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
4
The Problem on Nonlinear Oscillations
of a Plate in a Supersonic Gas Flow
Contents
....§1
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
....§2
Auxiliary Linear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
....§3
Theorem on the Existence and Uniqueness of Solutions . . 232
....§4
Smoothness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
....§5
Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . 246
....§6
Global Attractor and Inertial Sets . . . . . . . . . . . . . . . . . . . . . 254
....§7
Conditions of Regularity of Attractor . . . . . . . . . . . . . . . . . 261
....§8
On Singular Limit in the Problem
of Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
....§9
On Inertial and Approximate Inertial Manifolds . . . . . . . . . 276
....
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
In this chapter we use the ideas and the results of Chapters 1 and 3 to study in
details the asymptotic behaviour of a class of problems arising in the nonlinear theory of oscillations of distributed parameter systems. The main object is the following
second order in time equation in a separable Hilbert space H :
d
d2
------2- u + g ---- u + A 2 u + M æ A1 ¤ 2 u 2ö A u + L u = p ( t ) ,
è
ø
dt
dt
u t = 0 = u0 ,
du
-----= u1 ,
dt t = 0
(0.1)
(0.2)
where A is a positive operator with discrete spectrum in H , M ( s ) is a real function
(its properties are described below), L is a linear operator in H , p ( t ) is a given
bounded function with the values in H , and g is a nonnegative parameter. The
problem of type (0.1) and (0.2) arises in the study of nonlinear oscillations of a plate
in the supersonic flow of gas. For example, in Berger’s approach (see [1, 2]), the dynamics of a plate can be described by the following quasilinear partial differential
equation:
æ
¶t2 u + g ¶t u + D2 u + ç G è
ò
W
ö
Ñu 2 dx÷ D u + r ¶x u = p ( x , t ) ,
1
ø
x Î ( x1 , x2 ) Ì W Ì R2 ,
with boundary and initial conditions of the form
u ¶ W = Du ¶ W = 0 ,
u t = 0 = u0 ( x ) ,
(0.3)
t >0
¶t u
t=0
= u1 ( x ) .
(0.4)
Here D is the Laplace operator in the domain W ; g > 0 , r ³ 0 , and G are constants; and p ( x , t ) , u0 ( x ) , and u1 ( x ) are given functions. Equations (0.3)–(0.4)
describe nonlinear oscillations of a plate occupying the domain W on a plane which
is located in a supersonic gas flow moving along the x1 -axis. The aerodynamic pressure on the plate is taken into account according to Ilyushin’s “piston” theory (see,
e. g., [3]) and is described by the term r ¶x1 u . The parameter r is determined by
the velocity of the flow. The function u ( x , t ) measures the plate deflection at the
point x and the moment t . The boundary conditions imply that the edges of the
plate are hinged. The function p ( x , t ) describes the transverse load on the plate.
The parameter G is proportional to the value of compressive force acting in the
plane of the plate. The value g takes into account the environment resistance.
Our choice of problem (0.1) and (0.2) as the base example is conditioned by the
following circumstances. First, using this model we can avoid significant technical
difficulties to demonstrate the main steps of reasoning required to construct a solution and to prove the existence of a global attractor for a nonlinear evolutionary second order in time partial differential equation. Second, a study of the limit regimes
of system (0.3)–(0.4) is of practical interest. The point is that the most important
(from the point of view of applications) type of instability which can be found in the
system under consideration is the flutter, i.e. autooscillations of a plate subjected to
218
The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow
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aerodynamical loads. The modern look on the flutter instability of a plate is the following: there arises the Andronov-Hopf bifurcation leading to the appearance of
a stable limit cycle in the system. However, there are experimental and numerical
data that enable us to conjecture that an increase in flow velocity may result in the
complication of the dynamics and appearance of chaotic fluctuations [4]. Therefore,
the study of the existence and properties of the attractor of the given problem
enables us to better understand the mechanism of appearance of a nonlinear flutter.
4
§1
Spaces
As above (see Chapter 2), we use the scale of spaces Fs generated by a positive operator A with discrete spectrum acting in a separable Hilbert space H . We remind
(see Section 2.1) that the space Fs is defined by the equation
ì
Fs º D ( A s ) = í v :
î
¥
¥
å
ck ek :
åc
k=1
k=1
2
k
ü
lk2 s < ¥ ý ,
þ
where { e k } is the orthonormal basis of the eigenelements of the operator A in H ,
l 1 £ l2 £ ¼ are the corresponding eigenvalues and s is a real parameter
(for s = 0 we have Fs = H and for s < 0 the space Fs should be treated as a class
of formal series). The norm in Fs is given by the equality
¥
v s2 =
å
¥
ck l2k s
for
v=
k=1
åc e
k k.
k=1
Further we use the notation L2 ( 0 , T ; Fs ) for the set of measurable functions
on the segment [ 0 , T ] with the values in the space Fs such that the norm
v
L2 ( 0 , T ; Fs )
æT
= ç v(t)
ç
è0
ò
2
s
ö
dt ÷
÷
ø
1¤ 2
is finite. The notation L p ( 0 , T ; X ) has a similar meaning for 1 £ p £ ¥ .
We remind that a function u(t) with the values in a separable Hilbert space H
is said to be Bochner measurable on a segment [ 0 , T ] if it is a limit of a sequence of functions
N
uN ( t ) =
åu
N, k
cN , k ( t )
k=1
for almost all t Î [ 0 , T ] , where uN , k are elements of H and cN , k ( t ) are the characteristic functions of the pairwise disjoint Lebesgue measurable sets AN , k . One
Spaces
can prove (see, e.g., the book by K. Yosida [5]) that for separable Hilbert spaces under consideration a function u ( t ) is measurable if and only if the scalar function
( u ( t ) , h ) H is measurable for every h Î H . Furthermore, a function u ( t ) is said
to be Bochner integrable over [ 0 , T ] if
T
ò u (t) - u
2
N (t) H
dt ® 0 ,
N ® ¥,
0
where { uN ( t ) } is a sequence of simple functions defined above. The integral of the
function u(t) over a measurable set S Ì [ 0 , T ] is defined by the equation
òu (t) dt =
T
lim
N®¥
S
ò c (t) u
S
N (t)
dt ,
0
where c S ( t ) is the characteristic function of the set S and the integral of a simple
function in the right-hand side of the equality is defined in an obvious way.
For the function with the values in Hilbert spaces most facts of the ordinary Lebesgue integration theory remain true.
E x e r c i s e 1.1 Let u ( t ) be a function on [ 0 , T ] with the values in a separable Hilbert space H . If there exists a sequence of measurable functions un ( t ) such that un ( t ) ® u ( t ) almost everywhere, then u ( t )
is also measurable.
E x e r c i s e 1.2 Show that a measurable function u ( t ) with the values in H
is integrable if and only if u ( t ) Î L 1(0 , T ) . Therewith
ò u (t) dt
£
B
ò u (t) dt
B
for any measurable set B Ì [ 0 , T ] .
E x e r c i s e 1.3 Let a function u ( t ) be integrable over [ 0 , T ] and let B be
a measurable set from [ 0 , T ] . Show that
æ
( u ( t ) , h ) H dt = ç
ç
è
B
ò
ö
u ( t ) dt , h÷
÷
øH
B
ò
for any h Î H .
E x e r c i s e 1.4 Show that the space L 2 ( 0 , T ; Fs ) can be described as a set
of series
¥
h(t) =
åc
k=1
k ( t ) ek ,
219
220
The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow
where ck ( t ) are scalar functions that are square-integrable over
[ 0 , T ] and such that
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T
¥
å
l2k s
k=1
4
ò [ c (t) ]
k
2
dt < ¥ .
(1.1)
0
Below we also use the space C ( 0 , T ; Fs ) of strongly continuous functions on [ 0 , T ]
with the values in Fs and the norm
v C (0, T ; F ) =
s
max
t Î [0, T ]
v (t) s .
Let u ( t ) be a function with the values in Fs integrable over
[ 0 , T ] . Show that the function
E x e r c i s e 1.5
t
v (t) =
ò u ( t ) dt
0
lies in C ( 0 , T ; Fs ) . Moreover, v ( t ) is an absolutely continuous
function with the values in Fs , i.e. for any e > 0 there exists d > 0
such that for any collection of disjoint segments [ ak , bk ] Ì [ 0 , T ]
the condition å ( b k - ak ) < d implies that
k
å v (b ) - v (a )
k
k s
< e.
k
Show that for any absolutely continuous function v ( t ) on
[ 0 , T ] with the values in Fs there exists a function u ( t ) with the
values in Fs such that it is integrable over [ 0 , T ] and
E x e r c i s e 1.6
t
v (t) = v (0) +
ò u ( t ) dt ,
t Î [ 0, T ] .
0
(Hint: use the one-dimensional variant of this assertion).
The space
ì
WT = í v ( t ) : v ( t ) Î L 2 ( 0 , T ; F1 ) ,
î
with the norm
v W = æ v 22
è L (0, T ;
T
F1 )
+ v·
ü
v· ( t ) Î L 2 ( 0 , T ; H ) ý
þ
2
ö
L2 ( 0 , T ; H )ø
(1.2)
1¤ 2
plays an important role below. Hereinafter the derivative v· ( t ) = dv ¤ dt stands for
a function integrable over [ 0 , T ] and such that
Spaces
t
v (t) = h +
ò v· (t) dt
0
almost everywhere for some h Î H (see Exercises 1.5 and 1.6). Evidently, the space
WT is continuously embedded into C ( 0 , T ; H ) , i.e. every function u ( t ) from WT
lies in C ( 0 , T ; H ) and
max
t Î [0, T ]
u(t) £ C u W ,
T
where C is a constant. This fact is strengthened in the series of exercises given below.
Let pm be the projector onto the span of the set { ek : k =
= 1 , ¼ , m } and let v ( t ) Î WT . Show that pm v ( t ) is absolutely
continuous and possesses the property
E x e r c i s e 1.7
d (p ( )) = p · ( ) Î 2 ( , ;
---v t
L 0 T pm F1 ) .
mv t
dt m
E x e r c i s e 1.8
The equations
t
ò
pm v ( t ) 12¤ 2 = pm v ( s ) 12¤ 2 + 2 ( pm v ( t ) , pm v· ( t ) ) 1 ¤ 2 dt (1.3a)
s
and
( t - s ) pm v ( t )
2
1¤ 2
=
t
=
ò æè p
2
m v (t ) 1¤ 2
+ 2 ( t - s ) ( pm v ( t ) , pm v· ( t ) ) 1 ¤ 2ö dt
ø
(1.3b)
s
are valid for any 0 £ s £ t £ T and v ( t ) Î WT .
E x e r c i s e 1.9
Use (1.3) to prove that
sup
t Î [0, T ]
pm v ( t ) 1 ¤ 2 £ C T v W
T
(1.4a)
and
sup
t Î [0, T ]
( pm - pk ) v ( t )
1¤ 2
£ CT ( pm - p k ) v W .
(1.4b)
T
E x e r c i s e 1.10 Use (1.4) to prove that WT is continuously embedded into
C ( 0 , T ; F1 ¤ 2 ) and
max
t Î [0, T ]
v ( t ) 1 ¤ 2 £ CT v W .
T
221
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The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow
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The following three exercises result in a particular case of Dubinskii’s theorem
(see Exercise 1.13).
4
E x e r c i s e 1.11 Let { h k ( t ) }¥
be an orthonormal basis in L2 ( 0 , T ; R ) conk=0
sisting of the trigonometric functions
1 ,
h 0 ( t ) = ------T
h2 k - 1 ( t ) =
2pk
--2- sin ----------- x ,
T
T
h2 k ( t ) =
2pk
--2- cos ----------- x ,
T
T
k = 1 , 2 , ¼ . Show that f ( t ) Î L 2 ( 0 , T ; Fs ) if and only if
¥
f (t) =
¥
å åc
kj
hk ( t ) ej
(1.5)
k=0 j=1
and
¥
¥
å ål
2s
j
ck j
2
< ¥.
k=0 j=1
E x e r c i s e 1.12 Show that the space WT can be described as a set of series of
the form (1.5) such that
¥
å
k, j = 1
æ k 2 + l2ö c 2 < ¥ .
jø kj
è
E x e r c i s e 1.13 Use the method of the proof of Theorem 2.1.1 to show that
WT is compactly embedded into the space L 2 ( 0 , T ; Fs ) for any
s < 1.
E x e r c i s e 1.14 Show that WT is compactly embedded into C ( 0 , T ; H ) .
Hint: use Exercise 1.10 and the reasoning which is usually applied
to prove the Arzelà theorem on the compactness of a collection
of scalar continuous functions.
§2
Auxiliary Linear Problem
In this section we study the properties of a solution to the following linear problem:
ì
ï
ï
í
ï
ï
î
d2 u
du
---------- + g ------- + A 2 u + b ( t ) A u = h ( t ) ,
dt
d t2
u t = 0 = u0 ,
du
------= u1 .
dt t = 0
(2.1)
(2.2)
Auxiliary Linear Problem
Here A is a positive operator with discrete spectrum. The vectors h ( t ) , u 0 , u1
as well as the scalar function b ( t ) are given (for the corresponding hypotheses see
the assertion of Theorem 2.1).
The main results of this section are the proof of the theorem on the existence
and uniqueness of weak solutions to problem (2.1) and (2.2) and the construction
of the evolutionary operator for the system when h ( t ) º 0 . In fact, the approach we
use here is well-known (see, e.g., [6] and [7]).
A weak solution to problem (2.1) and (2.2) on a segment [ 0 , T ] is a function u ( t ) Î WT such that u ( 0 ) = u 0 and the equation
T
-
ò
T
( u· ( t ) + g u ( t ) , v· ( t ) ) d t +
0
0
T
= ( u1 + g u0 , v ( 0 ) ) +
ò (A u (t) + b (t) u (t) , A v (t) ) dt =
ò (h (t) , v (t) ) dt
(2.3)
0
holds for any function v ( t ) Î WT such that v ( T ) = 0 . As above, u· stands for the
derivative of u with respect to t .
E x e r c i s e 2.1 Prove that if a weak solution u ( t ) exists, then it satisfies
the equation
( u· ( t ) + g u ( t ) , w) = ( u1 + g u0 , w ) t
t
ò
- ( A u ( t ) + b ( t ) u ( t ) , Aw ) dt +
0
ò ( h ( t ) , w ) dt
0
T
j ( t ) dt
t
for every w Î F1 (Hint: take v ( t ) = ò
j ( t ) is a scalar function from C [ 0 , T ] ).
(2.4)
× w in (2.3), where
Theorem 2.1
Let u0 Î F 1 , u1 Î F 0 , and g ³ 0 . We als
also assume that b ( t ) is a bounded
continuous function on [ 0 , T ] and h ( t ) Î L ¥ ( 0 , T ; F 0 ) , where T is a positive number. Then problem (2.1) and (2.2) has a unique weak solution
u ( t ) on the segment [ 0 , T ] . This solution possesses the properties
(2.5)
u (t) Î C (0, T ; F ) ,
u· ( t ) Î C ( 0 , T ; F )
1
0
and satisfies the energy equation
t
1æ ·
--- u ( t ) 2 + Au ( t ) 2ö + g
ø
2è
t
1
= --- æ u 1
2è
2
+ Au0 2ö +
ø
ò
0
t
u· ( t ) 2 dt +
ò b (t) (A u (t) , u· (t) ) dt =
0
ò (h (t) , u· (t) ) dt .
0
(2.6)
223
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The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow
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Proof.
We use the compactness method to prove this theorem. At first we construct approximate solutions to problem (2.1) and (2.2). The approximate Galerkin solution
(to this problem) of the order m with respect to the basis { e k } is considered to be
the function
4
m
um ( t ) =
å g (t) e
k
(2.7)
k
k=1
satisfying the equations
( u·· + g u·
m
m
+ A 2 um + b ( t ) A um - h ( t ) , ej ) = 0 ,
( um ( 0 ) , ej ) = ( u 0 , ej ) ,
( u· m ( 0 ) , e j ) = ( u1 , e j ) ,
(2.8)
j = 1 , 2 , ¼ , m . (2.9)
Here gk ( t ) Î C 1 ( 0 , T ) and g· k ( t ) is absolutely continuous. Due to the orthogonality
of the basis { e k } equations (2.8) and (2.9) can be rewritten as a system of ordinary
differential equations:
g··k + g g· k + l k2 gk - b ( t ) l k gk = hk ( t ) º ( h ( t ) , e k ) e,
gk ( 0 ) = ( u 0 , ek ) ,
g· k ( 0 ) = ( u 1 , ek ) ,
k = 1, 2, ¼, m .
Lemma 2.1
Assume that g > 0 , a ³ 0 , b ( t ) is continuous, and c ( t ) is a measurable bounded function. Then the Cauchy problem
ì g·· + g g· + a ( a - b ( t ) ) g = c ( t ) ,
ï
í
ï g ( 0 ) = g0 , g· ( 0 ) = g1
î
t Î [ 0, T] ,
(2.10)
is uniquely solvable on any segment [ 0 , T ] . Its solution possesses the
property
t
æ
ö
12
·g ( t ) 2 + a 2 g ( t ) 2 £ ç g 2 + a 2 g 2 + ----÷ e b0 t ,
(
t
)
t
d
c
0 2g
ç 1
÷
è
ø
0
ò
t Î [ 0 , T ] , (2.11)
where b0 = max b ( t ) . Moreover, if b ( t ) Î C1 ( 0 , T ) , c ( t ) º 0 and for all
t
t Î [ 0 , T ] the conditions
g2
--- a + ----- £ b ( t ) £ 1
--- a ,
-1
a
2
2
·
--- - b ( t )ö ³ 0 ,
b(t) + g æ a
è4
ø
(2.12)
hold, then the following estimate is valid:
g
g· ( t ) 2 + a 2 g ( t ) 2 £ 3 ( g12 + a 2 g02 ) exp æ - --- tö .
è 2 ø
(2.13)
Auxiliary Linear Problem
Proof.
Problem (2.10) is solvable at least locally, i.e. there exists t such that a solution exists on the half-interval [ 0 , t ) . Let us prove estimate (2.11) for the interval of existence of solution. To do that, we multiply equation (2.10) by g· ( t ) .
As a result, we obtain that
d
1
--- ----- ( g· 2 + a 2 g 2 ) + g g· 2 = a b ( t ) g g· + c ( t ) g· .
2 dt
We integrate this equality and use the equations
--- max b ( t ) æ g· 2 + a 2 g 2ö ,
a b ( t ) g g· £ 1
2 t
è
ø
to obtain that
1- c 2 ,
c g· £ g g· 2 + ----4g
t
1g· ( t ) 2 + a 2 g ( t ) £ g12 + a 2 g02 + ----2g
ò
t
c2 ( t )
dt + b0
0
ò(g· (t)
2
+ a 2 g ( t ) 2 ) dt .
0
This and Gronwall’s lemma give us (2.11).
In particular, estimate (2.11) enables us to prove that the solution g ( t ) can
be extended on a segment [ 0 , T ] of arbitrary length. Indeed, let us assume the
contrary. Then there exists a point t such that the solution can not be extended
through it. Therewith equation (2.11) implies that
g· ( t ) 2 + a 2 g ( t ) 2 £ C ( T ; g0 , g1) ,
0 < t < t < T.
Therefore, (2.10) gives us that the derivative g·· ( t ) is bounded on [ 0 , t ) .
