lin { e 1 , ¼ e m } in H .
In the exercises given below it is assumed that
h ( t ) Î L¥ ( R , H ) ,
u0 Î D ( A1 / 2 ) ,
u1 Î H .
(7.12)
Show that problem (7.10) is uniquely solvable on any segment
[ s , s + T ] and um ( t ) Î Ls , T .
E x e r c i s e 7.3
E x e r c i s e 7.4
Show that the energy equality
t
1æ ·
--- um ( t ) 2 + A1 / 2 um ( t ) 2ö + 2 e
ø
2è
1
= --- æè pm u1
2
2
+ A1 / 2 pm u0 2ö +
ø
ò u·
m(t)
2
dt =
s
t
ò (h (t) , u·
m ( t ) ) dt
(7.13)
s
holds for any solution to problem (7.10).
E x e r c i s e 7.5
Using (7.11) and (7.13) prove the a priori estimate
A-1 / 2 u·· m ( t ) 2 + u· m ( t ) 2 + A1 / 2 u m ( t ) £ C ( T , u0 , u1 )
for the approximate Galerkin solution um(t) to problem (7.8).
E x e r c i s e 7.6 Using the linearity of problem (7.11) show that for every two
approximate solutions um ( t ) and u m¢ ( t ) the estimate
A-1 ¤ 2 ( u·· m ( t ) - u·· m¢ ( t ) ) 2 +
+ u· m ( t ) - u· m ¢ ( t ) + A1 / 2 ( um ( t ) - u m¢ ( t ) )
2
£
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3
2
£ CT ( pm - pm¢) u1
+ A1 / 2 ( pm - pm¢) u0
+
2
+ ess sup
t Î [ s , s + T]
( pm - pm¢) h ( t )
2
holds for all t Î [ s , s + T ] , where T > 0 is an arbitrary number.
E x e r c i s e 7.7 Using the results of Exercises 7.5 and 7.6 show that we can
pass to the limit n ® ¥ in equations (7.11) and prove the existence
and uniqueness of mild solutions to problem (7.8) on every segment
[ s , s + T ] under the condition (7.12).
E x e r c i s e 7.8 For a mild solution u ( t ) to problem (7.8) prove the energy
equation:
t
1
--- æ u· ( t ) 2 + A1 / 2 u ( t ) 2ö + 2 e
ø
2è
t
1
= --- æ u 1
2è
2
+ A1 / 2 u0 2ö +
ø
ò u· (t)
2
dt =
s
ò(h (t) , u· (t) ) dt .
(7.14)
s
In particular, the exercises above show that for h ( t ) º 0 problem (7.8) generates a
linear evolutionary semigroup e -t A in the space 0 = D ( A1 / 2 ) ´ H by the formula
e -t A ( u0 ; u 1 ) = ( u ( t ) ; u· ( t ) ) ,
(7.15)
where u ( t ) is a mild solution to problem (7.8) for h ( t ) º 0 . Equation (7.14) implies
that the semigroup e -t A is contractive for e ³ 0 .
E x e r c i s e 7.9 Assume that conditions (7.12) are fulfilled. Show that the
mild solution to problem (7.8) can be presented in the form
t
( u· ( t ) ; u ( t ) ) = e -( t - s ) A ( u 0 ; u 1 ) +
òe
-( t - s ) A ( 0 ;
h ( t ) ) dt , (7.16)
s
where the semigroup
e -t A
is defined by equation (7.15).
Let us now consider nonlinear problem (7.1) and define its mild solution as
a function U ( t ) º ( u ( t ) ; u· ( t ) ) Î C ( s , s + T ; 0 ) satisfying the integral equation
t
U(t) =
e -( t - s ) A U0
+
òe
-( t - s ) A B ( U ( t ) ,
t ) dt
s
on [ s , s + T ] . Here
B ( U ( t ) , t ) = ( 0 ; B ( u ( t ) , t ) ) and U0 = ( u0 ; u1 ) .
(7.17)
Inertial Manifold for Second Order in Time Equations
E x e r c i s e 7.10 Show that the estimates
B ( U, t ) 0 £ M ( 1 + U
0
),
B ( U1 , t ) - B ( U2 , t ) 0 £ M U1 - U2
hold in the space 0 =
D ( A1 / 2 )
0
´ H . Here M is a positive constant.
E x e r c i s e 7.11 Follow the reasoning used in the proof of Theorems 2.1 and
2.3 of Chapter 2 to prove the existence and uniqueness of a mild solution to problem (7.1) on any segment [ s , s + T ] .
Thus, in the space 0 there exists a continuous evolutionary family of operators
S ( t , s ) possessing the properties
S ( t , t) = I ,
S (t, t) ° S (t, s) = S (t, s) ,
and
S ( t , s ) U 0 = ( u ( t ) ; u· ( t ) ) ,
where u ( t ) is a mild solution to problem (7.1) with the initial condition U0 =
= ( u0 ; u1 ) .
Let condition e 2 > m N + 1 hold for some integer N . We consider the decomposition of the space 0 into the orthogonal sum
0 = 01 Å 02 ,
where
01 = Lin { ( ek ; 0 ) , ( 0 ; e k ) : k = 1 , 2 , ¼ N }
and 02 is defined as the closure of the set
Lin { ( e k ; 0 ) , ( 0 ; ek ) : k ³ N + 1 } .
E x e r c i s e 7.12 Show that the subspaces 01 and 02 are invariant with respect to the operator A . Find the spectrum of the restrictions of the
operator A to each of these spaces.
Let us introduce the following inner products in the spaces 01 and 02 (the purpose of this introduction will become apparent further):
á U , V ñ 1 = e 2 ( u0 , v0 ) - ( Au0 , v0 ) + ( e u 0 + u1 , e v0 + v1 ) ,
(7.18)
á U , V ñ 2 = ( Au 0 , v0 ) + ( e 2 - 2 m N + 1 ) ( u0 , v0 ) + ( e u 0 + u1 , e v0 + v1 ) .
Here U = ( u0 ; u1 ) and V = ( v0 ; v1 ) are elements from the corresponding subspace 0i . Using (7.18) we define a new inner product and a norm in 0 by the
equalities:
á U , V ñ = á U 1 , V1ñ 1 + á U 2 , V2ñ 2 ,
U = á U, U ñ 1/2 ,
where U = U1 + U2 and V = V1 + V2 are decompositions of the elements U and V
into the orthogonal terms Vi , Ui Î 0i , i = 1 , 2 .
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Lemma 7.1.
