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
5
Theory of Functionals
that Uniquely Determine Long-Time Dynamics
Contents
....§1
Concept of a Set of Determining Functionals . . . . . . . . . . . 285
....§2
Completeness Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
....§3
Estimates of Completeness Defect in Sobolev Spaces . . . . 306
....§4
Determining Functionals for Abstract Semilinear
Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
....§5
Determining Functionals for Reaction-Diffusion Systems . 328
....§6
Determining Functionals in the Problem
of Nerve Impulse Transmission . . . . . . . . . . . . . . . . . . . . . . 339
....§7
Determining Functionals for Second Order
in Time Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
....§8
On Boundary Determining Functionals . . . . . . . . . . . . . . . . 358
....
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
The results presented in previous chapters show that in many cases the asymptotic behaviour of infinite-dimensional dissipative systems can be described by a finite-dimensional global attractor. However, a detailed study of the structure of
attractor has been carried out only for a very limited number of problems. In this regard it is of importance to search for minimal (or close to minimal) sets of natural
parameters of the problem that uniquely determine the long-time behaviour of a system. This problem was first discussed by Foias and Prodi [1] and by Ladyzhenskaya
[2] for the 2D Navier-Stokes equations. They have proved that the long-time behaviour of solutions is completely determined by the dynamics of the first N Fourier
modes if N is sufficiently large. Later on, similar results have been obtained for
other parameters and equations. The concepts of determining nodes and determining local volume averages have been introduced. A general approach to the problem on the existence of a finite number of determining parameters has been
discussed (see survey [3]).
In this chapter we develop a general theory of determining functionals. This
theory enables us, first, to cover all the results mentioned above from a unified point
of view and, second, to suggest rather simple conditions under which a set of functionals on the phase space uniquely determines the asymptotic behaviour of the system by its values on the trajectories. The approach presented here relies on the
concept of completeness defect of a set of functionals and involves some ideas and
results from the approximation theory of infinite-dimensional spaces.
§1
Concept of a Set of Determining
Functionals
Let us consider a nonautonomous differential equation in a real reflexive Banach
space H of the type
du
-----= F (u, t) ,
dt
t > 0,
u t = 0 = u0 .
(1.1)
Let W be a class of solutions to (1.1) defined on the semiaxis R + º { t : t ³ 0 } such
that for any u ( t ) ÎW there exists a point of time t0 > 0 such that
2
( t0 , + ¥ , V ) ,
u ( t ) Î C ( t0 , + ¥ ; H ) Ç Lloc
(1.2)
where V is a reflexive Banach space which is continuously embedded into H . Hereinafter C ( a , b ; X ) is the space of strongly continuous functions on [ a , b ] with the
values in X and L2loc ( a , b ; X ) has a similar meaning. The symbols . H and . V
stand for the norms in the spaces H and V, . H £ . V .
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Theory of Functionals that Uniquely Determine Long-Time Dynamics
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The following definition is based on the property established in [1] for the Fourier modes of solutions to the 2D Navier-Stokes system with periodic boundary conditions.
Let L = { lj : j = 1 , ¼ , N } be a set of continuous linear functionals on V .
Then L is said to be a set of asymptotically ( V , H , W ) -determining functionals (or elements) for problem (1.1) if for any two solutions u , v Î W the condition
5
t+1
lim
t®¥
ò
l j ( u ( t ) ) - l j ( v ( t ) ) 2 dt = 0
for
j = 1, ¼, N
(1.3)
t
implies that
lim u ( t ) - v ( t ) H = 0 .
(1.4)
t®¥
Thus, if L is a set of asymptotically determining functionals for problem (1.1), the
asymptotic behaviour of a solution u ( t ) is completely determined by the behaviour
of a finite number of scalar values { lj ( u ( t ) ) : j = 1 , 2 , ¼ , N } . Further, if no ambiguity results, we will sometimes omit the spaces V , H , and W in the description of
determining functionals.
E x e r c i s e 1.1
Show that condition (1.3) is equivalent to
t+1
lim
t®¥
ò [N
L (u (t)
2
- v ( t ) ) ] dt = 0 ,
t
where NL ( u ) is a seminorm in H defined by the equation
NL ( u ) = max l ( u ) .
l ÎL
E x e r c i s e 1.2 Let u1 and u2 be stationary (time-independent) solutions to
problem (1.1) lying in the class W . Let L = { lj : j = 1 , ¼ , N } be a
set of asymptotically determining functionals. Show that condition
lj ( u1 ) = lj ( u2 ) for all j = 1 , 2 , ¼ , N implies that u 1 = u2 .
The following theorem forms the basis for all assertions known to date on the existence of finite sets of asymptotically determining functionals.
Theorem 1.1.
Let L = { lj : j = 1 , ¼ , N } be a family of continuous linear functionals
on V . Su
Suppose that there exists a continuous function V ( u , t ) on H ´ R +
with the values in R + which possesses the following properties:
a) there exist positive numbers a and s such that
V (u, t) ³ a × u
s
for all
u Î H , t Î R+ ;
(1.5)
Concept of a Set of Determining Functionals
b) for any two solutions u ( t ) , v ( t ) Î W to problem (1.1) there exist (i)
a point of time t0 > 0 , (ii) a function y ( t ) that is locally integrable
over the half-interval [ t0 , ¥ ) and such that
t+a
+
gy
= lim
t®¥
ò y(t) dt
> 0
(1.6)
t
and
t+a
+
Gy
= lim
t®¥
ò max {0, -y (t) } dt < ¥
(1.7)
t
for some a > 0 , and (iii) a positive constant C such that for all
t ³ s ³ t0 we have
t
V (u (t) - v(t) , t) +
òy(t) × V (u (t) - v((t) , t) ) dt
s
£
t
£ V (u (s) - v(s) , s) + C ×
ò
max
j = 1, ¼, N
lj ( u ( t ) ) - lj ( v ( t ) ) 2 dt .
(1.8)
s
Then L is a set of asymptotically ( V , H , W ) -determining functionals for
problem (1.1)..
It is evident that the proof of this theorem follows from a version of Gronwall's lemma
stated below.
Lemma 1.1.
Let y ( t ) and g ( t ) be two functions that are locally integrable over
some half-interval [ t0 , ¥ ) . Assume that (1.6) and (1.7) hold and g ( t )
is nonnegative and possesses the property
t+a
lim
t®¥
ò g ( t ) dt = 0 ,
a > 0.
(1.9)
t
Suppose that w ( t ) is a nonnegative continuous function satisfying the
inequality
t
w (t) +
ò y ( t ) × w ( t ) dt
t
£ w (s) +
s
ò g ( t ) dt
(1.10)
s
for all t ³ s ³ t0 . Then w ( t ) ® 0 as t ® ¥ .
It should be noted that this version of Gronwall's lemma has been used by many authors (see the references in the survey [3]).
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Proof.
Let us first show that equation (1.10) implies the inequality
ì t
ü
ï
ï
w ( t ) £ w ( s ) exp í - y ( s ) ds ý +
ï
ï
î s
þ
ò
5
ì t
ü
ï
ï
g ( t ) exp í - y ( s ) ds ý dt (1.11)
ï
ï
s
î t
þ
t
ò
ò
for all t ³ s ³ t0 . It follows from (1.10) that the function w ( t ) is absolutely
continuous on any finite interval and therefore possesses a derivative w· ( t ) almost everywhere. Therewith, equation (1.10) gives us
(1.12)
w· ( t ) + y ( t ) w ( t ) £ g ( t )
for almost all t . Multiplying this inequality by
ìt
ü
ï
ï
e ( t ) = exp í y ( s ) ds ý ,
ï
ï
îs
þ
ò
we find that
d
---(w (t) e (t) ) £ g (t) e (t)
dt
almost everywhere. Integration gives us equation (1.11).
Let us choose the value s such that
t+a
ò max {-y(s) , 0} ds
+
£ G+1 ,
G º Gy
(1.13)
t
and
t+a
ò y(s) ds ³ --2- ,
g
+
g º gy
t
(1.14)
t -t
for all t ³ s . It is evident that if t ³ t ³ s and k = ----------- , where [ . ] is the ina
teger part of a number, then
t
ka + t
ò y ( s ) ds = ò
t
g
³ --- k 2
t
y ( s ) ds +
ò
Thus, for all t ³ t ³ s
y ( s ) ds ³
ka +t
t
( k + 1) a + t
ka + t
ò
g
max { -y ( s ) , 0 } ds ³ --- k - ( G + 1 ) .
2
Concept of a Set of Determining Functionals
t
òy(s) ds
g
g
³ ------- ( t - t ) - æ 1 + G + ---ö .
è
2a
2ø
t
Consequently, equation (1.11) gives us that
ì g
ü
w ( t ) £ C ( G , g ) w ( s ) exp í - ------- ( t - s ) ý +
2
a
î
þ
t
ì
ü
g
- ( t - t ) ý dt
2a
òg (r) exp íî------þ
,
s
ì
gü
where C ( G , g ) = exp í 1 + G + --- ý . Therefore,
2
î
þ
t
lim w ( t ) £ C ( G, g ) × lim
t®¥
t®¥
ì
ü
g
- ( t - t ) ý dt .
2a
òg (t) exp íî------þ
(1.15)
s
It is evident that
t
G (t, s) º
ì
ü
g
- ( t - t ) ý dt
2a
ò g (t) exp íî------þ
£
s
N
s + (k + 1) a
k=0
s + ka
å
£
ò
ì g
ü
g ( t ) exp í - ------- ( t - t ) ý dt ,
î 2a
þ
t -s
t -s
where N = ----------- is the integer part of the number ----------- . Therefore,
a
a
N s + (k + 1) a
s+a
G ( t , s ) £ sup
s³s
ò g (t) dt × å ò
s+ a
=
sup
s³ s
ò
k=0
s
s + (N + 1) a
g ( t ) dt
s
ò
s
s+ a
a-----= 2
g × ssup
³s
ò
s
s + ka
ì g
ü
exp í - ------- ( t - t ) ý dt =
2
a
î
þ
g
ì g
ü --- ( N + 1 )
-1 .
g ( t ) dt × exp í - ------- ( t - s ) ý e 2
î 2a
þ
t -s
Since N = ---------- , this implies that
a
s+ a
lim G ( t , s ) £ C ( a , g ) × sup
t®¥
ì g
ü
exp í - ------- ( t - t ) ý dt =
2
a
î
þ
s³s
ò g ( t ) dt
s
(1.16)
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Theory of Functionals that Uniquely Determine Long-Time Dynamics
for any s such that equations (1.13) and (1.14) hold. Hence, equations (1.15)
and (1.16) give us that
s+ a
lim w ( t ) £ C ( G, g , a ) × sup
t®¥
s³s
ò g (t) dt .
s
If we tend s ® ¥ , then with the help of (1.9) we obtain
lim w ( t ) = 0 .
t®¥
This implies the assertion of Lemma 1.1.
In cases when problem (1.1) is the Cauchy problem for a quasilinear partial differential equation, we usually take some norm of the phase space as the function V ( u , t )
when we try to prove the existence of a finite set of asymptotically determining functionals. For example, the next assertion which follows from Theorem 1.1 is often
used for parabolic problems.
Corollary 1.1.
Let V and H be reflexive Banach spaces such that V is continuously
and densely embedded into H . Assume that for any two solutions
u1 ( t ) , u2 ( t ) Î W to problem (1.1) we have
t
u1 ( t ) - u2 ( t )
2
V
+
ò y(t) u (t) - u (t)
1
s
£
u1 ( s ) - u 2 ( s )
2
V
2
2
V
dt £
t
+K
ò u -u
1
2
2 H
dt
(1.17)
s
for t ³ s ³ t0 , where K is a constant and the function y ( t ) depends on
u1 ( t ) and u2( t ) in general and possesses properties (1.6) and (1.7).
Assume that the family L = { lj : j = 1 , ¼ , N } on V possesses the property
v H £ C × max l j ( v ) + eL v V
j=1¼N
(1.18)
for any v Î V , where C and eL are positive constants depending
on L . Then L is a set of asymptotically determining functionals for
problem (1.1), provided
t+a
2
1- × lim --1- ×
eL
< --K t®¥ a
ò y(t) dt º g
t
+ -1 -1
ya K
.
(1.19)
Concept of a Set of Determining Functionals
Proof.
Using the obvious inequality
( a + b ) 2 £ ( 1 + d ) a 2 + æ 1 + --1-ö b 2 ,
è
dø
we find from equation (1.18) that
d > 0,
2
2
£ ( 1 + d ) eL
vH
v V2 + C d × max lj ( v ) 2
j=1¼N
(1.20)
for any d > 0 . Therefore, equation (1.17) implies that
t
u1 ( t ) - u 2 ( t )
2
V
+
ò æèy(t) - (1 + d) K e öø u (t) - u (t)
2
L
s
£
u1 ( s ) - u 2 ( s )
2
V
1
2
2
V
dt £
t
+C
ò [N
L ( u1 ( t )
- u 2 ( t ) ) ] 2 dt ,
s
where NL ( v ) =
max
j=1¼N
lj ( v ) . Consequently, if for some d > 0 the function
˜
2
y ( t ) = y ( t ) - ( 1 + d ) K eL
possesses properties (1.6) and (1.7) with some constants g and G > 0 , then
Theorem 1.1 is applicable. A simple verification shows that it is sufficient to require that equation (1.19) be fulfilled. Thus, Corollary 1.1 is proved.
Another variant of Corollary 1.1 useful for applications can be formulated as follows.
Corollary 1.2.
Let V and H be reflexive Banach spaces such that V is continuously
embedded into H . Assume that for any two solutions u ( t ) , v ( t ) Î W to
problem (1.1) there exists a moment t0 > 0 such that for all t ³ s ³ t0
the equation
t
u (t) - v(t)
2
H
+n
ò u (t) - v(t)
2
V
dt £
s
t
£
u (s) - v(s)
2
H
+
ò f (t) × u (t) - v(t)
2
H
dt
(1.21)
s
holds. Here n > 0 and the positive function f ( t ) is locally integrable
over the half-interval [ t0 , ¥ ) and satisfies the relation
t+a
lim --1t®¥ a
ò f (t) dt
t
£ R
(1.22)
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Theory of Functionals that Uniquely Determine Long-Time Dynamics
for some a > 0 , where the constant R > 0 is independent of u ( t ) and
v ( t ) . Let L = { lj : j = 1 , ¼ , N } be a family of continuous linear functionals on V possessing the property
lj ( w ) + eL × w V
w H £ CL × max
j = 1, ¼, N
for any w Î V . Here CL and e L are positive constants. Then L is a set
of asymptotically ( V, H , W ) -determining functionals for problem (1.1),
provided that eL < n ¤ R .
Proof.
Equation (1.20) implies that
2
-2
× wH
- CL , d × max
w V2 ³ ( 1 + d ) -1 × e L
lj ( w ) 2
j = 1, ¼, N
for any d > 0 . Therefore, (1.21) implies that
t
u (t) - v(t)
2
H
+
ò y(t) × u (t) - v(t)
s
£
u (s) - v(s)
2
H
2
H
dt £
t
+ n CL, d ×
ò
max
j = 1, ¼, N
lj ( u ( t ) - v ( t ) ) 2 dt ,
s
-2
n ( 1 + d ) -1 eL
2
where y ( t ) =
with V ( u , t ) = u
- f ( t ) . Using (1.22) and applying Theorem 1.1
, we complete the proof of Corollary 1.2.
Other approaches of introduction of the concept of determining functionals are also
possible. The definition below is an extension to a more general situation of the property proved by O.A. Ladyzhenskaya [2] for trajectories lying in the global attractor
of the 2D Navier-Stokes equations.
Let W be a class of solutions to problem (1.1) on the real axis R such that
2
( -¥ , + ¥ ; V ) . A family L = { lj : j = 1 , ¼ , N } of continuous linear
W Ì Lloc
functionals on V is said to be a set of ( V , W ) -determining functionals (or
elements) for problem (1.1) if for any two solutions u , v Î W the condition
lj ( u ( t ) ) = lj ( v ( t ) ) for j = 1 , ¼ , N and almost all t Î R
(1.23)
implies that u ( t ) º v ( t ) .
It is easy to establish the following analogue of Theorem 1.1.
Theorem 1.2.
Let L = { lj : j = 1 , ¼ , N } be a family of continuous linear functionals
on V . Let W be a class of solutions to problem (1.1) on the real axis R such
that
Concept of a Set of Determining Functionals
2
(– ¥ , + ¥ ; V) .
W Ì C ( – ¥ , + ¥ ; H ) Ç Lloc
(1.24)
Assume that there exists a continuous function V ( u , t ) on H ´ R with the
values in R which possesses the following properties:
a) there exist positive numbers a and s such that
V (u, t) ³ a × u
s
for all
u ÎH,
t ÎR ;
(1.25)
b) for any u ( t ) , v ( t ) Î W
sup V ( u ( t ) - v ( t ) , t ) < ¥ ;
t ÎR
(1.26)
c) for any two solutions u ( t ) , v ( t ) Î W to problem (1.1) there exist (i)
a function y ( t ) locally integrable over the axis R with the properties
t+a
gy
º
ò y(t) dt > 0
(1.27)
ò max {0, -y(t) }dt < ¥
(1.28)
lim
t ® -¥
t
and
Gy
t+a
º
lim
t ® -¥
t
for some a > 0 , and (ii) a positive constant C such that equation
(1.8) holds for all t ³ s . Then L is a set of ( V , W ) -determining functionals for problem (1.1)..
Proof.
It follows from (1.23), (1.8), and (1.11) that the function w (t) = V (u (t) - v(t) , t )
satisfies the inequality
ì t
ü
ï
ï
(1.29)
w ( t ) £ w ( s ) × exp í - y ( t ) dt ý
ï
ï
î s
þ
for all t ³ s . Using properties (1.27) and (1.28) it is easy to find that there exist
numbers s* , a0 > 0 , and b0 > 0 such that
ò
s2
ò y ( t ) dt
³ a0 × ( s2 - s1 ) - b0 ,
s 1 £ s 2 £ s* .
s1
This equation and boundedness property (1.26) enable us to pass to the limit
in (1.29) for fixed t as s ® - ¥ and to obtain the required assertion.
Using Theorem 1.2 with V ( u , t ) = u 2 as above, we obtain the following assertion.
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Corollary 1.3.
Let V and H be reflexive Banach spaces such that V is continuously
embedded into H . Let W be a class of solutions to problem (1.1) on the
real axis R possessing property (1.24) and such that
sup u ( t ) H < ¥
for all
t ÎR
u(t) Î W .
(1.30)
Assume that for any u ( t ) , v ( t ) Î W and for all real t ³ s equation
(1.21) holds with n > 0 and a positive function f ( t ) locally integrable
over the axis R and satisfying the condition
s+a
--1s ® -¥ a
lim
ò f (t) dt
£ R
(1.31)
s
for some a > 0 . Here R > 0 is a constant independent of u ( t ) and v ( t ) .
Let L = { lj : j = 1 , ¼ , N } be a family of continuous linear functionals
on V possessing property (1.18) with e L < n ¤ R . Then L is a set of
asymptotically ( V , W ) -determining functionals for problem (1.1).
Proof.