Hence, the values
t
g· ( t ) = g0 +
ò
t
g·· ( t ) dt ,
0
g ( t ) = g0 +
òg· (t) dt
0
are continuous up to the point t . If we now apply the local theorem on existence to system (2.10) with the initial conditions at the point t that are equal to
g ( t ) and g· ( t ) , then we obtain that the solution can be extended through t .
This contradiction implies that the solution g ( t ) exists on an arbitrary segment
[0, T ] .
Let us prove estimate (2.13). To do that, we consider the function
g
g
--- æ g· 2 + a ( a - b ( t ) ) g 2ö + --- æ g g· + --- g 2ö .
V (t) = 1
è
2è
2 ø
ø 2
(2.14)
Using the inequality
1- g· 2 + --g- g 2 ,
1- g· 2 - --g- g 2 £ g g· £ ----- ----2g
2g
2
2
it is easy to find that the equation
1
--- æ g· 2 + a 2 g 2ö
--- æ g· 2 + a 2 g 2ö £ V ( t ) £ 3
ø
4è
4è
ø
(2.15)
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The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow
holds under the condition
--- a + b ( t ) - g 2 ³ 0 .
a 1
2
1
--- a - b ( t ) ³ 0 ,
2
Further we use (2.10) with c ( t ) º 0 to obtain that
g
·
dV
--- ( a b + a g ( a - b ) ) g 2 .
------ = - --- g· 2 - 1
2
2
dt
Consequently, with the help of (2.15) we get
dV --g------ + V £ 0
dt 2
under conditions (2.12). This implies that
g
V ( t ) £ V ( 0 ) exp æè - --- töø .
2
We use (2.15) to obtain estimate (2.13). Thus, Lemma 2.1 is proved.
E x e r c i s e 2.2 Assume that g £ 0 in Lemma 2.1. Show that problem (2.10)
is uniquely solvable on any segment [ 0 , T ] and the estimate
(b + 2 g + d) t
+
g· ( t ) + a 2 g ( t ) 2 £ æ g12 + a 2 g02ö e 0
è
ø
t
+ --1d
ò c (t)
2 ( b0 + 2 g + d ) ( t - t )
dt
e
0
is valid for g ( t ) and for any d > 0 , where b0 = max b ( t ) .
t
Lemma 2.1 implies the existence of a sequence of approximate solutions { um ( t ) }
to problem (2.1) and (2.2) on any segment [ 0 , T ] .
E x e r c i s e 2.3 Show that every approximate solution um is a solution to
problem (2.1) and (2.2) with u0 = u0 m , u 1 = u1 m , and h ( x , t ) =
= hm ( x , t ) , where
m
ui m = pm ui =
å (u , e ) e
i
k
k,
k=1
(2.16)
m
hm = pm h =
å (h(t) , e ) e
k
k,
k=1
and pm is the orthoprojector onto the span of elements { ek : k = 1 ,
2 , ¼ , m } in F0 = H .
Let us prove that the sequence of approximate solutions { u m } is convergent.
Auxiliary Linear Problem
At first we note that
m+l
D m , l ( t ) = u· m - u· m + l 2 + A ( um - u m + l ) 2 =
å
( g· k ( t ) 2 + l k2 gk ( t ) 2 )
k=m+1
for every t Î [ 0 , T ] . Therefore, by virtue of Lemma 2.1 we have that
Dm , l ( t ) £ e
b0 t
T
m+l
å
( u1 , ek )
2
+ l k2 ( u0 ,
k=m+1
1ek ) + ----2g
2
ò ( h ( t ) , e ) dt
2
k
0
for g > 0 . Moreover, in the case g = 0 , the result of Exercise 2.2 gives that
Dm , l ( t ) £ e
( b0 + 1 ) t
T
m+l
å
2
( u1 , ek ) +
l k2 ( u0 ,
2
ek ) +
k=m+1
ò ( h ( t ) , e ) dt
2
k
.
0
These equations imply that the sequences { u· m ( t ) } and { Au m ( t ) } are the Cauchy
sequences in the space C(0 , T ; H ) on any segment [ 0 , T ] . Consequently, there
exists a function u ( t ) such that
u· ( t ) Î C ( 0 , T ; H ) ,
lim
u ( t ) Î C ( 0 , T ; F1 ) ,
max æ u· m ( t ) - u· ( t ) 2 + um ( t ) - u ( t ) 1ö = 0 .
ø
m ® ¥ [0, T ] è
(2.17)
Equations (2.8) and (2.9) further imply that
T
-
ò
T
æ u· ( t ) + g u ( t ) , v· ( t )ö dt +
m
è m
ø
0
ò æèA u
ò (h
+ b ( t ) u m ( t ) , A v ( t ) ö dt =
ø
0
T
= ( u1 m + g u0 m , v ( 0 ) ) +
m(t)
m(t) ,
v ( t ) ) dt
0
for all functions v ( t ) from WT such that v ( T ) = 0 . Here u i m , and hm ( t ) , i = 0 , 1 ,
are defined by (2.16). We use equation (2.17) to pass to the limit in this equation and
to prove that the function u ( t ) satisfies equality (2.3). Moreover, it follows from
(2.17) that u ( 0 ) = u0 . Therefore, the function u ( t ) is a weak solution to problem
(2.1) and (2.2).
In order to prove the uniqueness of weak solutions we consider the function
ì s
ï
ï - u ( t ) dt , t < s ,
vs ( t ) = í
ï t
ï 0, s £ t £ T,
î
ò
(2.18)
for s Î [ 0 , T ] . Here u ( t ) is a weak solution to problem (2.1) and (2.2) for h = 0 ,
u0 = 0 , and u1 = 0 . Evidently vs ( t ) Î WT . Therefore,
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T
-
ò
T
( u· ( t ) + g u ( t ) , v· s ( t ) ) dt +
0
ò (A u (t) + b(t) u (t) , A v (t) ) dt = 0 .
s
0
Due to the structure of the function vs ( t ) we obtain that
4
s
1æ
--- u ( s ) 2 + Avs ( 0 ) 2ö + g
ø
2è
ò u(t)
2
dt = Js ( u , v ) ,
(2.19)
0
where
s
Js ( u , v ) =
ò ( b ( t ) u ( t ) , A v ( t ) ) dt .
s
0
It is evident that Avs ( t ) = Avs ( 0 ) - Avt ( 0 ) for t £ s . Therefore,
Js ( u , v )
æ
£ b0 çç Avs ( 0 )
è
s
ò
u ( t ) dt +
0
s
--- Avs ( 0 ) 2 + s b02
£ 1
4
ò u (t)
ö
u ( t ) × Avt ( 0 ) dt÷÷ £
ø
0
s
2
b
dt + ----02
0
ò
s
ò æè u (t)
2
+ Avt ( 0 ) 2ö dt .
ø
0
If we substitute this estimate into equation (2.19), then it is easy to find that
s
u(s)
2
+ Avs ( 0 )
2
£ CT
ò æè u (t)
2
+ Avt ( 0 ) 2ö dt ,
ø
0
where s Î [ 0 , T ] and CT is a positive constant depending on the length of the segment [ 0 , T ] . This and Gronwall’s lemma imply that u ( t ) º 0 .
Let us prove the energy equation. If we multiply equation (2.8) by g· j ( t ) and
summarize the result with respect to j , then we find that
d æ · 2
1
--- ---+ Au m 2ö + g u· m 2 + b ( t ) ( Au m , u· m ) = ( h , u· m ) .
u
ø
2 dt è m
After integration with respect to t we use (2.17) to pass to the limit and obtain (2.6).
Theorem 2.1 is completely proved.
E x e r c i s e 2.4
Prove that the estimate
t
æ
ö
b t
1
2
2
2
2
·
ç
£ u1 + Au0 + -----u ( t ) + Au ( t )
h ( t ) 2 dt÷ e 0 (2.20)
ç
÷
2g
è
ø
0
ò
is valid for a weak solution u ( t ) to problem (2.1) and (2.2). Here
b0 = max { b ( t ) : t ³ 0 } and g > 0 .
Auxiliary Linear Problem
E x e r c i s e 2.5 Let u ( t ) be a weak solution to problem (2.1) and (2.2). Prove
that A2 u ( t ) Î C ( 0 , T ; F -1 ) and
t
u· ( t ) + g u ( t ) = u1 + g u 0 -
ò (A u (t) + b (t) Au (t) - h (t) ) dt .
2
0
Here we treat the equality as an equality of elements in F-1 .
(Hint: use the results of Exercises 2.1 and 2.1.3).
E x e r c i s e 2.6 Let u ( t ) be a weak solution to problem (2.1) and (2.2) constructed in Theorem 2.1. Then the function u· ( t ) is absolutely continuous as a vector-function with the values in F-1 while the derivative u·· ( t ) belongs to the space L ¥ ( 0 , T ; F-1 ) . Moreover, the function
u ( t ) satisfies equation (2.1) if we treat it as an equality of elements
in F-1 for almost all t Î [ 0 , T ] .
In particular, the result of Exercise 2.6 shows that a weak solution satisfies equation
(2.1) in a stronger sense then (2.4).
We also note that the assertions of Theorem 2.1 and Exercises 2.4–2.6 with the
corresponding changes remain true if the initial condition is given at any other moment t0 which is not equal to zero.
Now we consider the case h ( t ) º 0 and construct the evolutionary operator of problem (2.1) and (2.2). To do that, let us consider the family of spaces
Hs = F1 + s ´ Fs ,
s ³ 0.
Every space Hs is a set of pairs y = ( u ; v ) such that u Î F 1 + s and v Î Fs .
We define the inner product in Hs by the formula
( y1 , y2 )
E x e r c i s e 2.7
Hs
= ( u1 , u2 ) 1 + s + ( v1 , v2 ) s .
Prove that Hs is compactly embedded into Hs for s1 > s .
1
In the space H0 we define the evolutionary operator U ( t ; t0 ) of problem (2.1) and
(2.2) for h ( t ) º 0 by the equation
U ( t ; t ) y = ( u ( t ) ; u· ( t ) ) ,
(2.21)
0
where u ( t ) is a solution to (2.1) and (2.2) at the moment t with initial conditions
that are equal to y = ( u0 ; u1 ) at the moment t0 .
The following assertion plays an important role in the study of asymptotic behaviour of solutions to problem (0.1) and (0.2).
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Theorem 2.2
Assume that the function b ( t ) is continuously differentiable in (2.1)
and such that
·
b0 = sup b ( t ) < ¥ ,
b1 = sup b ( t ) < ¥ .
t
t
Then the evolutionary operator U ( t ; t ) of problem (2.1) and (2.2) for
h ( t ) º 0 is a linear bounded operator in each space Hs for s ³ 0 and it
possesses the properties:
a) U ( t ; t ) U ( t ; s ) = U ( t ; s ) , t ³ t ³ s , U ( t ; t ) = I ;
b) for all s ³ 0 the estimate
U(t ; t) y
Hs
£
æ1
--- b0 ( t - t )ö
ø
2
H s exp è
y
(2.22)
is valid;
c) there exists a number N 0 depending on g , b0 , and b1 such that the
equation
g
( I - PN ) U ( t ; t ) y
H
£
s
3 ( I - PN ) y
H
s
- --- ( t - t )
,
e 4
t >t ,
(2.23)
holds for all N ³ N0 , where PN is the orthoprojector onto the subspace
LN = Lin { ( e k ; 0 ) , ( 0 ; ek ) : k = 1 , 2 , ¼ , N }
in the space H0 .
Proof.
Semigroup property a) follows from the uniqueness of a weak solution.
The boundedness property of the operator U ( t ; t ) follows from (2.22). Let us prove
relations (2.22) and (2.23). It is sufficient to consider the case t = 0 . According
to the definition of the evolutionary operator we have that
U ( t ; 0 ) y = ( u ( t ) ; u· ( t ) ) ,
y = (u ; u ) ,
0
1
where u ( t ) is the weak solution to problem (2.1) and (2.2) for h ( t ) º 0 . Due to
(2.17) it can be represented as a convergent series of the form
¥
u (t) =
å g (t) e
k
k.
k=1
Moreover, Lemma 2.1 implies that
b t
g· k ( t ) 2 + l k2 gk ( t ) 2 £ ( g· k ( 0 ) 2 + lk2 gk ( 0 ) 2 ) e 0 .
Since
¥
U(t ; 0) y
2
Hs
equation (2.24) implies (2.22).
=
å æèg· (t)
k
k=1
2
+ l k2 gk ( t ) 2ö l k2 s ,
ø
(2.24)
Auxiliary Linear Problem
Further we use equation (2.13) to obtain that
g
- --- t
g· k ( t ) 2 + l k2 gk ( t ) 2 £ 3 æ gk ( 0 ) 2 + l k2 gk ( 0 ) 2ö e 2 ,
è
ø
(2.25)
provided the conditions (cf. (2.12))
g2
--- l k + ------ £ b ( t ) £ 1
--- l k ,
-1
2
2
lk
l
·
b ( t ) + g æ -----k- - b ( t )ö ³ 0 ,
è 4
ø
are fulfilled. Evidently, these conditions hold if
4b
g2
lk ³ --------1- + 4 b0 + ----- ,
g
b0
where b0 = max b ( t )
t
and b1 = max b ( t ) . Since
t
¥
( I - PN ) U ( t ; 0 ) y
2
Hs
=
å
k=N+1
æ g· ( t ) 2 + k 4 g ( t ) 2ö k 2 s ,
k
è k
ø
equation (2.25) gives us (2.23) for all N ³ N0 - 1 , where N 0 is the smallest natural
number such that
4b
g2
l N ³ ---------1 + 4 b0 + ----- .
g
b0
0
(2.26)
Thus, Theorem 2.2 is proved.
E x e r c i s e 2.8 Show that a weak solution u ( t ) to problem (2.1) and (2.2)
can be represented in the form
t
( u ( t ) ; u· ( t ) ) = U ( t ; 0 ) y +
ò U ( t ; t ) ( 0 ; h ( t ) ) dt ,
(2.27)
0
where y = ( u0 ; u 1 ) and U ( t ; t ) is defined by (2.21).
E x e r c i s e 2.9 Use the result of Exercise 2.2 to show that Theorem 2.1 and
Theorem 2.2 (a, b) with another constant in (2.22) also remain true
for g < 0 . Use this fact to prove that if the hypotheses of Theorems
2.1 and 2.2 hold on the whole time axis, then problem (2.1) and (2.2)
is solvable in the class of functions
W = C ( R ; F1 ) Ç C 1 ( R ; F0 )
with g ³ 0 .
E x e r c i s e 2.10 Show that the evolutionary operator U ( t , t ) has a bounded
inverse operator in every space Hs for s ³ 0 . How is the operator
[ U ( t , t ) ] -1 for t > t related to the solution to equation (2.1) for
h ( t ) º 0 ? Define the operator U ( t , t ) using the formula U ( t , t ) =
= [ U ( t , t ) ] -1 for t < t and prove assertion (a) of Theorem 2.2 for
all t , t Î R .
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§ 3 Theorem on Existence
and Uniqueness of Solutions
In this section we use the compactness method (see, e.g., [8]) to prove the theorem
on the existence and uniqueness of weak solutions to problem (0.1) and (0.2) under
the assumption that
u0 Î F1 ,
u1 Î F0 ,
p ( t ) Î L ¥ ( 0 , T ; F0 ) ;
(3.1)
z
M ( z ) Î C1 ( R+ ) ,
M (z) º
ò M (x) dx ³ - a z - b ,
(3.2)
0
where 0 £ a < l1 , b Î R , l1 is the first eigenvalue of the operator A , and the
operator L is defined on D ( A ) and satisfies the estimate
L u £ C Au ,
u Î D (A) .
(3.3)
Similarly to the linear problem (see Section 2), the function u ( t ) Î WT is said
to be a weak solution to problem (0.1) and (0.2) on the segment [ 0 , T ]
if u ( 0 ) = u 0 and the equation
T
-
ò
T
( u· ( t ) + g u ( t ) , v· ( t ) ) dt +
0
ò æè Au (t) + M æè A
1 ¤ 2 u ( t ) 2ö u ( t ) ,
ò
Av ( t )ö dt +
ø
0
T
+
ø
T
( L u ( t ) , v ( t ) ) dt = ( u 1 + g u 0 , v ( 0 ) ) +
0
ò ( p (t) , v(t) ) dt
(3.4)
0
holds for any function v ( t ) Î WT such that v ( T ) = 0 . Here the space WT is defined
by equation (1.2).
E x e r c i s e 3.1 Prove the analogue of formula (2.4) for weak solutions to
problem (0.1) and (0.2).
The following assertion holds.
Theorem 3.1
Assume that conditions (3.1)–(3.3) hold. Then on every segment [ 0 , T ]
problem (0.1) and (0.2) has a weak solution u ( t ) . This solution is unique.
It possesses the properties
(3.5)
u (t) Î C (0, T ; F ) ,
u· ( t ) Î C ( 0 , T ; F )
1
0
Theorem on Existence and Uniqueness of Solutions
and satisfies the energy equality
t
E ( u ( t ) , u· ( t ) ) = E ( u0 , u1 ) +
ò æè-g u· (t)
2
+ ( - L u ( t ) + p ( t ) , u· ( t ))ö dt , (3.6)
ø
0
where
--- æ v 2 + Au 2 + M æ A1 ¤ 2 u 2ö ö .
E (u, v) = 1
è
øø
2è
(3.7)
We use the scheme from Section 2 to prove the theorem.
The Galerkin approximate solution of the order m to problem (0.1) and (0.2)
with respect to the basis ek is defined as a function of the form
m
um ( t ) =
å g (t) e
k
k
k=1
which satisfies the equations
( u·· m ( t ) + g u· m ( t ) , ej ) +
+ æ Aum ( t ) + M æ A1 ¤ 2 u 2ö um ( t ) , A ejö + ( L um ( t ) - p ( t ) , e j ) = 0
è
ø
è
ø
for j = 1 , 2 , ¼ , m with t Î ( 0 , T ] and the initial conditions
( um ( 0 ) , e j ) = ( u 0 , ej ) , ( u· m ( 0 ) , e j ) = ( u1 , e j ) ,
j = 1, 2, ¼, m .
(3.8)
(3.9)
Simple calculations show that the problem of determining of approximate solutions
can be reduced to solving the following system of ordinary differential equations:
m
m
æ
ö
lj gj ( t ) 2÷ gk +
( L ej , ek ) gj = pk ( t ) ,
g··k + g g· k + l k2 gk + l k M ç
èj=1
ø
j=1
å
gk ( 0 ) = g0 k = ( u0 , ek ) ,
å
g· k ( 0 ) = g1 k = ( u 1 , ek ) ,
(3.10)
k = 1 , 2 , ¼ , m . (3.11)
The nonlinear terms of this system are continuously differentiable with respect to
gj . Therefore, it is solvable at least locally. The global solvability follows from the
a priori estimate of a solution as in the linear problem. Let us prove this estimate.
We consider an approximate solution u m ( t ) to problem (0.1) and (0.2) on the
solvability interval ( 0 , t ) . It satisfies equations (3.8) and (3.9) on the interval
( 0 , t ) . We multiply equation (3.8) by g· j ( t ) and summarize these equations with respect to j from 1 to m . Since
d
---- M æ A1 ¤ 2 u 2ö = 2 M æ A1 ¤ 2 u 2ö ( Au ( t ) , u· ( t ) ) ,
è
ø
dt
è
ø
we obtain
d
---E ( u m ( t ) , u· m ( t ) ) = -g u· m ( t ) 2 - ( L u m ( t ) - p ( t ) , u· m ( t ) )
dt
(3.12)
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as a result, where E ( u , u· ) is defined by (3.7). Equation (3.3) implies that
£ C æ Au m
è
( L u m , u· m ) £ C Au m u· m
2
+ u· m 2ö .