The estimates
1 - e 2 - m Aq u ,
U 1 ³ -----0
N
mqN
1 -d
q
U 2 ³ -------------N , e A u0 ,
q
mN + 1
U = ( u0 ; u1 ) Î 0 1 ;
(7.19)
U = ( u0 ; u1 ) Î 02
(7.20)
hold for 0 £ q £ 1 ¤ 2 . Here
æ
m N + 1 minç 1 ,
è
dN , e =
e 2 - m N + 1ö
--------------------------÷ .
mN + 1 ø
(7.21)
Proof.
b
Let U = ( u0 ; u 1 ) Î 01 . It is evident that in this case Ab u 0 £ m N u0
for any b > 0 . Therefore,
U 12 ³ e 2 u 0 2 - A1 / 2 u0 2 ³ m-N2 q ( e 2 - mN ) Aq u 0 2 ,
i.e. equation (7.19) holds. Let U Î 02 . Then using the inequality
b
Ab u0 ³ m N + 1 u 0 ,
b > 0,
u0 Î Lin { ek : k ³ N + 1 }
(7.22)
for 0 < d £ 1 we find that
U 22 ³ d 2 A1 / 2 u 0 2 + ( e 2 - ( 1 + d 2 ) m N + 1 ) u0 2 .
If we take d = dN , e m-N1+/ 21 and use (7.22), then we obtain estimate (7.20). The
lemma is proved.
In particular, this lemma implies the estimate
-1
q
Aq u0 £ m N
+ 1 dN , e U
(7.23)
for any U = ( u0 ; u1 ) Î 0 , where 0 £ q £ 1 ¤ 2 and dN , e has the form (7.21).
E x e r c i s e 7.13 Prove the equivalence of the norm
by the inner product (7.4).
. and the norm generated
E x e r c i s e 7.14 Show that we can take dN , e = e 2 - m N + 1 for q = 0 in (7.20)
and (7.23).
E x e r c i s e 7.15 Prove that the eigenvectors { fk± } of the operator
(7.7)) possess the following orthogonal properties:
á fn+ , fk+ñ = á fn–, fk– ñ = á fn+, fk– ñ = 0 ,
á fk+ , fk–ñ = 0 ,
1 £ k £ N.
A (see
k ¹ n,
(7.24)
Inertial Manifold for Second Order in Time Equations
Note that the last of these equations is one of the reasons of introducing a new inner
product.
Let P0 be the orthoprojector onto the subspace 0i in 0 , i = 1 , 2 .
i
Lemma 7.2.
The equality
e -At P0 = e
-
-lN + 1 t
2
,
t ³ 0,
(7.25)
is valid. Here . is the operator norm which is induced by the corresponding vector norm.
Proof.
Let U Î 02 . We consider the function y ( t ) = e -At U 2 . Since 02 is invariant with respect to e -A t , the equation
y ( t ) = ( Au ( t ) , u ( t ) ) + ( e 2 - 2 m N + 1 ) ( u ( t ) , u ( t ) ) + u· + e u
2
holds, where u ( t ) is a solution to problem (7.8) for h ( t ) º 0 . After simple calculations we obtain that
dy
------- + 2 e y = 4 ( e 2 - m N + 1 ) ( u· + e u , u ) .
dt
It is evident that
2 e 2 - m N + 1 ( u· + e u , u ) £ ( e 2 - m N + 1 ) u 2 + u· + e u 2 £ y ( t ) .
Therefore,
dy
------- + 2 e y £ 2 e 2 - m N + 1 y .
dt
Consequently,
y (t) £ e
-
-lN + 1 t
y(0) ,
t > 0.
(7.26)
If we now notice that
exp { -A t } fN-+ 1 = e
-
-lN + 1 t
-
fN + 1 ,
then equation (7.26) implies (7.25). Thus, Lemma 7.2 is proved.
Let us consider the subspaces
±
01± = Lin { fk : k £ N } .
Equation (7.24) gives us that the subspaces are orthogonal to each other and therefore 01 = 01+ Å 01– . Using (7.24) it is easy to prove (do it yourself) that
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e At P
0
1
e -At P
+
01
3
£ e
-
lN t
£ e
,
+
-lN t
,
t ÎR,
(7.27)
t > 0.
(7.28)
We use the following pair of orthogonal (with respect to the inner product
á . , . ñ ) projectors in the space 0
P=P
01
,
Q = I -P = P
01+
+ P0
2
to construct the inertial manifold of problem (7.1) (or (7.5)). Lemma 7.2 and equations (7.27) and (7.28) imply the dichotomy equations
e At P £ e
-
lN t
,
t ÎR;
e -At Q £ e
-
-lN + 1 t
,
t > 0.
(7.29)
-
We remind that l N = e - e 2 - m k and e 2 > m N + 1 .
The assertion below plays an important role in the estimates to follow.
Lemma 7.3.
Let B ( U, t ) = ( 0 ; B ( u0 , t ) ) , where U = ( u0 ; u1 ) Î 0 and B ( u0 ) possesses properties (7.2). Then
B ( U, t ) £ M0 + KN U ,
U Î0,
B ( U1 , t ) - B ( U2 , t ) £ KN U1 - U2 ,
U1 , U2 Î 0 ,
(7.30)
where
mN + 1 ö
-------------------------÷ .
2
e - mN + 1 ø
æ
K N = M1 mqN-+11¤ 2 max ç 1 ,
è
The proof of this lemma follows from the structure of the mapping
estimates (7.2) and (7.23).
(7.31)
B ( U, t ) and from
E x e r c i s e 7.16 Show that one can take KN = M1 ( e 2 - m N + 1 ) -1 / 2 for q = 0 in
(7.30) (Hint: see Exercise 7.14).
Let us now consider the integral equation (cf. (2.1) for L = ¥ )
V ( t ) = Bps [ V ] ( t ) º
s
º
e -( t - s ) A p
-
ò
t
t
e -( t - t ) A P B ( V ( t ) ,
t ) dt +
òe
-¥
-( t - t ) A Q B ( V ( t ) ,
t ) dt
(7.32)
Inertial Manifold for Second Order in Time Equations
in the space C s of continuous vector-functions U ( t ) on ( - ¥ , s ] with the values
in 0 such that the norm
1
g = --- ( l-N + 1 + l-N ) ,
2
I U I º sup e -g ( s - t ) U ( t ) < ¥ ,
t< s
is finite. Here p Î P 0 and t Î ( - ¥ , s ) .
E x e r c i s e 7.17 Show that the right-hand side of equation (7.32) is a continuous function of the variable t with the values in 0 .
Lemma 7.4.
The operator Bps maps the space Cs into itself and possesses the properties
æ 1
4 KN
1 ö
s
- I VI
I Bp [ V ] I £ p + M0 ç -------- - + ------------- -÷ + ---------------------------lN + 1 ø l N + 1 - l-N
è lN
(7.33)
4 KN
s
s
- I V1 - V2 I .
I Bp [ V1] - Bp [ V2] I £ ----------------------------l N + 1 - l-N
(7.34)
and
Proof.