As in the proof of Corollary 1.2 equations (1.20) and (1.21) imply that
t
u (t) - v(t) H +
ò y (t) u (t) - v(t)
2
H
2
u (s) - v(s) H
dt £
s
-2
- f ( t ) and d is an arbitrary positive
for all t ³ s , where y ( t ) = n ( 1 + d ) -1 e L
number. Hence
u (t) - v (t)
2
H
£
u (s) - v (s)
2
H
ì t
ü
ï
ï
exp í - y ( t ) dt ý
ï
ï
î s
þ
ò
(1.32)
for all t ³ s . Using (1.31) it is easy to find that for any h > 0 there exists
Mh > 0 such that
s2
ò f ( t ) dt
£ ( R + h ) ( s2 - s1 + a )
s1
for all s 1 £ s 2 £ - Mh . This equation and boundedness property (1.30) enable
us to pass to the limit as s ® - ¥ in (1.32), provided e L < n ¤ R , and to obtain
the required assertion.
We now give one more general result on the finiteness of the number of determining
functionals. This result does not use Lemma 1.1 and requires only the convergence
of functionals on a certain sequence of moments of time.
Concept of a Set of Determining Functionals
Theorem 1.3.
Let V and H be reflexive Banach spaces such that V is continuously
embedded into H . Assume that W is a class of solutions to problem (1.1)
possessing property (1.2).. Assum
Assume
ume that there exist constants C , K ³ 0 ,
b > a > 0 , and 0 < q < 1 such that for any pair of solutions u1 ( t ) and
u2 ( t ) from W we have
u1 ( t ) - u2 ( t ) V £ C u1 ( s ) - u2 ( s ) V ,
s £ t £ s +b,
(1.33)
and
u1( t ) - u2 ( t ) V £ K u1( t ) - u 2 ( t ) H + q u1( s ) - u 2 ( s ) V ,
s + a £ t £ s + b (1.34)
for s large enough. Let L be a finite set of continuous linear functionals
on V possessing property (1.18) with e L < ( 1 - q ) K -1 . Assume that { tk } is a
sequence of positive numbers such that tk ® + ¥ and a £ tk + 1 - tk £ b .
Assume that
lim l ( u1 ( tk ) - u 2 ( tk ) ) = 0 ,
l ÎL .
k®¥
(1.35)
Then
u1 ( t ) - u 2 ( t ) V ® 0
t ® +¥.
as
(1.36)
It should be noted that relations like (1.33) and (1.34) can be obtained for a wide
class of equations (see, e.g., Sections 1.9, 2.2, and 4.6).
Proof.
Let u ( t ) = u1 ( t ) - u2 ( t ) . Then equations (1.34) and (1.18) give us
u ( tk ) V £ qL u ( tk - 1 ) V + C max lj ( u ( tk ) ) ,
j
where qL = q ( 1 - eL K ) -1 < 1 . Therefore, after iterations we obtain that
n
n
u ( tn ) V £ q L u ( t0 ) V + C ×
åq
k=1
n-k
L
max lj ( u ( tk ) ) .
j
Hence, equation (1.35) implies that u ( tn ) V ® 0 as n ® ¥ . Therefore, (1.36) follows from equation (1.33). Theorem 1.3 is proved.
Application of Corollaries 1.1–1.3 and Theorem 1.3 to the proof of finiteness of a set
L of determining elements requires that the inequalities of the type (1.17) and
(1.21), or (1.30) and (1.33), or (1.33) and (1.34), as well as (1.18) with the constant
eL small enough be fulfilled. As the analysis of particular examples shows, the fulfilment of estimates (1.17), (1.21), (1.30), (1.31), (1.33), and (1.34) is mainly connected with the dissipativity properties of the system. Methods for obtaining them
are rather well-developed (see Chapters 1 and 2 and the references therein) and in
many cases the corresponding constants n , R , K , and q either are close to opti-
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mal or can be estimated explicitly in terms of the parameters of equations. Therefore, the problem of description of finite families of functionals that asymptotically
determine the dynamics of the process can be reduced to the study of sets of functionals for which estimate (1.18) holds with eL small enough. It is convenient to
base this study on the concept of completeness defect of a family of functionals with
respect to a pair of spaces.
5
§ 2
Completeness Defect
Let V and H be reflexive Banach spaces such that V is continuously and densely
embedded into H . The completeness defect of a set L of linear functionals
on V with respect to H is defined as
ì
ü
eL ( V, H ) = sup í w H : w Î V , l ( w ) = 0 , l Î L , w V £ 1 ý .
(2.1)
î
þ
It should be noted that the finite dimensionality of Lin L is not assumed here.
E x e r c i s e 2.1 Prove that the value eL ( V, H ) can also be defined by one
of the following formulae:
ì
eL ( V, H ) = sup í w
î
H:
w Î V , l (w) = 0 ,
ü
w V = 1 ý , (2.2)
þ
ì w
ü
eL ( V, H ) = sup í -----------H- : w Î V , w ¹ 0 , l ( w ) = 0 ý , (2.3)
î wV
þ
ì
eL ( V, H ) = inf í C :
î
ü
w H £ C w V , w Î V , l ( w ) = 0 ý . (2.4)
þ
E x e r c i s e 2.2 Let L1 Ì L2 be two sets in the space V * of linear functionals
on V . Show that e L ( V, H ) ³ e L ( V, H ) .
1
ˆ
2
E x e r c i s e 2.3 Let L Ì V * and let L be a weakly closed span of the set L
in the space V * . Show that eL ( V, H ) = e ˆ ( V, H ) .
L
The following fact explains the name of the value eL ( V, H ) . We remind that a set L
of functionals on V is said to be complete if the condition l ( w ) = 0 for all l Î L implies that w = 0 .
E x e r c i s e 2.4 Show that for a set L of functionals on V to be complete it is
necessary and sufficient that eL ( V, H ) = 0 .
Completeness Defect
The following assertion plays an important role in the construction of a set of determining functionals.
Theorem 2.1.
Let eL = eL ( V, H ) be the completeness defect of a set L of linear functionals on V with respect to H . Then there exists a constant CL > 0 such
that
ì
ü
ˆ
u H £ eL u V + C L × sup í l ( u ) : l Î L , l £ 1 ý
(2.5)
*
î
þ
ˆ
for any element u Î V , where L is a weakly closed span of the set L in
V *.
Proof.
Let
L ^ = {v Î V : l (v ) = 0 , l Î L}
(2.6)
ˆ
be the annihilator of L . If u
then it is evident that l ( u ) = 0 for all l ÎL .
Therefore, equation (2.4) implies that
ÎL ^ ,
u H £ eL u V
u Î L^ ,
for all
(2.7)
i.e. for u Î L ^ equation (2.5) is valid.
Assume that u Ï L ^ . Since L ^ is a subspace in V , it is easy to verify that there
exists an element w Î L ^ such that
ì
ü
u - w V = dist V ( u , L ^ ) = inf í u - v V : v Î L ^ ý .
î
þ
(2.8)
Indeed, let the sequence { vn } Ì L ^ be such that
d º dist V ( u , L ^ ) = lim u - vn V .
n®¥
It is clear that { vn } is a bounded sequence in V . Therefore, by virtue of the reflexivity of the space V , there exist an element w from L ^ and a subsequence { vn }
k
such that vn weakly converges to w as k ® ¥ , i.e. for any functional f Î V * the
k
equation
f ( u - w ) = lim f ( u - vn )
k®¥
k
holds. It follows that
f (u - w) £
lim u - vn
k®¥
k V
× f
*
£ d f
*
.
Therefore, we use the reflexivity of V once again to find that
ì
ü
u - w V = sup í f ( u - w ) : f Î V * , f = 1 ý £ d .
*
î
þ
However, u - w V ³ d . Hence, u - w V = d . Thus, equation (2.8) holds.
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Equation (2.7) and the continuity of the embedding of V into H imply that
u H £ w H + u - w H £ eL w V + C u - w V .
It is clear that
w V £ u V + u -w V.
Therefore,
u H £ eL u V + ( eL + C ) u - w V .
(2.9)
Let us now prove that there exists a continuous linear functional l0 on the space V
possessing the properties
v ÎL^ .
l0 ( u - w ) = u - w V ,
l0 * = 1 ,
l0 ( v ) = 0 for
˜
To do that, we define the functional l 0 by the formula
˜
l 0 (m) = a u - w V ,
m = v + a (u - w) ,
(2.10)
on the subspace
ì
ü
M = ím = v + a (u - w) : v Î L ^ , a Î R ý .
î
þ
˜
˜
It is clear that l 0 is a linear functional on M and l 0 ( m ) = 0 for m Î L ^ . Let us calculate its norm. Evidently
mV º
1- v ,
v + a ( u - w ) V = a × u - w + --a
V
a ¹ 0.
Since w - a -1 v Î L ^ , equation (2.8) implies that
˜
m V ³ a × u - w V = l 0 (m) , m = v + a (u - w) ,
Consequently, for any m Î M
a ¹ 0.
˜
l 0 (m) £ m V .
˜
This implies that l 0 has a unit norm as a functional on M . By virtue of the Hahn-Ba˜
nach theorem the functional l 0 can be extended on V without increase of the norm.
Therefore, there exists a functional l0 on V possessing properties (2.10). Therewith
ˆ
ˆ
l0 lies in a weakly closed span L of the set L . Indeed, if l0 Ï L , then using the reflexivity of V and reasoning as in the construction of the functional l0 it is easy to
verify that there exists an element x Î V such that l0 ( x ) ¹ 0 and l ( x ) = 0 for all
l Î L . It is impossible due to (2.10) .
In order to complete the proof of Theorem 2.1 we use equations (2.9) and
(2.10). As a result, we obtain that
u H £ eL u V + ( eL + C ) l0 ( u - w ) .
ˆ
However, l0 Î L , l0 ( u - w ) = l0 ( u ) , and l0 = 1 . Therefore, equation (2.5)
*
holds. Theorem 2.1 is proved.
Completeness Defect
E x e r c i s e 2.5 Assume that L = { l j : j = 1 , ¼ , N } is a finite set in V * .
Show that there exists a constant CL such that
max
u H £ e L ( V, H ) u V + CL
lj ( u )
j = 1, ¼, N
for all u Î V .
(2.11)
In particular, if the hypotheses of Corollaries 1.1 and 1.3 hold, then Theorem 2.1 and
equation (2.11) enable us to get rid of assumption (1.18) by replacing it with the corresponding assumption on the smallness of the completeness defect eL ( V, H ) .
The following assertion provides a way of calculating the completeness defect when
we are dealing with Hilbert spaces.
Theorem 2.2.
Let V and H be separable Hilbert spaces such that V is compactly and
densely embedded into H . Let K be a selfadjoint positive compact operator
in the space V defined by the equality
(K u , v)V = (u, v)H ,
u, v Î V .
Then the completeness defect of a set L of functionals on V can be evaluated by the formula
e L ( V, H ) =
m max ( PL K PL ) ,
(2.12)
where PL is the orthoprojector in the space V onto the annihilator
L ^ = {v Î V : l (v) = 0 , l Î L}
and m max ( S ) is the maximal eigenvalue of the operator S .
Proof.
It follows from definition (2.1) that
eL ( V, H ) = sup { u
H:
u Î BL }
where BL = L ^ Ç { v : v V £ 1 } is the unit ball in L ^ . Due to the compactness
of the embedding of V into H , the set BL is compact in H . Therefore, there exists
an element u0 Î BL such that
eL ( V, H ) 2 = u0
2
H
= ( K u0 , u0 ) V º m .
Therewith u 0 is the maximum point of the function ( K u , u ) V on the set BL .
Hence, for any v Î L ^ and s Î R 1 we have
( K ( u0 + s v ) , u0 + s v )
------------------------------------------------------------V- £ m º ( K u0 , u0 ) V .
u 0 + s v V2
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It follows that
5
( K ( u0 + s v ) , u0 + s v ) V - m u0 + s v
2
V
£ 0.
It is also clear that u0 V = 1 . Therefore,
s2 { ( K v , v ) V - m v 2 } + 2 s { ( K u0 , v ) V - m ( u0 , v ) V } £ 0
for all s Î R . This implies that
( K u0 , v ) - m ( u0 , v ) V £ 0
for any v Î L ^ . If we take - v instead of v in this equation, then we obtain the opposite inequality. Therefore,
v Î L^ .
( K u0 , v ) - m ( u0 , v ) = 0 ,
Consequently,
PL K PL u0 = m u0 ,
2
i.e. m = ( K u0 , u0 ) V = u0 H
is an eigenvalue of the operator PL K PL . It is evident
that this eigenvalue is maximal. Thus, Theorem 2.2 is proved.
Corollary 2.1.
Assume that the hypotheses of Theorem 2.2 hold. Let { ej } be an orthonormal basis in the space V that consists of eigenvectors of the operator K :
K ej = m j e j ,
( ei , ej ) V = di j ,
m1 ³ m2 ³ ¼ ,
mN ® 0 .
Then the completeness defect of the system of functionals
L = { lj Î V * : lj ( v ) = ( v , e j ) V : j = 1 , 2 , ¼ , N }
is given by the formula eL ( V, H ) =
mN + 1 .
To prove this assertion, it is just sufficient to note that PL is the orthoprojector
onto the closure of the span of elements { e j : j ³ N + 1 } and that PL commutes
with K .
E x e r c i s e 2.6 Let A be a positive operator with discrete spectrum in the
space H :
A ek = lk ek ,
l1 £ l2 £ ¼ ,
lN ® + ¥ ,
( ek , ej ) H = dk j ,
and let Fs = D ( A s ) , s Î R , be a scale of spaces generated by the
operator A (see Section 2.1). Assume that
L = { lj : lj ( v ) = ( v , ej ) H : j = 1 , 2 , ¼ , N } .
Prove that eL ( Fs , Fs ) =
-( s - s )
lN +1
(2.13)
for all s > s .
It should be noted that the functionals in Exercise 2.6 are often called modes .
Completeness Defect
Let us give several more facts on general properties of the completeness defect.
Theorem 2.3.
Assume that the hypotheses of Theorem 2.2 hold. Assume that L is a set
of linear functionals on V and K L is a family of linear bounded operators
R that map V into H and are such that R u = 0 for all u Î L ^ . Let
ì
ü
eVH ( R ) = sup í u - R u H : u V £ 1 ý
(2.14)
î
þ
be the global approximation error in H arising from the approximation
of elements v Î V by elements R v . Then
ì
ü
eL ( V ; H ) = min í e VH ( R ) : R ÎK L ý .
î
þ
(2.15)
Proof.
Let R ÎK L . Equation (2.14) implies that
H
u - R u H £ eV ( R ) u V ,
u ÎV.
Therefore, for u Î L ^ we have u H £ eVH ( R ) u V , i.e. eL ( V ; H ) £ e VH ( R ) for all
R Î K L . Let us show that there exists an operator R 0 Î K L such that
eL ( V ; H ) = e VH ( R0 ) . Equation (2.12) implies that
ì
ü
= sup í K 1 ¤ 2 PL u V : u V £ 1 ý ,
î
þ
1
2
^
¤
where PL is the orthoprojector in the space V onto L and K PL L ( V, V ) is the
norm of the operator K 1 ¤ 2 PL in the space L ( V, V ) of bounded linear operators in V .
Therefore, the definition of the operator K implies that
eL ( V ; H ) = K 1 ¤ 2 PL
L (V ; V)
ì
eL ( V ; H ) = sup í PL u
î
H
ü
: u V £ 1 ý = e VH ( I - PL ) .
þ
(2.16)
It is evident that the orthoprojector QL = I - PL belongs to K L (it projects onto
the subspace that is orthogonal to L ^ in V ). Theorem 2.3 is proved.
E x e r c i s e 2.7 Assume that L = { l j : j = 1 , ¼ , N } is a finite set. Show that
the family K L consists of finite-dimensional operators R of the
form
N
Ru =
å l (u) j ,
j
j
u ÎV,
j=1
where { j j : j = 1 , ¼ , N } is an arbitrary collection of elements
of the space V (they do not need to be distinct). How should the
choice of elements { jj } be made for the operator QL from the
proof of Theorem 2.3?
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Theorem 2.3 will be used further (see Section 3) to obtain upper estimates of
the completeness defect for some specific sets of functionals. The simplest situation
is presented in the following example.
5
E x a m p l e 2.1
Let H = L 2 ( 0 , l ) and let V = ( H 2 Ç H 01 ) ( 0 , l ) . As usual, here H s ( 0 , l ) is
the Sobolev space of the order s and H 0s ( 0 , l ) is the closure of the set C 0¥ ( 0 , l )
in H s ( 0 , l ) . We define the norms in H and V by the equalities
l
u
2
H
= u
2
º
ò(u (x) ) dx ,
2
u V2 = u ¢¢ 2 .
0
Let h = l ¤ N , xj = j h , j = 1 , ¼ , N - 1 . Consider a set of functionals
L = { l ( u ) = u ( xj ) : j = 1 , ¼ , N - 1 }
on V . Assume that R is a transformation that maps a function u Î V into its
linear interpolating spline
N -1
s (x) =
å u (x ) c æè --hx- - jöø .
j
j=1
Here c ( x ) = 1 - x for x £ 1 and c ( x ) = 0 for x > 1 . We apply Theorem 2.3 and obtain
ì
eL ( V ; H ) £ sup í u - s : u Î V ,
î
ü
u ¢¢ £ 1 ý .
þ
We use an easy verifiable equation
x
u ( x ) - s ( x ) = - --1h
xj + 1
x
ò dt ò dx ò u¢¢(y) dy ,
xj
xj
x Î [ xj , xj + 1 ] ,
t
to obtain the estimate
u-s
h 2- u ¢¢ .
£ -----3
This implies that eL ( H ; V ) £ h 2 ¤ 3 .
The assertion on the interdependence of the completeness defect eL and the Kolmogorov N -width made below enables us to obtain effective lower estimates
for eL ( V ; H ) .
Let V and H be separable Hilbert spaces such that V is continuously and
densely embedded into H . Then the Kolmogorov N -width of the embedding
of V into H is defined by the relation
Completeness Defect
ì H
ü
šN = š N ( V ; H ) = inf í e V ( F ) : F Î FN ý ,
î
þ
(2.17)
where F N is the family of all N -dimensional subspaces F of the space V and
ì
ü
eVH ( F ) = sup í dist H ( v , F ) : v V £ 1 ý
î
þ
is the global error of approximation of elements v Î V in H by elements of the subspace F . Here
ì
ü
distH ( v , F ) = inf í v - f H : f Î F ý .
î
þ
In other words, the Kolmogorov N -width šN of the embedding of V into H is the
minimal possible global error of approximation of elements of V in H by elements
of some N -dimensional subspace.
Theorem 2.4.
Let V and H be separable Hilbert spaces such that V is continuously
and densely embedded into H . Then
ì
ü
šN ( V ; H ) = min í eL ( V ; H ) : L Ì V * ; dim Lin L = N ý = m N + 1 , (2.18)
î
þ
where { m j } is the nonincreasing sequence of eigenvalues of the operator K
defined by the equality ( u , v ) H = ( K u , v ) V .
The proof of the theorem is based on the lemma given below as well as on the fact
that ( u , v) H = ( K u , v ) V , where K is a compact positive operator in the space V
(see Theorem 2.2). Further the notation { e j }¥
stands for the proper basis of the
j=1
operator K in the space V while the notation m j stands for the corresponding eigenvalues:
K ej = mj ej ,
m1 ³ m2 ³ ¼ ,
mn ® 0 ,
( ei , ej ) V = di j .