ø
Condition (3.2) gives us the estimate
1
--- æ Au 2 + u· m 2ö £ C1 + C 2 E ( u m , u· m )
m
2è
ø
4
(3.13)
with the constants independent of m . Therefore, due to Gronwall’s lemma equation
(3.12) implies that
C t
2
2
£ æ C0 + C1 E ( um ( 0 ) , u· m ( 0 ) )ö e 2 ,
u· m ( t ) + Aum ( t )
è
ø
(3.14)
with the constants C0 , C1 , and C 2 depending on the problem parameters only.
E x e r c i s e 3.2 Use equation (3.14) to prove the global solvability of Cauchy
problem (3.10) and (3.11).
It is evident that
u· m ( 0 ) £ u 1
1 - Au ( 0 )
£ --------m
l1
A1 ¤ 2 um ( 0 )
and
1- u
£ --------l1 0
1
.
Therefore,
--- æ u 1 2 + u 0 2 + C M ( u0 )ö ,
E ( um ( 0 ) , u· m ( 0 ) ) £ 1
2è
1
1 ø
where CM ( r ) = max {M ( z ) : 0 £ z £ r 2 ¤ l 1 } . Consequently, equation (3.14) gives
us that
sup
æ u· ( t )
m
t Î [0, T ] è
2
+ Aum ( t ) 2ö £ CT
ø
(3.15)
for any T > 0 , where C T does not depend on m . Thus, the set of approximate solutions { um ( t ) } is bounded in WT for any T > 0 . Hence, there exist an element
u ( t ) Î WT and a sequence { mk } such that umk ( t ) ® u ( t ) weakly in WT . Let us
show that the weak limit point u ( t ) possesses the property
u· ( t ) 2 + Au ( t ) 2 £ CT
(3.16)
for almost all t Î [ 0 , T ] . Indeed, the weak convergence of the sequence { u m } to
k
the function u in WT means that u· mk and Au m weakly (in L 2 ( 0 , T ; H ) ) conk
verge to u· and Au respectively. Consequently, this convergence will also take
place in L 2 ( a , b ; H ) for any a and b from the segment [ 0 , T ] . Therefore, by
virtue of the known property of the weak convergence we get
b
ò
a
b
æ u· ( t )
è
2
+ Au ( t ) 2ö dt £
ø
lim
k®¥
ò æè u·
a
mk ( t )
2
+ Au m ( t ) 2ö dt .
ø
k
Theorem on Existence and Uniqueness of Solutions
With the help of (3.15) we find that
b
ò æè u· (t)
2
+ Au ( t ) 2ö dt £ CT ( b - a ) .
ø
a
Therefore, due to the arbitrariness of a and b we obtain estimate (3.16).
Lemma 3.1
For any function v ( t ) Î L 2 ( 0 , T ; H )
lim JT ( um , v ) = JT ( u , v ) ,
k
k®¥
where
T
JT ( u , v ) =
òM æè A
1 ¤ 2 u ( t ) 2ö ( u ( t ) ,
ø
v ( t ) ) dt .
0
Proof.
Since
M æ A1 ¤ 2 u m 2ö - M æ A1 ¤ 2 u 2ö £
è
è
ø
k ø
1
£
ò M æèx A
˜
1¤ 2 u
mk
+ ( 1 - x ) A1 ¤ 2 u ö dx × A1 ¤ 2 ( um - u ) ,
ø
k
0
˜
where M ( z ) = 2 z M' ( z 2 ) Î C ( R+ ) , due to (3.15) and (3.16) we have
(1)
M æ A1 ¤ 2 u m 2ö - M æ A1 ¤ 2 u 2ö £ CT A1 ¤ 2 ( um ( t ) - u ( t ) ) ,
è
è
ø
k ø
k
˜
( 1)
where the constant CT is the maximum of the function M ( z ) on the sufficiently large segment [ 0 , aT ] , determined by the constant C T from inequalities
(3.15) and (3.16). Hence,
T
dk º
ò M æè A
1¤ 2 u
m(t)
2ö
- M æ A1 ¤ 2 u ( t ) 2ö × ( um ( t ) , v ( t ) ) dt £
è
ø
k
ø
0
£ CT v
L2 ( 0 , T ; H )
æ
ç
ç
è
T
òA
1¤ 2 ( u
0
mk ( t )
- u ( t ))
2
ö
dt÷
÷
ø
1¤ 2
.
The compactness of the embedding of WT into L 2 ( 0 , T ; F 1 ¤ 2 ) (see Exercise 1.13) implies that d k ® 0 as k ® ¥ . It is evident that
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T
JT ( um , v ) - JT ( u , v )
k
£ dk +
ò (u
mk ( t )
- u ( t ) , v ( t ) ) × M æ A1 ¤ 2 u ( t ) 2ö dt .
è
ø
0
Because of the weak convergence of um to u this gives us the assertion of the
k
lemma.
4
E x e r c i s e 3.3
Prove that the functional
T
L [ u] =
ò (L u (t) , v (t) ) dt
0
is continuous on WT for any v Î L 2 ( 0 , T ; H ) .
Let us prove that the limit function u ( t ) is a weak solution to problem (0.1) and
(0.2).
Let pl be the orthoprojector onto the span of elements ek , k = 1 , 2 , ¼ , l
in the space H . We also assume that
˜
WT = { v Î WT : v ( T ) = 0 }
and
˜
˜
˜
W Tl º p l WT = { p l v : v Î WT } .
˜
It is clear that an arbitrary element of the space W Tl has the form
l
vl ( x , t ) =
å h (t) e
k
k,
k=1
where h k ( t ) is an absolutely continuous real function on [ 0 , T ] such that
h (T) = 0 ,
h· ( t ) Î L 2 ( 0 , T ) .
k
k
If we multiply equation (3.8) by h j ( t ) , summarize the result with respect to j from
1 to l , and integrate it with respect to t from 0 to T , then it is easy to find that
T
ò
T
- ( u· m + g um , v· l ) dt +
0
ò
T
æ Au + M æ A1 ¤ 2 u 2ö u , Av ö dt +
m
m ø m
lø
è
è
0
ò (L u
m,
v l ) dt =
0
T
= ( u1 + g u0 , vl ( 0 ) ) +
ò (p, v ) dt
l
0
for m ³ l . The weak convergence of the sequence um to u in WT as well as Lemk
ma 3.1 and Exercise 3.3 enables us to pass to the limit in this equality and to show
Theorem on Existence and Uniqueness of Solutions
˜
that the function u ( t ) satisfies equation (3.4) for any function v Î WTl , where
l = 1 , 2 , ¼ . Further we use (cf. Exercise 2.1.11) the formula
T
lim
l®¥
ò æè p v· (t) - v· (t)
l
2
+ pl Av ( t ) - Av ( t ) 2ö dt = 0
ø
0
˜
˜
for any function v ( t ) Î WT in order to turn from the elements v of WTl to the func˜
tions v from the space WT .
E x e r c i s e 3.4
Prove that u ( t ) t = 0 = u 0 .
Thus, every weak limit point u ( t ) of the sequence of Galerkin approximations { um }
in the space WT is a weak solution to problem (0.1) and (0.2).
If we compare equations (3.4) and (2.3), then we find that every weak solution
u ( t ) is simultaneously a weak solution to problem (2.1) and (2.2) with b ( t ) = 0 and
h ( t ) = - M æ A1 ¤ 2 u ( t ) 2ö Au ( t ) - L u ( t ) + p ( t ) .
è
ø
(3.17)
It is evident that h ( t ) Î L ¥ ( 0 , T ; F0 ) . Therefore, due to Theorem 2.1 equations
(3.5) are valid for the function u ( t ) .
To prove energy equality (3.6) it is sufficient (due to (2.6)) to verify that for
h ( t ) of form (3.17) the equality
t
ò (h (t) , u· (t) ) dt =
0
t
=
ò
--- M æ A1 ¤ 2 u ( t ) 2ö
( - L u ( t ) + p ( t ) , u· ( t ) ) dt - 1
è
ø
2
0
t=t
(3.18)
t=0
holds. Here u ( t ) is a vector-function possessing property (3.5). We can do that by
first proving (3.18) for the function of the form pl u and then passing to the limit.
E x e r c i s e 3.5 Let u ( t ) be a weak solution to problem (0.1) and (0.2). Use
equation (3.6) to prove that
C t
u· ( t ) 2 + Au ( t ) 2 £ ( C0 + C1 E ( u 0 , u1 ) ) e 2 ,
t > 0,
(3.19)
where C0 , C 1 , C 2 > 0 are constants depending on the parameters
of problem (0.1) and (0.2).
Let us prove the uniqueness of a weak solution to problem (0.1) and (0.2). We assume that u1 ( t ) and u 2 ( t ) are weak solutions to problem (0.1) and (0.2) with the
initial conditions { u01 , u 11 } and { u 02 , u12 } , respectively. Then the function
u ( t ) = u1 ( t ) - u2 ( t )
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is a weak solution to problem (2.1) and (2.2) with the initial conditions u 0 = u01 - u02 , u1 = u11 - u12 , the function b ( t ) = M ( A1 ¤ 2 u1 ( t ) 2 ) , and the right-hand
side
4
h ( t ) = M æ A1 ¤ 2 u2 ( t ) 2ö - M æ A1 ¤ 2 u1 ( t ) 2ö Au2 ( t ) + L ( u 2 ( t ) - u 1 ( t ) ) .
è
ø
è
ø
We use equation (3.19) to verify that
h (t)
£ G T ( E ( u 01 , u11 ) + E ( u02 , u12 ) ) A ( u1 ( t ) - u2 ( t ) ) ,
t Î [ 0, T ] ,
where G T ( x ) is a positive monotonely increasing function of the parameter x .
Therefore, equation (2.20) implies that
u· 1 ( t ) - u· 2 ( t ) 2 + u 1 ( t ) - u2 ( t ) 12 £
æ
£ CT ç u 11 - u1 2
ç
è
ö
u 1 ( t ) - u 2 ( t ) 12 dt÷ ,
÷
ø
0
t
2
+
u01 - u02 12
+
ò
where CT > 0 depends on T and the problem parameters and is a function of the
variables E ( u0 j , u1 j ) , j = 1 , 2 . We can assume that C T is the same for all initial
data such that E ( u 0 j , u 1 j ) £ R , j = 1 , 2 . Using Gronwall’s lemma we obtain that
aC
u· 1 ( t ) - u· 2 ( t ) 2 + u1 ( t ) - u 2 ( t ) 12 £ C1 æ u 11 - u12 2 + u 01 - u02 12ö e 2 , (3.20)
è
ø
where t Î [ 0 , T ] and C > 0 is a constant depending only on T , the problem parameters and the value R > 0 such that u 0 j 12 + u1 j 2 £ R . In particular, this estimate implies the uniqueness of weak solutions to problem (0.1) and (0.2). The proof
of Theorem 3.1 is complete.
E x e r c i s e 3.6 Show that a weak solution u ( t ) satisfies equation (0.1) if we
consider this equation as an equality of elements in F-1 for almost all t .
Moreover, u·· ( t ) Î C ( 0 , T ; F-1) (Hint: see Exercise 2.5).
E x e r c i s e 3.7 Assume that the hypotheses of Theorem 3.1 hold. Let u ( t ) be
a weak solution to problem (0.1) and (0.2) on the segment [ 0 , T ]
and let um ( t ) be the corresponding Galerkin approximation of the
order m . Show that
um ( t ) ® u ( t ) weakly in L 2 ( 0 , T ; F1 ) ,
u· m ( t ) ® u· ( t ) weakly in L 2 ( 0 , T ; F0 ) ,
um ( t ) ® u ( t ) strongly in L 2 ( 0 , T ; Fs ) ,
as m ® ¥ .
s < 1,
Theorem on Existence and Uniqueness of Solutions
In conclusion of the section we note that in case of stationary load p ( t ) º p Î H we
can construct an evolutionary operator St of problem (0.1) and (0.2) in the space
H º H0 = F1 ´ F0 supposing that
S y = ( u ( t ) ; u· ( t ) )
t
for y = ( u0 , u1 ) , where u ( t ) is a weak solution to problem (0.1) and (0.2) with the
initial conditions y = ( u0 ; u1 ) . Due to the uniqueness of weak solutions we have
St ° St = St + t ,
S0 = I ,
t, t ³ 0 .
By virtue of (3.20) the nonlinear mapping St is a continuous mapping of H . Equation (3.5) implies that the vector-function St y is strongly continuous with respect
to t for any y Î H . Moreover, for any R > 0 and T > 0 there exists a constant
C ( R , T ) > 0 such that
St y1 - St y2
H
£ C ( R , T ) × y1 - y2
for all t Î [ 0 , T ] and for all yj Î { y Î H :
y
H
H
(3.21)
£ R} .
E x e r c i s e 3.8 Use equation (3.21) to show that ( t , y ) ® S t y is a continuous mapping from R+ ´ H into H .
E x e r c i s e 3.9 Prove the theorem on the existence and uniqueness of solutions to problem (0.1) and (0.2) for g £ 0 . Use this fact to show that
the collection of operators { St } is defined for negative t and forms
a group (Hint: cf. Exercises 2.9 and 2.10).
E x e r c i s e 3.10 Prove that the mapping S t is a homeomorphism in H for every t > 0 .
E x e r c i s e 3.11 Let p ( t ) Î L ¥ ( R + , H ) be a periodic function: p ( t ) =
= p ( t + t0 ) , t0 > 0 . Define the family of operators Sm by the formula
Sm y = ( u ( m t0 ) ; u· ( m t0 ) ) ,
m = 0, 1, 2, ¼ ,
in the space H = F1 ´ F 0 . Here u ( t ) is a solution to problem (0.1)
and (0.2) with the initial conditions y = ( u0 , u 1 ) . Prove that the
pair ( H , Sm ) is a discrete dynamical system. Moreover, Sm = S1m
and S1 is a homeomorphism in H .
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§ 4
Smoothness of Solutions
In the study of smoothness properties of solutions constructed in Section 3 we use
some ideas presented in paper [9]. The main result of this section is the following assertion.
Theorem 4.1
Let the hypotheses of Theorem 3.1 hold. We assume that M ( z ) Î
l
+ ) and the load p ( t ) lies in C ( 0 , T ; F 0 ) for some l ³ 1 . Then
for a weak solution u ( t ) to problem (0.1) and (0.2) to possess the properties
Î C l + 1 (R
u ( k ) ( t ) Î C ( 0 , T ; F2 ) ,
k = 0, 1, 2, ¼, l - 1 ,
u ( l ) ( t ) Î C ( 0 , T ; F1 ) ,
u( l + 1 ) ( t ) Î C ( 0 , T ; F0 ) ,
(4.1)
it is necessary and sufficient that the following compatibility conditions
are fulfilled:
u ( k ) ( 0 ) Î F2 ,
k = 0, 1, 2, ¼, l - 1 ;
u( l) ( 0 ) Î F1 .
(4.2)
u(j) (t )
Here
is a strong derivative of the function u ( t ) with respect to t
of the order j and the values u ( k ) ( 0 ) are recurrently defined by the initial
conditions u0 and u1 with the help of equation (0.1)::
u(0) ( 0 ) = u0 ,
u( 1 ) ( 0 ) = u1 ,
ì
u ( k ) ( 0 ) = - í g u ( k - 1 ) ( 0 ) + A2 u ( k - 2 ) ( 0 ) + Lu ( k - 2 ) ( 0 ) +
î
k-2
d
- æ M æ A1 ¤ 2 u ( t ) 2ö Au ( t ) - p ( t )ö
+ -------------ø
ø
dt k - 2 è è
ü
ý,
t = 0þ
(4.3)
where k = 2 , 3 , ¼ .
Proof.
It is evident that if a solution u ( t ) possesses properties (4.1) then compatibility
conditions (4.2) are fulfilled. Let us prove that conditions (4.2) are sufficient for
equations (4.1) to be satisfied. We start with the case l = 1 . The compatibility conditions have the form: u0 Î F2 , u 1 Î F 1 . As in the proof of Theorem 3.1 we consider
the Galerkin approximation
m
um ( t ) =
å g (t) e
k
k
k=1
of the order m for a solution to problem (0.1) and (0.2). It satisfies the equations
Smoothness of Solutions
u·· m ( t ) + g u· m ( t ) + A 2 um ( t ) +
+ M æ A 1 ¤ 2 um ( t ) 2ö A um ( t ) + pm L u m ( t ) = pm p ( t ) ,
è
ø
um ( 0 ) = pm u 0 ,
(4.4)
u· m ( 0 ) = pm u1 ,
where pm is the orthoprojector onto the span of elements e1 , ¼ em . The structure
of equation (3.10) implies that um ( t ) Î C 2 ( 0 , T ; F2 ) . We differentiate equation
(4.4) with respect to t to obtain that vm ( t ) = u· m ( t ) satisfies the equation
v··m + g v· m + A2 vm + M æ A1 ¤ 2 um 2ö Avm + pm L vm =
è
ø
= -2 M ¢ æ A1 ¤ 2 um 2ö ( Au m , u· m ) Aum + pm p· ( t )
è
ø
(4.5)
and the initial conditions
vm ( 0 ) = pm u 1 ,
ì
ü
v· m ( 0 ) = -pm í g u 1 + A2 u 0 + M æ A1 ¤ 2 pm u 0 2ö Au0 + L pm u0 - p ( 0 ) ý . (4.6)
è
ø
î
þ
It is clear that
v· m ( 0 ) = pm u ( 2 ) ( 0 ) + M æè A1 ¤ 2 u0 2öø - M æè A1 ¤ 2 pm u0 2öø Apm u0 + L ( u 0 - pm u 0 ) ,
where u ( 2 ) ( 0 ) is defined by (4.3). Therefore,
vm ( 0 ) - pm u ( 2 ) ( 0 )
£ C ( u0 1 ) u0 - pm u 0 .
1
The compatibility conditions give us that u0 Î F 2 and hence u ( 2 ) ( 0 ) Î F 0 . Thus,
the initial condition v· m ( 0 ) possesses the property
v· m ( 0 ) - u ( 2 ) ( 0 ) ® 0 ,
m ® ¥.
·
We multiply (4.5) by vm ( t ) scalarwise in H to find that
d æ ·
1
--- ---v ( t ) 2 + Avm ( t ) 2ö + g v· m ( t ) 2 = ( Fm ( t ) , v· m ( t ) ) ,
ø
2 dt è m
(4.7)
(4.8)
where
Fm ( t ) = - M æè A1 ¤ 2 um 2öø Avm - pm L vm -
-2 M ¢ æ A1 ¤ 2 u m 2ö ( Aum , u· m ) Aum + pm p· ( t ) .
è
ø
Using a priori estimates (3.14) for u m ( t ) we obtain
Fm ( t )
£ CT æ 1 + Avm ( t ) ö ,
è
ø
t Î [ 0, T ] .