Let us prove (7.34). Evidently, equations (7.29) and (7.30) imply that
s
Bsp [ V1 ( t ) ]
òe
£ KN
ò
-
e
lN ( t - t )
V1 ( t ) - V2 ( t ) dt +
t
t
+ KN
- Bps [ V2 ( t ) ]
-
-lN + 1 ( t - t )
V1 ( t ) - V2 ( t ) dt .
-¥
Since
V1 ( t ) - V2 ( t ) £ e g ( s - t ) I V1 - V2 I ,
it is evident that
Bps [ V1 ] ( t ) - Bps [ V2 ] ( t ) £ q e g ( s - t ) I V1 - V2 I
with
ì
ï
q = KN í
ï
î
s
òe
t
-
( lN - g ) ( t - t )
t
dt +
òe
-
-( lN + 1 - g ) ( t - t )
-¥
-
-
ü
ï
dt ý .
ï
þ
Simple calculations show that q £ 4 KN ( l N + 1 - l N ) -1 . Consequently, equation (7.34) holds. Equation (7.33) can be proved similarly. Lemma 7.4 is proved.
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Thus, if for some q < 1 the condition
3
4 KN
l-N + 1 - l-N ³ ---------(7.35)
qholds, then equation (7.32) is uniquely solvable in Cs and its solution V can be estimated as follows:
1 - ------------1 -ö ö
-1
+
.
I V I £ ( 1 - q ) æè p + M 0 æè -------l-N l-N + 1ø ø
(7.36)
Therefore, we can define a collection of manifolds { Ms } in the space 0 by the formula
Ms = { p + F ( p , s ) : p Î P 0 } ,
(7.37)
where
s
òe
F(p , s) =
-( t - t ) A Q B ( V ( t ) ,
t ) dt .
(7.38)
-¥
Here V ( t ) is a solution to integral equation (7.32). The main result of this section is
the following assertion.
Theorem 7.1.
Assume that
e 2 > m N + 1 and
4 KN
l-N + 1 - l-N ³ ---------q-
(7.39)
-
for some 0 < q < 1 , where l k = e - e 2 - m k and KN is defined by formula
(7.31).. Then the function F ( p , s ) given by equality (7.38) satisfies the Lipschitz condition
q
F ( p1 , s ) - F ( p2 , s ) £ -------------------- p1 - p2
(7.40)
2 (1 - q)
and the manifold Ms is invariant with respect to the evolutionary operator S ( t , t ) generated by the formula
S ( t , t ) U = ( u ( t ) ; u· ( t ) ) ,
t ³ s,
0
in 0 , where u ( t ) is a solution to problem (7.1) with the initial condition
U0 = ( u0 ; u1 ) . Moreover, if 0 < q < 2 - 2 , then there exist initial conditions
U0* = ( u*0 ; u 1* ) Î Ms such that
S ( t , s ) U0 - S ( t , s ) U 0* £ Cq e - g ( t - s ) Q U0 - F( PU0 , s )
1 for t ³ s , where g = --- ( lN + l N + 1 ) .
2
The proof of the theorem is based on Lemma 7.4 and estimates (7.29) and (7.30).
It almost entirely repeats the corresponding reasonings in Sections 2 and 3. We give
the reader an oppotunity to recover the details of the reasonings as an exercise.
Inertial Manifold for Second Order in Time Equations
Let us analyse condition (7.39). Equation (7.31) implies that (7.39) holds if
e2 ³ 2 m N + 1 ,
m N + 1 - mN
4
--------------------------- ³ --- M1 mqN -+ 11 ¤ 2 .
q
2
2 e - mN
(7.41)
However, if we assume that
8 2
m N + 1 - m N ³ ----------- M1 mqN + 1 ,
q
(7.42)
then for condition (7.41) to be fulfilled it is sufficient to require that
2 m N + 1 £ e 2 £ 2 m N + 1 + mN .
(7.43)
Thus, if for some N conditions (7.42) and (7.43) hold, then the assertions of Theorem 7.1 are valid for system (7.1). This enables us to formulate the assertion on the
existence of IM as follows.
Theorem 7.2.
Assume that the eigenvalues m N of the operator A possess the properties
mN
- > 0 and m N ( k ) + 1 = c 0 k r ( 1 + o ( 1 ) ) ,
r > 0 , k ® ¥ , (7.44)
inf ------------N mN +1
for some sequence { N ( k ) } which tends to infinity and satisfies the estimate
8 2q
m N( k ) + 1 - m N( k ) ³ ---------q M1 m N ( k ) + 1 ,
0< q< 2- 2 .
Then there exists e0 > 0 such that the assertions of Theorem 7.1 hold for all
e ³ e0 .
Proof.
Equation (7.44) implies that there exists k0 such that the intervals
[ 2 mN ( k ) + 1 , 2 mN ( k ) + 1 + mN ( k ) ] ,
k ³ k0 ,
cover some semiaxis [ e0 , + ¥ ) . Indeed, otherwise there would appear a subsequence { N ( kj ) } such that
mN( k ) < 2 ( mN ( k
j
j + 1) + 1
- mN ( k ) + 1 )
j
But that is impossible due to (7.44). Consequently, for any e ³ e0 there exists
N = Ne such that equations (7.42), (7.43) as well as (7.39) hold.
u) =
-----E x e r c i s e 7.18 Consider problem (7.3) with the function f ( x , t , u , ¶
¶x
= f ( x , t , u ) possessing the property
f ( x , t , u 1 ) - f ( x , t , u 2 ) £ L u1 - u2 .
Use Theorem 7.1 to find a domain in the plane of the parameters
( e , L ) for which one can guarantee the existence of an inertial manifold.
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§ 8 Approximate Inertial Manifolds
for Second Order in Time Equations
As seen from the results of Section 7, in order to guarantee the existence of IM for
a problem of the type
2u
d--------u
- + g d------ + A u = B (u) ,
dt
d t2
u t = 0 = u0 ,
(8.1)
du
------= u1 ,
dt t = 0
we have to require that the parameter g = 2 e > 0 be large enough and the spectral
gap condition (see (7.41)) be valid for the operator A . Therefore, as in the case with
parabolic equations there arises a problem of construction of an approximate inertial
manifold without any assumptions on the behaviour of the spectrum of the operator
A and the parameter g > 0 which characterizes the resistance force.
Unfortunately, the approach presented in Section 6 is not applicable to the
equation of the type (8.1) without any additional assumptions on g . First of all, it is
connected with the fact that the regularizing effect which takes place in the case of
parabolic equations does not hold for second order equations of the type (8.1) (in
the parabolic case the solution at the moment t > 0 is smoother than its initial condition).