ì 1
ü
- e i ý is an orthonormalized basis in the space H .
It is evident that í --------m
î i þ
Lemma 2.1.
Assume that the hypotheses of Theorem 2.4 hold. Then
ì
ü
min í eL ( V ; H ) : L Ì V * , dim Lin L = N ý =
î
þ
mN +1 .
(2.19)
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Proof.
By virtue of Corollary 2.1 it is sufficient to verify that
eL ( V ; H ) ³
mN + 1
for all L = { lj : j = 1 , ¼ , N } , where lj are linearly independent functionals
on V . Definition (2.1) implies that
[ eL ( V ; H ) ] 2 ³ u
2
H
¥
=
å m (u, e )
2
j V
j
(2.20)
j=1
for all u Î V such that u V = 1 and lj ( u ) = 0 , j = 1 , ¼ , N . Let us substitute in (2.20) the vector
N +1
u=
åc
j
ej ,
j=1
where the constants c j are choosen such that lj ( u ) = 0 for j = 1 , ¼ , N and
u V = 1 . Therewith equation (2.20) implies that
N +1
[ eL ( V ; H ) ] 2 ³
å
m j c j2 ³ m N + 1
j=1
N +1
åc
2
j
= mN + 1 u
2
V
= mN + 1 .
j=1
Thus, Lemma 2.1 is proved.
We now prove that šN = min { eL } . Let us use equation (2.16)
ì
eL ( V ; H ) = sup í u - QL u
î
H
:
ü
u V £ 1ý .
þ
(2.21)
Here QL is the orthoprojector onto the subspace QL V orthogonal to L ^ in V . It is
evident that QL V is isomorphic to Lin L . Therefore, dim QL V = N . Hence, equation (2.21) gives us that
ì
eL ( V ; H ) ³ sup í dist H ( u , QL V ) : u
î
V
ü
£ 1 ý ³ šN
þ
(2.22)
for all L Î V * such that dim Lin L = N . Conversely, let F be an N -dimensional
subspace in V and let { fj : j = 1 , ¼ , N } be a orthonormalized basis in the space H .
Assume that
L F = { lj : lj ( v ) = ( fj , v ) H : j = 1 , ¼ , N } .
Let Q H , F be the orthoprojector in the space H onto F . It is clear that
N
QH , F u =
^
å (u, f )
j H
fj .
j=1
Therefore, if u Î L F , then QH , F u = 0 . It is clear that Q H , F is a bounded operator
from V into H . Using Theorem 2.3 we find that
Completeness Defect
ì
eL ( V ; H ) £ e VH ( Q H , F ) = sup í u - Q H , F u
F
î
H
:
ü
u V £ 1ý .
þ
However, u - Q H , F u H = dist H ( u , F ) . Hence,
min { eL } £ eL ( V ; H ) £ e VH ( F )
F
(2.23)
for any N -dimensional subspace F in V . Equations (2.22) and (2.23) imply that
ì
ü
šN ( V ; H ) = min í eL ( V ; H ) : L Ì V * , dim LinL = N ý .
î
þ
This equation together with Lemma 2.1 completes the proof of Theorem 2.4.
E x e r c i s e 2.8 Let Vk and Hk be reflexive Banach spaces such that Vk is continuously and densely embedded into H k , let L k be a set of linear
functionals on Vk , k = 1 , 2 . Assume that
L = L1 È L 2 Ì ( V1 ´ V2 ) * ,
where
ì
ü
L k = í l Î ( V1 ´ V2 ) * : l ( v1 , v2 ) = l ( vk ) , l Î L k ý ,
î
þ
k = 1, 2 .
Prove that
ì
ü
eL ( V1 ´ V2 , H1 ´ H2 ) = max í eL ( V1 , H1 ) , eL ( V2 , H2 ) ý .
1
2
î
þ
E x e r c i s e 2.9 Use Lemma 2.1 and Corollary 2.1 to calculate the Kolmogorov
N -width of the embedding of the space Fs = D ( A s ) into Fs = D ( As )
for s > s , where A is a positive operator with discrete spectrum.
E x e r c i s e 2.10 Show that in Example 2.1 š N ( V, H ) = l 2 × [ p ( N + 1 ) ] -2 .
Prove that p -2 h 2 £ eL ( V ; H ) £ h 2 ¤ 3 .
E x e r c i s e 2.11 Assume that there are three reflexive Banach spaces V Ì
Ì W Ì H such that all embeddings are dense and continuous.
Let L be a set of functionals on W . Prove that eL ( V, H ) £
£ eL ( V, W ) × eL ( W, H ) (Hint: see (2.3)).
E x e r c i s e 2.12 In addition to the hypotheses of Exercise 2.11, assume that
the inequality
q
× u V1 - q , u Î V ,
u W £ aq u H
holds for some constants aq > 0 and q Î ( 0 , 1 ) . Show that
--1-
1
----------
[ a-q1 eL ( V, W ) ] q £ eL ( V, H ) £ [ aq eL ( W, H ) ] 1- q .
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5
§3
Estimates of Completeness Defect
in Sobolev Spaces
In this section we consider several families of functionals on Sobolev spaces that are
important from the point of view of applications. We also give estimates of the corresponding completeness defects. The exposition is quite brief here. We recommend
that the reader who does not master the theory of Sobolev spaces just get acquainted with the statements of Theorems 3.1 and 3.2 and the results of Examples 3.1 and
3.2 and Exercises 3.2–3.6.
We remind some definitions (see, e.g., the book by Lions-Magenes [4]). Let W
be a domain in R n . The Sobolev space H m ( W ) of the order m ( m = 0 , 1 , 2 , ¼ )
is a set of functions
ì
H m (W ) = í f Î L2 (W ) : D j f (x ) Î L2 (W ) ,
î
where j = ( j1 , ¼ , jn ) , jk = 0 , 1 , 2 , ¼ ,
ü
j £ mý ,
þ
j = j1 + ¼ + jn and
¶j f
D j f ( x ) = ------------------------------------------------------ .
j
j
j
¶ x1 1 × ¶ x2 2 ¼ ¶ xn n
(3.1)
The space H m ( W ) is a separable Hilbert space with the inner product
(u , v )m =
å òD
ju
× D j v dx .
j £ m W
H 0m ( W )
Further we also use the space
which is the closure (in H m ( W ) ) of the set
¥
C 0 ( W ) of infinitely differentiable functions with compact support in W and the
space H s ( Rn ) which is defined as follows:
ì
H s ( Rn ) = í u ( x ) Î L2 ( Rn ) :
î
ò (1 + y
2 s
) û ( y ) 2 dy º u
2
s
Rn
ü
< ¥ý ,
þ
where s ³ 0 , û ( y ) is the Fourier transform of the function u ( x ) ,
û ( y ) =
òe
ixy
u ( x ) dx ,
Rn
2
y12
+ y22 ,
+¼
y =
and x y = x1 y1 + ¼ + xn yn . Evidently this definition coincides with the previous one for natural s and W = R n .
E x e r c i s e 3.1 Show that the norms in the spaces H s ( R n ) possess the property
u s ( q ) £ u sq × u s1 - q ,
1
2
0 £ q £ 1,
s ( q ) = q s1 + ( 1 - q ) s2 ,
s1 , s2 ³ 0 .
Estimates of Completeness Defect in Sobolev Spaces
We can also define the space H s ( W ) as restriction (to W ) of functions from H s ( R n )
with the norm
ì
ü
: v ( x ) = u ( x ) in W , v Î H s ( R n ) ý
u s , W = inf í v
s , Rn
î
þ
and the space H 0s ( W ) as the closure of the set C 0¥ ( W ) in H s ( W ) . The spaces H s ( W )
and H 0s ( W ) are separable Hilbert spaces. More detailed information on the Sobolev
spaces can be found in textbooks on the theory of such spaces (see, e.g., [4], [5]).
The following version of the Sobolev integral representation will be used further.
Lemma 3.1.
Let W be a domain in R n and let l ( x ) be a function from L¥( R n ) such
that
ò l (x) dx = 1 .
supp l Ì Ì W ,
(3.2)
Rn
Assume that W is a star-like domain with respect to the support supp l
of the function l . This means that for any point x Î W the cone
Vx = { z = t x + ( 1 - t ) y ; 0 £ t £ 1 , y Î supp l }
(3.3)
belongs to the domain W . Then for any function u ( x ) Î H m ( W ) the representation
å -----a!- ò (x - y)
m
u ( x ) = Pm - 1 ( x ; u ) +
a =m
a
K ( x , y ) D a u ( y ) dy
(3.4)
Vx
is valid, where
Pm - 1 ( x , u ) =
å
a < m
a = ( a1 , ¼ , an ) ,
1----a!
ò l (y) ( x - y)
D a u ( y ) dy ,
a! = a1! ¼ an ! ,
òs
-n - 1 l æx
è
a
y-x
+ ------------ö ds .
s ø
Proof.
If we multiply Taylor’s formula
å
a <m
+ m
å
(x - y)a
--------------------- D a u ( y ) +
a!
(x - y)a
--------------------a!
a =m
1
òs
0
m - 1 D a u (x
a
za = z1 1 ¼ zn n ,
0
u (x) =
(3.5)
W
1
K (x, y) =
a
+ s ( y - x ) ) ds
(3.6)
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by l ( y ) and integrate it over y , then after introducing a new variable
z = x + s ( y - x ) we obtain the assertion of the lemma.
Integral representation (3.4) enables us to obtain the following generalization of the
Poincaré inequality.
5
Lemma 3.2.
Let the hypotheses of Lemma 3.1 be valid for a bounded domain
W Ì R n and for a function l ( x ) . Then for any function u ( x ) Î H 1 ( W )
the inequality
u - á uñ l
s
£ ------n- d n + 1 l
n
L2 ( W )
L¥ ( W )
× Ñu
(3.7)
L2( W )
is valid, where á uñ l = ò l ( x ) u ( x ) dx , s n is the surface measure of
W
the unit sphere in R n and d = diam W º sup { x - y : x , y Î W } .
Proof.
We use formula (3.4) for m = 1 :
n
u ( x ) = á uñ l +
å ò K (x, y) (x
j=1
j
¶u
- yj ) × -------- ( y ) dy .
¶ yj
(3.8)
Vx
It is clear that l ( x + ( 1 ¤ s ) ( y - x ) ) = 0 when s -1 x - y ³ d º diam W . Therefore,
1
K (x, y) =
ò
--- ( y - x )ö ds .
s -n - 1 l æ x + 1
è
ø
s
d -1 x - y
Consequently,
d
K ( x , y ) £ -----------------n- ,
x -y
dn
d º d ( l , n ) = ----n l
L¥( W )
.
(3.9)
Thus, it follows from (3.8) that
u ( x ) - á uñ l
Ñu ( y )
- dy £
£ d -----------------------x - y n -1
ò
W
æ
£ dç
è
ò
W
ö 1/2 æ
Ñu ( y ) 2
-----------------------d
y
÷ ç
x - y n -1 ø è
ò
W
ö
dy
-----------------------n - 1÷
x-y
ø
1/2
.
(3.10)
Let Br ( x ) = { y : x - y £ r } . Then it is evident that
dy
ò -----------------------x-y
n -1
W
£
ò
Bd ( x )
dy
------------------------ £ sn × d ,
x - y n -1
(3.11)
Estimates of Completeness Defect in Sobolev Spaces
where sn is the surface measure of the unit sphere in R n . Therefore, equation
(3.10) implies that
u ( x ) - á uñ l 2 £ d 2 × sn × d
ò
W
Ñu ( y ) 2
------------------------ dy .
x - y n -1
After integration with respect to x and using (3.11) we obtain (3.7). Lemma 3.2
is proved.
Lemma 3.3.
Assume that the hypotheses of Lemma 3.1 are valid for a bounded domain W from R n and for a function l ( x ) . Let mn = [ n ¤ 2 ] + 1 , where
[ . ] is a sign of the integer part of a number. Then for any function
m
u ( x ) Î H n ( W ) we have the inequality
max u ( x ) - Pm - 1 ( x ; u )
n
£
x ÎW
c m + n--£ ----n- d n 2 l
n
å
L¥( W )
a = mn
mn
-------- D a u L2( W ) ,
a!
(3.12)
where Pm - 1( x ; u ) is defined by formula (3.5), cn = [ s n ¤ ( 2 mn - n ) ] 1 / 2 ,
s n is the surface measure of the unit sphere in R n , and d = diam W .
Proof.
Using (3.4) and (3.9) we find that
u ( x ) - Pm - 1 ( x ; u ) £
n
£
å
d mn
-----------a!
å
d mn
------------ D a u
a!
a = mn
£
a = mn
ò
x-y
å
a = mn
mn - n
mn
-------a!
ò x-y
mn
K (x , y) × D a u (y) dy £
Vx
× D a u (y) dy £
W
æ
×
L2( W ) ç
è
ò
x -y
2 ( mn - n )
W
ö 1/2
d y÷ .
ø
As above, we obtain that
ò x-y
W
d
2 ( mn - n )
d y £ sn
òr
2 mn - n - 1
d r £ sn d
0
Thus, equation (3.12) is valid. Lemma 3.3 is proved.
2 mn - n
× ( 2 mn - n ) -1 .
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Lemma 3.4.
Assume that the hypotheses of Lemma 3.3 hold. Then for any function
m
u ( x ) Î H n ( W ) and for any x* Î W the inequality
u - u (x )
5
*
L2 ( W )
£ Cn × ( dn l
æ
º ç
è
ö 1/2
£
u ( x ) - u ( x ) 2 dx÷
*
ø
ò
W
mn
L¥( W )
å d å -----a!-
) ×
1
j
j=1
Dau
a =j
L2 (W )
(3.13)
is valid, where Cn is a constant that depends on n only and d is the
diameter of the domain W .
Proof.
It is evident that
u - u (x )
* L2 ( W )
£
u - á uñ l
L2 ( W )
+ d n / 2 á uñ l - u ( x ) ,
*
(3.14)
where
á uñ l =
ò l(y) u (y) dy .
W
The structure of the polynomial Pm - 1 ( x , u ) implies that
á uñ l - u ( x ) £
£ u ( x ) - Pm
n -1
£ u ( x ) - Pm
(x, u) +
n -1
å
1----a!
å
da
1 £ a £ mn - 1
(x, u) +
1 £ a £ mn - 1
ò l (y) (x - y)
a
D a u (y) dy £
W
-------- l 2
× D a u L2( W )
L (W)
a!
for all x Î W . Therefore, estimate (3.13) follows from (3.14) and Lemmata 3.2
and 3.3. Lemma 3.4 is proved.
These lemmata enable us to estimate the completeness defect of two families of
functionals that are important from the point of view of applications. We consider
these families of functionals on the Sobolev spaces in the case when the domain is
strongly Lipschitzian, i.e. the domain W Ì R n possesses the property: for every
x Î ¶ W there exists a vicinity U such that
U Ç W = { x = ( x1 , ¼ , xn ) : xn < f ( x1 , ¼ , xn - 1 ) }
Estimates of Completeness Defect in Sobolev Spaces
in some system of Cartesian coordinates, where f ( x ) is a Lipschitzian function.
For strongly Lipschitzian domains the space H s( W ) consists of restrictions to W
of functions from H s ( R n ) , s > 0 (see [5] or [6]).
Theorem 3.1.
Assume that a bounded strongly Lipschitzian domain W in R n can be
divided into subdomains { W j : j = 1 , 2 , ¼ , N } such that
N
W=
ÈW
j
Wi Ç Wj = Æ
,
j=1
for
i ¹ j.
(3.15)
Here the bar stands for the closure of a set. Assume that l j ( x ) is a function
in L¥ ( W j ) such that
supp l j Ì Ì W j ,
ò l (x) dx = 1
(3.16)
j
Wj
and W j is a star-like domain with respect to supp l j . We define the set L
of generalized local volume averages corresponding to the collection
T = { ( Wj ; lj ) : j = 1 , 2 , ¼ , N }
as the family of functionals
ì
L = í lj ( u ) =
î
ò l (x) u (x) dx ,
j
Wj
ü
j = 1, 2, ¼, N ý .
þ
(3.17)
Then the estimate
eL
(H s(W) ,
H s (W) )
ìC (n, s) (L d)s - s , s > 1 , 0 £ s £ s ,
ï
£ í
(3.18)
s
---ï C L1 - s × d s - s , 0 £ s £ s £ 1 , s ¹ 0 ,
î n
ì
ü
holds, where L = max í djn lj ¥
: j = 1, 2, ¼, N ý ,
L
(
W
)
î
þ
d = max dj ,
j
dj º diam W = sup { x - y : x , y Î W j } ,
C ( n , s ) and Cn are constants.
Proof.
Let us define the interpolation operator R T for the collection T by the formula
(R T u ) ( x ) = lj ( u ) =
ò l (x) u (x) dx ,
j
Wj
x Î Wj ,
j = 1, 2, ¼, N .
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It is easy to check that
u - RT u
£ Cn L u
L2 ( W )
L2( W )
u Î L2 ( W ) .
,
It further follows from Lemma 3.2 that
u - lj ( u )
5
L2( Wj )
s
£ -----n- djn + 1 lj
n
L¥( Wj )
Ñu
L2( Wj )
.
This implies the estimate
u - RT u
L2 ( W )
s
£ -----n- d L u
n
H1( W )
.
Using the fact (see [4, 6]) that
u
H s( W )
£ C u
1- s
×
L2( W )
u s1
H (W)
0 £ s £ 1,
,
and the interpolation theorem for operators [4] we find that
u - RT u
L2( W )
s
sn
- d Lö u
£ ( Cn L) 1 - s æ ----èn
ø
H s (W)
for all 0 £ s £ 1 . Consequently, Theorem 2.3 gives the equation
eL ( H s ( W ) ; L2 ( W ) ) £ Cn L d s ,
0 £ s £ 1.
(3.19)
Using the result of Exercise 2.12 and the interpolation inequalities (see, e.g., [4,6])
u
H s (q) ( W)
£C u
q
×
s
H 1( W )
u 1 -s q
H 2(W )
,
s ( q ) = s1q + s2 ( 1 - q ) ,
0 £ q £ 1 , (3.20)
it is easy to obtain equation (3.18) from (3.19).
Let us illustrate this theorem by the following example.
E x a m p l e 3.1
Let W = ( 0 , l ) n be a cube in R n with the edge of the length l . We construct
a collection T which defines local volume averages in the following way.
Let K = ( 0 , 1 ) n be the standard unit cube in R n and let w be a measurable set
in K with the positive Lebesgue measure, mes w > 0 . We define the function
l ( w , x ) on K by the formula
-1
ì
l ( w , x ) = í [ mes w ] ,
î 0,
x Îw ,
x ÎK\w .
Assume that
x
ì
ü
W j = h × ( j + K ) º í x = ( x1 , ¼ , xn ) : ji < -----i < ji + 1 , i = 1 , 2 , ¼ , n ý ,
h
î
þ
1 l æ w , --x- - jö ,
l j ( x ) = ----h ø
hn è
x Î Wj ,
Estimates of Completeness Defect in Sobolev Spaces
for any multi-index j = ( j 1 , ¼ , jn ) , where jn = 0 , 1 , ¼ , N - 1 , h = l ¤ N .