(4.9)
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Thus, equation (4.8) implies that
4
d
---- æ v· m ( t ) 2 + Avm ( t ) 2ö £ CT æ 1 + v· m ( t ) 2 + Avm ( t ) 2ö ,
dt è
ø
è
ø
t Î [ 0, T] .
Equation (4.7) and the property u 1 Î F 1 give us that the estimate
v· ( 0 ) 2 + Av ( 0 ) 2 < C
m
m
holds uniformly with respect to m . Therefore, we reason as in Section 3 and use
Gronwall’s lemma to find that
Avm ( t ) 2 + v· m ( t ) 2 £ C T ,
t Î [0, T ] .
(4.10)
Au· m ( t ) 2 + u·· m ( t ) 2 £ C T ,
t Î [0, T ] .
(4.11)
Consequently,
Equation (4.4) gives us that
A2 u m ( t ) £ u·· m( t ) + g u· m ( t ) + M ( A1 ¤ 2 um( t ) 2 ) Au m ( t ) + L um( t ) + p ( t ) .
Therefore, (3.14) and (4.11) imply that
A2 u m ( t )
£ CT ,
t Î [0, T ] .
(4.12)
Thus, the sequence { u m ( t ) } of approximate solutions to problem (0.1) and (0.2)
possesses the properties (cf. Exercise 3.7):
ü
um ( t ) ® u ( t ) weakly in L 2 ( 0 , T ; D ( A2 ) ) ; ï
ï
u· m ( t ) ® u· ( t ) weakly in L 2 ( 0 , T ; D ( A) ) ; ý
ï
u·· m ( t ) ® u·· ( t ) weakly in L 2 ( 0 , T ; H ) ; ïþ
(4.13)
where u ( t ) is a weak solution to problem (0.1) and (0.2). Moreover (see Exercise 1.13),
T
lim
m®¥
ò u·
m(t)
- u· ( t )
2
s
+ um ( t ) - u ( t )
2
1+ s
dt = 0
(4.14)
0
for every s < 1 . If we use these equations and arguments similar to the ones given in
Section 3, then it is easy to pass to the limit and to prove that the function w ( t ) = u· ( t )
is a weak solution to the problem
·· + g w· + A2 w + M æ A1 ¤ 2 u ( t ) 2ö Aw + L w =
ìw
è
ø
ï
ï
ï
= - 2 M ¢ æ A1 ¤ 2 u ( t ) 2ö ( Au , u· ) Au + p· ( t ) ,
í
è
ø
ï
ï
ï w ( 0 ) = u , w· ( 0 ) = u ( 2 ) ( 0 ) ,
1
î
(4.15)
Smoothness of Solutions
where u ( 2 ) ( 0 ) is defined by (4.3). Therefore, Theorem 2.1 gives us that
u· ( t ) = w ( t ) Î C ( 0 , T ; D ( A ) ) Ç C 1 ( 0 , T ; H ) .
This implies equation (4.1) for l = 1 .
Further arguments are based on the following assertion.
Lemma 4.1
Let u ( t ) be a weak solution to the linear problem
ì ··
·
2
ï u ( t ) + g u ( t ) + A u ( t ) + b ( t ) Au = h ( t ) ,
(4.16)
í
ï u ( 0 ) = u0 ,
î
where b ( t ) is a scalar continuously differentiable function, h ( t ) Î
Î C 1 ( 0 , T ; F0 ) and u 0 Î F2 , u1 Î F1 . Then
(4.17)
u ( t ) Î C ( 0 , T ; F2 ) Ç C1 ( 0 , T ; F1 ) Ç C2 ( 0 , T ; F0 )
and the function v ( t ) = u· ( t ) is a weak solution to the problem obtained
by the formal differentiation of (4.16) with respect to t and equiped
with the initial conditions v ( 0 ) = u1 and v· ( 0 ) = u·· ( 0 ) º - ( g u1 + A2 u0 +
+ b ( 0 ) Au0 - h ( 0 ) ) .
Proof.
Let u m ( t ) be the Galerkin approximation of the order m of a solution to
problem (4.16) (see (2.7)). It is clear that um ( t ) is thrice differentiable with respect to t and vm ( t ) = u· m ( t ) satisfies the equation
·
v··m + g v· m + A2 vm + b ( t ) Avm = - b ( t ) Au m + pm h ( t ) ,
t > 0,
and the initial conditions
vm ( 0 ) = pm u1 ,
v· m ( 0 ) = -pm ( g u 1 + A2 u0 + b ( 0 ) Au 0 - h ( 0 ) ) .
Therefore, as above, it is easy to prove the validity of equations (4.10)–(4.14)
for the case under consideration and complete the proof of Lemma 4.1.
Assume that the hypotheses of Lemma 4.1 hold with b ( t ) Î
Î C l ( 0 , T ) and h ( t ) Î C l ( 0 , T ; F0 ) for some l ³ 1 . Let the compatibility conditions (4.2) be fulfilled with u ( 0 ) ( 0 ) = u0 , u ( 1 ) ( 0 ) =
= u1 , and
E x e r c i s e 4.1
ì
u ( k ) ( 0 ) = -í g u ( k - 1 ) ( 0 ) + A2 u ( k - 2 ) ( 0 ) +
î
k -2
+
åC
j=0
j
( j)
(k - 2 - j) (0 )
k - 2 b (0) A u
ü
+ L u(k - 2) ( 0 ) - h( k - 2) ( 0 ) ý
þ
+
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for k = 2 , 3 , ¼ , l . Show that the weak solution u ( t ) to problem
(4.16) possesses properties (4.1) and the function vk ( t ) = u ( k ) ( t ) is
a weak solution to the equation obtained by the formal differentiation of (4.16) k times with respect to t . Here k = 0 , 1 , ¼ , l .
In order to complete the proof of Theorem 4.1 we use induction with respect to
l . Assume that the hypotheses of the theorem as well as equations (4.2) for
l = n + 1 hold. Assume that the assertion of the theorem is valid for l = n ³ 1 .
Since equations (4.1) hold for the solution u ( t ) with l = n , we have
d k- ì M æ A1 ¤ 2 u ( t ) 2ö Au ( t ) ü = M æ A1 ¤ 2 u ( t ) 2ö Au( k ) ( t ) + G ( t ) ,
------í
ý
k
ø
è
ø
dt k î è
þ
where Gk ( t ) Î C 1 ( 0 , T ; F 0 ) , k = 1 , ¼ , n . Therefore, we differentiate equation
(0.1) n - 1 times with respect to t to obtain that v ( t ) = u ( n - 1 ) ( t ) is a weak solution
to problem (4.16) with
b ( t ) = M æ A1 ¤ 2 u ( t ) 2ö
è
ø
h ( t ) = - Gn - 1 ( t ) + p ( n - 1 ) ( t ) .
Consequently, Lemma 4.1 gives us that w ( t ) = v· ( t ) is a weak solution to the problem
which is obtained by the formal differentiation of equation (0.1) n times with respect to t :
and
ì ··
2ö
·
æ
ï w + g w + A2 w + M è A1 ¤ 2 u ( t ) ø Aw = p ( n ) ( t ) - Gn ( t ) ,
í
ï w ( 0 ) = u ( n ) ( 0 ) , w· ( 0 ) = u ( n + 1 ) ( 0 ) .
î
However, the hypotheses of Lemma 4.1 hold for this problem. Therefore (see (4.17)),
u ( n ) ( t ) = w ( t ) Î C ( 0 , T ; F2 ) Ç C 1 ( 0 , T ; F1 ) Ç C 2 ( 0 , T ; F0 ) ,
i.e. equations (4.1) hold for l = n + 1 . Theorem 4.1 is proved.
E x e r c i s e 4.2 Show that if the hypotheses of Theorem 4.1 hold, then the
function v ( t ) = u ( k ) ( t ) is a weak solution to the problem which is
obtained by the formal differentiation of equation (0.1) k times with
respect to t , k = 1 , 2 , ¼ , l .
E x e r c i s e 4.3 Assume that the hypotheses of Theorem 4.1 hold and L º 0
in equation (0.1). Show that if the conditions
p ( t) Î Ck ( [ 0 , T ] ; F l - k ) ,
k = 0, 1, ¼, l ,
(4.18)
are fulfilled, then a solution u ( t ) to problem (0.1) and (0.2) possesses the properties
u ( t) Î Ck ( [ 0 , T ] ; F l + 1 - k ) ,
k = 0, 1, ¼, l + 1 .
Smoothness of Solutions
E x e r c i s e 4.4 Assume that the hypotheses of Theorem 4.1 hold. We define
the sets
ì
ü
Vk = í ( u 0 , u1 ) Î H : equation (4.2) holds with l = k ý
î
þ
(4.19)
in the space H = F1 ´ F0 . Prove that
V1 = F2 ´ F1 and V1 É V2 É ¼ É Vl .
E x e r c i s e 4.5
Show that every set Vk given by equality (4.19) is invariant:
( u ; u ) Î V Þ ( u ( t ) ; u· ( t ) ) Î V ,
k = 1, ¼, l .
0
1
k
k
Here u ( t ) is a weak solution to problem (0.1) and (0.2).
E x e r c i s e 4.6 Assume that L º 0 in equation (0.1) and the load p ( t ) possesses property (4.18). Show that for k = 1 , 2 , ¼ , l the set Vk of
form (4.19) contains the subspace Fk + 1 ´ Fk .
E x e r c i s e 4.7 Assume that the hypotheses of Theorem 3.1 hold and the operator L (in equation (0.1)) possesses the property
L u s £ C u 1 + s for some 0 < s < 1 .
Let u 0 Î F1 + s and let u 1 Î Fs . Show that the estimate
u· m ( t ) s2 + um ( t ) 12 + s £ CT ,
t Î [0, T ] ,
(4.20)
(4.21)
is valid for the approximate Galerkin solution u m ( t ) to problem
(0.1) and (0.2). Here the constant C T does not depend on m
(Hint: multiply equation (3.8) by l2j s g· j ( t ) and summarize the result with respect to j ; then use relation (3.14) to estimate the nonlinear term).
E x e r c i s e 4.8 Show that if the hypotheses of Exercise 4.7 hold, then problem (0.1) and (0.2) possesses a weak solution u ( t ) such that
u ( t ) Î C ( 0 , T ; F1 + s ) Ç C1 ( 0 , T ; Fs ) Ç C 2 ( 0 , T ; F- 1 + s ) ,
where s Î ( 0 , 1 ) is the number from Exercise 4.7.
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§ 5
Dissipativity and Asymptotic
Compactness
In this section we prove the dissipativity and asymptotic compactness of the dynamical system ( H , St ) generated by weak solutions to problem (0.1) and (0.2) for
g > 0 in the case of a stationary load p ( t ) º p Î H = F0 . The phase space is H =
= F1 ´ F0 . The evolutionary operator is defined by the formula
S y = ( u ( t ) , u· ( t ) ) ,
(5.1)
t
where u ( t ) is a weak solution to problem (0.1) and (0.2) with the initial condition
y = ( u0 ; u1 ) .
Theorem 5.1
Assume that in addition to (3.2) the following conditions are fulfilled:
fulfilled:
a) there exist numbers aj > 0 such that
z
ò
z M ( z ) - a 1 M ( x ) dx ³ a 2 z 1 + a - a 3 ,
z ³ 0
(5.2)
0
with a constant a > 0 ;
b) there exist 0 £ q < 1 and C > 0 such that
Lu £ C Aq u ,
u Î D ( Aq ) .
(5.3)
Then the dynamical system ( H , St ) generated by problem (0.1) and (0.2)
for g > 0 and for p ( t ) º p Î H is dissipative.
dissipative.
To prove the theorem it is sufficient to verify (see Theorem 1.4.1 and Exercise 1.4.1)
that there exists a functional V ( y ) on H which is bounded on the bounded sets of
the space H , differentiable along the trajectories of system (0.1) and (0.2), and
such that
2
- D1 ,
V (y) ³ a y H
(5.4)
d- (V ( S )) + b (
---V St y ) £ D 2 ,
ty
dt
(5.5)
where a , b > 0 and D 1 , D 2 ³ 0 are constants. To construct the functional V ( y )
we use the method which is widely-applied for finite-dimensional systems (we used
it in the proof of estimate (2.13)).
Let
V(y) = E (y) + n F(y) ,
where y = ( u0 ; u 1 ) Î H . Here E ( y ) = E ( u 0 ; u 1 ) is the energy of system (0.1)
and (0.2) defined by the formula (3.7),
Dissipativity and Asymptotic Compactness
g
F ( y ) = ( u0 ; u1 ) + --- u0
2
2
,
and the parameter n > 0 will be chosen below. It is evident that
1- u
- ----2g 1
2
1- u
£ F ( y ) £ ----2g 1
2
+ g u0
2
.
For 0 < n < g this implies estimate (5.4) and the inequality
V ( y ) £ b1 E ( y ) + b2
(5.6)
with the constants b1 , b2 > 0 independent of n . This inequality guarantees the
boundedness of V ( y ) on the bounded sets of the space H .
Energy equality (3.6) implies that the function E ( y ( t ) ) , where y ( t ) = St y , is
continuously differentiable and
d
----- E ( y ( t ) ) = - g u· ( t ) 2 + ( -L u ( t ) + p , u· ( t ) ) .
dt
Therefore, due to (5.3) we have that
g
d---E ( y ( t ) ) £ - --- u· 2 + C1 Aq u 2 + C 2 .
2
dt
We use interpolation inequalities (see Exercises 2.1.12 and 2.1.13) to obtain that
Aq u 2 £ d Au 2 + C d A1 ¤ 2 u 2 ,
d > 0.
Thus, the estimate
g
d---E ( y ( t ) ) £ - --- u· 2 + --e- Au 2 + Ce A1 ¤ 2 u 2 + C2
2
2
dt
(5.7)
holds for any e > 0 .
Lemma 5.1
Let u ( t ) be a weak solution to problem (0.1) and (0.2) and let y ( t ) =
= ( u ( t ) , u· ( t ) ) . Then the function F ( y ( t ) ) is continuously differentiable and
d---F ( y ( t ) ) = u· ( t ) 2 + ( u·· ( t ) + g u· ( t ) , u ( t ) ) .
dt
(5.8)
We note that since u·· ( t ) Î C ( R + , F-1 ) (see Exercise 3.6), equation (5.8) is correctly defined.
Proof.
It is sufficient to verify that
t
( u ( t ) , u· ( t ) ) = ( u 0 , u1 ) +
ò íî(u·· (t) , u (t) ) + u· (t)
0
ì
2ü
ý dt .
þ
(5.9)
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Let pl be the orthoprojector onto the span of elements { e k , k = 1 , 2 , ¼ , l } in
F0 . Then it is evident that the vector-function ul ( t ) = pl u ( t ) is twice continuously differentiable with respect to t . Therefore,
t
( ul ( t ) , u· l ( t ) ) = ( ul ( 0 ) , u· l ( 0 ) ) +
ì
ò íî(u·· (t) , u (t) ) + u· (t)
l
l
l
0
2ü
ý dt .
þ
The properties of the projector p l (see Exercise 2.1.11) enable us to pass to
the limit l ® ¥ and to obtain (5.9). Lemma 5.1 is proved.
Since u ( t ) is a solution to equation (0.1), relation (5.8) implies that
ì
ü
2
d
----- F ( y ( t ) ) = u· 2 - í Au 2 + M æ A1 ¤ 2 u ö × A1 ¤ 2 u 2 + ( Lu - p , u ) ý .
è
ø
dt
î
þ
Therefore, equation (5.2) and the evident inequality
--- Au
( Lu - p , u ) £ 1
2
2
+ C1 u
2
+ p
2
give us that
d
----- F ( y ( t ) ) £
dt
--- Au 2 - a1M æ A1 ¤ 2 u 2ö - a2 A1 ¤ 2 u 2 + 2 a + C1 u 2 + C 2 .
u· 2 - 1
è
ø
2
Hence, (5.6) and (5.7) enable us to obtain the estimate
d--- ( g - d b1 - 2 n ) u· 2 ---V (y (t) ) + d V (y (t) ) £ -1
2
dt
db
--- ( n - d b1 - e ) Au 2 - æ n a1 - --------1-ö M æ A1 ¤ 2 u 2ö + R ( u ; n , e , d ) ,
-1
è
ø
è
2
2 ø
where d > 0 and
R ( u ; n , e , d ) = - n a2 A1 ¤ 2 u 2 + 2 a + Ce A1 ¤ 2 u 2 + n C1 u 2 + Cd .
Therefore, for any 0 < n < g ¤ 2 estimate (5.5) holds, provided d and e are chosen
appropriately. Thus, Theorem 5.1 is proved.
E x e r c i s e 5.1 Prove that if the hypotheses of Theorem 5.1 hold, then the assertion on the dissipativity of solutions to problem (0.1) and (0.2)
remains true in the case of a nonstationary load p ( t ) Î L ¥ ( R+ , H ) .
Theorem 5.2
Let the hypotheses of Theorem 5.1 hold and assume that for some s > 0
(5.10)
p Î Fs ,
L D ( A) Ì D ( As ) ,
As L u £ C Au .
Then there exists a positively invariant bounded set Ks in the space Hs =
= F1 + s ´ Fs which is closed in H and such that
Dissipativity and Asymptotic Compactness
g
- --- ( t - t B )
ì
ü
sup í dist H ( St y , Ks ) : y Î B ý £ C e 4
î
þ
for any bounded set B in the space H , t > tB .
(5.11)
Due to the compactness of the embedding of Hs in H for s > 0 , this theorem and
Lemma 1.4.1 imply that the dynamical system ( H , St ) is asymptotically compact.
Proof.
Since the system ( H , St ) is dissipative, there exists R > 0 such that for all
y Î B and t ³ t0 = t0 ( B )
u· ( t ) 2 + Au ( t ) 2 £ R 2 ,
(5.12)
where u ( t ) is a weak solution to problem (0.1) and (0.2) with the initial conditions
y = ( u0 ; u1 ) Î B . We consider u ( t ) as a solution to linear problem (2.1) and (2.2)
with b ( t ) = M ( A1 ¤ 2 u ( t ) 2 ) and h ( t ) = - L u ( t ) + p . It is easy to verify that b ( t ) is a
continuously differentiable function and
·
b ( t ) + b ( t ) £ CR ,
t ³ t0 .
Moreover, equation (2.27) implies that
St y = U ( t , t0 ) y ( t 0 ) + G ( t , t0 ; y ) ,
(5.13)
where
t
ò
G ( t , t0 ; y ) = - U ( t , t ) ( 0 ; L u ( t ) - p ) dt .
t0
Here U ( t , t ) is the evolutionary operator of the homogeneous problem (2.1) and
(2.2) with h ( t ) = 0 and b ( t ) = M ( A1 ¤ 2 u ( t ) 2 ) . By virtue of Theorem 2.2 there
exists N0 ³ 0 such that
ì g
ü
í - --4- ( t - t ) ý ,
î
þ
where N ³ N0 , t ³ t ³ t0 , and PN is the orthoprojector H onto
( 1 - PN ) U ( t , t ) h
Hs
£
3 h
H s exp
(5.14)
LN = Lin { ( e k ; 0 ) , ( 0 ; ek ) , k = 1 , 2 , ¼ , N } .
This implies that
t
( 1 - PN ) G ( t , t0 ; y )
0
£
Hs
3
ì g
ò exp íî---4- (t - t) ýþ L u (t) - p
ü
s
dt .
t0
Therefore, we use (5.10) to obtain that
( 1 - PN ) G ( t , t0 ; y )
0
Hs
£ C (R) .