In this section (see also [17]) we suggest an iteration scheme that enables us to
construct an approximate IM as a solution to a class of linear problems. For the sake
of simplicity, we restrict ourselves to the case of autonomous equations ( B(u , t) º
º B(u) ) . The suggested scheme is based on the equation in functional derivatives
such that the function giving the original true IM should satisfy it. This approach was
developed for the parabolic equation in [9] (see also [8]). Unfortunately, this approach has two defects. First, approximate IMs have the power order (not the exponential one as in Section 6) and, second, we cannot prove the convergence of
approximate IMs to the exact one when the latter exists.
Thus, in a separable Hilbert space H we consider a differential equation of the type
(8.1) where g is a positive number, A is a positive selfadjoint operator with discrete
spectrum and B ( . ) is a nonlinear mapping from the domain D ( A1 / 2 ) of the operator
A1 / 2 into H such that for some integer m ³ 2 the function B ( u ) lies in C m as a
mapping from D ( A1 / 2 ) into H and for every r > 0 the following estimates hold:
á B ( k ) ( u ) ; w 1 , ¼ , w kñ
£ Cr
k
ÕA
j=1
1/2 w
j
,
(8.2)
Approximate Inertial Manifolds for Second Order in Time Equations
k
á B ( k ) ( u ) - B ( k ) ( u* ) ; w1 , ¼ , wkñ
£ C r A1 / 2 ( u - u* )
ÕA
1/2 w
j
, (8.3)
j=1
where k = 0 , 1 , ¼ , m , . is a norm in the space H , A1 / 2 u £ r , A1 / 2 u* £ r ,
and wj Î D ( A1 / 2 ) . Here B ( k ) ( u ) is the Frechét derivative of the order k of B ( u )
and á B ( k ) ( u ) ; w1 , ¼ , wkñ is its value on the elements w1 , ¼ , wk .
Let Lm , R be a class of solutions to problem (8.1) possessing the following
properties of regularity:
I) for k = 0 , 1 , ¼ , m - 1 and for all T > 0
u(k) (t ) Î C(0 , T ; D ( A) )
and
u ( m ) ( t ) Î C ( 0 , T ; D ( A1 / 2 ) ) ,
u( m + 1) ( t ) Î C ( 0 , T ; H ) ,
where C ( 0 , T ; V ) is the space of strongly continuous functions on [ 0 , T ]
with the values in V , hereinafter u ( k ) ( t ) = ¶ tk u ( t ) ;
II) for any u Î Lm , R the estimate
u ( k + 1 ) ( t ) 2 + A1 / 2 u ( k ) ( t ) 2 + Au ( k - 1 ) ( t ) 2 £ R 2
(8.4)
t * , where t *
holds for k = 1 , ¼ , m and for t ³
depends on u0 and u1 only.
In fact, the classes Lm , R are studied in [18]. This paper contains necessary and
sufficient conditions which guarantee that a solution belongs to a class Lm , R .
It should be noted that in [18] the nonlinear wave equation of the type
¶t2 u + g ¶ t u - Du + g ( u ) = f ( x ) ,
u ¶W = 0,
u t = 0 = u0 ( x ) ,
¶t u
x ÎW, t > 0 ,
t=0
= u1 ( x ) ,
(8.5)
serves as the main example. Here g > 0 , f ( x ) Î C ¥ ( W ) and the conditions set on
the function g ( s ) from C ¥ ( R ) are such that we can take g ( u ) = sin u or g ( u ) =
= u 2 p + 1 , where p = 0 , 1 , 2 , ¼ for d = dim W £ 2 and p = 0 , 1 for d = 3 .
In this example the classes Lm , R are nonempty for all m . Other examples will be
given in Chapter 4.
We fix an integer N and assume P = PN to be the projector in H onto the subspace generated by the first N eigenvectors of the operator A . Let Q = I - P . If we
apply the projectors P and Q to equation (8.1), then we obtain the following system of two equations for p ( t ) = PU ( t ) and q ( t ) = Q u ( t ) :
¶ t2 p + g ¶ t p + A p = PB ( p + q ) ,
¶ t2 q + g ¶ t q + A q = Q B ( p + q ) .
(8.6)
The reasoning below is formal. Its goal is to obtain an iteration scheme for the determination of an approximate IM. We assume that system (8.6) has an invariant manifold of the form
M = { ( p + h ( p , p· ) ; p· + l ( p , p· ) ) : p , p· Î PH }
(8.7)
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in the phase space D ( A1 / 2 ) ´ H . Here h and l are smooth mappings from PH ´ P H
into QD ( A ) . If we substitute q ( t ) = h ( p ( t ) , ¶t p ( t ) ) and ¶ t q ( t ) = l ( p ( t ) , ¶ t p ( t ) )
in the second equality of (8.6), then we obtain the following equation:
á dp l ; p· ñ + á dp· l ; -g p· - A p + PB ( p + h ( p , p· ) )ñ +
3
+ g l ( p , p· ) + A h ( p , p· ) = Q B ( p + h ( p , p· ) ) .
The compatibility condition
l ( p ( t ) , ¶t p ( t ) ) = ¶t h ( p ( t ) , ¶t p ( t ) )
gives us that
l ( p , p· ) = á d p h ; p· ñ + á d p· h ; -g p· - A p + PB ( p + h ( p , p· ) )ñ .
Hereinafter dp f and dp· f are the Frechét derivatives of the function f ( p , p· ) with
respect to p and p· ; á dp f ; wñ and á dp· f ; wñ are values of the corresponding derivatives on an element w .
Using these formal equations, we can suggest the following iteration process to
determine the class of functions { hk ; lk } giving the sequence of approximate IMs
with the help of (8.7):
A hk ( p , p· ) = QB ( p + h k - 1 ( p , p· ) ) - g ln ( k ) ( p , p· ) - á dp lk - 1 ; p· ñ -
- á dp· lk - 1 ; -g p· - A p + PB ( p + hk - 1 ( p , p· ) )ñ ,
(8.8)
where k = 1 , 2 , 3 , ¼ and the integers n ( k ) should be choosen such that k - 1 £
£ n ( k ) £ k . Here lk ( p , p· ) is defined by the formula
; p· ñ + á d h
; -g p· - Ap + P B ( p + h
( p , p· ) )ñ , (8.9)
l ( p , p· ) = á d h
k
p k-1
p· k - 1
where k = 1 , 2 , 3 , ¼ . We also assume that
h0 ( p , p· ) º l0 ( p , p· ) º 0 .
k-1
(8.10)
Find the form of h1 ( p , p· ) and l1 ( p , p· ) for n ( 1 ) = 0 and for
n (1) = 1 .
E x e r c i s e 8.1
The following assertion contains information on the smoothness properties of the
functions hn and ln which will be necessary further.
Theorem 8.1.