It is clear that the hypotheses of Theorem 3.1 are valid for the collection
T = { W j , l j } . Moreover,
dj = diam W j =
n×h,
nn / 2 L = -------------mes w
and hence in this case we have
s-s
h
eL ( H s ( W ) ; H s ( W ) ) £ Cn , s æ ---------------- ö
è mes w ø
(3.21)
for the set L of functionals of the form (3.17) when s ³ 1 and 0 £ s £ s .
It should be noted that in this case the number of functionals in the set L is
equal to NL = N n . Thus, estimate (3.21) can be rewritten as
æ l ö
eL ( H s ( W ) , H s ( W ) ) £ C n , s ç --------------mes w÷
è
ø
s-s
s-s
æ 1 ö ----------- ÷ n .
× ç ------N
è L ø
However, one can show (see, e.g., [6]) that the Kolmogorov NL -width of the embedding of H s ( W ) into H s ( W ) has the same order in NL , i.e.
s-s
æ 1 ö ----------- n .
šN ( H s ( W ) , H s ( W ) ) = c 0 ç ------L
NL ÷
è ø
Thus, it follows from Theorem 2.4 that local volume averages have a completeness defect that is close (when the number of functionals is fixed) to the minimal. In the example under consideration this fact yields a double inequality
c1 h s - s £ eL ( H s ( W ) , H s ( W ) ) £ c2 h s - s ,
s < s,
(3.22)
where c 1 and c2 are positive constants that may depend on s , n , w , and W .
Similar relations are valid for domains of a more general type.
Another important example of functionals is given in the following assertion.
Theorem 3.2.
Assume the hypotheses of Theorem 3.1.. Let us choose a point xj (called
a node) in every set W j and define a set of functionals on H m ( W ) ,
m = [ n ¤ 2 ] + 1 , by
ì
ü
L = í l j ( u ) = u ( xj ) : xj Î W j , j = 1 , ¼ , N ý .
î
þ
(3.23)
Then for all s ³ m and 0 £ s £ s the estimate
eL ( H s ( W ) ; H s ( W ) ) £ C ( n , s ) ( d L ) s - s
is valid for the completeness defect of this set of functionals, where
(3.24)
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ì
ü
d = max í dj : j = 1 , 2 , ¼ , N ý ,
î
þ
ì
L = max í djn l j
î
5
L¥( W )
d j = diam W ,
ü
: j = 1, 2, ¼, N ý .
þ
Proof.
Let u Î H m ( W ) and let lj ( u ) = u ( xj ) = 0 , j = 1 , 2 , ¼ , N . Then using
(3.13) for W = W j and x = xj we obtain that
*
u
£ Cn æ djn lj
è
L2 ( W j )
m
åd
ö
L ¥ ( W j )ø
l
j
l=1
u l, W ,
j
where
u l, W =
j
å -----a!1- D
au
L2( Wj )
a =l
.
It follows that
m
u
£ C (n) × L ×
L2 ( W )
åd
l
u
l=1
Hl ( W )
for all u Î H m ( W ) such that lj ( u ) = u ( xj ) = 0 , j = 1 , 2 , ¼ , N . Using interpolation inequality (3.20) we find that
m
u
£ Cn × L ×
L2 ( W )
å
dl u
l=1
l
1 - ----m
L2( W )
× u
l/m
H m( W )
.
(3.25)
By virtue of the inequality
q
x p- + y
----x y £ -----q- ,
p
1- + 1
--p
-q- = 1 ,
x, y ³ 0 ,
we get
L × dl × u
l
1 - ----m
×
L2 ( W )
u
l/m
H m (W)
m
l- ) d -----------m-l u
£ ( 1 - ---m
= u
l
1 - ----m
2
L
m
----æ
× çd m L l u
è
m
L2 ( W )
l
ö ---m
£
÷
m
H
ø
m
----- ----l
l m l
+ ---m- × d d L u
H m (W)
for l = 1 , 2 , ¼ , m - 1 and for all d > 0 . We substitute these inequalities in (3.25)
to obtain that
m-1
u
L2 ( W )
æ
+ Cn ç
è
£ Cn
å
l=1
m
l- -----------m -l × u
( 1 - ---)
d
m
L2 ( W )
+
m-1
m m
ö m
l- - ----l- ----l---d
L
+
L
÷ d u H m (W) .
m
ø
l=1
å
(3.26)
Estimates of Completeness Defect in Sobolev Spaces
We choose d = d ( n , m ) such that
m-1
Cn
å
l=1
m
l ö -----------æ 1 - ---m-l £ 1
--- .
m-ø d
è
2
Then equation (3.26) gives us that
m
u
L2 ( W )
£ C (n) ×
å
l=1
m
l- ----l- m
---L ×d u m
m
H (W)
for all u Î H m ( W ) such that u ( xj ) = 0 , j = 1 , 2 , ¼ , N . Hence, the estimate
m
eL ( H m ( W ) ; L 2 ( W ) ) £ C ( n ) d m ×
å
m
l=1
l- ----l---L
m
is valid. Since L ³ 1 , this implies inequality (3.24) for s = 0 and s = m =
= [ n ¤ 2 ] + 1 . As in Theorem 3.1 further arguments rely on Lemma 2.1 and interpolation inequalities (3.20). Theorem 3.2 is proved.
E x a m p l e 3.2
We return to the case described in Example 3.1. Let us choose nodes xj Î W j
and assume that w = K . Then for a set L of functionals of the form (3.23) we
have
eL ( H s ( W ) ; H s ( W ) ) £ Cn , s h s - s
for all s ³ [ n ¤ 2 ] + 1 , 0 £ s £ s and for any location of the nodes xj inside the
W j . In the case under consideration double estimate (3.22) is preserved.
In the exercises below several one-dimensional situations are given.
E x e r c i s e 3.2
Prove that
pN
eL ( H 1 ( 0 , l ) ; L2 ( 0 , l ) ) = š N ( H 1 ( 0 , l ) ; L2 ( 0 , l ) ) = 1 + æ --------ö
è l ø
1
2 - --2-
for
ì
L = í l jc ( u ) =
î
E x e r c i s e 3.3
l
òu (x) cos ------jlp- x dx ,
0
ü
j = 0, 1, ¼, N - 1 ý .
þ
Verify that
pN
eL ( H01 ( 0 , l ) ; L2 ( 0 , l ) ) = š N - 1 ( H01 ( 0 , l ) ; L2 ( 0 , l ) ) = 1 + æ --------ö
è l ø
for
ì
L = í ljs ( u ) =
î
l
òu (x) sin ----------l - dx ;
0
jp x
ü
j = 1, 2, ¼, N - 1 ý .
þ
1
2 - --2
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E x e r c i s e 3.4
Let
1
( 0 , l ) = { u Î H1( 0 , l) : u ( 0) = u ( l) } .
H per
Show that
1
eL ( Hper
(0 , l) ; L2 (0 , l ) ) =
5
2 p Nö 2
1 ( 0 , l ) ; L 2 ( 0 , l ) ) = 1 + æ -----------= š2 N - 1 ( H per
è l ø
---1
2
,
where L consists of functionals l2c k and l2s k for k = 0 , 1 , ¼ , N -1
(the functionals ljc and ljs are defined in Exercises 3.2 and 3.3).
E x e r c i s e 3.5
Consider the functionals
h
lj ( u ) = --1h
ò u ( x + t ) dt ,
j
0
xj = j h ,
h = ---l- ,
N
j = 0, 1, ¼, N -1 ,
on the space L 2 ( 0 , l ) . Assume that an interpolation operator Rh
maps an element u Î L 2 ( 0 , l ) into a step-function equal to lj ( u )
on the segment [ xj , xj + 1 ] . Show that
u - Rh u
L2 ( 0 , l )
£ h u¢
L2 ( 0 , l )
.
Prove the estimate
h ---------------------£ eL ( H 1 ( 0 , l ) ; L 2 ( 0 , l ) ) £ h .
2
2
p +h
E x e r c i s e 3.6
Consider a set L of functionals
lj ( u ) = u ( xj ) ,
xj = j h ,
h = ---l- ,
N
j = 0, 1, ¼, N -1 ,
on the space H 1 ( 0 , l ) . Assume that an interpolation operator Rh
maps an element u Î H 1 ( 0 , l ) into a step-function equal to lj ( u )
on the segment [ xj , xj + 1 ] . Show that
u - Rh u
L2 (0 , l)
h
£ ------- u ¢
2
L2( 0 , l)
.
Prove the estimate
h h
--------------------£ eL ( H 1 ( 0 , l ) ; L 2 ( 0 , l ) ) £ ------- .
2
p2 + h2
Determining Functionals for Abstract Semilinear Parabolic Equations
§ 4
Determining Functionals for Abstract
Semilinear Parabolic Equations
In this section we prove a number of assertions on the existence and properties of
determining functionals for processes generated in some separable Hilbert space H
by an equation of the form
du
------- + A u = B ( u , t ) ,
dt
t > 0,
u t = 0 = u0 .
(4.1)
Here A is a positive operator with discrete spectrum (for definition see Section 2.1)
and B ( u , t ) is a continuous mapping from D ( A1/ 2 ) ´ R into H possessing the properties
B ( 0 , t ) £ M0 ,
B ( u1 , t ) - B ( u2 , t )
£ M ( r ) A1/ 2 ( u1 - u2 )
(4.2)
for all t and for all uj Î D ( A1/ 2 ) such that A1/ 2 uj £ r , where r is an arbitrary
positive number, M0 and M ( r ) are positive numbers.
Assume that problem (4.1) is uniquely solvable in the class of functions
W = C ( [ 0 , + ¥ ) ; H ) Ç C ( [ 0 , + ¥ ) ; D ( A1/ 2 ) )
and is pointwise dissipative, i.e. there exists R > 0 such that
A1/ 2 u ( t ) £ R
t ³ t0 ( u )
(4.3)
for all u ( t ) Î W . Examples of problems of the type (4.1) with the properties listed
above can be found in Chapter 2, for example.
The results obtained in Sections 1 and 2 enable us to establish the following assertion.
when
Theorem 4.1.
For the set of linear functionals L = { lj : j = 1 , 2 , ¼ , N } on the space
V = D ( A1/ 2 ) with the norm . V = A1/ 2 . H to be ( V , V ; W ) -asymptotically
determining for problem (4.1) under conditions (4.2) and (4.3), it is sufficient that the completeness defect eL ( V, H ) satisfies the inequality
eL º eL ( V, H ) < M ( R ) -1 ,
(4.4)
where M ( r ) and R are the same as in (4.2) and (4.3)..
Proof.
We consider two solutions u1 ( t ) and u2 ( t ) to problem (4.1) that lie in W .
By virtue of dissipativity property (4.3) we can suppose that
A1/ 2 uj ( t ) £ R ,
t ³ 0,
j = 1, 2 .
(4.5)
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Let u ( t ) = u1 ( t ) - u2 ( t ) . If we consider u ( t ) as a solution to the linear problem
du
------- + Au = f ( t ) º B ( u 1 ( t ) , t ) - B ( u 2 ( t ) , t ) ,
dt
then it is easy to find that
5
t
1
--- u ( t ) 2 +
2
ò
t
òA
--- u ( s ) 2 + M ( R )
( Au ( t ) , u ( t ) ) dt £ 1
2
1/ 2 u ( t )
s
× u ( t ) dt
s
for all t ³ s ³ 0 . We use (2.11) to obtain that
M ( R ) × A1/ 2 u × u
£ eL × M ( R ) × A1/ 2 u
2
+ d A1/ 2 u
2
+ C (R , L , d ) [ N(u) ]2
for any d > 0 , where
N ( u ) = max { l j ( u ) : j = 1 , 2 , ¼ , N } .
Therefore,
t
u ( t ) 2 + 2 ( 1 - d - eL M ( R ) )
òA
1/ 2 u ( t ) 2
dt £
s
t
£
u(s) 2 + C (R, L , d)
ò [ N(u(t) ) ]
2
dt .
(4.6)
s
Using (4.4) we can choose the parameter d > 0 such that 1 - d - eL M ( R ) > 0 .
Thus, we can apply Theorem 1.1 and find that under condition (4.4) equation
t +1
lim
t®¥
ò [ N (u (t) - u (t) ) ]
1
2
2
dt = 0
t
implies the equality
lim u1 ( t ) - u 2 ( t ) = 0 .
t®¥
(4.7)
In order to complete the proof of the theorem we should obtain
lim
t®¥
A1/ 2 ( u1 ( t ) - u 2 ( t ) ) = 0
(4.8)
from (4.7). To prove (4.8) it should be first noted that
lim Aq ( u1 ( t ) - u 2 ( t ) ) = 0
t®¥
(4.9)
for any 0 £ q < 1 ¤ 2 . Indeed, the interpolation inequality (see Exercise 2.1.12)
Aq u
£
u 1 - 2 q × A1/ 2 u 2 q ,
0 £ q < 1¤ 2 ,
and dissipativity property (4.5) enable us to obtain (4.9) from equation (4.7). Now
we use the integral representation of a weak solution (see (2.2.3)) and the method
applied in the proof of Lemma 2.4.1 to show (do it yourself) that
Determining Functionals for Abstract Semilinear Parabolic Equations
1
--- < b < 1 ,
2
Ab u j ( t ) £ C ( R , M0 ) ,
t ³ t0 .
Therefore, using the interpolation inequality
A1/ 2 u
--- ,
0< d< 1
2
A1 ¤ 2 - d u × A1 ¤ 2 + d u ,
£
we obtain (4.8) from (4.9). Theorem 4.1 is proved.
E x e r c i s e 4.1 Show that if the hypotheses of Theorem 4.1 hold, then equation (4.9) is valid for all 0 £ q < 1 .
The reasonings in the proof of Theorem 4.1 lead us to the following assertion.
Corollary 4.1.
Assume that the hypotheses of Theorem 4.1 hold. Then for any two weak
(in D ( A1/ 2 ) ) solutions u1 ( t ) and u2 ( t ) to problem (4.1) that are bounded on the whole axis,
ì
ü
sup í A1/ 2 u i ( t ) : -¥ < t < ¥ ý £ R ,
î
þ
i = 1, 2 ,
(4.10)
the condition lj ( u1 ( t ) ) = lj ( u2 ( t ) ) for lj Î L and t Î R implies that
u1 ( t ) º u 2 ( t ) .
Proof.
In the situation considered equation (4.6) implies that
t
u (t)
2
+ bL
òA
1/ 2 u ( t ) 2 dt
£ u (s)
2
s
for all t > s and some bL > 0 . It follows that
u (t)
£ e
- bL ( t - s )
u (s) ,
t ³ s.
Therefore, if we tend s ® - ¥ , then using (4.10) we obtain that u ( t ) = 0
for all t Î R , i.e. u1 ( t ) º u2 ( t ) .
It should be noted that Corollary 4.1 means that solutions to problem (4.1) that are
bounded on the whole axis are uniquely determined by their values on the functionals
lj . It was this property of the functionals { lj } which was used by Ladyzhenskaya [2]
to define the notion of determining modes for the two-dimensional Navier-Stokes
system. We also note that a more general variant of Theorem 4.1 can be found in [3].
Assume that problem (4.1) is autonomous, i.e. B ( u , t ) º
º B ( u ) . Let A be a global attractor of the dynamical system ( V, St )
generated by weak (in V = D ( A1/ 2 ) ) solutions to problem (4.1) and
assume that a set of functionals L = { lj : j = 1 , ¼ , N } possesses
E x e r c i s e 4.2
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property (4.4). Then for any pair of trajectories
lying in the attractor A the condition lj ( u1 ( t ) ) =
that u1 ( t ) º u 2 ( t ) for all t Î R and j = 1 , 2 , ¼ ,
u1 ( t ) and u2 ( t )
lj ( u 2 ( t ) ) implies
N.
Theorems 4.1 and 2.4 enable us to obtain conditions on the existence of N determining functionals.
Corollary 4.2.
Assume that the Kolmogorov N -width of the embedding of the space
V = D ( A1/ 2 ) into H possesses the property š N ( V ; H ) < M ( R ) -1 . Then
there exists a set of asymptotically ( V, V ; W ) -determining functionals
for problem (4.1) consisting of N elements.
Theorem 2.4, Corollary 2.1, and Exercise 2.6 imply that if the hypotheses of Theorem
4.1 hold, then the family of functionals L given by equation (2.13) is a ( V, V ; W ) determining set for problem (4.1), provided l N + 1 > M ( R ) 2 . Here M ( R ) and R are
the constants from (4.2) and (4.3). It should be noted that the set L of the form
(2.13) for problem (4.1) is often called a set of determining modes . Thus, Theorem 4.1 and Exercise 2.6 imply that semilinear parabolic equation (4.1) possesses
a finite number of determining modes.
When condition (4.2) holds uniformly with respect to r , we can omit the requirement of dissipativity (4.3) in Theorem 4.1. Then the following assertion is valid.
Theorem 4.2.
Assume that a continuous mapping B ( u , t ) from D ( A1/ 2 ) ´ R into H
possesses the properties
B ( 0 , t ) £ M0 ,
B ( u1 , t ) - B ( u 2 , t )
£ M A1/ 2 ( u1 - u2 )
(4.11)
for all uj Î D ( A1/ 2 ) . Then a set of linear functionals L = { lj : j = 1 , 2 , ¼ , N }
on V = D ( A1/ 2 ) is asymptotically ( V, V ; W ) -determining for problem (4.1),,
provided eL = eL ( V, H ) < M -1 .
Proof.
If we reason as in the proof of Theorem 4.1, we obtain that if eL < M -1 , then
for an arbitrary pair of solutions u1 ( t ) and u2 ( t ) emanating from the points u1 and
u 2 at a moment s the equation (see (4.6))
t
u (t)
2
+b
ò
s
t
A1/ 2 u ( t ) 2
dt £
u (s)
2
+C
ò [ N ( u ( t ) ) ] dt
2
(4.12)
s
is valid. Here u ( t ) = u1 ( t ) - u2 ( t ) , b = b ( eL , M ) and C ( L , M ) are positive constants, and
N ( u ) = max { lj ( u ) : j = 1 , 2 , ¼ , N } .
Determining Functionals for Abstract Semilinear Parabolic Equations
Therefore, using Theorem 1.1 we conclude that the condition
t +1
ò [ N (u(t) ) ] dt = 0
2
lim
t®¥
(4.13)
t
implies that
ì
lim í u ( t )
t®¥î
t +1
2
+
ò
t
ü
A1/ 2 u ( t ) 2 dt ý = 0 .
þ
Since u ( t ) = u1 ( t ) - u2 ( t ) is a solution to the linear equation
du
------- + A u = f ( t ) º B ( u1 ( t ) , t ) - B ( u2 ( t ) , t ) ,
dt
it is easy to verify that
(4.14)
(4.15)
t
A1/ 2 u ( t ) 2
A1/ 2 u ( s ) 2
£
--+1
2
ò
f ( t ) 2 dt
(4.16)
s
for t ³ s . It should be noted that equation (4.16) can be obtained with the help
of formal multiplication of (4.15) by u· ( t ) with subsequent integration. This conversion can be grounded using the Galerkin approximations. If we integrate equation
(4.16) with respect to s from t - 1 to t , then it is easy to see that
t
A1/ 2 u ( t )
£
t
ò
A1/ 2 u ( s ) 2
--ds + 1
2
t -1
ò
f ( t ) 2 dt .
t -1
Using the structure of the function f ( t ) and inequality (4.11), we obtain that
t
A1/ 2 u ( t )
M2
£ æ 1 + -------ö
è
2ø
ò
A1/ 2 u ( t ) 2 dt .
t -1
Consequently, (4.14) gives us that
lim A1/ 2 ( u1 ( t ) - u2 ( t ) ) = 0 .
t®¥
Therefore, Theorem 4.2 is proved.