(5.15)
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It is also easy to find that
4
PN St y
0
£ N 0s St y
Hs
H
£ R N0s ,
t ³ t0 .
Consequently, there exists a number Rs depending on the radius of dissipativity R
and the parameters of the problem such that the value
St y - ( 1 - PN ) U ( t , t0 ) St y0 = PN St y + ( 1 - PN ) G ( t , t0 ; y )
0
0
0
0
(5.16)
lies in the ball
Bs = { y : y
Hs
£ Rs }
for t ³ t0 . Therefore, with the help of (5.12) and (2.23) we have that
distH ( St y , Bs ) £
( 1 - PN ) U ( t , t 0 ) St y
0
0
£ R 3e
g
- --- ( t - t 0 )
4
.
(5.17)
Let K s = g + ( Bs ) º Èt ³ 0 St ( Bs ) . Evidently equation (5.11) is valid. Moreover, K s
is positively invariant. The continuity of St y with recpect to the both variables
( t ; y ) in the space H (see Exercise 3.8) and attraction property (5.17) imply that
K s is a closed set in H . Let us prove that Ks is bounded in Hs . First we note that
K s is bounded in H . Indeed, by virtue of the dissipativity we have that St y £ R
for all y Î Bs and t ³ ts º t ( Bs ) . Since S t y is continuous with respect to the variables ( t ; y ) , its maximum is attained on the compact [ 0 , ts ] ´ Bs . Thus, there exists
R > 0 such that y £ R for all y Î Ks . Let us return to equality (5.16) for t0 = 0
and y0 Î Bs . It is evident that the norm of the right-hand side in the space Hs
is bounded by the constant C = C ( s , R ) . However, equation (5.14) implies that
( 1 - PN ) U ( t , 0 ) y0
0
Hs
£
3 Rs ,
t ³ 0,
y0 Î Bs .
Therefore, equation (5.16) leads to the uniform estimate
St y
Hs
£ Rs ,
t ³ 0,
y Î Bs .
Thus, the set K s is bounded in Hs . Theorem 5.2 is proved.
E x e r c i s e 5.2 Show that for any 0 £ s £ s a bounded set of Hs is attracted
by K s at an exponential rate with respect to the metric of the space
Hs . Thus, we can replace dist H by dist H in (5.11).
s
E x e r c i s e 5.3 Prove that if the hypotheses of Theorem 5.2 hold, then the assertion on the asymptotic compactness of solutions to problem (0.1)
and (0.2) remains true in the case of nonstationary load p ( t ) Î
Î L ¥ ( R+ , Fs ) (see also Exercise 5.1).
E x e r c i s e 5.4 Prove that the hypotheses of Theorem 5.1 and 5.2 hold for
problem (0.3) and (0.4) for any 0 < s < 1 ¤ 4 , provided that g > 0
and p ( x , t ) º p ( x ) lies in the Sobolev space H 01 ( W ) .
Dissipativity and Asymptotic Compactness
Let us consider the dissipativity properties of smooth solutions (see Section 4) to
problem (0.1) and (0.2).
Theorem 5.3
Let the hypotheses of Theorem 5.1 hold. Assume that M ( z ) Î C l + 1 ( R+ )
and the initial conditions y0 = ( u0 ; u1 ) are such that equations (4.1) (and
hence (4.2))) are valid for the solution ( u ( t ) ; u· ( t ) ) = St y0 . Then there exists
Rl > 0 such that for any initial data y0 = ( u0 ; u1 ) possessing the property
u ( k + 1 ) ( 0 ) 2 + Au ( k ) ( 0 ) 2 + A2 u ( k - 1 ) ( 0 ) 2 < r 2 ,
k = 1 , 2 , ¼ , l , (5.18)
the solution ( u ( t ) ; u· ( t ) ) = St y0 admits the estimate
u ( k + 1 ) ( t ) 2 + Au( k ) ( t ) 2 + A2 u ( k - 1 ) ( t ) 2 < Rl2
(5.19)
for all k = 1 , 2 , ¼ , l as soon as t ³ t0 ( r ) .
We use induction to prove the theorem. The proof is based on the following assertion.
Lemma 5.2
Assume that the hypotheses of Theorem 5.3 hold for l = 1 . Then the dynamical system ( H1 , St ) generated by problem (0.1) and (0.2) in the
space H1 = F 2 ´ F1 is dissipative.
Proof.
Let ( u ( t ) ; u· ( t ) ) = St y 0 be a semitrajectory of the dynamical system
( H1 , St ) and let y0 = ( u0 ; u1 ) Î H1 = F2 ´ F1 . If the hypotheses of the lemma hold, then the function w ( t ) = u· ( t ) is a weak solution to problem (4.15) obtained by formal differentiation of (0.1) with respect to t (as we have shown in
Section 4). By virtue of Theorem 2.1 the energy equality of the form (2.6) holds
for the function w ( t ) . We rewrite it in the differential form:
d
----- F ( t ; w ( t ) , w· ( t ) ) + g w· ( t ) 2 = Y ( u ( t ) , w ( t ) ) ,
dt
(5.20)
--- æ w· 2 + Aw 2 + M æ A1 ¤ 2 u ( t ) 2ö × A1 ¤ 2 w 2ö
F ( t ; w , w· ) = 1
è
ø
ø
2è
(5.21)
where
and
Y ( u ( t ) , w ( t ) ) = -( L w ( t ) , w· ( t ) ) +
+ M ¢ æ A1 ¤ 2 u ( t ) 2ö ( Au ( t ) , u· ( t ) )
è
ø
A1 ¤ 2 w ( t ) 2 - 2 ( Au ( t ) , w· ( t ) ) .
The dissipativity property of ( H , St ) given by Theorem 5.1 leads to the estimates
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M æè A1 ¤ 2 u ( t ) 2öø A1 ¤ 2 w 2 £ C R Aw ( t ) ,
Y(u (t) , w (t) ) £
L w w· + CR ( Aw ( t ) + w· ( t ) )
for all ( u0 ; u 1 ) with the property
Au 0 2 + u1 2 £ r 2
and for all t ³ t0 ( r ) large enough. Hereinafter R is the radius of dissipativity
of the system ( H , St ) . These estimates imply that for t ³ t0 ( r ) we have
1
--- æ w· 2 + Aw 2ö - a 1 £ F ( t ; w , w· ) £ a2 æ w ( t ) 2 + w· ( t ) 2ö + a3 (5.22)
è
ø
4è
ø
and
g
d
----- F ( t ; w , w· ) + --- w· 2 £ b Aq w 2 + a4 Aw + a 5 ,
2
dt
(5.23)
where the constants a j > 0 depend on R . Here q < 1 . Similarly, we use Lemma 5.1 to find that
g
d ì
----- í ( w ( t ) , w· ( t ) ) + --- w ( t )
2
dt
î
2ü
ý £
þ
--- Aw ( t ) 2 + C R
w· ( t ) 2 - 1
2
for t ³ t0 ( r ) . Consequently, the function
ì
ü
g
V ( t ) = F ( t ; w , w· ) + n í ( w , w· ) + --- w 2 ý
2
î
þ
possesses the properties
d-----V+ w V £ CR ,
dt
w > 0,
and
1
--- æ w· 2 + Aw 2ö - a1 £ V £ a2 æ w 2 + w· 2ö + a3
ø
è
ø
4è
for t ³ t0 ( r ) and for n > 0 small enough. This implies that
w· ( t ) 2 + Aw ( t ) 2 £ C1 æ w· ( t0 ) 2 + Aw ( t0 ) 2ö e -w t + C2
è
ø
(5.24)
for t ³ t0 = t0 ( r ) , provided that
Au0 2 + u1 2 £ r 2 .
(5.25)
If we use (5.4)–(5.6), then it is easy to find that
Au ( t ) 2 + u· ( t ) 2 £ Cr , R ,
t > 0,
under condition (5.25). Using the energy equality for the weak solutions to
problem (4.15) we conclude that
Dissipativity and Asymptotic Compactness
d
----- æ Aw ( t ) 2 + w· ( t ) 2ö £ C 1 ( r , R ) × Aw × w· + C 2 ( r , R ) .
ø
dt è
Therefore, standard reasoning in which Gronwall’s lemma is used leads to
Aw ( t ) 2 + w· ( t ) 2 £ æ Aw ( 0 ) 2 + w· ( 0 ) 2 + aö e b t ,
è
ø
provided that equation (5.25) is valid. Here a and b are some positive constants depending on r and R . This and equation (5.24) imply that
w· ( t ) 2 + Aw ( t ) 2 £ C1 ( r , R ) æ( 1 + w· ( 0 ) 2 ) + Aw ( 0 ) 2ö e -w t + C 2 , (5.26)
è
ø
where C2 depends on R only. Since
w· ( 0 ) 2 + Aw ( 0 ) 2 £ Cr ,
r 2
u1 2 + Au0 2 £ æ ------ö
è l 1ø
provided that
Au 1 2 + A2 u 0 2 £ r 2 ,
equation (5.26) gives us the estimate
u·· ( t ) 2 + Au· ( t ) 2 £ bR2 ,
t ³ t 0 (r) .
This easily implies the dissipativity of the dynamical system ( H 1 , St ) . Thus,
Lemma 5.2 is proved.
E x e r c i s e 5.5 Prove that the dynamical system ( H1 , S t ) generated by
problem (0.1) and (0.2) with the initial data y 0 = ( u 0 ; u1 ) Î H1 =
= F2 ´ F1 is asymptotically compact provided that equations (5.10)
hold.
In order to complete the proof of Theorem 5.3, we should note first that Lemma 5.2
coincides with the assertion of the theorem for l = 1 and second we should use the
fact that the derivatives u ( k ) ( t ) are weak solutions to the problem obtained by differentiation of the original equation. The main steps of the reasoning are given in the
following exercises.
E x e r c i s e 5.6 Assume that the hypotheses of Theorem 5.3 hold for l = n + 1
and its assertion is valid for l = n . Show that w ( t ) = u ( n + 1 ) ( t ) is
a weak solution to the problem of the form
·· ( t ) + g w· ( t ) + A2 w ( t ) + M A1 ¤ 2 u 2 Aw + L w = G
ìw
n + 1(t) ,
ï
í
ï w ( 0 ) = u ( n + 1 ) ( 0 ) , w· ( 0 ) = u ( n + 2 ) ( 0 ) ,
î
where
Gn + 1 ( t )
£ C ( Rn )
for all
t > t0 ( r ) .
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E x e r c i s e 5.7 Use the result of Exercise 5.6 and the method given in the
proof of Lemma 5.2 to prove that w ( t ) = u ( n + 1 ) ( t ) can be estimated as follows:
w· ( t ) 2 + Aw ( t ) 2 £
4
£ C1 æ w· ( 0 )
è
2
+ Aw ( 0 )
2
+ C2ö e -w t + C3 ,
ø
(5.27)
where w > 0 , the numbers C j depend on r and R n , j = 1 , 2 , and
the constant C 3 depends on Rn only.
E x e r c i s e 5.8 Use the induction assumption and equation (5.27) to prove
the assertion of Theorem 5.3 for l = n + 1 .
§ 6
Global Attractor and Inertial Sets
The above given properties of the evolutionary operator St generated by problem
(0.1) and (0.2) in the case of stationary load p ( t ) º p enable us to apply the general assertions proved in Chapter 1 (see also [10]).
Theorem 6.1
Assume that conditions (3.2),, (5.2),, (5.3),, and (5.10) are fulfilled. Then
the dynamical system ( H , St ) generated by problem (0.1) and (0.2) possesses a global attractor A of a finite fractal dimension. This attractor is
a connected compact set in H and is bounded in the space Hs = F 1+ s ´ Fs ,
where s > 0 is defined by condition (5.10)..
Proof.
By virtue of Theorems 5.1, 5.2, and 1.5.1 we should prove only the finite dimensionality of the attractor. The corresponding reasoning is based on Theorem 1.8.1
and the following assertions.
Lemma 6.1.
Assume that conditions (3.2), (5.2), and (5.3) are fulfilled. Let p Î H .
Then for any pair of semitrajectories { St yj : t ³ 0 } , j = 1 , 2 , possessing the property St yj £ R for all t ³ 0 the estimate
St y1 - St y2
H
£ exp ( a0 t ) y1 - y2
H
holds with the constant a0 depending on R .
,
t ³ 0 ,
(6.1)
Global Attractor and Inertial Sets
Proof.
If u1 ( t ) and u 2 ( t ) are solutions to problem (0.1) and (0.2) with the initial
conditions y1 = ( u01 ; u 11 ) and y 2 = ( u 02 ; u 12 ) , then the function v ( t ) =
= u1 ( t ) - u2 ( t ) satisfies the equation
v·· + g v· + A2 v = F ( u 1 , u 2 , t ) ,
(6.2)
where
F ( u 1 , u 2 , t ) = M æ A1 ¤ 2 u2 ( t ) 2ö Au2 ( t ) - M æ A1 ¤ 2 u1 ( t ) 2ö Au1 ( t ) - L v ( t ) .
è
ø
è
ø
It is evident that the estimate
F ( u1 , u2 , t )
2
holds, provided that yi ( t ) H
implies that
£ C R × A ( u1 ( t ) - u2 ( t ) )
= u· i ( t ) 2 + ui ( t ) 12 £ R 2 . Therefore, (2.20)
t
St y1 - St y2
2
H
£
y1 - y2
2
H
+ a1
ò S y -S y
t 1
2
t 2 H dt
.
0
Gronwall’s lemma gives us equation (6.1).
Lemma 6.2
Assume that the hypotheses of Theorem 6.1 hold. Let Ks be the compact
positively invariant set constructed in Theorem 5.2. Then for any
y1 , y2 Î Ks the inequality
QN ( St y1 - St y2 )
H
£ a2
g
- --- t
4
e
L s a tö
æ
3
ç 1 + ------------÷ y1 - y2
s - e
lN + 1
è
ø
H
(6.3)
is valid, where Q N = 1 - PN , N ³ N 0 , the orthoprojector PN and the
number N 0 are defined as in (5.14), L s and ai are positive constants
which depend on the parameters of the problem.
Proof.
It is evident that
Q N St yi = ( qN ui ( t ) ; qN u· i ( t ) ) ,
where qN is the orthoprojector onto the closure of the span of elements { e k :
k = N +1 , N + 2 , ¼} in F0 = H . Moreover, the function wN ( t ) = qN ( u1(t) - u2(t))
is a solution to the equation
·· + g w· + A2 w + M æ A1 ¤ 2 u 2ö Aw = F ( u , u , t ) ,
w
N
N
1 ø
1
2
N
N
N
è
where
FN ( u 1 , u 2 , t ) = - M æ A1 ¤ 2 u 1 2ö - M æ A1 ¤ 2 u 2 2ö qN Au 2 - qN L ( u 1 - u 2 ) .
è
ø
è
ø
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Let us estimate the value FN . Since y
of Theorem 5.2), we have
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H
s
£ R s for y Î K s (see the proof
qN Au2 = qN u2 1 £ l-Ns+ 1 qN u 2 1 + s £ l-Ns+ 1 R s .
Using equation (5.10) we similarly obtain that
4
qN L v
£ l-Ns+ 1 As L v
£ C l-Ns+ 1 Av .
Therefore,
qN L ( u1 ( t ) - u2 ( t ) )
£ C l-Ns+ 1 St y1 - St y2
H
.
Consequently,
FN ( u1 , u2 , t )
C ( Rs )
£ ---------------St y1 - St y2
s
lN + 1
H
.
(6.4)
Using equation (2.27) we obtain that
t
QN ( St y1 - St y2 ) = U ( t , 0 ) QN ( y1 - y2 ) +
òU(t, t) (0 ; F (t)) dt ,
N
0
where U ( t , t ) is the evolutionary operator of homogeneous problem (2.1) with
b ( t ) = M ( A1 ¤ 2 u1 ( t ) 2 ) and h ( t ) = 0 . Therefore, (2.23) and (6.4) imply that
QN ( St y1 - St y2 )
£
g ì
- --- t ï
4
3e
H
í y1 - y2
ï
î
£
H
t
C ( Rs )
+ ---------------lsN + 1
ò
g
--- t
e4
St y1 - St y2
0
ü
ï
dt ý .
ï
þ
(6.5)
We substitute (6.1) in this equation to obtain (6.3). Lemma 6.2 is proved.
Let us choose t0 and N ³ N0 such that
ì g ü
a 2 exp í - --- t0 ý = --d- ,
L s l-Ns+ 1 exp { a3 t0 } £ 1 ,
2
î 4 þ
d < 1.
Then Lemmata 6.1 and 6.2 enable us to state that
St y1 - S t y2
0
0
H
£ l y1 - y2
H
and
QN ( St y1 - St y2 )
0
0
a t
H
£ d y1 - y2
H
,
where l = e 0 0 and the elements y1 and y2 lie in the global attractor A . Hence,
we can use Theorem 1.8.1 with M = A , V = St , and P = PN . Therefore, the fractal
0
dimension of the attractor A is finite. Thus, Theorem 6.1 is proved.
Global Attractor and Inertial Sets
Theorem 5.2 and Lemmata 6.1 and 6.2 enable us to use Theorem 1.9.2 to obtain an
assertion on the existence of the inertial set (fractal exponential attractor) for the
dynamical system ( H , St ) generated by problem (0.1) and (0.2).
Theorem 6.2
Assume that the hypotheses of Theorem 6.1 hold. Then there exists
a compact positively invariant set Aexp Ì H of the finite fractal dimension such that
-n ( t - tB )
ì
ü
sup í dist H ( St y , A exp ) : y Î B ý £ C e
î
þ
for any bounded set B in H and t ³ tB . Here C and n are positive numbers. The inertial set A exp is bounded in the space Hs .
To prove the theorem we should only note that relations (5.11), (6.1), and (6.3)
coincide with conditions (9.12)–(9.14) of Theorem 1.9.2.
Using (1.8.3) and (1.9.18) we can obtain estimates (involving the parameters of the
problem) for the dimensions of the attractor and the inertial set by an accurate observing of the constants in the proof of Theorems 5.1 and 5.2 and Lemmata 6.1 and
6.2. However, as far as problem (0.3) and (0.4) is concerned, it is rather difficult
to evaluate these estimates for the values of parameters that are very interesting
from the point of view of applications. Moreover, these estimates appear to be quite
overstated. Therefore, the assertions on the finite dimensionality of an attractor and
inertial set should be considered as qualitative results in this case. In particular, this
assertions mean that the nonlinear flutter of a plate is an essentially finite-dimensional phenomenon. The study of oscillations caused by the flutter can be reduced to
the study of the structure of the global attractor of the system and the properties
of inertial sets.
E x e r c i s e 6.1 Prove that the global attractor of the dynamical system generated by problem (0.1) and (0.2) is a uniformly asymptotically stable
set (Hint: see Theorem 1.7.1).
We note that theorems analogous to Theorems 6.1 and 6.2 also hold for a class of retarded perturbations of problem (0.1) and (0.2). For example, instead of (0.1) and
(0.2) we can consider (cf. [11–13]) the following problem
u·· + g u· + A2 u + M æ A1/ 2 u 2ö Au + L u + q ( ut ) = p ,
è
ø
u t = 0 = u0 , u· t = 0 = u1 , u t Î ( -r , 0 ) = j ( t ) .