Assume that the class of functions { hn ; ln } is defined according to
(8.8)–(8.10).. Then for each n the functions hn and ln belong to the class
C m as mappings from PH ´ PH into Q H and for all integers a , b ³ 0 such
that a + b £ m the estimates
Approximate Inertial Manifolds for Second Order in Time Equations
A á D a , b hn ( p , p· ) ; w1 , ¼ , wa ; w· 1 , ¼ , w· bñ £
b
a
£ Ca , b , R
Õ
Awi ×
ÕA
·
1/2 w
i
,
(8.11)
i=1
i=1
A1 / 2 á D a , b ln ( p , p· ) ; w1 , ¼ , wa ; w· 1 , ¼ , w· bñ
b
a
£ Ca , b, R
Õ
Awi ×
ÕA
·
1 / 2w
£
(8.12)
i
i=1
i=1
are valid for all p and p· from P H such that Ap £ R and A1 / 2 p· £ R .
Hereinafter D a , b f is the mixed Frechét derivative of the function f of the
order a with respect to p and of the order b with respect to p· ; the values
wj and w· j are from PH . Moreover, if a = 0 or b = 0 , then the corresponding products in (8.11) and (8.12) should be omitted.
Proof.
We use induction with respect to n . It follows from (8.10) and (8.2) that estimates (8.11) and (8.12) are valid for n = 0 , 1 . Assume that (8.11) and (8.12) hold
for all n £ k - 1 . Then the following lemma holds.
Lemma 8.1.
Let Fn ( p , p· ) = B ( p + h n ( p , p· ) ) and let
Fna , b ( w ) = á D a , b Fn ( p , p· ) ; w1 , ¼ , wa ; w· 1 , ¼ , w· bñ .
Then for n £ k - 1 and for all integers a , b ³ 0 such that a + b £ m
the estimate
Fna , b ( w )
b
a
£ C
Õ Aw × Õ A
i
i=1
holds, where wj , w· j , p , p· Î P H and
·
1/2 w
i=1
Ap £ R ,
(8.13)
i
A1 / 2 p· £ R .
Proof.
It is evident that Fna , b ( w ) is the sum of terms of the type
B ns ( y ) = á B ( s ) ( p + hn ( p , p· ) ) ; y1 , ¼ , ysñ ,
s ³ 0.
Here ys is one of the values of the form:
y * = w s + á d p h n ; w sñ ,
y** = á D s , t h n ; w1 , ¼ , wa ; w· 1 , ¼ , w· bñ .
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Equation (8.2) implies that
3
Therefore, the induction hypothesis gives us (8.13).
s
Bns ( y ) £ CR
ÕA
1/2 y
j
.
j=1
Let us prove (8.12). The induction hypothesis implies that it is sufficient to estimate
the derivatives of the second term in the right-hand side of (8.9). It has the form
(8.14)
ád · h
; D ( p , p· )ñ ,
p k -1
k
where
Dk ( p , p· ) = -g p· - A p + PB ( p + hk - 1 ( p , p· ) ) .
The Frechét derivatives of value (8.14)
á D a , b á dp· hk - 1 ; Dk ( p , p· )ñ ; w1 , ¼ , wa ; w· 1 , ¼ , w· bñ
are sums of the terms of the type
G ( s , t ) º á D s , t + 1 hk - 1 ( p , p· ) ; wj , ¼ , wj ; w· i , ¼ , w· i , ys , tñ ,
t
1
s
1
where
; w· r , ¼ , w· r
ñ.
ys , t = á D a - s , b - t Dk ( p , p· ) ; ww , ¼ , ww
b -t
1
a -s
1
Here 0 £ s £ a , 0 £ t £ b and the sets of indices possess the following properties:
{ j1 , ¼ , js } Ç { w1 , ¼ , wa - s } = Æ ,
{ j 1 , ¼ , j s } È { w1 , ¼ , w a - s } = { 1 , 2 , ¼ , a } ;
{ i1 , ¼ , i t } Ç { r1 , ¼ , rb - t } = Æ ,
{ i1 , ¼ , it } È { r1 , ¼ , rb -t } = { 1 , 2 , ¼ , b } .
The induction hypothesis implies that
s
AG ( s , t )
£ C
Õ
q=1
t
Awj ×
q
Õ
A1 / 2 w· i × A1 / 2 ys , t .
q
q=1
Using the induction hypothesis again as well as Lemma 8.1 and the inequality
A1 / 2 Ph £ l1N/ 2 Ph ,
we obtain an estimate of the following form (if s = a or t = b , then the corresponding product should be considered to be equal to 1):
A1 / 2 ys , t
£ C ( 1 + l1N/ 2 )
a-s
Õ
q=1
b-t
Aww ×
q
ÕA
q=1
·
1/2 w
rq
.
Approximate Inertial Manifolds for Second Order in Time Equations
Hereinafter lk is the k -th eigenvalue of the operator A . Thus, it is possible to state
that
AG ( s , t ) £ C ( 1 + l1 / 2 )
Aw ×
A1 / 2 w· .
(8.15)
N
Õ
i
i
Õ
i
i
Using the inequality
Q h £ l-Ns+ 1 As Q h ,
s > 0,
(8.16)
and equation (8.15) it is easy to find that estimates (8.12) are valid for n = k . If we
use (8.8), (8.12) and follow a similar line of reasoning, we can easily obtain (8.11).
Theorem 8.1 is proved.
Theorem 8.1 and equation (8.4) imply the following lemma.
Lemma 8.2.
Assume that u ( t ) is a solution to problem (8.1) lying in Lm , R , m ³ 1 .
Let p ( t ) = Pu ( t ) and let
qs ( t ) = hs ( p ( t ) , ¶t p ( t ) ) ,
Then the estimates
qs ( t ) = ls ( p ( t ) , ¶t p ( t ) ) .
(8.17)
A1 / 2 qs( j ) ( t ) 2 + Aqs( j ) ( t ) 2 £ CR , m
with 0 £ j £ m -1 and
qs( m ) ( t ) 2 + A1 / 2 qs( m ) ( t ) 2 £ C R , m
are valid for t large enough.
Proof.
It should be noted that qs( j ) ( t ) is the sum of terms of the form
á D a , b hs ( p , ¶t p) , p
( i1 )
(t) , ¼, p
( ia )
(t) ; p
( t1 + 1 )
(t) , ¼, p
( tb + 1 )
( t )ñ ,
where a , b , i1 , ¼ , ia , t1 , ¼ , tb are nonnegative integers such that
1 £ a +b £ j,
i1 + ¼ + ia + t1 + ¼ + tb = j .
Similar equation also holds for qs( i ) ( t ) . Further one should use Theorem 8.1 and
the estimates
p ( k + 1 ) ( t ) 2 + A1 / 2 p ( k ) ( t ) 2 + Ap ( n - 1 ) ( t ) 2 £ R 2 ,
t ³ t* ,
1 £ k £ m,
which follow from (8.4).