Further considerations in this section are related to the problems possessing inertial
manifolds (see Chapter 3). In order to cover a wider class of problems, it is convenient to introduce the notion of a process.
Let H be a real reflexive Banach space. A two-parameter family { S ( t , t ) ;
t ³ t ; t , t Î R } of continuous mappings acting in H is said to be evolutionary ,
if the following conditions hold:
(a) S ( t , s ) × S ( s , t ) = S ( t , t ) , t ³ s ³ t , S ( t , t ) = I .
(b) S ( t , s ) u0 is a strongly continuous function of the variable t .
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A pair ( H , S ( t , t ) ) with S ( t , t ) being an evolutionary family in H is often called
a process . Therewith the space H is said to be a phase space and the family
of mappings S ( t , t ) is called an evolutionary operator. A curve
5
is said to be a trajectory of the process emanating from the point u0 at the moment s .
It is evident that every dynamical system ( H , St ) is a process. However, main
examples of processes are given by evolutionary equations of the form (1.1). Therewith the evolutionary operator is defined by the obvious formula
gs [ u0 ] = { u ( t ) º S ( t + s , s ) u0 : t ³ 0 }
S ( t , s ) u0 = u ( t , s ; u0 ) ,
where u ( t ) = u ( t , s ; u0 ) is the solution to problem (1.1) with the initial condition
u 0 at the moment s .
E x e r c i s e 4.3 Assume that the conditions of Section 2.2 and the hypotheses
of Theorem 2.2.3 hold for problem (4.1). Show that weak solutions
to problem (4.1) generate a process in H .
Similar to the definitions of Chapter 3 we will say that a process ( V ; S ( t , t ) ) acting
in a separable Hilbert space V possesses an asymptotically complete finite-dimensional inertial manifold { Mt } if there exist a finite-dimensional orthoprojector P in
the space V and a continuous function F ( p , t ) : P V ´ R ® ( 1 - P ) V such that
F ( p1 , t ) - F ( p2 , t )
(a)
£ L p1 - p2
V
(4.17)
V
for all p j Î P V , t Î R , where L is a positive constant;
(b) the surface
Mt = { p + F ( p , t ) : p Î P V } Ì V
(4.18)
is invariant: S ( t , t ) Mt Ì Mt ;
(c) the condition of asymptotical completeness holds: for any s Î R and
u 0 Î V there exists u*0 Î M s such that
S ( t , s ) u0 - S ( t , s ) u*0 V £ C e -g ( t - s ) ,
t > s,
(4.19)
where C and g are positive constants which may depend on u0 and
s ÎR.
E x e r c i s e 4.4 Show that for any two elements u1 , u2 Î M s , s Î R the following inequality holds
( 1 + L 2 ) -1 / 2 u1 - u2
E x e r c i s e 4.5
V
£
P ( u1 - u 2 ) V £
u1 - u 2 V .
Using equation (4.19) prove that
( 1 - P ) S ( t , s ) u0 - F ( P S ( t , s ) u0 , t )
for t ³ s .
V
£ ( 1 + L ) × C e -g ( t - s )
Determining Functionals for Abstract Semilinear Parabolic Equations
E x e r c i s e 4.6 Show that for any two trajectories u j ( t ) = S( t , s ) uj , j = 1 , 2 ,
of the process ( V ; S ( t , t ) ) the condition
lim P ( u1 ( t ) - u2 ( t ) ) V = 0
t®¥
implies that
lim u1 ( t ) - u 2 ( t ) V = 0 .
t®¥
In particular, the results of these exercises mean that Ms is homeomorphic to a subset in R N , N = dim P , for every s Î R . The corresponding homeomorphism
r : V ® R N can be defined by the equality r u = { ( u , fj ) V }N , where { f j } is a baj=1
sis in PV . Therewith the set of functionals { lj ( u ) = ( u , j j ) V : j = 1 , ¼ , N } appears
to be asymptotically determining for the process. The following theorem contains
a sufficient condition of the fact that a set of functionals { lj } possesses the properties mentioned above.
Theorem 4.3.
Assume that V and H are separable Hilbert spaces such that V is continuously and densely embedded into H . Let a process ( V ; S ( t , t ) ) possess
an asymptotically complete finite-dimensional inertial manifold { M t } .
Assume that the orthoprojector P from the definition of { M t } can be continuously extended to the mapping from H into V , i.e. there exists a constant
L = L ( P ) > 0 such that
(4.20)
Pv V £ L × v H ,
v ÎV.
If L = { lj : j = 1 , ¼ , N } is a set of linear functionals on V such that
eL ( V ; H ) < ( 1 + L2 ) -1/ 2 L-1 ,
(4.21)
then the following conditions are valid:
1) there exist positive constants c1 and c 2 depending on L such that
c1 u1 - u 2 H £ max { lj ( u1 - u2 ) : j = 1 , ¼ , N } £ c2 u1 - u 2 H (4.22)
for all u1 , u 2 Î Mt , t Î R ; i.e. the mapping r L acting from V into
R N according to the formula rL u = { lj ( u ) }jN= 1 is a Lipschitzian homeomorphism from Mt into R N for every t Î R ;
2) the set of functionals L is determining for the process ( V ; S ( t , t ) ) in
the sense that for any two trajectories uj ( t ) = S ( t , s ) uj the condition
lim ( lj ( u1 ( t ) ) - lj ( u2 ( t ) ) ) = 0
t®¥
implies that
lim u1 ( t ) - u2 ( t ) V = 0 .
t®¥
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Proof.
Let u1 , u2 Î Mt . Then
uj = P uj + F ( P uj , t ) ,
Therewith equation (4.17) gives us that
j = 1, 2 .
u1 - u2 V £ ( 1 + L 2 ) 1/ 2 Pu1 - P u2 V ,
uj Î Mt .
(4.23)
Consequently, using Theorem 2.1 and inequality (4.20) we obtain that
u1 - u 2 H £ CL NL ( u1 - u2 ) + eL ( 1 + L 2 ) 1/ 2 × L u1 - u2 H ,
where NL ( u ) = max { lj ( u ) : j = 1 , ¼ , N } and eL = eL ( V ; H ) . Therefore, equation (4.21) implies that
u1 - u2 H £ C1 ( L ) × NL ( u1 - u 2 ) ,
where
(4.24)
C1 ( L ) = CL × ( 1 - eL ( 1 + L 2 ) 1/ 2 L ) -1 .
On the other hand, (4.20) and (4.23) give us that
(4.25)
lj ( u1 - u2 ) £ CL u1 - u2 V £ CL ( 1 + L 2 ) 1/ 2 L u1 - u2 H .
Equations (4.24) and (4.25) imply estimate (4.22). Hence, assertion 1 of the theorem is proved.
Let us prove the second assertion of the theorem. Let u j ( t ) = S ( t , s ) u j , t ³ s ,
be trajectories of the process. Since
uj ( t ) = ( Puj ( t ) + F ( Puj ( t ) , t ) ) + ( ( 1 - P ) uj ( t ) - F ( P u j ( t ) , t ) ) ,
using (4.17) it is easy to find that
u1 ( t ) - u2 ( t ) V £ ( 1 + L 2 ) 1/ 2 P ( u 1 ( t ) - u2 ( t ) ) V +
+
å
( 1 - P) u j ( t ) - F ( Puj ( t ) , t )
V
.
j = 1, 2
The property of asymptotical completeness (4.19) implies (see Exercise 4.5) that
( 1 - P) u j ( t ) - F ( P uj ( t ) , t )
£ C e -g ( t - s ) ,
t ³ s.
Therefore, equation (4.20) gives us the estimate
u1 ( t ) - u2 ( t ) V £ ( 1 + L 2 ) 1/ 2 L u1 ( t ) - u2 ( t ) H + C e -g ( t - s ) .
It follows from Theorem 2.1 that
u1 ( t ) - u 2 ( t ) H £ CL NL ( u1 ( t ) - u2 ( t ) ) + eL u1 ( t ) - u2 ( t ) V .
Therefore, provided (4.21) holds, equation (4.26) implies that
u1 ( t ) - u 2 ( t ) V £ AL × NL ( u1 ( t ) - u2 ( t ) ) + BL e -g ( t - s ) ,
t ³ s,
(4.26)
Determining Functionals for Abstract Semilinear Parabolic Equations
where AL and BL are positive numbers. Hence, the condition
lim NL ( u1 ( t ) - u2 ( t ) ) = 0
t®¥
implies that
lim u1 ( t ) - u 2 ( t ) V = 0 .
t®¥
Thus, Theorem 4.3 is proved.
Assume that the hypotheses of Theorem 4.3 hold. Let
u1 , u2 Î Ms be such that lj ( u1 ) = lj ( u 2 ) for all lj ÎL . Show that
S ( t , s ) u1 º S ( t , s ) u2 for t ³ s .
E x e r c i s e 4.7
E x e r c i s e 4.8 Prove that if the hypotheses of Theorem 4.3 hold, then inequality (4.22) as well as the equation
c1 u1 - u2 V £ max { lj ( u1 - u 2 ) : j = 1 , 2 , ¼ , N } £ c2 u1 - u 2 V
is valid for any u1 , u2 Î Mt and t Î R , where c1 , c2 > 0 are constants depending on L .
Let us return to problem (4.1). Assume that B ( u , t ) is a continuous mapping from
D ( Aq ) ´ R into H , 0 £ q < 1 , possessing the properties
B ( u0 , t ) £ M æ 1 + Aq u0 ö ,
è
ø
B ( u1 , t ) - B ( u2 , t ) £ M Aq ( u1 - u2 )
for all u j Î D ( Aq ) , 0 £ q < 1 . Assume that the spectral gap condition
2M
ln + 1 - l n ³ -------- ( ( 1 + k ) lqn + 1 + lqn )
q
holds for some n and 0 < q < 2 - 2 . Here { l n } are the eigenvalues of the operator A indexed in the increasing order and k is a constant defined by (3.1.7). Under
these conditions there exists (see Chapter 2) a process ( D ( Aq ) ; S ( t , s ) ) generated
by problem (4.1). By virtue of Theorems 3.2.1 and 3.3.1 this process possesses an
asymptotically complete finite-dimensional inertial manifold { Mt } and the corresponding orthoprojector P is a projector onto the span of the first n eigenvectors
of the operator A . Therefore,
Aq P u £ lqn - s As u ,
-¥ < s < q .
Therewith the Lipschitz constant L for F ( p , t ) can be estimated by the value
q ¤ ( 1 - q ) . Thus, if L is a set of functionals on V = D ( Aq ) , then in order to apply
Theorem 4.3 with H = D ( As ) , - ¥ < s < q , it is sufficient to require that
1- q
1 - .
eL ( D ( Aq ) ; D ( As ) ) £ ------------------------------------- × -----------q-s
2
2 - 2 q + 2 q ln
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Due to Theorem 2.4 this estimate can be rewritten as follows:
5
ln + 1 ö q - s
1- q
eL ( D ( Aq ) ; D ( As ) ) £ ------------------------------------ × æ ------------× šn ( D ( Aq ) ; D ( As ) ) ,
è ln ø
2 - 2 q + 2 q2
where šn ( D ( Aq ) ; D ( As ) ) is the Kolmogorov n -width of the embedding of D ( Aq )
into D ( As ) , - ¥ < s < q , 0 £ q < 1 .
It should be noted that the assertion similar to Theorem 4.3 was first established for the Kuramoto-Sivashinsky equation
ut + ux x x x + ux x + ux u = 0 ,
x Î (0, L) ,
t > 0,
with the periodic boundary conditions on [ 0 , L ] in the case when { lj } is a set of
uniformly distributed nodes on [ 0 , L ] , i.e.
L- , j = 0 , 1 , ¼ , N - 1 .
h = --N
For the references and discussion of general case see survey [3].
lj ( u ) = u ( j h ) , where
In conclusion of this section we give one more theorem on the existence of determining functionals for problem (4.1). The theorem shows that in some cases we can
require that the values of functionals on the difference of two solutions tend to zero
only on a sequence of moments of time (cf. Theorem 1.3).
Theorem 4.4.
As before, assume that A is a positive operator with discrete spectrum:
Aek = lk ek ,
0 < l1 £ l2 £ ¼ ,
lim lk ® ¥ .
k®¥
Assume that B ( u , t ) is a continuous mapping from D ( Aq ) ´ R into H for
some 0 < q < 1 and the estimate
B ( u1 , t ) - B ( u2 , t ) £ M Aq ( u1 - u2 ) ,
u1 , u2 Î D ( Aq ) ,
holds. Let L = { lj } be a finite set of linear functionals on D ( Aq ) . Then for
any 0 < a < b there exists n = n ( a , b , q , l 1 , M ) such that the condition
--- l-nq
eL º eL ( D ( Aq ) , H ) < 1
2
implies that the set of functionals L is determining for problem (4.1)
in the sense that if for some pair of solutions { u1 ( t ) ; u2(t) } and for some sequence { tk } such that
lim tk = ¥ ,
k®¥
a £ tk + 1 - tk £ b ,
k ³ 1,
the condition
lim lj ( u1 ( tk ) - u2 ( tk ) ) = 0 ,
k®¥
lj Î L ,
holds, then
lim
t®¥
Aq ( u1 ( t ) - u 2 ( t ) ) = 0 .
(4.27)
Determining Functionals for Abstract Semilinear Parabolic Equations
Proof.
Let u ( t ) = u1 ( t ) - u2 ( t ) . Then the results of Chapter 2 (see Theorem 2.2.3 and
Exercise 2.2.7) imply that
Aq u ( t ) £ a1 e
a 2(t - s)
Aq u ( s ) ,
(4.28)
a 3 a ( t - s) ü q
ì -l
( t - s)
-e 2
( 1 - Pn ) Aq u ( t ) £ í e n + 1
+ -----------ý A u (s)
1- q
l
î
þ
n +1
(4.29)
for t ³ s ³ 0 , where a 1 , a2 , and a3 are positive numbers depending on q , l1 , and
M and Pn is the orthoprojector onto Lin { e1 , ¼ , en } . It follows form (4.29) that
( 1 - Pn ) Aq u ( t )
£ qa , b Aq u ( s ) ,
s +a £ t £ s + b ,
(4.30)
where
qa , b = e
-ln + 1 a
a3
a b
-e 2 .
+ ------------1- q
ln + 1
Let us choose n = n ( a , b , q , l 1 , M ) such that qa , b £ 1 ¤ 2 . Then equation (4.30)
gives us that
Aq u ( t )
--- Aq u ( s ) ,
£ lqn u ( t ) + 1
2
s+a £ t £ s+b.
This inequality as well as estimate (4.28) enables us to use Theorem 1.3 with V =
= D ( Aq ) and to complete the proof of Theorem 4.4.
E x e r c i s e 4.9 Assume that n is chosen such that q a , b < 1 in the proof
of Theorem 4.4. Show that the condition Pn ( u1 ( tk ) - u2 ( tk ) ) ® 0
as k ® ¥ implies (4.27).
The results presented in this section can also be proved for semilinear retarded
equations. For example, we can consider a retarded perturbation of problem (4.1)
of the following form
ì
ï
í
ï
î
du
------- + A u = B ( u , t ) + Q ( ut , t ) ,
dt
u t Î [ -r , 0 ] = j ( t ) Î Cr º C ( [ -r , 0 ] , D ( A1 ¤ 2 ) ) ,
where, as usual (see Section 2.8), u t is an element of Cr defined with the help
of u ( t ) by the equality ut ( q ) = u ( t + q ) , q Î [ - r , 0 ] , and Q is a continuous mapping from Cr ´ R into H possessing the property
0
Q ( v1 , t ) - Q ( v2 , t ) 2 £ M1 ×
òA
1 / 2 (v ( q )
1
- v 2 ( q ))
2
dq
-r
for any v1 , v2 Î Cr . The corresponding scheme of reasoning is similar to the method used in [3], where the second order in time retarded equations are considered.
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§ 5 Determining Functionals
for Reaction-Diffusion Systems
In this section we consider systems of parabolic equations of the reaction-diffusion
type and find conditions under which a finite set of linear functionals given on the
phase space uniquely determines the asymptotic behaviour of solutions. In particular, the results obtained enable us to prove the existence of finite collections of determining modes, nodes, and local volume averages for the class of systems under
consideration. It also appears that in some cases determining functionals can be given only on a part of components of the state vector. As an example, we consider
a system of equations which describes the Belousov-Zhabotinsky reaction and the
Navier-Stokes equations.
Assume that W is a smooth bounded domain in R n , n ³ 1 , H s ( W ) is the
Sobolev space of the order s on W , and H 0s ( W ) is the closure (in H s ( W ) ) of the set
of infinitely differentiable functions with compact support in W . Let . s be a norm
in H s ( W ) and let . and ( . , . ) be a norm and an inner product in L 2 ( W ) , respectively. Further we also use the spaces
H s = (H s (W) )m º H s (W) ´ ¼ ´ H s (W) ,
m ³ 1.
Notations L 2 and H0s have a similar meaning. We denote the norms and the inner
products in L 2 and H s as in L2 ( W ) and H s ( W ) .
We consider the following system of equations
¶ t u = a ( x , t ) Du - f ( x , u , Ñu ; t ) ,
u ¶W = 0 ,
x ÎW,
t > 0,
(5.1)
u ( x , 0 ) = u0 ( x ) ,
as the main model. Here u ( x , t ) = ( u1 ( x , t ) ; ¼ ; u m ( x , t ) ) , D is the Laplace operator, Ñu k = ( ¶x1 uk , ¼ , ¶ xn u k ) , and a ( x , t ) is an m -by- m matrix with the elements from L ¥ (W ´ R+ ) such that for all x Î W and t Î R +
--- × ( a + a * ) ³ m 0 × I ,
a+ ( x , t ) º 1
2
We also assume that the continuous function
m0 > 0 .
(5.2)
f = ( f1 ; ¼ ; fm ) : W ´ R ( n + 1 ) m ´ R+ ® R m
is such that problem (5.1) has solutions which belong to a class W of functions
on R + with the following properties:
a) for any u Î W there exists t0 > 0 such that
u ( t ) Î C ( t0 , + ¥ ; H 2 Ç H01 ) ,
¶ t u ( t ) Î C ( t0 , + ¥ ; L 2 ) ,
(5.3)
where C ( a , b ; X ) is the space of strongly continuous functions on
[ a , b ] with the values in X ;
Determining Functionals for Reaction-Diffusion Systems
b) there exists a constant k > 0 such that for any u , v Î W there exists
t1 > 0 such that for t > t1 we have
f ( u , Ñu ; t ) - f ( v , Ñv ; t )
£ k × æ u - v + Ñu - Ñv ö .