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Here A , M , and L are the same as in Theorems 6.1 and 6.2, the symbol u t denotes
the function on [ - r , 0 ] which is given by the equality ut ( s ) = u ( t + s ) for s Î
Î [ - r , 0 ] , the parameter r is a delay value, and q ( . ) is a linear mapping from
L 2 ( -r , 0 ; F1 + a ) into H possessing the property
0
4
Aa q ( v ) 2 £ c0
òA
1 + a v (s) 2
ds
-r
for c 0 small enough and for all a Î [ 0 , a0 ] , where a0 is a positive number. Such
a formulation of the problem corresponds to the case when we use the model of the
linearized potential gas flow (see [11–14]) to take into account the aerodynamic
pressure in problem (0.3) and (0.4).
The following assertion gives the time smoothness of trajectories lying in the attractor of problem (0.1) and (0.2).
Theorem 6.3
Assume that conditions (3.2) and (5.2) are fulfilled and the linear operator L possesses the property
As L u
£ C As + 1 ¤ 2 u ,
u Î D (A)
(6.6)
for all s Î [ - 1 ¤ 2 , 1 ¤ 2 ) . Let p Î F1 ¤ 2 . Then the assertions of Theorem 6.1
are valid for any s Î ( 0 , 1 ¤ 2 ) . Moreover, if M ( z ) Î C l + 1 ( R+ ) for some
l ³ 1 , then the trajectories y = ( u ( t ) , u· ( t ) ) lying in the global attractor A
of the system ( H , St ) generated by problem (0.1) and (0.2) for g > 0 possess the property
u ( k + 1 ) ( t ) 2 + Au ( k ) ( t ) 2 + A2 u ( k - 1 ) ( t ) 2 £ Rl2
(6.7)
for all - ¥ < t < ¥ , k = 1 , 2 , ¼ , l , where Rl is a constant depending on
the problem parameters only.
only.
Proof.
It is evident that conditions (5.3) and (5.10) follow from (6.6). Therefore, we
can apply Theorem 6.1 which guarantees the existence of a global attractor A . Let
us assume that M ( z ) Î C l + 1 ( R+ ) , l ³ 1 . Let y ( t ) = ( u ( t ) , u· ( t ) ) be a trajectory in
A , -¥ < t < ¥ . We consider a function u m ( t ) = pm u ( t ) , where pm is the orthoprojector onto the span of the basis vectors { e 1 , ¼ , e m } in F 0 for m large enough.
It is clear that um ( t ) Î C 2 ( R ; F2 ) and satisfies the equation
2
u·· m + g u· m + A2 um + M æ A1 ¤ 2 u ( t ) ö Aum + pm L u = pm p .
è
ø
(6.8)
Equation (6.6) for s = - 1 ¤ 2 implies that pm L u is a continuously differentiable
function. It is also evident that M ( A1 ¤ 2 u ( t ) 2 ) Î C1 ( R ) . Therefore, we differentiate equation (6.8) with respect to t to obtain the equation
Global Attractor and Inertial Sets
·· + g w· + A2 w + M æ A1 ¤ 2 u ( t ) 2ö Aw = - p L u· ( t ) + F ( t )
w
m
m
m
m
m
m
è
ø
for the function w ( t ) = u· ( t ) = p u· ( t ) . Here
m
m
m
2
Fm ( t ) = -2 M ¢ æ A 1 ¤ 2 u ( t ) ö ( A u ( t ) , u· ( t ) ) A um ( t ) .
è
ø
·
Since any trajectory y = ( u ( t ) , u ( t ) ) lying in the attractor possesses the property
u· ( t ) 2 + Au ( t ) 2 £ R 02 ,
-¥ < t < ¥ ,
(6.9)
it is clear that
Fm ( t ) £ CR ,
0
-¥ < t < ¥ .
(6.10)
Relation (6.9) also implies that the function
b ( t ) = M æ A1 ¤ 2 u ( t ) 2ö
è
ø
(6.11)
possesses the property
·
b ( t ) + b ( t ) £ CR ,
0
-¥ < t < ¥ .
Therefore, as in the proof of Theorem 2.2, we find that there exists N 0 such that
( 1 - PN ) U ( t , t ) y
HS
£ C ( 1 - PN ) y
HS
g
- --- ( t - t )
4
e
(6.12)
for all real s , where Hs = F 1 + s ´ F s , PN is the orthoprojector onto
LN = Lin { ( e k , 0 ) ; ( 0 , e k ) : k = 1 , 2 , ¼ , N } ,
N ³ N0 , and U ( t , t ) is the evolutionary operator of the problem
··
·
ì u + g u + A2 u + b ( t ) Au = 0 ,
í
·
î u t = 0 = u0 , u t = 0 = u1 ,
with b ( t ) of the form (6.10). Moreover, z m ( t ) = ( wm ( t ) , w· m ( t ) ) can be presented
in the form
t
zm ( t ) = U ( t , t0 ) zm ( t0 ) +
ò U(t , t) (0 ;
- pm L u· ( t ) + Fm ( t ) ) dt .
t0
Then for m > N > N0 we have
( 1 - PN ) zm ( t )
t
+C
ò
t0
H
g
- --- ( t - t )
4
e
-1 ¤ 2
£ Ce
g
- --- ( t - t0 )
4
zm ( t0 )
H
pm L u· ( t ) + Fm ( t ) -1 ¤ 2 dt .
-1 ¤ 2
+
(6.13)
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Therefore, we use (6.9), (6.10), and (6.6) for s = - 1 ¤ 2 to obtain that
4
We tend t0 ® - ¥ in this inequality to find that
( 1 - PN ) zm ( t )
H
-1 ¤ 2
£ CR
01 m
g
- --- ( t - t 0 )
4
e
t
+ CR
0
ò
g
- --- ( t - t )
4
e
dt .
t0
sup ( 1 - PN ) z m ( t )
t ÎR
H-1 ¤ 2
£ C,
where C > 0 does not depend on m . It further follows from (6.9) and equation (0.1)
that
sup PN zm ( t )
t ÎR
H
-1 ¤ 2
£ C,
where the constant C can depend on N . Hence,
2
2
sup æè p m A1 ¤ 2 u· ( t ) + A-1 ¤ 2 pm u·· ( t ) öø £ C .
t ÎR
We tend m ® ¥ to find that any trajectory y ( t ) = ( u ( t ) , u· ( t ) ) lying in the attractor possesses the property
u· ( t ) 1 ¤ 2 = A1 ¤ 2 u· ( t )
£ CR ,
-¥ < t < ¥ .
0
By virtue of (6.6) we have
Lu· ( t )
£ CR ,
0
-¥ < t < ¥ .
Therefore, we reason as above to find that equation (6.13) implies
g
( 1 - PN ) zm ( t )
H
£ CR
- --- ( t - t0 )
+ CR .
e 4
m
0
0
Similarly we get
u·· ( t ) 2 + Au· ( t ) 2 £ R12
for all t Î (- ¥,¥) . Consequently, using equation (0.1) we obtain estimate (6.7) for
k = 1 . In order to prove (6.7) for the other values of k we should use induction with
respect to k and similar arguments. We offer the reader to make an independent detailed study as an exercise.
In addition to the hypotheses of Theorem 6.3 we assume that
L º 0 and r Î Fl º D ( A l ) . Prove that the global attractor A of
the system ( H , St ) lies in F l + 1 ´ Fl .
E x e r c i s e 6.2
Conditions of Regularity of Attractor
§7
Conditions of Regularity of Attractor
Unfortunately, the structure of the global attractor of problem (0.1) and (0.2) can be
described only under additional conditions that guarantee the existence of the Lyapunov function (see Section 1.6). These conditions require that L º 0 and assume
the stationarity of the transverse load p ( t ) . For the Berger system (0.3) and (0.4)
these hypotheses correspond to r = 0 and p ( x , t ) º p ( x ) , i.e. to the case of plate
oscillations in a motionless stationary medium.
Thus, let us assume that the operator L is identically equal to zero and
p ( t ) º p in (0.1). Assume that the hypotheses of Theorem 3.1 hold. Then energy
equality (3.6) implies that
t2
E ( y ( t2 ) ) - E ( y ( t1 ) ) = - g
ò u· (t)
t2
2
dt +
t1
ò (p, u· (t) )
2
dt ,
(7.1)
t1
where y ( t ) = St y = ( u ( t ) , u· ( t ) ) , the function u ( t ) is a weak solution to problem
(0.1) and (0.2) with the initial conditions y = ( u0 ; u 1 ) , and E ( y ) is the energy
of the system defined by formula (3.7).
Let us prove that the functional Y ( y ) = E ( y ) - ( p , u 0 ) with y = ( u0 ; u1 )
is a Lyapunov function (for definition see Section 1.6) of the dynamical system
( H , St ) . Indeed, it is evident that the functional Y ( y ) is continuous on H . By virtue of (7.1) it is monotonely increasing. If E ( y ( t0 ) ) = E ( y ) for some t0 > 0 , then
t0
ò u· (t)
2
dt = 0 .
0
Therefore, u· ( t ) = 0 for t Î [ 0 , t0 ] , i.e. u ( t ) = u is a stationary solution to problem (0.1) and (0.2). Hence, y = ( u ; 0 ) is a fixed point of the semigroup St .
Therefore, Theorems 1.6.1 and 6.1 give us the following assertion.
Theorem 7.1
Assume that g > 0 , L º 0 , and p Î Fs for some s > 0 . We also assume
that the function M ( z ) satisfies conditions (3.2) and (5.2).. Then the global
attractor A of the dynamical system ( H , St ) generated by problem (0.1)
and (0.2) has the form
A = M+ (N ) ,
(7.2)
where N is the set of fixed points of the semigroup St , i.e.
ì
ü
2
N = í ( u ; 0 ) : u Î F1 , A2 u + M æ A1 ¤ 2 u ö Au = p ý ,
è
ø
î
þ
(7.3)
and M+ (N ) is the unstable set emanating from N (for definition see Section 1.6).
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E x e r c i s e 7.1 Let p º 0 . Prove that if the hypotheses of Theorem 7.1 hold,
then any fixed point z of problem (0.1) and (0.2) either equals to zero, z = ( 0 ; 0 ) , or has the form z = ( c × ek ; 0 ) , where the constant
c is the solution to the equation M ( c 2 lk ) + l k = 0 .
4
E x e r c i s e 7.2 Assume that p º 0 and M ( z ) = - G + z . Then problem (0.1)
and (0.2) has a unique fixed point z0 = ( 0 ; 0 ) for G £ l1 .
If l n < G £ l n + 1 , then the number of fixed points is equal to 2 n + 1
and all of them have the form z k = ( wk ; 0 ) , k = 0 , ± 1 , ¼ , ±n ,
where
w0 = 0 ,
G- l
w±k = ± --------------k- × ek ,
lk
k = 1, 2, ¼, n .
E x e r c i s e 7.3 Show that if the hypotheses of Exercise 7.2 hold, then the
energy E ( zk ) of each fixed point z k has the form
E ( z0 ) = 0 ,
--- ( G - l k ) 2 ,
E ( z ±k ) = - 1
4
k = 1, 2, ¼, n ,
for l n < G £ l n + 1 .
E x e r c i s e 7.4 Assume that the hypotheses of Theorem 7.1 hold. Show that
if the set
ì
ü
Dc = í y = ( u0 ; u1 ) Î H : Y ( y ) º E ( y ) - ( p , u0 ) £ c ý
î
þ
(7.4)
is not empty, then it is a closed positively invariant set of the dynamical system ( H , St ) generated by weak solutions to problem (0.1)
and (0.2).
Assume that the hypotheses of Theorem 7.1 hold and the set
Dc defined by equality (7.4) is not empty. Show that the dynamical
system ( Dc , St ) possesses a compact global attractor Ac = M+ (Nc ) ,
where Nc is the set of fixed points of St satisfying the condition
Y(z) £ c .
E x e r c i s e 7.5
E x e r c i s e 7.6 Show that if the hypotheses of Theorem 7.1 hold, then the
global minimal attractor Amin (for definition see Section 1.3)
of problem (0.1) and (0.2) coincides with the set N of the fixed
points (see (7.3)).
Further we prove that if the hypotheses of Theorem 7.1 hold, then the attractor A
of problem (0.1) and (0.2) is regular in generic case. As in Section 2.5, the corresponding arguments are based on the results obtained by A. V. Babin and M. I. Vishik
(see also Section 1.6). These results prove that in generic case the number of fixed
points is finite and all of them are hyperbolic.
Conditions of Regularity of Attractor
Lemma 7.1.
Assume that conditions (3.2) and (5.2) are fulfilled. Then the problem
L [ u ] º A2 u + M æè A1 ¤ 2 u 2öø Au = p ,
u Î D ( A2 + s ) ,
(7.5)
possesses a solution for any p Î Fs , where s ³ 0 . If B is a bounded set
in Fs , then its preimage L -1 (B ) is bounded in F 2 + s = D ( A2 + s ) . If B
is a compact in Fs , then L -1 (B ) is a compact in D ( A2 + s ) , i.e. the
mapping L is proper.
Proof.
We follow the line of arguments given in the proof of Lemma 2.5.3. Let us
consider the continuous functional
ì
ü
--- í ( Au , Au ) + M æ A1 ¤ 2 u 2ö ý - ( p , u )
W (u) = 1
è
ø
2î
þ
on F 1 = D ( A ) , where M ( z ) =
Equation (3.2) implies that
(7.6)
ò M (x) dx is a primitive of the function M (z) .
z
0
--- æ Au 2 - a A1 ¤ 2 u 2 - bö - A-1 p × Au ³
W (u) ³ 1
ø
2è
1
a-ö Au
³ --- æ 1 - ----4è
l 1ø
2
a-ö -1 A-1 p 2 .
--- - æ 1 - -----b
2 è
l 1ø
(7.7)
Thus, the functional W ( u ) is bounded below. Let us consider it on the subspace
pm F1 , where p m is the orthoprojector onto Lin { e1 , ¼ , em } as before. Since
W ( u ) ® + ¥ as Au ® ¥ , there exists a minimum point um on the subspace
pm F1 . This minimum point evidently satisfies the equation
A2 um + M æ A1 ¤ 2 u m 2ö Aum = pm p .
è
ø
(7.8)
Equation (7.7) gives us that
ì
ü
2
Au m 2 £ c1 + c2 inf í W ( u ) : u Î p m F1 ý + c 3 A- 1 p
î
þ
with the constants being independent of m . Therefore, it follows from (7.8)
that A2 um £ C R , provided p £ R . This estimate enables us to pass to the
limit in (7.8) and to prove that if s = 0 , then equation (7.5) is solvable for any
p Î Fs . Equation (7.5) implies that
A2 + s um £ CR
for
p s £ R,
i.e. L -1 (B ) is bounded in D ( A2 + s ) if B is bounded. In order to prove that the
mapping L is proper we should reason as in the proof of Lemma 2.5.3. We give
the reader an opportunity to follow these reasonings individually, as an exercise.
Lemma 7.1 is proved.
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Lemma 7.2
Let u Î F 1 . Then the operator L ¢ [ u ] defined by the formula
L ¢ [ u ] w = A 2 w + 2 M ¢ æ A1 ¤ 2 u 2ö ( Au , w ) Au + M æ A1 ¤ 2 u 2ö Aw
è
ø
è
ø
(7.9)
with the domain D (L ¢ [ u ] ) = D ( A2 ) is selfadjoint and dim Ker L ¢ [ u ] <
< ¥.
Proof.
It is clear that L ¢ [ u ] is a symmetric operator on D ( A2 ) . Moreover, it is
easy to verify that
L ¢ [ u ] w - A2 w
£ C ( u ) × Aw ,
w Î D ( A2 ) ,
(7.10)
i.e. L ¢ [W ] is a relatively compact perturbation of the operator A2 . Therefore,
L ¢ [ u ] is selfadjoint. It is further evident that
Ker L ¢ [ u ] = Ker I + A -2 æè L ¢ [ u ] - A 2öø .
However, due to (7.10) the operator A -2 ( L ¢ [ u ] - A 2 ) is compact. Therefore,
dim Ker L ¢ [ u ] < ¥ . Lemma 7.2 is proved.
E x e r c i s e 7.7 Prove that for any u Î F1 the operator L ¢ [ u ] is bounded
below and has a discrete spectrum, i.e. there exists an orthonormal
basis { fk } in H such that
L ¢ [ u ] fk = mk fk ,
k = 1, 2, ¼ ,
m1 £ m2 £ ¼
lim mn = ¥ .
n®¥
E x e r c i s e 7.8 Assume that u = c 0 e k , where c0 is a constant and ek is an
0
0
element of the basis { e k } of eigenfunctions of the operator A . Show
that L ¢ [ u ] e k = mk ek for all k = 1 , 2 , ¼ , where
mk = l2k 1 + 2 dk k c02 M ¢ ( c 02 lk ) + M ( c02 l k ) lk .
0
0
0
Here d k k = 1 for k = k0 and d k k = 0 for k ¹ k0 .
0
0
As in Section 2.5, Lemmata 7.1 and 7.2 enable us to use the Sard-Smale theorem
(see, e.g., the book by A. V. Babin and M. I. Vishik [10]) and to state that the set
ì
ü
R s = í h Î Fs : $ [ L ¢ [ u ] ] -1 for all u Î L -1 [ h ] ý
î
þ
of regular values of the operator L is an open everywhere dense set in Fs for
s ³ 0.
Show that the set of solutions to equation (7.5) is finite for
p ÎR s (Hint: see the proof of Lemma 2.5.5).
E x e r c i s e 7.9
Conditions of Regularity of Attractor
Let us consider the linearization of problem (0.1) and (0.2) on a solution u Î D ( A 2 )
to problem (7.5):
·· + g w· + L ¢ [ u ] w = 0 ,
w
(7.11)
w t = 0 = w0 , w· t = 0 = w1 .
Here L ¢ [ u ] is given by formula (7.9).
E x e r c i s e 7.10 Prove that problem (7.11) has a unique weak solution on any
segment [ 0 , T ] if w 0 Î F 1 , w1 Î F 0 , and the function M ( z ) Î
Î C 2 ( R+ ) possesses property (3.2).
Thus, problem (7.11) defines a strongly continuous linear evolutionary semigroup
Tt [ u ] in the space H = F1 ´ F0 by the formula
(7.12)
Tt [ u ] ( w0 ; w1 ) = ( w ( t ) ; w· ( t ) ) ,
where w ( t ) is a weak solution to problem (7.11).
E x e r c i s e 7.11 Let { f k } be the orthonormal basis of eigenelements of the
operator L ¢ [ u ] and let mk be the corresponding eigenvalues. Then
each subspace
Hk = Lin { ( fk ; 0 ) , ( 0 ; fk ) } Ì H
is invariant with respect to Tt [ u ] . The eigenvalues of the restriction
of the operator Tt [ u ] onto the subspace Hk have the form
ì g
ü
g2
exp í - æ --- ± ----- - m kö t ý .
è2
ø
4
î
þ
Lemma 7.3
Let L º 0 . Assume that M ( z ) Î C 2 ( R + ) possesses property (3.2). Then
the evolutionary operator St of problem (0.1) and (0.2) is Frechét differentiable at each fixed point y = ( u ; 0 ) . Moreover, St¢ [ u ] = Tt [ u ] ,
where Tt [ u ] is defined by equality (7.12).
Proof.