Let us define the induced trajectories of the system by the formula
Us ( t ) = ( u s ( t ) ; us ( t ) ) ,
where s = 0 , 1 , 2 , ¼ and
us ( t ) = p ( t ) + qs ( t ) ,
us ( t ) = ¶t p ( t ) + qs ( t ) .
(8.18)
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Here p ( t ) = Pu ( t ) , u ( t ) is a solution to problem (8.1); qs ( t ) and qs ( t ) are defined
with the help of (8.17). Assume that u ( t ) lies in L m , R . Then Lemma 8.2 implies
that the induced trajectories can be estimated as follows:
A1 / 2 us( j ) ( t ) + Aus( j ) ( t ) £ C R , s ,
3
0 £ j £ m -1;
us( m ) ( t ) 2 + A1 / 2 u s( m ) ( t ) 2 £ CR , s
for t large enough. Using (8.3), (8.4), and the last estimates, it is easy to prove the
following assertion (do it yourself).
Lemma 8.3.
Let
Es ( t ) = B ( p ( t ) + q ( t ) ) - B ( p ( t ) + qs ( t ) ) .
Then
j
E s( j ) ( t ) £ CR , j
åA
1 / 2 ( q( i) ( t )
- qs( i ) ( t ) )
i=0
for j = 0 , 1 , ¼ , m and for t large enough.
The main result of this section is the following assertion.
Theorem 8.2.
Let u ( t ) be a solution to problem (8.1) lying in Lm , R with m ³ 2 .
Assume that hn ( p , p· ) and ln ( p , p· ) are defined by (8.8)–(8.10).. Then the estimates
A ¶ tj ( u ( t ) - u n ( t ) ) £ Cn , R l-Nn+/ 12 ,
(8.19)
A1 / 2 ¶ tj ( ¶t u ( t ) - un ( t ) ) £ Cn , R l-Nn+/ 12 ,
(8.20)
are valid for n £ m - 1 and for t large enough. Here 0 £ j £ m - n - 1 ,
u n ( t ) and un ( t ) are defined by (8.18),, and l N + 1 is the ( N + 1 ) -th eigenvalue of the operator A .
Proof.
Let us consider the difference between the solution u ( t ) and the trajectory induced by this solution:
cs ( t ) = u ( t ) - us ( t ) ,
cs ( t ) = ¶t u ( t ) - us ( t ) ,
s ³ 0,
where u s ( t ) and us ( t ) are defined by formula (8.18). Since c 0 ( t ) = q ( t ) , equation
(8.4) implies that
(j + 1)
A1 / 2 c 0
(j)
( t ) + A c0 ( t ) £ C ,
j = 0, 1, 2, ¼, m -1 ,
(8.21)
Approximate Inertial Manifolds for Second Order in Time Equations
for t large enough. Equations (8.8)–(8.10) also give us that
A c 1( t ) = - c 0² ( t ) - g c 0¢ ( t ) + Q E0 ( t ) .
We use Lemma 8.3 and equation (8.21) to find that
(j)
A c 1 ( t ) £ C l-N1+/ 21 ,
j = 0, 1, ¼, m - 2 ,
for t large enough. Therefore, equation (8.19) holds for n = 0 , 1 and for t large
enough. From equations (8.6), (8.8), and (8.9) it is easy to find that
A ck = - ¶ t ck - 1 - g ¶ t c n ( k ) -1 - g á dp· hn ( k ) - 1 ; P En ( k ) - 1ñ -
- á dp· lk - 1 ; P Ek - 1ñ + Q Ek - 1
and
c k = ¶ t c k - 1 + á dp· hk -1 ; PEk -1ñ .
(8.22)
Lemma 8.4.
The estimates
j
A ¶ tj á dp· h n ; PE nñ
£ C l1N/ 2
åA
1 / 2 c( s ) ( t )
n
(8.23)
s=0
and
j
A1 / 2
¶tj á dp· ln ;
PE n ñ
£
C l1N/ 2
åA
1 / 2 c( s ) ( t )
n
(8.24)
s=0
are valid for t large enough and for each n ³ 0 , where j = 0 , 1 , ¼ , m - 1 .
Proof.
Let fn = h n or fn = ln . It is clear that the value ¶ tj á dp· fn ; PEnñ is the
algebraic sum of terms of the form:
g
á D a , b +1 f n ; p 1 , ¼ , p
ga
; p
s1
, ¼, p
sb
, ¶ts PE nñ .
Therefore, Theorem 8.1 and Lemma 8.3 imply (8.23) and (8.24). Lemma 8.4
is proved.
We use Lemmata 8.3 and 8.4 as well as inequality (8.16) to obtain that
A c k( j ) ( t )
£
ì
ck , j l-N1+ 1 í
å
î
j
+ dk j l-N1+/ 21
j+2
å Ac
s=0
(s)
k - 1( t )
,
(s)
A c k - 1( t )
j +1
+
å Ac
s=0
(s)
n ( k ) - 1( t )
ü
ý+
þ
(8.25)
s=0
where j = 0 , 1 , ¼ , m - 2 and the numbers c k , j and dk , j do not depend on N .
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If we now assume that (8.19) holds for n £ k - 1 , then equation (8.25) implies
(8.19) for n = k and for k £ m - 1 . Using (8.22) and (8.23) we obtain equation
(8.20). Theorem 8.2 is proved.
3
Corollary 8.1
Let the manifold Mn have the form (8.7) with h ( p , p· ) = hn ( p , p· ) and
l ( p , p· ) = ln( p , p· ) . We also assume that U( t ) = ( u ( t ) ; u· ( t ) ) , where u( t )
is the solution to problem (0.1) from the class L m , R . Then
dist
-n ¤ 2
( U( t ) , M n ) £ C n l N + 1 ,
D(A) ´ D(A1 ¤ 2 )
n = 0, 1, 2 ¼ m - 1 .
Thus, the thickness of the layer that attracts the trajectories in the phase space has
the power order with respect to lN + 1 unlike the semilinear parabolic equations
of Section 6.
E x a m p l e 8.1
Let us consider the nonlinear wave equation (8.5). Let d = dim W £ 2 . We assume the following (cf. [18]) about the function g ( s ) :
s
lim
s ®¥
s -1
ò g ( s ) ds ³ 0 ;
0
there exists C1 > 0 such that
æ
lim
s ®¥
s -1 ç sg ( s )
ç
è
ö
- C1 g ( s ) ds÷ ³ 0 ;
÷
ø
0
s
ò
for any m there exists b ( m ) > 0 such that
g ( m ) ( s ) £ C2 ( 1 + s b ( m ) ) .