è
ø
(5.4)
It should be noted that if a ( x , t ) is a diagonal matrix with the elements from
C 2 ( W ´ R+ ) and f is a continuously differentiable mapping such that
f (x , u , p ; t) - f (x, v, q ; t)
Î Rm ,
£ k×( u-v + p-q )
(5.5)
Î Rn m ,
p, q
x Î W , and t Î R+ , then under natural compafor all u , v
tibility conditions problem (5.1) has a unique classical solution [7] which evidently
possesses properties (5.3) and (5.4). In cases when the dynamical system generated
by equations (5.1) is dissipative, the global Lipschitz condition (5.5) can be
weakened. For example (see [8]), if a is a constant matrix and
n
f ( x , u , Ñu ; t ) = f ( x , u ) +
å b (x) ¶
j
j=1
xj u
+ g (x) ,
where bj = ( b1j , ¼ , bjm ) Î L ¥ , g = ( g1 , ¼ , gm ) Î L 2 , and
is continuously differentiable and satisfies the conditions
f ( x , u ) u ³ m1 u
p0
,
f ( x , u ) £ m2 u
p
¶f
-------k- £ C × ( 1 + u 1 ) ,
¶uj
1 £ k £ m,
p0 - 1
+C,
f = ( f1 , ¼ , f m )
p0 > 2 ,
1 £ j £ n,
where m 1 , m2 > 0 and p1 < min ( 4 ¤ n , 2 ¤ ( n - 2 ) ) for n > 2 , then any solution
to problem (5.1) with the initial condition from L 2 is unique and possesses properties (5.3) and (5.4).
Let us formulate our main assertion.
Theorem 5.1.
A set L = { lj : j = 1 , ¼ , N } of linearly independent continuous linear
1
functionals on H 2 Ç H0 is an asymptotically determining set with respect
1
to the space H0 for problem (5.1) in the class W if
c0 m0 æ
27 k- ö 2ö -1 / 2
eL ( H 2 Ç H01 , L 2 ) < -----------× 1 + ------ × æ ----º p ( k , m0 ) ,
4 è m0 ø ø
k 2 è
(5.6)
where c0-1 = sup { w 2 : w Î ( H 2 Ç H01 ) ( W ) , Dw £ 1 } , and m 0 and k are
constants from (5.2) and (5.4).. This means that if inequality (5.6) holds,
then for some pair of solutions u , v Î W the equation
t +1
lim
t®¥
ò
t
lj ( u ( t ) ) - lj ( v ( t ) ) 2 dt = 0 ,
j = 1, ¼, N ,
(5.7)
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Theory of Functionals that Uniquely Determine Long-Time Dynamics
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implies their asymptotic closeness in the space H 1 :
5
lim u ( t ) - v ( t ) 1 = 0 .
(5.8)
t®¥
Proof.
Let u , v Î W . Then equation (5.1) for w = u - v gives us that
¶ t w = a ( x , t ) Dw - ( f ( x , u , Ñu ; t ) - f ( x , v , Ñv ; t ) ) .
If we multiply this by D w in L 2 scalarwise and use equation (5.4), then we find that
d
1
--- × ----- Dw ( t ) 2 + m 0 Dw ( t ) 2 £ k æ w ( t ) + ( Ñw ) ( t ) ö × Dw ( t )
è
ø
2 dt
for t > 0 large enough. Therefore, the inequality Ñw 2 = ( w , Dw ) £ w × Dw
enables us to obtain the estimate
d
k2 æ
27
k- ö 2 ö w 2 .
-------------- æ --------- Ñw 2 + m 0 Dw 2 £ 2
+
1
m0 è
4 è m0 ø ø
dt
(5.9)
Theorem 2.1 implies that
2
× w 22
w 2 £ C ( N , d ) max lj ( w ) 2 + ( 1 + d ) × eL
j
(5.10)
for all w Î H 2 Ç H01 and for any d > 0 , where C ( N , d ) > 0 is a constant and eL º
º eL ( H 2 Ç H01 , L 2 ) . Consequently, estimate (5.9) gives us that
2
( 1 + d ) eL
æ
ö
d
-÷ × Dw ( t ) 2 £ C × max lj ( w ) 2 ,
----- Ñw ( t ) 2 + m 0 × ç 1 - ------------------------2
j
dt
p ( k , m0 ) ø
è
where p ( k , m 0 ) is defined by equation (5.6). It follows that if estimate (5.6) is valid,
then there exists b > 0 such that
t
Ñw ( t )
2
£ e
-b ( t - t0 )
Ñw ( t0 )
2
+ C×
òe
-b ( t - t )
max lj ( w ( t ) ) 2 dt
j
t0
for all t ³ t0 , where t0 is large enough. Therefore, equation (5.7) implies (5.8).
Thus, Theorem 5.1 is proved.
E x e r c i s e 5.1 Assume that u ( t ) and v ( t ) are two solutions to equation (5.1)
defined for all t Î R . Let (5.3) and (5.4) hold for every t0 Î R and
let
sup æ Ñu ( t ) + Ñv ( t ) ö < ¥ .
ø
t < 0è
Prove that if the hypotheses of Theorem 5.1 hold, then equalities
lj ( u ( t ) ) = lj ( v ( t ) ) for j = 1 , ¼ , N and t < s for some s £ ¥ imply that u ( t ) º v ( t ) for all t < s .
Determining Functionals for Reaction-Diffusion Systems
Let us give several examples of sets of determining functionals for problem (5.1).
E x a m p l e 5.1 (determining modes,, m ³ 1 )
Let { e k } be eigenelements of the operator - D in L 2 with the Dirichlet boundary conditions on ¶ W and let 0 < l 1 £ l2 £ ¼ be the corresponding eigenvalues. Then the completeness defect of the set
ì
L = í lj :
î
lj ( w ) =
ü
ò w (x) × e (x) dx , j = 1, ¼, N ýþ
j
W
-1
can be easily estimated as follows: eL ( H 2 Ç H01 , L 2 ) £ n × l N
+ 1 (see Exercise 2.6). Thus, if N possesses the property l N + 1 > n × p ( k , m 0 ) -1 , then L
is a set of asymptotically ( H 2 Ç H01 , H01 , W ) -determining functionals for problem (5.1).
Considerations of Section 5.3 also enable us to give the following examples.
E x a m p l e 5.2 (determining generalized local volume averages)
Assume that the domain W is divided into local Lipschitzian subdomains
{ W j : j = 1 , ¼ , N } , with diameters not exceeding some given number h > 0 .
Assume that on every domain W j a function l j ( x ) Î L ¥ ( W j ) is given such
that the domain W j is star-like with respect to supp lj and the conditions
ò l (x) dx = 1 ,
j
Wj
e s s sup l j ( x )
x Î Wj
L
£ -----,
hn
hold, where the constant L > 0 does not depend on h and j . Theorem 3.1 implies that
eL ( H 2 ( W ) Ç H 01 ( W ) ; L 2 ( W ) ) £ cn h 2 × L 2
h
for the set of functionals
ì
L h = í lj ( u ) =
î
ü
ò l (x) u (x) dx , j = 1, 2, ¼, N ýþ .
j
Wj
Therewith the reasonings in the proof of Theorem 3.1 imply that cn = s n2 n -2 ,
where sn is the area of the unit sphere in R n . For every lj Î L h we define the
functionals lj( k ) on H 2 by the formula
(k)
lj ( w ) = lj ( wk ) ,
w = ( w1 , ¼ , wm ) Î H 2 ,
k = 1 , 2 , ¼ , m . (5.11)
Let
(k)
Lh( m ) = { lj ( u ) : k = 1 , 2 , ¼ , m , lj Î Lh } .
(5.12)
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One can check (see Exercise 2.8) that
e
(m ) ( H
Lh
2Ç
H01 ; L 2 ) £ cm × h 2 × L 2 .
(m)
Therefore, if h is small enough, then Lh
mining functionals for problem (5.1).
is a set of asymptotically deter-
E x a m p l e 5.3 (determining nodes, n £ 3 )
Let W be a convex smooth domain in R n , n £ 3 . Let h > 0 and let
Wj = W Ç { x = ( x1 , ¼ , xn ) : jk h < xk < ( jk + 1 ) h , k = 1 , 2 , 3 } ,
where j = ( j1 , ¼ , jn ) Î Z n Ç W . Let us choose a point xj in every subdomain
W j and define the set of nodes
Lh = { lj ( u ) = u ( xj ) : j Î Zn Ç W } ,
n £ 3.
(5.13)
Theorem 3.2 enables us to state that
eL ( H 2 ( W ) Ç H 01 ( W ) ; L2 ( W ) ) £ c × h 2 ,
h
(m)
where c is an absolute constant. Therefore, the set of functionals L h defined
by formulae (5.11) and (5.12) with L h given by equality (5.13) possesses the
property
e
L
(m) ( H
h
2
Ç H01 ; L 2 ) £ c × h 2 .
(m)
Consequently, Lh is a set of asymptotically determining functionals for problem (5.1) in the class W , provided that h is small enough.
It is also clear that the result of Exercise 2.8 enables us to construct mixed determining functionals: they are determining nodes or local volume averages depending
on the components of a state vector. Other variants are also possible.
However, it is possible that not all the components of the solution vector u ( x , t )
appear to be essential for the asymptotic behaviour to be uniquely determined.
A theorem below shows when this situation can occur.
Let I be a subset of { 1 , ¼ , m } . Let us introduce the spaces
HIs = { w = ( w1 , ¼ , wm ) Î H s : wk º 0 , k Ï I } ,
s ÎR.
We identify these spaces with ( H s ( W ) ) I , where I is the number of elements of
the set I . Notations L2I and H0s , I have the similar meaning. The set L of linear
functionals on HI2 is said to be determining if p*I L is determining, where p I is the
s
natural projection of H s onto HI .
Determining Functionals for Reaction-Diffusion Systems
Theorem 5.2.
Let a = diag ( d 1 , ¼ , dm ) be a diagonal matrix with constant elements
and let { I , I ¢} be a partition of the set { 1 , ¼ , m } into two disjoint subsets.
Assume that there exist positive constants w , k * , qi , where i = 1 , ¼ , m ,
such that for any pair of solutions u , v Î W the following inequality holds
(hereinafter w = u - v ):
):
åq
i
i ÎI
+
ì di
- Dw i
í - ---î 2
åq
i
i Î I¢
£ -w
2
ü
+ ( fi ( u , Ñu ; t ) - fi ( v , Ñv ; t ) , D wi ) ý +
þ
ì
í - di Ñwi
î
å
ü
- ( fi ( u , Ñu ; t ) - fi ( v , Ñv ; t ) , wi ) ý £
þ
2
wi 2 + k *
i ÎI¢
åw
2
i
.
(5.14)
i ÎI
Then a set = { lj : j = 340 ( 1 ) , ¼ , Nof linearly independent continuous linear
1
functionals on HI2 Ç H0 , I is an asymptotically determining set with respect
1
to the space H0 for problem (5.1) in the class W if
eL º eL ( HI2 Ç H01, I , LI2 ) < c 0 × min
i ÎI
di qi
----------- ,
2 k*
(5.15)
where c0 > 0 is defined as in (5.6).. This means that if two solutions
u , v Î W possess the property
t +1
lim
t®¥
ò
l j ( p I u ( t ) ) - l j ( p I v ( t ) ) 2 dt = 0
j = 1, ¼, N ,
for
(5.16)
t
s
where pI is the natural projection of H s onto HI , then equation (5.8)
holds.
Proof.
Let u , v Î W and w = u - v . Then
¶ t wi = d i Dwi - ( fi ( x , u , Ñu ; t ) - fi ( x , v , Ñv ; t ) )
(5.17)
for i = 1 , ¼ , m . In L2 ( W ) we scalarwise multiply equations (5.17) by - qi Dwi
for i Î I and by qi wi for i Î I ¢ and summarize the results. Using inequality (5.14)
we find that
d1
--- × -----F(w (t) ) + 1
2 dt
2
åd q
i i
Dw i
2
+w
å
wi 2 £ k*
i ÎI¢
i ÎI
i ÎI
where
F(w) =
åq
i ÎI
i
Ñwi
2
+
åq
i ÎI¢
åw
i
wi 2 .
2
i
,
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As in the proof of Theorem 5.1 (see (5.10)) we have
5
åw
2
i
i ÎI
+ d- e 2 ×
----------£ C ( N , d ) max lj ( pI w ) 2 + 1
L
j
c20
å Dw
2
i
i ÎI
for every d > 0 . Therefore, provided (5.15) holds, we obtain that
d
----- F ( w ( t ) ) + b F ( w ( t ) ) £ C × max lj ( pI w ( t ) ) 2
j
dt
with some constant b > 0 . Equation (5.16) implies that F ( w ( t ) ) ® 0 as t ® 0 .
Hence,
lim w ( t ) = 0 .
(5.18)
t®¥
However, equation (5.9) and the inequality
Ñw
w Î H 2 Ç H01 ,
£ C × Dw ,
imply that
t
Ñw ( t )
2
£ e
- b ( t - t0 )
Ñw ( t0 )
2
+C
òe
-b ( t - t0 )
w ( t ) 2 dt
t0
for all t ³ t0 with t0 large enough and for b > 0 . Therefore, equation (5.8) follows
from (5.18). Theorem 5.2 is proved.
The abstract form of Theorem 5.2 can be found in [3].
As an application of Theorem 5.2 we consider a system of equations which describe
the Belousov-Zhabotinsky reaction. This system (see [9], [10], and the references therein) can be obtained from (5.1) if we take n £ 3 , m = 3 , a ( x , t ) = diag ( d1 , d 2 , d3 )
and
f ( x , u , Ñu ; t ) º f ( u ) = ( f1 ( u ) ; f2 ( u ) ; f3 ( u ) ) ,
where
f1 ( u ) = -a ( u2 - u1 u2 + u1 - b u12 ) ,
1
f2 ( u ) = - --a- ( g u3 - u2 - u1 u2 ) ,
f3 ( u ) = -d ( u1 - u3 ) .
Here a , b , g , and d are positive numbers. The theorem on the existence of classical solutions can be proved without any difficulty (see, e.g., [7]). It is well-known
[10] that if a3 > a1 > max ( 1 , b -1 ) and a2 > g a3 , then the domain
D = { u º ( u1 , u2 , u3 ) : 0 £ uj £ aj , j = 1 , 2 , 3 } Ì R 3
is invariant (if the initial condition vector lies in D for all x Î W , then u ( x , t ) Î D
for x Î W and t > 0 ). Let W º WD be a set of classical solutions the initial conditions of which have the values in D . It is clear that assumptions (5.3) and (5.4) are
valid for W . Simple calculations show that the numbers w , k * , q2 , and q3 can be
Determining Functionals for Reaction-Diffusion Systems
chosen such that equation (5.14) holds for I = { 1 } , I ¢ = { 2 , 3 } , and q1 = 1 . Indeed, let smooth functions u ( x ) and v ( x ) be such that u ( x ) , v ( x ) Î D for all
x Î W and let w ( x ) = u ( x ) - v ( x ) . Then it is evident that there exist constants
Cj > 0 such that
d
F1 º ( f1 ( u ) - f1 ( v ) , Dw1 ) £ -----1 D w1 2 + C1 × æ w1 2 + w2 2ö ,
è
ø
2
1- w
F2 º -( f2 ( u ) - f2 ( v ) , w2 ) £ - -----2a 2
2
+ C2 × æ w1
è
F3 º -( f3 ( u ) - f3 ( v ) , w3 ) £ - --d- w3
2
2
2
+ w3 2ö ,
ø
+ C3 × w1 2 .
Consequently, for any q2 , q3 > 0 we have
d
F1 + q2 F2 + q3 F3 £ -----1- Dw1 2 + ( C 1 + q2 C 2 + q3 C3 ) w1
2
q
q3 d ö
- w 2.
+ æ C 1 - ------2- ö w2 2 + æ q2 C2 - --------è
è
2 ø 3
2a ø
2
+
It follows that there is a possibility to choose the parameters q2 and q3 such that
d
F1 + q2 F2 + q3 F3 £ -----1- Dw1 2 + k * w1 2 - w æ w2 2 + w3 2ö
è
ø
2
*
with positive constants k and w . This enables us to prove (5.14) and, hence, the
validity of the assertions of Theorem 5.2 for the system of Belousov-Zhabotinsky
equations. Therefore, if L = { lj : j = 1 , ¼ , N } is a set of linear functionals on
H 2 ( W ) Ç H01 ( W ) such that eL ( ( H 2 Ç H 01 ) ( W ) , L 2 ( W ) ) is small enough, then the
condition
t +1
lim
t®¥
ò
lj ( u1 ( t ) ) - lj ( v1 ( t ) ) 2 d t = 0 ,
j = 1, ¼, N ,
t
for some pair of solutions u ( t ) = ( u1 ( t ) , u 2 ( t ) , u 3 ( t ) ) and v ( t ) = ( v1 ( t ) , v2 ( t ) ,
v3 ( t ) ) which lie in W implies that
u ( t ) - v ( t ) 1 ® 0 as t ® ¥ .
In particular, this means that the asymptotic behaviour of solutions to the BelousovZhabotinsky system is uniquely determined by the behaviour of one of the components of the state vector. A similar effect for the other equations is discussed in the
sections to follow.
The approach presented above can also be used in the study of the Navier-Stokes
system. As an example, let us consider equations that describe the dynamics of a viscous incompressible fluid in the domain W º T 2 = ( 0 , L ) ´ ( 0 , L ) with periodic
boundary conditions:
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¶ t u - n D u + ( u , Ñ ) u + Ñp = F ( x , t ) ,
Ñu = 0 ,
x ÎT2,
x Î T2 ,
t > 0,
u ( x , 0 ) = u0 ( x ) ,
(5.19)
where the unknown velocity vector u ( x , t ) = ( u1 ( x , t ) ; u2 ( x , t ) ) and pressure
p ( x , t ) are L -periodic functions with respect to spatial variables, n > 0 , and
F ( x , t ) is the external force.
Let us introduce some definitions. Let V be a space of trigonometric polynomials v ( x ) of the period L with the values in R 2 such that div v = 0 and
òT 2 v ( x )2 d x = 0 . Let H be the closure of V in L2 , let P be the orthoprojector onto
H in L , let A = -P Du = -Du and B ( u , v ) = P ( u , Ñ) v for all u and v from
D ( A) = H Ç H 2 . We remind (see, e.g., [11]) that A is a positive operator with discrete spectrum and the bilinear operator B ( u , v ) is a continuous mapping from
D ( A) ´ D ( A) into H . In this case problem (5.19) can be rewritten in the form
¶t u + n Au + B ( u , u ) = P F ( t ) ,
u t = 0 = u0 Î H .
(5.20)
It is well-known (see, e.g., [11]) that if u0 Î H and P F ( t ) Î L ¥ ( R+ ; H ) , then problem (5.20) has a unique solution u ( t ) such that
u ( t ) Î C ( R+ ; H ) Ç C ( t0 , + ¥ ; D ( A) ) ,
t0 > 0 .
(5.21)
One can prove (see [9] and [12]) that it possesses the property
t+a
lim --1t®¥ a
ò
t
2
1 ö
-----2- æ 1 + -------------Au ( t ) 2 dt £ F
a n l1ø
n è
(5.22)
for any a > 0 . Here l 1 = ( 2 p ¤ L ) 2 is the first eigenvalue of the operator A in H
and
F = lim P F ( t ) .
t®¥
Lemma 5.1.