Let
z ( t ) = St [ y + h ] - y - Tt [ u ] h ,
where h = ( h0 ; h1 ) Î H , y = ( u ; 0 ) , and u is a solution to equation (7.5).
It is clear that z ( t ) = ( v ( t ) ; v· ( t ) ) , where v ( t ) º u ( t ) - u - w ( t ) is a weak solution to problem
··
ì v + g v + A2 v = F (u (t ) , u , w (t) ) ,
í
·
î v t=0 = 0, v t=0 = 0 .
(7.13)
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Here
F ( u ( t ) , u , w ( t ) ) = M æ A1 ¤ 2 u 2ö Au - M æ A1 ¤ 2 u ( t ) 2ö Au ( t ) +
è
ø
è
ø
+ M æ A1 ¤ 2 u 2ö Aw ( t ) + 2 M ¢ æ A1 ¤ 2 u ö ( Au , w ( t ) ) Au ,
è
ø
è
ø
where u ( t ) is a weak solution to problem (0.1) and (0.2) with the initial conditions y0 = y + h º ( u + h0 ; h1 ) and w ( t ) is a solution to problem (7.11) with
w0 = h0 and w1 = h1 . It is evident that
F ( u ( t ) , u , w ( t ) ) = -M æ A1 ¤ 2 u 2ö Av + F1 ( t ) - M ¢ æ A1 ¤ 2 u 2ö F2 ( t ) , (7.14)
è
ø
è
ø
where
ì
F1 ( t ) = -í M æ A1 ¤ 2 u ( t ) 2ö - M æ A1 ¤ 2 u 2ö è
ø
è
ø
î
ü
- M ¢ æ A1 ¤ 2 u 2ö æ A1 ¤ 2 u(t) 2 - A1 ¤ 2 u 2ö ý Au ( t ) ,
è
øè
ø
þ
F2 ( t ) = æ A1 ¤ 2 u ( t ) 2 - A1 ¤ 2 u 2ö Au ( t ) - 2 ( Au , w ( t ) ) Au .
è
ø
It is also evident that the value F1 ( t ) can be estimated in the following way
F1 ( t )
ì
ü
£ c 1 × max í M ¢¢ ( z ) : z Î [ 0 , c 2 ] ý
î
þ
2
A1 ¤ 2 u ( t ) 2 - A1 ¤ 2 u 2
for t Î [ 0 , T ] and for h H £ R , where the constants c 1 and c 2 depend on T ,
R , and u . This implies that
F1 ( t )
£ C (T, R, u) A(u(t) - u)
2
.
(7.15)
Let us rewrite the value F2 ( t ) in the form
F2 ( t ) = ( u ( t ) + u , A ( u ( t ) - u ) ) × A ( u ( t ) - u ) +
+ A1 ¤ 2 ( u ( t ) - u )
2
Au + 2 ( u , Av ( t ) ) Au .
Consequently, the estimate
F2 ( t )
£ C1 ( T , R , u ) A ( u ( t ) - u )
holds for t Î [ 0 , T ] and for h
us that
F(u (t) , u, w (t) )
H
2
+ C 2 ( u ) Av ( t )
(7.16)
£ R . Therefore, equations (7.14)–(7.16) give
£ C1 Av ( t ) + C 2 A ( u ( t ) - u )
2
on any segment [ 0 , T ] . Here C1 = C1 ( u ) and C 2 = C 2 ( T , R , u ) . We use continuity property (3.20) of a solution to problem (0.1) and (0.2) with respect to
the initial conditions to obtain that
Conditions of Regularity of Attractor
A(u (t ) - u)
£ CT , R h
t Î [ 0, T ] ,
H,
h
H
£ R.
Therefore,
F(u (t) , u, w (t) )
£ C1 Av ( t ) + C2 h
2
H
for t Î [ 0 , T ] and for h H £ R . Hence, the energy equality for the solutions
to problem (7.13) gives us that
d
----- æ Av ( t ) 2 + v· ( t ) 2ö £ a æ Av ( t ) 2 + v· ( t ) 2ö + C h 4 .
ø
è
ø
dt è
Therefore, Gronwall’s lemma implies that
Av ( t ) 2 + v· ( t ) 2 £ C h 4 ,
t Î [0, T ] .
This equation can be rewritten in the form
St [ y + h ] - y - Tt [ u ] h
H
£ C h
2
.
Thus, Lemma 7.3 is proved.
E x e r c i s e 7.12 Use the arguments given in the proof of Lemma 7.3 to verify
that under condition (3.2) for M ( z ) Î C 2 ( R+ ) the evolutionary
operator St of problem (0.1) and (0.2) in H belongs to the class C 1
and
St¢ [ y1 ] - St¢ [ y2 ]
L ( H, H )
£ C y1 - y2
H
for any t > 0 and yj Î H .
E x e r c i s e 7.13 Use the results of Exercises 7.7 and 7.11 to prove that for
a regular value p of the mapping L [ u ] the spectrum of the operator Tt [ u ] does not intersect the unit circumference while the eigensubspace E + which corresponds to the spectrum outside the unit
disk does not depend on t and is finite-dimensional.
The results presented above enable us to prove the following assertion (see Chapter V of the book by A. V. Babin and M. I. Vishik [10]).
Theorem 7.2
Assume that the hypotheses of Theorem 7.1 hold. Then there exists an
open dense set Rs in Fs such that the dynamical system ( H , St ) possesses
a regular global attractor A for every p Î Rs , i. e.
N
A=
È M (z ) ,
j=1
+
j
where M+ ( z j ) is the unstable manifold of the evolutionary operator St emanating from the fixed point zj . Moreover, each set M+ ( zj ) is a finite-dimensional surface of the class C 1 .
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In the case of a zero transverse load ( p º 0 ) Theorem 7.2 is not applicable in general. However, this case can be studied by using the structure of the problem. For example, we can guarantee finiteness of the set of fixed points if we assume (see Exercise 7.1) that the equation M ( c 2 l k ) + lk = 0 , first, is solvable with respect to c
only for a finite number of the eigenvalues l k and, second, possesses not more than
a finite number of solutions for every k . The solutions to equation (7.5) are either
u º 0 , or u = c 0 e k0 , where c 0 and k0 satisfy M ( c02 lk ) + lk = 0 . The eigen0
0
values of the operator L ¢ [ u ] have the form
4
m k = lk2 + M ( 0 ) l k
if
u º 0
and
m k = l k lk - l k + 2 dk k l k c 02 M ¢ ( c02 l k )
0
0
0
if
0
u = c0 ek .
0
Therefore, the result of Exercise 7.11 implies that the fixed points are hyperbolic
if all the numbers mk are nonzero, i.e. if
M ( 0 ) ¹ -lk ,
k = 1, 2, ¼ ;
lk ¹ lk ,
0
k ¹ k0 ;
M ( c02 lk ) ¹ 0
0
M ( c02
l k ) + lk = 0 . In particular, if M ( z ) = -G + z ,
for all c 0 and k0 such that
0
0
then for any real G there exists a finite number of fixed points (see Exercise 7.2)
and all of them are hyperbolic, provided that G ¹ l k for all k and the eigenvalues
l j satisfying the condition l j < G are simple. Moreover, we can prove that for
l n < G < ln + 1 the unstable manifold M+ ( zk ) , k = 0 , ±1 , ¼ , ±n , emanating
from the fixed point z k (see Exercise 7.2) possesses the property
dim M+ ( z 0 ) = n ,
§8
dim M+ ( z k ) = k - 1 .
On Singular Limit in the Problem
of Oscillations of a Plate
In this section we consider problem (0.1) and (0.2) in the following form:
m u·· + g u· + A2 u + M æ A1 ¤ 2 u 2ö Au + L u = p , t > 0 ,
è
ø
= u , u·
=u .
u
t=0
0
t=0
1
(8.1)
(8.2)
Equation (8.1) differs from equation (0.1) in that the parameter m > 0 is introduced. It stands for the mass density of the plate material. The introduction of a new
time t ¢ = t ¤ m transforms equation (8.1) into (0.1) with the medium resistance parameter g ¢ = g ¤ m instead of g . Therefore, all the above results mentioned above
remain true for problem (8.1) and (8.2) as well.
On Singular Limit in the Problem of Oscillations of a Plate
The main question discussed in this section is the asymptotic behaviour of the
solution to problem (8.1) and (8.2) for the case when the inertial forces are small
with respect to the medium resistance forces m ” g . Formally, this assumption
leads to a quasistatic statement of problem (8.1) and (8.2):
g u· + A2 u + M æè A1 ¤ 2 u 2öø Au + L u = p , t > 0 ,
(8.3)
u t = 0 = u0 .
(8.4)
Here we prove that the global attractor of problem (8.1) and (8.2) is close to the global attractor of the dynamical system generated by equations (8.3) and (8.4)
in some sense.
Without loss of generality we further assume that g = 1 . We also note that
problem (8.3) and (8.4) belongs to the class of equations considered in Chapter 2.
Assume that conditions (3.2) and (3.3) are fulfilled and
p Î F0 = H . Show that problem (8.3) and (8.4) has a unique mild
(in F 1 = D ( A ) ) solution on any segment [ 0 , T ] , i.e. there exists
a unique function u ( t ) Î C ( 0 , T ; F1 ) such that
E x e r c i s e 8.1
2
u ( t ) = e -A t u0 -
t
-
òe
-A2 ( t - t )
0
ì æ 1¤ 2
ü
2ö
í M è A u ( t ) ø Au ( t ) + L u ( t ) - p ý dt .
î
þ
(Hint: see Theorem 2.2.4 and Exercise 2.2.10).
Let us consider the Galerkin approximations of problem (8.3) and (8.4):
u· m ( t ) + A2 u m ( t ) + M æ A1 ¤ 2 u m ( t ) 2ö Aum + pm L um ( t ) = pm p ,
è
ø
(8.5)
um ( 0 ) = pm u0 ,
(8.6)
where pm is the orthoprojector onto the first m eigenvectors of the operator A and
um ( t ) Î Lin { e1 , ¼ , em } .
Assume that conditions (3.2) and (3.3) are fulfilled and
p Î F0 = H . Then problem (8.5) and (8.6) is solvable on any segment [ 0 , T ] and
E x e r c i s e 8.2
max A ( u ( t ) - um ( t ) ) ® 0 ,
[0, T ]
m ® ¥.
(8.7)
Theorem 8.1
Let p Î H and assume that conditions (3.2),, (5.2),, and (5.3) are fulfilled. Then the dynamical system ( F 1 , St ) generated by weak solutions to
problem (8.3) and (8.4) possesses a compact connected global attractor A .
This attractor is a bounded set in F 1 + b for 0 £ b < 1 and has a finite fractal dimension.
dimension.
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Proof.
First we prove that the system ( F1 , St ) is dissipative. To do that we consider
the Galerkin approximations (8.5) and (8.6). We multiply (8.5) by um ( t ) scalarwise
and find that
1 d
--- ----- um ( t ) 2 + Aum ( t ) 2 + M æ A1 ¤ 2 um ( t ) 2ö A1 ¤ 2 u m ( t ) 2 +
è
ø
2 dt
4
+ ( L um ( t ) , um ( t ) ) = ( p , um ( t ) ) .
Using equation (5.2) we obtain that
d
1
--- ----- um ( t ) 2 + Au m ( t ) 2 + a1 M æ A1 ¤ 2 um ( t ) 2ö £
è
ø
2 dt
£ a3 - a2 A1 ¤ 2 u m ( t )
2+2a
- ( L um ( t ) , um ( t ) ) + ( p , um ( t ) ) .
We use equation (5.3) and reason in the same way as in the proof of Theorem 5.1 to
find that
d
1
--- ----- um 2 + b0 æ Au m 2 + M æ A1 ¤ 2 um 2ö ö + b1 A1 ¤ 2 u m 2 + 2 a £ b2
è
è
øø
2 dt
(8.8)
with some positive constants bj , j = 0 , 1 , 2 . Multiplying equation (8.5) by u· m ( t )
we obtain that
d1
--- ---P ( um ( t ) ) + u· m ( t ) 2 + ( L um ( t ) , u· m ( t ) ) = ( p , u· m ( t ) ) ,
2 dt
where
P ( u ) = Au
2
+ M æ A1 ¤ 2 u 2ö .
è
ø
It follows that
d
----- P ( um ( t ) ) + u· m ( t ) 2 £ 2 p 2 + C Aq u m ( t ) 2 .
dt
If we summarize (8.8) and (8.9), then it is easy to find that
(8.9)
ü
ì
ü
d- ì
---u 2 + P ( u m ) ý + b0 í um 2 + P ( um ) ý £ C .
d t íî m
þ
î
þ
This implies that
ì
ü -b t
um ( t ) 2 + P ( um ( t ) ) £ í pm u0 2 + P ( pm u0 ) ý e 0 + C .
î
þ
We use (8.7) to pass to the limit as m ® ¥ and to obtain that
ì
ü -b t
u ( t ) 2 + P ( u ( t ) ) £ í u0 2 + P ( u0 ) ý e 0 + C .
î
þ
This implies the dissipativity of the dynamical system ( F1 , St ) generated by problem (8.3) and (8.4). In order to complete the proof of the theorem we use Theorem
2.4.1.
On Singular Limit in the Problem of Oscillations of a Plate
We note that the dissipativity also implies that the dynamical system ( F1 , St ) possesses a fractal exponential attractor (see Theorem 2.4.2).
E x e r c i s e 8.3 Assume that the hypotheses of Theorem 8.1 hold and L º 0 .
Show that for generic p Î H the attractor of the dynamical system
( F1 , S t ) generated by equations (8.3) and (8.4) is regular (see the
definition in the statement of Theorem 7.2). Hint: see Section 2.5.
We assume that M ( z ) Î C 2 ( R + ) and conditions (3.2), (5.2), (5.3), and (5.10) are
m
fulfilled. Let us consider the dynamical system ( H1 , St ) generated by problem
(8.1) and (8.2) in the space H1 = F 2 ´ F 1 º D ( A2 ) ´ D ( A ) . Lemma 5.2 and Exercise 5.5 imply that ( H1 , Stm) possesses a compact global attractor A m for any
m > 0.
The main result of this section is the following assertion on the closeness of
attractors of problem (8.1) and (8.2) and problem (8.3) and (8.4) for small m > 0 .
Theorem 8.2
Assume that M ( z ) Î C 2 ( R + ) and conditions (3.2),, (5.2),, (5.3),, and
(5.10) concerning M ( z ) , L , and p are fulfilled. Then the equation
lim sup { dist H ( y , A * ) : y ÎA m } = 0
m®0
(8.10)
m
is valid, where A m is a global attractor of the dynamical system ( H1 , St )
generated by problem (8.1) and (8.2),,
ì
ü
A* = í ( z0 ; z1 ) : z0 ÎA , z1 = - A 2 z0 - M æ A1 ¤ 2 z0 2ö Az0 - L z0 + p ý .
è
ø
î
þ
Here A is a global attractor of problem (8.3) and (8.4) in F 1 and
dist H ( y , A ) is the distance between the element y and the set A in the
space H = F1 ´ F 0 . We remind that g = 1 in equations (8.1) and (8.3)..
The proof of the theorem is based on the following lemmata.
Lemma 8.1
m
The dynamical system ( H1 , St ) is uniformly dissipative in H with
respect to m Î ( 0 , m 0 ] for some m 0 > 0 , i.e. there exists m 0 > 0 and
R > 0 such that for any set B Ì H1 which is bounded in H we have
ì
ü
m
St B Ì í y = ( u0 ; u1 ) : m u1 2 + Au0 2 £ R 2 ý
î
þ
for all t ³ t ( B , m ) , m Î ( 0 , m 0 ] .
(8.11)
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Proof.
We use the arguments from the proof of Theorem 5.1 slightly modifying
them. Let
4
where
V (y) = E (y) + n F(y) ,
y = ( u0 ; u1 ) ,
--- æ m u1 2 + Au0 2 + M æ A1 ¤ 2 u0 2ö ö
E (y) = 1
è
øø
2è
and
--- u0 2 .
F ( y ) = m ( u0 , u1 ) + 1
2
As in the proof of Theorem 5.1 it is easy to find that the inequalities
d
--- u 2 + --e- Au 2 + Ce A1 ¤ 2 u 2 + C1
----- E ( y ( t ) ) £ - 1
2
2
dt
(8.12)
and
d--- Au 2 - a1 M æ A1 ¤ 2 u 2ö ---F ( y ( t ) ) £ m u· 2 - 1
è
ø
2
dt
- a2 A1 ¤ 2 u
2+ 2a
+ C2 u
2
+ C3
(8.13)
m
are valid for y ( t ) = ( u ( t ) ; u· ( t ) ) = St y0 . Here e > 0 is an arbitrary number,
the constants aj and cj do not depend on m . Moreover, it is also evident that
--- b 0 æ m u1 2 + Au0 2 + M æ A1 ¤ 2 u0 2ö ö + b1
V (y) £ 1
è
øø
2 è
(8.14)
for m Î ( 0 , m 0 ] and for any m 0 . Here b0 and b 1 do not depend on m Î
Î ( 0 , m 0 ] and n £ 1 . Equations (8.12)–(8.14) lead us to the inequality
d
--- ( 1 - ( d b 0 + n ) m ) u· 2 ----- V ( y ( t ) ) + d V ( y ( t ) ) £ - 1
2
dt
--- ( n - d b0 - e ) Au
-1
2
2
db
- æ n a1 - ---------0-ö M æ A1 ¤ 2 u 2ö + D ,
è
ø
è
2 ø
where the constant D does not depend on m Î ( 0 , m 0 ] . If we choose m 0 small
enough, then we can take d > 0 and n > 0 independent of m Î ( 0 , m 0 ] and
such that
d
----- V ( y ( t ) ) + d V ( y ( t ) ) £ D 1 ,
dt
(8.15)
where D 1 > 0 does not depend on m Î ( 0 , m 0 ] . Moreover, we can assume
(due to the choice of m 0 ) that
V ( y ( t ) ) ³ b 3 æè m u· ( t ) 2 + Au ( t ) 2öø - C ,
(8.16)
where b 3 and C do not depend on m Î ( 0 , m 0 ] . Using equations (8.14)–(8.16)
we obtain the assertion of Lemma 8.1.
On Singular Limit in the Problem of Oscillations of a Plate
Lemma 8.2
Let u ( t ) be a solution to problem (8.1) and (8.2) such that Au ( t ) £ R
for all t ³ 0 . Then the estimate
t
1
--2
ò u· (t)
2 -b ( t - t )
e
dt £ æ m u· ( 0 ) 2 + Au ( 0 ) 2ö e -b t + C ( R , b )
è
ø
0
is valid for t ³ 0 , where b is a positive constant such that b m £ 1¤ 2 .
Proof.
It is evident that the estimate
M æè A1 ¤ 2 u ( t ) 2öø Au ( t ) + L u ( t ) - p
£ CR
holds, provided Au ( t ) £ R . Therefore, equaiton (8.1) easily implies the estimate
d- æ ·
1
--- ------ u· ( t ) 2 £ C R
m u ( t ) 2 + Au ( t ) 2ö + 1
ø 2
2 dt è
for the solution u ( t ) . We multiply this inequality by 2 exp ( b t ) . Then by virtue
of the fact that Au ( t ) £ R we have
d- b t æ ·
--- u· ( t ) 2 e b t £ C R , b e b t
---m u ( t ) 2 + Au ( t ) 2ö + 1
e
è
ø 2
dt
for m b £ 1 ¤ 2 . We integrate this equation from 0 to t to obtain the assertion
of the lemma.