(8.26)
Under these assumptions the solution u ( t ) lies in Lm , R for R > 0 large
enough if and only if the initial data satisfy some compatibility conditions [18].
Moreover, the global attractor ) of system (8.5) exists and any trajectory lying
in ) possesses properties (8.4) for all t Î R and k = 1 , 2 , ¼ , [18]. It is easy
to see that Theorem 8.2 is applicable here (the form of A , B ( . ) and H is evident in this case). In particular, Theorem 8.2 gives us that for a trajectory
·
U ( t ) = ( u ( t ) ; ¶t u ( t ) ) of problem (8.5) which lies in the global attractor ) the
estimate
ì
j
í A ¶t ( u ( t ) - un ( t ) )
î
2
ü
+ A1 / 2 ¶ tj ( ¶ t u ( t ) - u n ( t ) ) 2 ý
þ
1/2
-n / 2
£ Cn , R , j lN + 1
holds for all n = 1 , 2 , ¼ , all j = 1 , 2 , ¼ , and all t Î R . Here u n ( t ) and
un ( t ) are defined with the help of (8.18). Therewith
Idea of Nonlinear Galerkin Method
sup { dist ( U ,
Mn ) : U Î) } £ cn l-Nn+/ 21 ,
n = 1, 2, ¼ ,
(8.27)
where Mn is a manifold of the type (8.7) with h = hn ( p , p· ) and l = ln ( p , p· ) .
Here dist ( U , Mn ) is the distance between U and Mn in the space
D ( A ) ´ D ( A1/ 2 ) . Equation (8.27) gives us some information on the location of
the global attractor in the phase space.
Other examples of usage of the construction given here can be found in papers [17]
and [19] (see also Section 9 of Chapter 4).
§9
Idea of Nonlinear Galerkin Method
Approximate inertial manifolds have proved to be applicable to the computational
study of the asymptotic behaviour of infinite-dimensional dissipative dynamical systems (for example, see the discussion and the references in [8]). Their usage leads
to the appearance of the so-called nonlinear Galerkin method [20] based on the replacement of the original problem by its approximate inertial form. In this section we
discuss the main features of this method using the following example of a second order in time equation of type (8.1):
2u
d--------u
- + g d------ + A u = B (u) ,
dt
d t2
d
u-----= u1 .
dt t = 0
u t = 0 = u0 ,
(9.1)
If all conditions on A and B ( . ) given in the previous section are fulfilled, then
Theorem 8.2 is valid. It guarantees the existence of a family of mappings { hk ; lk }
from PH ´ PH into QH possessing the properties:
1) there exist constants Mj º Mj ( n , r ) and Lj º Lj ( n , r ) , j = 1 , 2 , such
that
Ah ( p , p· ) £ M ,
A1/ 2 l ( p , p· ) £ M e,
(9.2)
0
n
0
1
n
0
0
2
A ( hn ( p1 , p· 1 ) - hn ( p2 , p· 2 ) )
£ L1 æ A ( p1 - p2 ) + A1/ 2 ( p· 1 - p· 2 ) ö ,
è
ø
(9.3)
A1/ 2 ( ln ( p1 , p· 1 ) - ln ( p2 , p· 2 ) )
£ L 1 æè A ( p 1 - p2 ) + A1/ 2 ( p· 1 - p· 2 ) öø
(9.4)
for all pj and p· j from PH such that
Ap 2 + A1/ 2 p· 2 £ r2 ,
j
j
j = 0, 1 ,
r > 0;
2) for any solution u ( t ) to problem (9.1) which lies in L m , R for m ³ 2 the estimate
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Inertial Manifolds
ì
í A ( u ( t ) - un ( t ) )
î
2
ü1 / 2
-n / 2
+ A1/ 2 ( ¶t u ( t ) - un ( t ) ) 2 ý
£ Cn , R lN + 1
þ
(9.5)
is valid (see Theorem 8.2) for all n £ m - 1 and t large enough. Here
un ( t ) = p ( t ) + hn ( p ( t ) , ¶t p ( t ) ) ,
3
(9.6)
un ( t ) = ¶ t p ( t ) + ln ( p ( t ) , ¶ t p ( t ) ) ,
l N + 1 is the ( N + 1 ) -th eigenvalue of A , and R is the constant from (8.4).
The family { hk ; lk } is defined with the help of a quite simple procedure (see (8.8)
and (8.9)) which can be reduced to the process of solving of stationary equations of
the type A v = g in the subspace QH . Moreover,
h ( p , p· ) º l ( p , p· ) º 0 ,
h ( p , p· ) = A -1 Q B ( p ) ,
l ( p , p· ) º 0 . (9.7)
0
0
1
1
In particular, estimates (9.5) and (9.6) mean (see Corollary 8.1) that trajectories
U ( t ) = ( u ( t ) ; ¶ t u ( t ) ) of system (9.1) are attracted by a small (for N large enough)
vicinity of the manifold
M = { ( p + h ( p , p· ) ; p· + l ( p , p· ) ) : p , p· Î PH } .
(9.8)
n
n
n
The sequence of mappings { hn ( p , p· ) } generates a family of approximate inertial
forms of problem (9.1):
¶t2 p + g ¶t p + A p = P B ( p + hn ( p , ¶ t p ) ) .
(9.9)
A finite-dimensional dynamical system in P H which approximates (in some sense)
the original system corresponds to each form. For n = 0 equation (9.9) transforms
into the standard Galerkin approximation of problem (9.1) (due to (9.7)). If n > 0 ,
then we obtain a class of numerical methods which can be naturally called the nonlinear Galerkin methods. However, we cannot use equation (9.9) in the computational study directly. The point is that, first, in the calculation of hn ( p , p· ) we have
to solve a linear equation in the infinite-dimensional space QH and, second, we can
lose the dissipativity property. Therefore, we need additional regularization. It can
be done as follows. Assume that fn ( p , p· ) stands for one of the functions hn ( p , p· )
or ln ( p , p· ) . We define the value
1/ 2
fn* ( p , p· ) º f N , M , n ( p , p· ) = c æ R -1 æ Ap 2 + A1/ 2 p· 2ö ö PM fn ( p , p· ) , (9.10)
è
è
ø ø
where c ( s ) is an infinitely differentiable function on R+ such that a) 0 £ c ( s ) £ 1;
b) c ( s ) = 1 for 0 £ s £ 1 ; c) c ( s ) = 0 for s ³ 2 ; R is the radius of dissipativity
(see (8.4) for k = 0 ) of system (9.1); PM is the orthoprojector in H onto the subspace generated by the first M eigenvectors of the operator A , M > N . We consider
the following N -dimensional evolutionary equation in the subspace PN H :
Idea of Nonlinear Galerkin Method
¶ t2 p* + g ¶t p* + A p* = PN B ( p* + h*n ( p* , ¶t p* ) ) ,
p* t = 0 = PN u0 ,
¶t p*
t=0
(9.11)
= PN u1 .