Let u , v Î D ( A) and let w = u - v . Then
( B ( u , u ) - B ( v , v ) , Aw ) £
2 w ¥ Ñw Au ,
( B ( u , u ) - B ( v , v ) , Aw ) £ c L Ñwk × Ñw
1/ 2
× Dw
1/ 2
(5.23)
× Au ,
(5.24)
where . ¥ is the L ¥ -norm, wk is the k -th component of the vector w ,
k = 1 , 2 , and c is an absolute constant.
Proof.
Using the identity (see [12])
( B ( u , w ) , Aw ) + ( B ( w , v ) , Aw ) + ( B ( w , w ) , Au ) = 0
for u , v Î D ( A) and w = u - v , it is easy to find that
( B ( u , u ) - B ( v , v ) , Aw ) = -( B ( w , w ) , Au ) .
Determining Functionals for Reaction-Diffusion Systems
Therefore, it is sufficient to estimate the norm B ( w , w ) . The incompressibility condition Ñw = 0 gives us that
( w , Ñ) w = ( - w1 ¶ 2 w2 + w2 ¶ 2 w1 , w1 ¶1 w2 - w2 ¶1 w1 ) ,
where ¶ i is the derivative with respect to the variable xi . Consequently,
( w , Ñ) w 2 £ 2 æ ( w1 , Ñ) w2 2 + ( w2 , Ñ) w1 2ö .
(5.25)
è
ø
This implies (5.23). Let us prove (5.24). For the sake of definiteness we let
k = 1 . We can also assume that w Î V . Then (5.25) gives us that
( w , Ñ) w
£
2 æè w1
L4
Ñw 2
L4
+ w2
¥
Ñw1 öø .
We use the inequalities (see, e.g., [11], [12])
v
L¥
£ a0 v
1/ 2
£ a1 v
1/ 2
L4
× Dv
1/ 2
and
v
× Ñv
1/ 2
,
where a0 and a1 are absolute constants (their explicit equations can be found
in [12]). These inequalities as well as a simply verifiable estimate v £
£ ( L ¤ 2 p ) × Ñv imply that
( w , Ñ) w
£
--L- ( a12 + a0 ) Ñw1 × Ñw 1/ 2 × Dw 1/ 2 .
p
This proves (5.24) for k = 1 .
Theorem 5.3.
1. A set L = { lj : j = 1 , 2 , ¼ , N } of linearly independent continuous
linear functionals on D ( A) = H 2 Ç H is an asymptotically determining set
with respect to H 1 for problem (5.20) in the class of solutions with property
(5.21) if
-1
eL º eL ( D ( A) , H ) < c1 G º c1n 2 æè lim P F ( t ) öø ,
t®¥
(5.26)
where c1 is an absolute constant.
2. Let L = { lj : j = 1 , ¼ , N } be a set of linear functionals on H 2 ( W )
and let
eL¢ º eL ( H 2 ( W ) , L 2 ( W ) ) < c2 n 4 L-3 æ lim P F ( t ) ö
èt®¥
ø
-2
,
where c 2 is an absolute constant. Then every set p*k L is an asymptotically
determining set with respect to H 1 for problem (5.20) in the class of solutions with property (5.21).. Here p k is the natural projection onto the k -th
component of the velocity vector, pk ( u 1 ; u2 ) = u k , k = 1 , 2 .
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Proof.
Let u ( t ) and v ( t ) be solutions to problem (5.20) possessing property (5.21).
Then equations (5.20) and (5.23) imply that
5
d
1
--- × ----- Ñ w 2 + n Dw 2 £
2 dt
2 w ¥ × Ñw × Au
for w = u - v . As above, Theorem 2.1 gives us the estimate
w £ C h ( w ) + eL Dw ,
h ( w ) = max lj ( w ) .
j
Therefore,
w ¥ £ a0 × w 1/ 2 × Dw 1/ 2 £ c1 h ( w ) × Dw 1/ 2 + a0 eL × Dw .
The inequality
Ñw
2
£ w × Dw
1/ 2
enables us to obtain the estimate (see Exercise 2.12) eL ( D ( A) , D ( A1/ 2 ) ) £ eL and
hence
Dw
2
-1
³ -Cd h ( w ) 2 + ( 1 - d ) eL
× Ñw
2
for every d > 0 . Therefore, we use dissipativity properties of solutions (see [8] and
[9] as well as Chapter 2) to obtain that
d---Ñw ( t ) 2 + a ( t ) × Ñw 2 £ C × h ( w ) 2 + h ( w) ,
dt
where
-1
a ( t ) = n ( 1 - d ) eL
- 2 a02 eL n -1 Au ( t ) 2 .
Equation (5.22) for a = ( n l 1 ) -1 implies that the function a ( t ) possesses properties (1.6) and (1.7), provided (5.26) holds. Therefore, we apply Lemma 1.1 to obtain
that Ñw ( t ) ® 0 as t ® ¥ , provided
t +1
lim
t®¥
ò [ h(w(t) ) ]
2
dt = 0 .
t
In order to prove the second part of the theorem, we use similar arguments. For
the sake of definiteness let us consider the case k = 1 only. It follows from (5.20)
and (5.24) that
d1
--- × ---Ñw 2 + n Dw 2 £ c L Ñw1 1/ 2 × Ñw × Dw 1/ 2 × Au .
2 dt
(5.27)
The definition of completeness defect implies that
Ñw1
1/2
£ C h ( w1) + eL¢ Dw1 1/ 2 £ C h ( w1 ) + e L
¢ Dw
where h ( w1 ) = max lj ( w1 ) .Therefore,
j
1/ 2
,
Determining Functionals in the Problem of Nerve Impulse Transmission
c L Ñw1 1/ 2 Ñw × Dw 1/ 2 Au £
¢ × Ñw × Dw × Au + C h ( w1 ) Ñw × Dw
£ c L × eL
c 2- e ¢ ö × Ñw 2 × Au
-------£ C [ h ( w1 ) ] 2 + æ š + L
è
n Lø
2
--- Dw
+ n
2
1/ 2
× Au £
2
for any š > 0 , where the constant C depends on š , n , L , and L . Consequently,
from (5.27) we obtain that
2
d
--------- e L
----- Ñw 2 + n Dw 2 £ C [ h ( w1) ] 2 + 2 æ š + Lc
¢ ö × Ñw 2 Au 2 .
n
è
ø
dt
2
1
¢ and find that
Therefore, we can choose š = d × L × c × n × eL
d
2 p-ö 2 - 2 ( 1 + d ) L
c 2- ¢ × Au 2
------------ Ñw 2 + n æ -----n eL
è Lø
dt
Ñw
2
£ C [ h ( w1 ) ] 2 ,
where d > 0 is an arbitrary number. Further arguments repeat those in the proof
of the first assertion. Theorem 5.3 is proved.
It should be noted that assertion 1 of Theorem 5.3 and the results of Section 3 enable
us to obtain estimates for the number of determining nodes and local volume averages that are close to optimal (see the references in the survey [3]). At the same time,
although assertion 2 uses only one component of the velocity vector, in general it
makes it necessary to consider a much greater number of determining functionals in
comparison with assertion 1. It should also be noted that assertion 2 remains true if
instead of wk we consider the projections of the velocity vector onto an arbitrary
a priori chosen direction [3]. Furthermore, analogues of Theorems 1.3 and 4.4 can be
proved for the Navier-Stokes system (5.19) (the corresponding variants of estimates
(4.28) and (4.29) can be derived from the arguments in [2], [8], and [9]).
§ 6 Determining Functionals
in the Problem of Nerve Impulse
Transmission
We consider the following system of partial differential equations suggested by
Hodgkin and Huxley for the description of the mechanism of nerve impulse transmission:
¶ t u - d0 ¶ x2 u + g ( u , v1 , v2 , v3 ) = 0 ,
¶ t vj - dj ¶x2 vj + kj ( u ) × ( vj - hj ( u ) ) = 0 ,
x Î (0, L) ,
x Î (0, L) ,
t > 0,
t > 0,
(6.1)
j = 1 , 2 , 3 . (6.2)
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Here d 0 > 0 , dj ³ 0 , and
5
g ( u , v1 , v2 , v3 ) = - g1 v13 v2 ( d 1 - u ) - g 2 v34 ( d 2 - u ) - g3 ( d 3 - u ) ,
(6.3)
where g j > 0 and d1 > d 3 > 0 > d 2 . We also assume that kj ( u ) and hj ( u ) are the
given continuously differentiable functions such that kj ( u ) > 0 and 0 < h j ( u ) < 1 ,
j = 1 , 2 , 3 . In this model u describes the electric potential in the nerve and vj is
the density of a chemical matter and can vary between 0 and 1 . Problem (6.1) and
(6.2) has been studied by many authors (see, e.g., [9], [10], [13] and the references
therein) for different boundary conditions. The results of numerical simulation given
in [13] show that the asymptotic behaviour of solutions to this problem can be quite
complicated. In this chapter we focus on the existence and the structure of determining functionals for problem (6.1) and (6.2). In particular, we prove that the
asymptotic behaviour of densities vj is uniquely determined by sets of functionals
defined on the electric potential u only. Thus, the component u of the state vector
( u , v1 , v2 , v3 ) is leading in some sense.
We equip equations (6.1) and (6.2) with the initial data
u t = 0 = u0 ( x ) ,
vj
t=0
= vj 0 ( x ) ,
j = 1, 2, 3 ,
(6.4)
and with one of the following boundary conditions:
= dj vj
= 0,
u x = 0 = u x = L = dj vj
x=0
x=L
¶x u
x=0
= ¶x u
x=L
= dj × ¶x vj
x=0
= dj × ¶x vj
x=L
u ( x + L , t ) - u ( x , t ) = dj × ( vj ( x + L , t ) - vj ( x , t ) ) = 0 ,
t > 0,
= 0,
t > 0,
(6.5a)
(6.5b)
x Î R 1 , t > 0 , (6.5c)
where j = 1 , 2 , 3 . Thus, we have no boundary conditions for the function vj ( x , t )
when the corresponding diffusion coefficient dj is equal to zero for some j = 1 , 2 , 3 .
Let us now describe some properties of solutions to problem (6.1)–(6.5). First
of all it should be noted (see, e.g., [10]) that the parallelepiped
ì
ü
D = í U º ( u , v1 , v2 , v3 ) : d 2 £ u £ d 1 , 0 £ vj £ 1 , j = 1 , 2 , 3 ý Ì R 4
î
þ
is a positively invariant set for problem (6.1)–(6.5). This means that if the initial data
U0 ( x ) = ( u 0 ( x ) , v10 ( x ) , v20 ( x ) , v30 ( x ) ) belongs to D for almost all x Î [ 0 , L ] ,
then
U ( t ) º ( u ( x , t ) , v1 ( x , t ) , v2 ( x , t ) , v3 ( x , t ) ) Î D
for x Î [ 0 , L ] and for all t > 0 for which the solution to problem (6.1)–(6.5) exists.
Let H0 º [H ] 4 º [ L 2 ( 0 , L ) ] 4 be the space consisting of vector-functions
U ( x ) º ( u , v1 , v2 , v3 ) , where u Î L2 ( 0 , L ) , vj Î L2 ( 0 , L ) , j = 1 , 2 , 3 .
We equip it with the standard norm. Let
Determining Functionals in the Problem of Nerve Impulse Transmission
ì
ü
H0 ( D) = í U ( x ) Î H0 : U ( x ) Î D for almost all x Î ( 0 , L ) ý .
î
þ
Depending upon the boundary conditions (6.5 a, b, or c) we use the following notations H1 = [ V1 ] 4 and H 2 = [ V2 ] 4 , where
V1 = H01 ( 0 , L ) , or
H1( 0 , L ) ,
or
1
(0, L)
H per
(6.6)
and
V2 = H 2 ( 0 , L ) Ç H 01 ( 0 , L ) , or
ì
í u ( x ) Î H 2 ( 0 , L ) : ¶x u
î
x = 0, x = L
ü
= 0 ý , or
þ
2
(0, L) ,
Hper
(6.7)
respectively. Hereinafter H s ( 0 , L ) is the Sobolev space of the order s on ( 0 , L ) ,
1
are subspaces in H 1 ( 0 , L ) corresponding to the boundary conditions
H 01 and H per
(6.5a) and (6.5c). The norm in H s ( 0 , L ) is defined by the equality
L
u s2 = ¶xs u 2 + u 2 =
ò æè ¶ u (x)
s
x
2
+ u ( x ) 2ö dx ,
ø
s = 1, 2, ¼
0
We use notations . and ( . , . ) for the norm and the inner product in H º
º L2 ( 0 , L ) . Further we assume that C ( 0 , T ; X ) is the space of strongly continuous functions on [ 0 , T ] with the values in X . The notation L p ( 0 , T ; X ) has a similar meaning.
Let dj > 0 for all j = 1 , 2 , 3 . Then for every vector U0 Î H 0 ( D) problem
(6.1)–(6.5) has a unique solution U ( t ) Î H 0 ( D) defined for all t (see, e.g., [9], [10]).
This solution lies in
C ( 0 , T ; H0 ( D) ) Ç L 2 ( 0 , T ; H1 )
for any segment [ 0 , T ] and if U0 Î H0 ( D) Ç H1 , then
U ( t ) Î C ( 0 , T ; H 0 ( D) Ç H1) Ç L2 ( 0 , T ; H 2 ) .
(6.8)
Therefore, we can define the evolutionary semigroup St in the space H0 ( D) Ç H 1
by the formula
St U0 = U ( t ) º ( u ( x , t ) , v1 ( x , t ) , v2 ( x , t ) , v3 ( x , t ) ) ,
where U ( t ) is a solution to problem (6.1)–(6.5) with the initial conditions
U0 º ( u0 ( x ) , v10 ( x ) , v20 ( x ) , v30 ( x ) ) .
The dynamical system ( H 0 ( D) Ç H1 ; St ) has been studied by many authors.
In particular, it has been proved that it possesses a finite-dimensional global attractor [9].
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If d1 = d 2 = d3 = 0 , then the corresponding evolutionary semigroup can be
defined in the space V1 = ( V1 ´ [ H 1 ( 0 , L ) ] 3 ) Ç H 0 ( D) . In this case for any segment
[ 0 , T ] we have
5
if U0 Î V1 . This assertion can be easily obtained by using the general methods
of Chapter 2.
St U0 = U ( t ) Î C ( 0 , T ; V1 ) Ç L 2 ( 0 , T ; V2 ´ ( H 1 ( 0 , L ) ) 3 ) ,
(6.9)
The following assertion is the main result of this section.
Theorem 6.1.
Let L = { lj : j = 1 , ¼ , N } be a finite set of continuous linear functionals on the space Vs , s = 1 , 2 (see (6.6) and (6.7)).
). Assume that
d0
(1)
e L º eL ( V1 , H ) £ -----------------(6.10)
d0 + K1
or
d0
( 2)
,
e L º eL ( V2 , H ) £ -----------------------2
d0 + K 22
(6.11)
where
3
K1 = ( d1 - d2 ) ×
bj ( Aj + Bj )
å ---------------------------k
*
j
j=1
(6.12)
and
2 1/ 2
3
æ
æ b j ( A j + B j )ö ö
--------------------------K2 = 2 × ç ( g1 + g 2 ) 2 + 5 ( d1 - d2 ) 2 ×
ç
÷ ÷
2 kj*
è
è
ø ø
j=1
å
(6.13)
with b1 = 3 g1 , b 2 = g1 , b 3 = 4 g2 , and
ì
ü
A j = max í ¶u kj ( u ) : d 2 £ u £ d 1 ý ,
î
þ
ì
ü
Bj = max í ¶u ( kj hj ) ( u ) : d 2 £ u £ d 1 ý ,
î
þ
ì
ü
kj* = min í kj ( u ) : d 2 £ u £ d 1 ý .
î
þ
Then L is an asymptotically determining set with respect to the space H0
for problem (6.1)–(6.5) in the sense that for any two solutions
U ( t ) = ( u ( x , t ) , v1(x , t) , v2 (x , t) , v3 (x , t) )
and
U * ( t ) = ( u* ( x , t ) , v*1 ( x , t ) , v*2 ( x , t ) , v*3 ( x , t ) )
satisfying either (6.8) with dj > 0 , or (6.9) with dj = 0 , j = 1 , 2 , 3 , the condition
Determining Functionals in the Problem of Nerve Impulse Transmission
t +1
lim
t®¥
ò
lj ( u ( t ) ) - lj ( u* ( t ) ) 2 dt = 0
j = 1, ¼, N
for
(6.14)
t
implies that
3
ì
lim í u ( t ) - u* ( t ) 2 +
vj ( t ) - v*j ( t )
t®¥î
j=1
å
2ü
ý =0 .
þ
(6.15)
Proof.
Assume that (6.10) is valid. Let
U ( t ) = ( u ( x , t ) , v1 ( x , t ) , v2 ( x , t ) , v3 ( x , t ) )
and
U * ( t ) = ( u* ( x , t ) , v*1 ( x , t ) , v*2 ( x , t ) , v*3 ( x , t ) )
be solutions satisfying either (6.8) with dj > 0 , or (6.9) with d j = 0 , j = 1 , 2 , 3 .
It is clear that
G ( U, U * ) º g ( u , v1 , v2 , v3 ) - g ( u* , v*1 , v*2 , v*3 ) =
= ( g1 v31 v2 + g2 v34 + g3 ) ( u - u* ) + h ( U , U * ) ,
where
h ( U , U * ) = - g1 ( v13 v2 - v*1 3 v*2 ) ( d 1 - u*) - g2 ( v34 - v*34 ) ( d2 - u*) .
Since U , U * Î D , it is evident that
3
h(U,
U *)
£
åa
j
vj - v*j ,
j=1
where aj = ( d 1 - d 2 ) × b j . Let w =
from (6.1) that
u - u*
and y j = vj - v*j , j = 1 , 2 , 3 . It follows
¶ t w - d0 ¶ x2 w + G ( U , U * ) = 0 ,
x Î (0 , L) ,
t > 0.
(6.16)
If we multiply (6.16) by w in L 2 ( 0 , L ) , then it is easy to find that
d1
--- × ---w 2 + d0 ¶x w 2 + g3 w 2 £
2 dt
3
åa
j
yj × w .
(6.17)
j=1
Equation (6.2) also implies that
d
1
--- × ----- y j 2 + d j ¶x y j 2 + k*j y j 2 £ ( Aj + Bj ) yj × w .
2 dt
Thus, for any 0 < e < 1 and qj > 0 , j = 1 , 2 , 3 , we obtain that
(6.18)
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3
3
ö
d æ
1
--- × ----- ç w 2 +
qj y j 2÷ + d0 ¶x w 2 + g 3 w 2 + e
k*j qj y j 2 £
2 dt è
ø
j=1
j=1
å
å
3
£ -
5
å
3
( 1 - e ) kj* qj y j
2
j=1
3
£
+
å [ a + q (A + B )] y
j
j
j
j
× w
j
£
j=1
[ aj + qj ( Aj + B j ) ]
2
× w
å ----------------------------------------------4 (1 - e) q k
*
j j
j=1
2
.