Lemma 8.3
Let u ( t ) be a solution to problem (8.1) and (8.2) with the initial conditions ( u 0 ; u1 ) Î H1 = F 2 ´ F 1 and such that Au ( t ) £ R for t ³ 0 . Then
the estimate
-b t
£ C1 æ 1 + w· ( 0 ) 2 + Aw ( 0 ) 2ö e 0 + C 2
(8.17)
è
ø
is valid for the function w ( t ) = u· ( t ) . Here m Î ( 0 , m 0 ] , m 0 is small
enough, b0 > 0 , and the numbers C1 and C 2 do not depend on
m Î ( 0 , m0 ] .
m w· ( t )
2
+ Aw ( t )
2
Proof.
Let us consider the function
--- æ m w· ( t ) 2 + Aw ( t ) 2ö + n æ m ( w· , w ) + 1
--- w 2ö
W (t) = 1
ø
è
ø
2è
2
for n > 0 . It is clear that
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4
1
--- m ( 1 - n m ) w· 2 + 1
--- Aw 2 £ W ( t ) £
2
2
--- m ( 1 + n m ) w·
£ 1
2
2
--- + n l-12 ö Aw
+ æ1
è2
ø
2
.
(8.18)
Since the function w ( t ) is a weak solution to the equation obtained by the differentiation of (8.1) with respect to t (cf. (4.15)):
·· ( t ) + w· ( t ) + A2 w ( t ) + M æ A1 ¤ 2 u ( t ) 2ö Aw ( t ) + L w ( t ) = F ( t ) ,
mw
è
ø
where
F( t ) = -2 M ¢ æè A1 ¤ 2 u ( t ) 2öø ( Au ( t ) , w ( t ) ) Au ( t ) ,
then we have that
d
----- W ( t ) = - ( 1 - n m ) w· 2 - M ( A 1 ¤ 2 u ( t ) 2 ) ( A w , w· ) - ( L w , w· ) + ( F , w· ) dt
ì
- n í A2 w
î
2
+ M ( A1 ¤ 2 u ( t) 2 ) A1 ¤ 2 w
2
ü
+ (L w, w) - (F, w) ý .
þ
It follows that
(1)
( 2)
d
--- - n mö w· 2 - 1
--- ( n - C R ) Aw 2 + n CR w 2 .
----- W ( t ) £ - æ 1
è2
ø
2
dt
(1)
We take n = CR + 2 and choose m 0 small enough to obtain with the help
of (8.18) that
d---W ( t ) + b0 W ( t ) £ C w ( t ) 2 ,
dt
t ³ 0,
m Î ( 0 , m0 ] ,
where the constants b 0 > 0 and C do not depend on m . Consequently,
t
W(t) £ W(0) e
-b0 t
+C
ò u· (t)
2 -b0 ( t - t )
e
dt .
0
Therefore, estimate (8.17) follows from equation (8.18) and Lemma 8.2. Thus,
Lemma 8.3 is proved.
Lemma 8.3 and equations (8.11) imply the existence of a constant R1 such that for
any bounded set B in H1 there exists t0 = t0 ( B , m ) such that
m u·· ( t )
2
+ Au· ( t )
2
£ R12 ,
m Î ( 0 , m0 ) ,
(8.19)
where u ( t ) is a solution to problem (8.1) and (8.2) with the initial conditions from
B . However, due to (8.1) equations (8.11) and (8.19) imply that A2 u ( t ) £ C for
t ³ t0 ( B , m ) . Thus, there exists R 2 > 0 such that
On Singular Limit in the Problem of Oscillations of a Plate
m u·· ( t )
2
+ Au· ( t )
2
+ A2 u ( t )
2
£ R22 ,
t ³ t0 ( B , m ) ,
(8.20)
where u ( t ) is a solution to system (8.1) and (8.2) with the initial conditions from the
bounded set B in H1 , R2 does not depend on m Î ( 0 , m0 ) , and m 0 is small
enough. Equation (8.20) and the invariance property of the attractor A m imply the
estimate
m u·· m ( t )
2
+ Au· m ( t )
2
+ A2 um ( t )
2
£ R 22
(8.21)
for any trajectory Stm y0 = ( um ( t ) ; u· m ( t ) ) lying in A m for all t Î (- ¥, ¥) .
Let us prove (8.10). It is evident that there exists an element ym = ( u0 m ; u1 m )
from A m such that
d ( y m ) º dist H ( ym , A * ) = sup { dist H ( y , A * ) : y ÎA m } .
Let ym ( t ) = ( u m ( t ) ; u· m ( t ) ) be a trajectory of system (8.1) and (8.2) lying in the attractor A m and such that ym ( 0 ) = ym . Equation (8.21) implies that there exist
a subsequence { ym n ( t ) } and an element y ( t ) = ( u ( t ) ; u· ( t )) Î L ¥ ( - ¥ , ¥ ; H1 )
such that for any segment [ a , b ] the sequence ymn( t ) converges to y ( t ) in the
* - weak topology of the space L ¥ ( a , b ; H1 ) as m n ® 0 . Equation (8.21) gives us
that the subsequence { Aumn ( t ) } is uniformly continuous and uniformly bounded
in H . Therefore (cf. Exercise 1.14),
lim
max
m n ® 0 t Î [ a , b]
A ( um ( t ) - u ( t ) ) = 0
(8.22)
n
for any a < b . However, it follows from (8.21) that m u·· m ( t ) ® 0 as m ® 0 . Therefore, we pass to the limit m ® 0 in equation (8.1) and obtain that the function u ( t )
is a bounded (on the whole axis) solution to problem (8.3) and (8.4). Hence, it lies in
the attractor A of the system ( F 1 , St ) . With the help of (8.21) and (8.22) it is easy
to find that
d ( ym ) £
n
ym - y0
n
H
® 0,
mn ® 0 ,
where
y0 = æ u ( 0 ) ; - Au ( 0 ) - M æ A1 ¤ 2 u ( 0 ) 2ö Au 0 - L u0 + pö Î A * .
è
ø
è
ø
Thus, Theorem 8.2 is proved.
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4
§9
On Inertial and Approximate
Inertial Manifolds
The considerations of this section are based on the results presented in Sections 3.7,
3.8, and 3.9. For the sake of simplicity we further assume that p ( t ) º g Î H .
Theorem 9.1
Assume that conditions (3.2),, (5.2),, and (5.3) are fulfilled. We also assume that eigenvalues of the operator A possess the properties
lN
- >0
inf ------------N lN +1
lN ( k ) + 1 = c0 k r ( 1 + o ( 1 ) ) ,
and
r > 0, k ® ¥,
(9.1)
for some sequence { N ( k ) } ® ¥ . Then there exist numbers g0 > 0 and k0 > 0
such that the conditions
g > g0
and
2
2
lN
( k ) + 1 - lN ( k ) ³ k0 lN ( k ) + 1
(9.2)
imply that the dynamical system ( H , St ) generated by problem (0.1) and
(0.2) possesses a local inertial manifold, i.e. there exists a finite-dimensional manifold M in H = F1 ´ F0 of the form
M = { y = w + F ( w ) : w Î P H , F ( w ) Î ( 1 - P) H } ,
(9.3)
where F ( . ) is a Lipschitzian mapping from P H into ( 1 - P ) H and P is a
finite-dimensional projector in H . This manifold possesses the properties:
1) for any bounded set B in H and for t ³ t0 ( B )
sup { dist ( St y , M ) : y Î B } £ C exp { -b ( t - t0 ( B ) ) } ;
(9.4)
2) there exists R > 0 such that the conditions y ÎM and St y H £ R for
t Î [ 0 , t0 ] imply that St y ÎM for t Î [ 0 , t0 ] ;
3) if the global attractor of the system ( H , St ) exists, then the set M
contains it (see Theorem 6.1)..
Proof.
Conditions (3.2), (5.2), and (5.3) imply (see Theorem 5.1) that the dynamical
system ( H , St ) is dissipative, i.e. there exists R > 0 such that
t ³ t0 ( B )
(9.5)
˜
for any bounded set B Î H . This enables us to use the dynamical system ( H , St )
generated by an equation of the type
··
·
2
ì u + g u + A u = BR ( u ) ,
(9.6)
í
= u , u·
=u ,
îu
St y
£ R,
H
t=0
y ÎB,
0
t=0
1
to describe the asymptotic behaviour of solutions to problem (0.1) and (0.2). Here
BR ( u ) = c æ ( 2 R )
è
-1
ì
ü
Au ö × í g - M æ A1 ¤ 2 u 2ö Au - L u ý
ø
è
ø
î
þ
On Inertial and Approximate Inertial Manifolds
and c ( s ) is an infinitely differentiable function on R+ possessing the properties
c ¢( s ) £ 2 ;
0 £ c (s) £ 1 ;
c(s) = 1 ,
0 £ s £ 1;
c (s) = 0 ,
s ³ 2.
It is easy to find that there exists a constant CR such that
BR ( u ) £ CR
and
BR ( u1 ) - BR ( u 2 ) £ CR A ( u1 - u 2 ) .
˜
Therefore, we can apply Theorem 3.7.2 to the dynamical system ( H , St ) generated
by equation (9.6). This theorem guarantees the existence of an inertial manifold of
˜
the system ( H , St ) if the hypotheses of Theorem 9.1 hold. However, inside the dissipativity ball { y : y H £ R } problem (9.6) coincides with problem (0.1) and (0.2).
This easily implies the assertion of Theorem 9.1.
E x e r c i s e 9.1 Show that the hypotheses of Theorem 9.1 hold for the problem on oscillations of an infinite panel in a supersonic flow of gas:
p
ì
æ
ö
ï ¶2 u + g ¶ u + ¶4 u + ç G - ¶ u ( x , t ) 2 dx÷ ¶ 2 u + r¶ u = g ( x ) , x Î ( 0 , p ) , t > 0 ,
t
x
x
x
ï t
ç
÷ x
í
è
ø
0
ï
ïu
= ¶x2 u x = 0 , x = p = 0 , u t = 0 = u0(x) , ¶t u
= u1(x) .
î x = 0, x = p
t=0
ò
Here G and r are real parameters and g ( x) Î L2 ( 0 , p ) .
It is evident that the most essential assumption of Theorem 9.1 that restricts its application is condition (9.2). In this connection the following assertion concerning the
case when problem (0.1) and (0.2) possesses a regular attractor is of some interest.
Theorem 9.2
Assume that in equation (0.1) we have L º 0 and p ( t ) º g Î
Î Lin { e 1 , ¼ , eN } for some N 0 . We also assume that conditions (3.2) and
0
(5.2) are fulfilled. Then there exists N1 ³ N0 such that for all N ³ N1 the
subspace
HN = Lin { ( ek ; 0 ) , ( 0 ; e k ) : k = 1 , 2 , ¼ , N }
(9.7)
is an invariant and exponentially attracting set of the dynamical system
( H , St ) generated by problem (0.1) and (0.2)::
g
dist ( St y , HN ) £ CB ( 1 - PN ) y
H
- --- ( t - t 0 ( B ) )
e 4
,
y ÎB
(9.8)
for t ³ t0 ( B ) and for any bounded set B in H . Here PN is the orthoprojector
onto HN .
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Proof.
Since pN g = g for N ³ N0 , where pN is the orthoprojector onto the span of
{ e1 , ¼ , eN } , the uniqueness theorem implies the invariance of HN . Let us prove
0
attraction property (9.8). It is sufficient to consider a trajectory ( u ( t ) , u· ( t ) ) lying in
the ball of dissipativity { y : y H £ R } . Evidently the function v ( t ) = ( 1 - p N ) u ( t )
satisfies the equation
4
ì ··
2ö
·
æ 1¤ 2
2
ï v + g v + A v + M è A u ( t ) ø Av = 0 ,
í
ïv
= ( 1 - pN ) u0 , v· t = 0 = ( 1 - pN ) u1 .
î t=0
(9.9)
It is also clear that the conditions
M æ A1 ¤ 2 u ( t ) 2ö
è
ø
£ b0 ( R )
and
d
----- M æ A1 ¤ 2 u ( t ) 2ö
ø
dt è
£ b1 ( R )
hold in the ball of dissipativity. This fact enables us to use Theorem 2.2 with b ( t ) =
= M ( A1 ¤ 2 u ( t ) 2 ) . In particular, equation (2.23) guarantees the existence of a number N1 ³ N0 which depends on g , b0 ( R ) , and b1 ( R ) and such that
y(t)
H
= ( 1 - PN ) y ( t )
H
£
3 ( 1 - PN ) y ( 0 )
g
- --- t
4
e
,
t > 0,
for all N ³ N1 , where PN is the orthoprojector onto HN and y ( t ) = ( v ( t ) , v· ( t ) ) .
This implies estimate (9.8). Theorem 9.2 is proved.
E x e r c i s e 9.2 Assume that the hypotheses of Theorem 9.2 hold. Show that
for any semitrajectory St y there exists an induced trajectory in
HN , i.e. there exists y Î HN such that
St y - St y
H
£ CR
for t ³ t0 and for some t0 = t0 ( y
H)
g
- --- ( t - t0 )
4
e
.
E x e r c i s e 9.3 Write down an inertial form of problem (0.1) and (0.2) in the
subspace HN , provided the hypotheses of Theorem 9.2 hold. Prove
that the inertial form coincides with the Galerkin approximation of
the order N of problem (0.1) and (0.2).
E x e r c i s e 9.4 Show that if the hypotheses of Theorem 9.2 hold, then the
global attractor of problem (0.1) and (0.2) coincides with the global
attractor of its Galerkin approximation of a sufficiently large order.
Let us now turn to the question on the construction of approximate inertial manifolds for problem (0.1) and (0.2). In this case we can use the results of Section 3.8
and the theorems on the regularity proved in Sections 4 and 5.
On Inertial and Approximate Inertial Manifolds
E x e r c i s e 9.5
Assume that M ( z ) Î C m + 1 ( R+ ) , m ³ 1 and
B ( u ) = p - M æ A1 ¤ 2 u 2ö Au - L u ,
è
ø
where p Î H and u Î D ( A) = F 1 . Show that the mapping B ( . )
has the Frechét derivatives B ( k ) up to the order l inclusive. Moreover, the estimates
á B ( k ) [ u ] ; w 1 , ¼ , w kñ
£ CR
k
Õ Aw
j
(9.10)
j=1
and
á B ( k ) [ u ] - B ( k ) [ u* ] ; w1 , ¼ , wkñ
£
k
£ CR A ( u - u* )
Õ Aw
j
(9.11)
j=1
are valid, where k = 0 , 1 , ¼ , m , Au £ R , Au* £ R , and
wj Î D ( A) . Here á B ( k ) [ u ] ; w1 , ¼ , wkñ is the value of the
Frechét derivative on the elements w1 , ¼ , wk .
We consider equations (9.10) and (9.11) as well as Theorem 5.3 which guarantees
nonemptiness of the classes L m , R corresponding to the problem considered when
R > 0 is large enough. They enable us to apply the results of Section 3.8.
Let P be the orthoprojector onto the span of elements { e 1 , ¼ , eN } in H and
let Q = 1 - P . We define the sequences { hn ( p , p· ) }n¥= 0 and { ln ( p , p· ) }n¥= 0 of mappings from PH ´ PH into Q H by the formulae
h ( p , p· ) = l ( p , p· ) = 0 ,
(9.12)
0
0
A2 hk ( p , p· ) = a0 - Mk - 1 ( p , p· ) Ahk - 1 - QL ( p + hk - 1 ) - g ln ( k ) -
- á dp lk - 1 ; p· ñ + á dp· lk - 1 ; g p· + A2 p - b0 + Mk - 1 ( p , p· ) Ap + PL ( p + hk - 1 )ñ , (9.13)
lk ( p , p· ) = á d p h k - 1 ; p· ñ -
- á dp· hk - 1 ; g p· + A2 p - b0 + M k - 1 ( p , p· ) Ap + PL ( p + hk - 1 )ñ .
(9.14)
Here M k ( p , p· ) = M æ A1 ¤ 2 p 2 + A1 ¤ 2 hk ( p , p· ) 2ö , d p and d p· are the Frechét deè
ø
rivatives with respect to the corresponding variables, a0 = Q g , b0 = Pg , where
g º p ( t ) is a stationary transverse load in (0.1), k = 1 , 2 , ¼ , m , the numbers
n(k) are chosen to fulfil the inequality k - 1 £ n ( k ) £ k .
E x e r c i s e 9.6
Evaluate the functions h1 ( p , p· ) and l1 ( p , p· ) .
279
280
The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow
C
h
a
p
t
e
r
Theorem 3.8.2 implies the following assertion.
4
Theorem 9.3
Assume that p ( t ) º g Î H , M ( z ) Î C m + 1 ( R+ ) , m ³ 2 , and conditions
(3.2),, (5.2),, and (5.3) are fulfilled. Then for all k = 1 , ¼ , m the collection of
–(9.14) possesses the properties
mappings ( hn , ln ) given by equalities (9.12)–
1) there exist constants M j = Mj ( n , r ) and L j ( n , r ) , j = 1 , 2 such that
A2 hn ( p0 , p· 0 ) £ M1 ,
A2 ( hn ( p1 , p· 1 ) - hn ( p 2 , p· 2 ) )
Aln ( p0 , p· 0 ) £ M2 ,
£ L1 ( A2 ( p1 - p2 ) + A ( p· 1 - p· 2 ) ) ,
A ( ln ( p1 , p· 1 ) - ln ( p2 , p· 2 ) ) £ L2 ( A2 ( p1 - p2 ) + A ( p· 1 - p· 2 ) ) ,
for all pj and p· j from PH and such that
j = 0, 1, 2 , r > 0 ;
A2 pj 2 + Ap· j 2 £ r ,
2) for any solution u ( t ) to problem (0.1) and (0.2) which satisfies compatibility conditions (4.3) with l = m the estimate
ì 2
í A ( u ( t ) - un ( t ) )
î
2
ü
+ A ( u ( t ) - un ( t ) ) 2 ý
þ
1¤ 2
£ Cn l-Nn+ 1
is valid for n £ m - 1 and for t large enough. Here
u ( t ) = p ( t ) + h ( p ( t ) , p· ( t ) ) ,
n
n
u n ( t ) = p· ( t ) + ln ( p ( t ) , p· ( t ) ) ,
lN is the N -th eigenvalue of the operator A and the constant Cn depends on the radius of dissipativity.
In particular, Theorem 9.3 means that the manifold
M = { ( p + h ( p , p· ) ; p· + l ( p , p· ) ) :
n
n
n
p , p· Î PH }
attracts sufficiently smooth trajectories of the dynamical system ( H , St ) generated
by problem (0.1) and (0.2) into a small vicinity (of the order Cn l-Nn+ 1 ) of Mn .
E x e r c i s e 9.7 Assume that the hypotheses of Theorem 6.3 hold (this theorem guarantees the existence of the global attractor A consisting of
smooth trajectories of problem (0.1) and (0.2)). Prove that
sup { dist ( y , Mn ) : y ÎA } £ Cn l-Nn+ 1
for all n £ m - 1 (the number m is defined by the condition
M ( z ) Î C m + 1 ( R+) ).
E x e r c i s e 9.8 Prove the analogue of Theorem 3.9.1 on properties of the nonlinear Galerkin method for problem (0.1) and (0.2).
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