E x e r c i s e 9.1 Prove that problem (9.11) has a unique solution for t > 0 and
the corresponding dynamical system is dissipative in PN H ´ PN H .
We call problem (9.11) a nonlinear Galerkin ( n , N , M ) -approximation of problem
(9.1). The following assertion is valid.
Theorem 9.1.
Assume that the mappings hn ( p , p· ) and ln ( p , p· ) satisfy equations
(9.2)–(9.5) for n £ m - 1 and for some m ³ 2 . Moreover, we assume that
(9.5) is valid for all t > 0 . Let h*n and ln* be defined by (9.10) with the help
of h n and ln and let
u*n ( t ) = p* ( t ) + h*n ( p* ( t ) , ¶ t p* ( t ) ) ,
un* ( t ) = ¶ t p* ( t ) + ln* ( p* ( t ) , ¶ t p* ( t ) ) ,
where p* ( t ) is a solution to problem (9.11).. Then the estimate
ì 1/ 2
í A ( u ( t ) - u*n ( t ) )
î
-( n + 1 ) ¤ 2
£ ( a1 l N + 1
2
ü
+ ¶ t u ( t ) - u*n ( t ) 2 ý
þ
1/2
£
-1 ¤ 2
+ a2 lM
+ 1 ) exp ( b t )
(9.12)
holds, where u ( t ) is a solution to problem (9.1) which lies in Lm , R for
m ³ 2 and possesses property (8.4) for k = 1 and for all t > 0 . Here n £
£ m - 1 , a1 , a2 and b are positive constants independent of M and N ,
l k is the k -th eigenvalue of the operator A .
Proof.
Let p ( t ) = PN u ( t ) . We consider the values
u ( t ) - u*n ( t ) = p ( t ) - p* ( t ) + [ Q N u ( t ) - hn ( p ( t ) , ¶ t p ( t ) ) ] +
+ [ hn ( p ( t ) , ¶ t p ( t ) ) - h*n ( p* ( t ) , ¶ t p* ( t ) ) ]
and
¶ t u ( t ) - u*n ( t ) = ¶t ( p ( t ) - p* ( t ) ) + [ Q N ¶ t u ( t ) - ln ( p ( t ) , ¶t p ( t ) ) ] +
+ [ ln ( p ( t ) , ¶ t p ( t ) ) - ln* ( p* ( t ) , ¶t p* ( t ) ) ] .
The equalities
PM h n ( p ( t ) , ¶ t p ( t ) ) = h*n ( p ( t ) , ¶t p ( t ) )
211
212
Inertial Manifolds
C
h
a
p
t
e
r
and
PM ln ( p ( t ) , ¶ t p ( t ) ) = l*n ( p ( t ) , ¶t p ( t ) )
are valid for the class of solutions under consideration. Therefore, we use (9.5) to
find that
3
A1/ 2 ( u ( t ) - u*n ( t ) ) £
Cn , R
C2
- + ---------------------£ C 1 æ A1/ 2 ( p ( t ) - p* ( t ) ) + ¶ t p ( t ) - ¶ t p* ( t ) ö + ------------è
ø
l1M/ 2+ 1 l(Nn++11 ) ¤ 2
(9.13)
and
¶ t u ( t ) - ¶ t u*n ( t ) £
C4
Cn , R
£ C3 æè A1/ 2 ( p ( t ) - p* ( t ) ) + ¶t p ( t ) - ¶ t p* ( t ) öø + -------------- + ---------------------- . (9.14)
1/ 2
(n + 1) ¤ 2
lM + 1 ln + 1
Therefore, we must compare the solution p* ( t ) to problem (9.11) with the value
p ( t ) = PN u ( t ) which satisfies the equation
¶t2 p + g ¶ t p + A p = Q N B ( p + QN u )
(9.15)
with the same initial conditions as the function p* ( t ) . Let r ( t ) = p ( t ) - p* ( t ) . Then
it follows from (9.11) and (9.15) that
¶ t2 r ( t ) + g ¶ t r ( t ) + A r ( t ) = F ( t , p* , u ) ,
r (0) = 0,
(9.16)
¶t r ( 0 ) = 0 ,
where
F ( t , p*, u ) = QN [ B ( u ( t ) ) - B ( u*n ( t ) ) ] .
Due to the dissipativity of problems (9.11) and (9.15) we use (9.13) to obtain
F ( t , p*, u )
£ C R æ A1/ 2 r ( t )
è
2
+ r· ( t ) 2ö
ø
1/2
-( n + 1 ) ¤ 2
+ Cn , R lN + 1
+ C l-M1+/ 21
for the class of solutions under consideration. Therefore, equation (9.16) implies
that
-( n + 1 )
d æ ·
1
--- ---+ C l-M1+ 1 .
r ( t ) 2 + A1/ 2 r ( t ) 2ö £ C R æ r· ( t ) 2 + A1/ 2 r ( t ) 2ö + Cn , R l N + 1
ø
è
ø
2 dt è
Hence, Gronwall’s lemma gives us that
C t
r· ( t ) 2 + A1/ 2 r ( t ) 2 £ ( C n , R l-N(+n1+ 1 ) + C l-M1+ 1 ) e R .
This and equations (9.13) and (9.14) imply estimate (9.12). Theorem 9.1 is proved.
If we take n = 0 and N = M in Theorem 9.1, then estimate (9.12) changes into the
accuracy estimate of the standard Galerkin method of the order N . Therefore, if the
Idea of Nonlinear Galerkin Method
+1
parameters N , M , and n are compatible such that l M + 1 £ ln
N + 1 , then the error of
the corresponding nonlinear Galerkin method has the same order of smallness as in
the standard Galerkin method which uses M basis functions. However, if we use the
nonlinear method, we have to solve a number of linear algebraic systems of the order
M - N and the Cauchy problem for system (9.11) which consists of N equations.
In particular, in order to determine the value h 1 ( p , p· ) we must solve the equation
A h ( p , p· ) = ( P - P ) Q B ( p )
1
M
N
2
for n = 1 and choose the numbers N and M such that lM + 1 £ l N
+ 1 . Moreover,
if l k @ c 0 k s ( 1 + o ( 1 ) ) , s > 0 , as k ® ¥ , then the values N and M must be compatible such that M £ cs N 2 .
We note that Theorem 9.1 as well as the corresponding variant of the nonlinear
Galerkin method can be used in the study of the asymptotic properties of solutions
to the nonlinear wave equation (8.5) under some conditions on the nonlinear term
g ( u ) . Other applications of Theorem 9.1 can also be pointed out.
213
214
C
h
a
p
t
e
r
3
Inertial Manifolds
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