(6.19)
Theorem 2.1 gives us that
(1)
w 2 £ CL , h × max lj ( w ) 2 + ( 1 + h ) [ eL ] 2 × æ ¶x w 2 + w 2ö
è
ø
j
for any h > 0 . Therefore,
æ
ö
1
³ ç ----------------------------------- - 1÷ × w
( 1) 2
è ( 1 + h ) [ eL ]
ø
2
¶x w
2
- CL , h × max lj ( w ) 2 .
(6.20)
j
Consequently, it follows from (6.19) that
d- æ
1
--- × ---w 2+
2 d t çè
3
+e
åk q
*
j j
3
ö
qj y j 2÷ + d0 × w ( L , e , q , h ) w
ø
j=1
å
2
yj
2
+
£ CL , h × max lj ( w ) 2
(6.21)
j
j=1
for any 0 < e < 1 , qj > 0 , and h > 0 , where
3 [ a + q (A + B )]2
g
j
j
j
j
1
------------------------------------------------ .
w ( L , e , q , h ) = -----3- + ---------------------------------- - 1 *d
(1) 2
d0
(
e
)
q
4
1
k
( 1 + h ) [ eL ]
j j 0
j=1
å
We choose qj = aj × ( Aj + B j ) -1 and obtain that
K1
g
1
w ( L , e , q , h ) = -----3- + ----------------------------------- - 1 - ------------------------ .
d0 ( 1 + h ) [ e ( 1) ] 2
( 1 - e ) d0
L
It is easy to see that if (6.10) is valid, then there exist 0 < e < 1 and h > 0 such that
w ( L , e , q , h ) > 0 . Therefore, equation (6.21) gives us that
3
w(t)
2
+
åq
j
yj ( t )
2
£
j=1
æ
£ ç w0
è
where
w*
3
2
+
åq
j=1
2ö
j
yj 0 ÷
ø
t
*
× e -w t
+ CL , h ×
òe
- w*( t - t )
max lj ( w ( t ) ) 2 dt ,
j
0
> 0 . Thus, if (6.10) is valid, then equation (6.14) implies (6.15).
Determining Functionals in the Problem of Nerve Impulse Transmission
Let us now assume that (6.11) is valid. Since
-( G ( U , U * ) , ¶ x2 w ) £ - g3 ¶x w
2
+ æ ( g 1 + g2 ) × w + h ( U , U * ) ö × ¶ x2 w ,
è
ø
then it follows from (6.16) that
d
d
1
--- × ----- ¶x w 2 + -----0- ¶ x2 w 2 + g3 ¶x w 2 £
2 dt
2
2- × ( g + g ) 2 × w
£ ----d0 1 2
3
2
+ 2×
å
j=1
aj2
------ × y j 2 .
d0
(6.22)
Therefore, we can use equation (6.18) and obtain (cf. (6.19)) that
3
æ
d
1
--- × ----- ç ¶x w 2 +
qj yj
2 dt ç
j=1
è
å
ö
3
d
+ -----0- ¶ x2 w 2 + g3 ¶x w 2 + e
kj* qj yj 2 £
2
÷
j +1
ø
å
2÷
3
æ
aj2 ( A j + B j ) 2 ö
2- × ( g + g ) 2 + 9
----------------------------------------÷ × w
£ ç ----×
2
4
ç d0 1 2
*]2 d ÷
(
e
)
[
1
k
0ø
j
j=1
è
å
2
for any 0 < e < 1 and qj = 3 aj2 × [ ( 1 - e ) k*j d0 ] -1 . As above, we find that
¶x2 w
2
æ
ö
1
³ ç ----------------------------------- 1÷ × w
( 2) 2
è ( 1 + h ) [ eL ]
ø
2
- CL , h × max lj ( w ) 2
j
for any h > 0 . Therefore,
3
3
æ
ö
d
1
--- × ----- ç ¶x w 2 +
qj yj 2÷ + g3 ¶x w 2 + e
kj* qj yj 2 £ CL , h × max lj ( w ) 2 ,
2 dt ç
÷
j
j=1
j+1
è
ø
provided that
å
å
3
9 aj2 ( Aj + Bj ) 2
1
4- × ( g + g ) 2 ----------------------------------- - 1 - ------------------------------------------------ ³ 0 .
1
2
2
(2) 2
* 2 2
d02
( 1 + h ) [ eL ]
j = 1 2 ( 1 - e ) [ kj ] d0
å
As in the first part of the proof, we can now conclude that if (6.11) is valid, then (6.14)
implies the equations
lim ¶x w ( t ) = 0
t®¥
and
lim yj ( t ) = 0 ,
t®¥
j = 1, 2, 3 .
(6.23)
It follows from (6.17) that
d3
---w 2 + g 3 w 2 £ ----g3
dt
3
åa
2
j
yj ( t )
2
.
j=1
Therefore, as above, we obtain equation (6.15) for the case (6.11). Theorem 6.1
is proved.
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As in the previous sections, modes, nodes and generalized local volume averages can
be choosen as determining functionals in Theorem 6.1.
5
E x e r c i s e 6.1 Let { e k } be a basis in L 2 ( 0 , L ) which consists of eigenvectors of the operator ¶ x2 with one of the boundary conditions (6.5).
Show that the set
ì
L = í lj ( u ) =
î
L
ò e (x) u (x) dx ,
j
0
ü
j = 1, ¼, N ý
þ
is determining (in the sense of (6.14) and (6.15)) for problem (6.1)–
(6.5) for N large enough.
E x e r c i s e 6.2 Show that in the case of the Neumann boundary conditions
(6.5b) it is sufficient to choose the number N in Exercise 6.1 such
that
LN > -p
K
------j ,
d0
j = 1 or 2 .
(6.24)
Obtain a similar estimate for the other boundary conditions (Hint:
see Exercises 3.2–3.4).
E x e r c i s e 6.3
Let
ì
L- , j = 1 , ¼ , N ü . (6.25)
L = í lj : lj ( u ) = u ( xj ) , xj = j h , h = --ý
N
î
þ
Show that for every w Î V1 the estimate
2
¶x w 2 ³ -----------------------2- w 2 - CN , h max lj ( w ) 2
(1 + h) h
(6.26)
holds for any h > 0 (Hint: see Exercise 3.6).
E x e r c i s e 6.4 Use estimate (6.26) instead of (6.20) in the proof of Theorem
6.1 to show that the set of functionals (6.25) is determining for problem (6.1)–(6.5), provided that N > L × ( 2 K 1 ) ¤ d0 .
E x e r c i s e 6.5 Obtain the assertions similar to those given in Exercises 6.3
and 6.4 for the following set of functionals
ì
L = í lj ( w ) = --1h
î
h
ò l æè --ht- öø u (x + t) dt ,
j
0
L- , j = 0 , ¼ , N - 1 ü ,
xj = j h , h = --ý
N
þ
Determining Functionals in the Problem of Nerve Impulse Transmission
where the function l ( x ) Î L ¥ ( R 1 ) possesses the properties
¥
ò l (x) dx = 1 ,
supp l ( x ) Ì [ 0 , 1 ] .
-¥
It should be noted that in their work Hodgkin and Huxley used the following expressions (see [13]) for kj ( u ) and h j ( u ) :
aj ( u )
hj ( u ) = ------------------------------------- ,
aj ( u ) + bj ( u )
kj ( u ) = a j ( u ) + bj ( u ) ,
where
a1 ( u ) = e ( - 0.1 u + 2.5 ) ,
0.07
a 2 ( u ) = ---------------------------- ,
e ( - 0.05 u )
b 1 ( u ) = 4 exp ( - u ¤ 18 ) ;
1
b2 ( u ) = --------------------------------------------------- ;
1 + exp ( - 0.1 u + 3 )
a3 ( u ) = 0.1 e ( - 0.1 u + 1 ) ,
b 3 ( u ) = 0.125 exp ( -u ¤ 80 ) .
Here e ( z ) = z ¤ ( e z - 1 ) . They also supposed that d 1 = 115 , d 2 = - 12 , g1 = 120 ,
and g2 = 36 . As calculations show, in this case K1 £ 5.2 × 10 4 and K2 £ 7.4 × 10 4 .
Therefore (see Exercise 6.4), the nodes { xj = j h , h = l ¤ N , j = 0 , 1 , 2 , ¼ , N }
are determining for problem (6.1)–(6.5) when N ³ 2.3 × 10 2 × L ¤ d0 . Of course,
similar estimates are valid for modes and generalized volume averages.
Thus, for the asymptotic dynamics of the system to be determined by a small
number of functionals, we should require the smallness of the parameter L ¤ d0 .
However, using the results available (see [14]) on the analyticity of solutions to
problem (6.1)–(6.5) one can show (see Theorem 6.2 below) that the values of all
components of the state vector U = ( u , v1 , v2 , v3 ) in two sufficiently close nodes
uniquely determine the asymptotic dynamics of the system considered not depending on the value of the parameter L ¤ d0 . Therewith some regularity conditions for
the coefficients of equations (6.1) and (6.2) are necessary.
Let us consider the periodic initial-boundary value problem (6.1)–(6.5c). Assume that dj > 0 for all j and the functions kj ( u ) and h j ( u ) are polynomials such
that kj ( u ) > 0 and 0 £ hj ( u ) £ 1 for u Î [ d 2 , d 1 ] . In this case (see [14]) every
solution
U ( t ) = ( u ( x , t ) , v1 ( x , t ) , v2 ( x , t ) , v3 ( x , t ) )
possesses the following Gevrey regularity property: there exists t* > 0 such
that
¥
3
æ
ö
2÷
ç F (u (t)) 2 +
(
(
)
)
× et
F
v
t
l j
ç l
÷
l = -¥ è
j=1
ø
å
å
l
£ C
(6.27)
for some t > 0 and for all t ³ t* . Here Fl ( w ) are the Fourier coefficients of the
function w ( x ) :
347
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Theory of Functionals that Uniquely Determine Long-Time Dynamics
L
Fl ( w ) = --1- ×
L
ì
ü
ò w (x) × exp íîi 2---------Lp -l x ýþ d x ,
l = 0 , ±1 , ±2 , ¼
0
In particular, property (6.27) implies that every solution to problem (6.1)–(6.5c) becomes a real analytic function for all t large enough. This property enables us to
prove the following assertion.
Theorem 6.2.
Let dj > 0 for all j and let kj ( u ) and hj ( u ) be the polynomials possessing the properties
kj ( u ) > 0 ,
0 £ hj ( u ) £ 1
u Î [ d2 , d1 ] .
for
Let x1 and x2 be two nodes such that 0 £ x1 < x2 £ L and
x2 - x1 < 2 d0 ¤ K1 , where K1 is defined by formula (6.12).. Then for every
two solutions
U ( t ) = ( u ( x , t ) , v1 ( x , t ) , v2 ( x , t ) , v3 ( x , t ) )
and
U * ( t ) = ( u* ( x , t ) , v*1 ( x , t ) , v*2 ( x , t ) , v*3 ( x , t ) )
to problem (6.1)–(6.5 c) the condition
3
ì
ü
max í u ( xl , t ) - u* ( xl , t ) +
vj ( x l , t ) - v*j ( xl , t ) ý = 0
t ® ¥ l = 1, 2 î
þ
j=1
implies their asymptotic closeness in the space H0 :
å
lim
3
ì
lim í u ( t ) - u* ( t ) 2 +
vj ( t ) - v*j ( t )
t ® ¥î
j=1
å
2ü
ý = 0.
þ
(6.28)
Proof.
Let w = u - u* and let yj = vj - v*j , j = 1 , 2 , 3 . We introduce the notations:
x2
D = { x : x1 £ x £ x2 } ,
D = x2 - x1 , and
2
=
wD
ò w (x)
x1
Let
3
ì
ü
yj ( xl , t ) ý .
m ( t , D ) = max í w ( xl , t ) +
l = 1, 2 î
þ
j=1
å
As in the proof of Theorem 6.1, it is easy to find that
d
2
1
--- × ----- w D2 + d0 ¶x w D2 + g3 w D
£
2 dt
2
dx .
Determining Functionals in the Problem of Nerve Impulse Transmission
å
£
3
w ( xl , t ) × ¶x w ( xl , t ) +
l = 1, 2
åa
j
j=1
yj
D
× w
D
and
d
2
1
--- × ----- yj D
+ kj* yj D2
2 dt
£
å
yj ( xl , t ) × ¶x yj ( xl , t ) + ( Aj + B j ) yj
l = 1, 2
D
× w
D.
It follows from (6.27) that
3
sup
max
t > 0 x Î [ 0 , L]
ì
ü
¶x yj ( x , t ) ý < ¥ .
í ¶x w ( x , t ) +
î
þ
j=1
å
Therefore, for any 0 < e < 1 and qj = aj × ( Aj + B j ) -1 , j = 1 , 2 , 3 , we have
3
3
æ
ö
d- ç
1
2÷ + d ¶ w 2 + g w 2 + e
--- × ---qj yj D
k*j qj yj D2 £
w ( t ) D2 +
0 x
3
D
D
2 dt ç
÷
j+1
j=1
è
ø
å
£ K1 × ( 1 - e ) -1 × w
å
2
D
+ m (t, D) ,
where K 1 is defined by formula (6.12). Simple calculations give us that
x2
w
2
D
º
ò
D2
w ( x ) d x £ ( 1 + h ) --------- ×
2
2
x1
x2
ò ¶ w (x)
2
x
d x + C h × D × w ( x1 ) 2
x1
for any h > 0 . Consequently, if D
h > 0 such that
2
< 2 d0 ¤ K1 , then there exist 0 < e < 1 and
3
3
æ
ö
æ
ö
d
2
2÷
----- ç w D2 +
qj yj D
+ w×ç w D
+
qj yj D2 ÷ £ C × m ( t , D ) ,
÷
ç
÷
dt ç
j=1
j=1
è
ø
è
ø
å
å
where w is a positive constant. As in the proof of Theorem 6.1, it follows that condition m ( t , D ) ® 0 as t ® ¥ implies that
3
æ
2
2ö
qj yj D÷ = 0 .
(6.29)
lim U ( t ) - U *(t) D º lim ç w D +
t ® ¥è
t®¥
ø
j=1
Let us now prove (6.28). We do it by reductio ad absurdum. Assume that there exists
a sequence tn ® + ¥ such that
å
lim
n®¥
U ( tn ) - U *(tn ) > 0 .
(6.30)
Let { Vn } and { Vn* } be sequences lying in the attractor A of the dynamical system
generated by equations (6.1)–(6.5c) and such that
U ( tn ) - Vn ® 0 ,
U *(tn ) - Vn* ® 0 .
(6.31)
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Using the compactness of the attractor we obtain that there exist a sequence { n k }
and elements V, V * Î A such that Vn ® V and Vn* ® V * . Since
5
k
Vn - Vn* D £
k
U ( tn ) - U *(tn ) D + U ( tn ) - Vn + U *(tn ) - Vn* ,
it follows from (6.29) and (6.31) that
V - V * D = lim
k®¥
= 0.
Vn - Vn*
k
k D
Therefore, V ( x ) = V *(x) for x Î D . However, the Gevrey regularity property implies that elements of the attractor are real analytic functions. The theorem on the
uniqueness of such functions gives us that V ( x ) º V *(x) for x Î [ 0 , L ] . Hence,
Vn - Vn* ® 0 as k ® ¥ . Therefore, equation (6.31) implies that
k
k
lim
k®¥
U ( tn ) - U * ( tn ) = 0 .
k
k
This contradicts assumption (6.30). Theorem 6.2 is proved.
It should be noted that the connection between the Gevrey regularity and the existence of two determining nodes was established in the paper [15] for the first time.
The results similar to Theorem 6.2 can also be obtained for other equations (see the
references in [3]). However, the requirements of the spatial unidimensionality
of the problem and the Gevrey regularity of its solutions are crucial.
§ 7 Determining Functionals
for Second Order in Time Equations
In a separable Hilbert space H we consider the problem
u t = s = u0 ,
u· t = s = u1 ,
u·· + g u· + A u = B ( u , t ) ,
(7.1)
where the dot over u stands for the derivative with respect to t , A is a positive
operator with discrete spectrum, g > 0 is a constant, and B ( u , t ) is a continuous
mapping from D ( Aq ) ´ R into H with the property
£ M ( r ) × Aq ( u1 - u2 )
B ( u1 , t ) - B ( u2 , t )
(7.2)
for some 0 £ q < 1 ¤ 2 and for all u j Î D ( A1/ 2 ) such that A1/ 2 uj £ r . Assume
that for any s Î R , u 0 Î D ( A1/ 2 ) , and u1 Î H problem (7.1) is uniquely solvable in
the class of functions
C ( [ s , + ¥ ) ; D ( A1/ 2 ) ) Ç C 1 ( [ s , + ¥ ) , H )
(7.3)
and defines a process ( H ; S ( t , t ) ) in the space H = D ( A1/ 2 ) ´ H with the evolutionary operator given by the formula
S ( t , s ) ( u ; u ) = ( u ( t ) ; u· ( t ) ) ,
(7.4)
0
1
Determining Functionals for Second Order in Time Equations
where u ( t ) is a solution to problem (7.1) in the class (7.3). Assume that the process
( H ; S ( t , t ) ) is pointwise dissipative, i.e. there exists R > 0 such that
S ( t , s ) y0
H
£ R,
t ³ s + t0 ( y0 )
(7.5)
for all initial data y0 = ( u 0 ; u1 ) Î H . The nonlinear wave equation (see the book
by A. V. Babin and M. I. Vishik [8])
ì 2
¶ u
¶u
ï --------- D u = f (u ) , x Î W , t > 0 ,
ï ¶t 2- + g -----¶t
í
¶u
ïu
= u1 ( x ) ,
ï ¶ W = 0 , u t = 0 = u0 ( x ) , -----¶t t = 0
î
is an example of problem (7.1) which possesses all the properties listed above. Here
W is a bounded domain in R d and the function f ( u ) Î C 1 ( R ) possesses the properties:
u
-( l1
- e ) u2
- C1 £
ò f (v) d v
--- ( l 1 - e ) u 2 ,
£ C2 u f ( u ) + C3 + 1
2
0
f ¢ ( u ) £ C4 ( 1 + u b ) ,
where l 1 is the first eigenvalue of the operator - D with the Dirichlet boundary
conditions on ¶ W , C j > 0 and e > 0 are constants, b £ 2 ( d - 2 ) -1 for d ³ 3 and
b is arbitrary for d = 2 .
Theorem 7.1.
Let L = { lj : j = 1 , ¼ , N } be a set of continuous linear functionals
on D ( A1/ 2 ) . Assume that
e º eL
( D ( A1/ 2 ) ;
H) <
g
-------------------------------------------------------M ( R ) ( 4 + 4 l-11 g 2 ) 3 / 2
1
---------------1-2q
,
(7.6)
where R is the radius of dissipativity (see (7.5)),
), M ( r ) and q Î [ 0 , 1 ¤ 2 )
are the constants from (7.2),, and l 1 is the first eigenvalue of the operator
A . Then L is an asymptotically determining set for problem (7.1) in the
sense that for a pair of solutions u1 ( t ) and u2 ( t ) from the class (7.3) the
condition
t +1
lim
t®¥
ò
lj ( u1 ( t ) - u2 ( t ) ) dt = 0
for
j = 1, ¼, N
(7.7)
t
implies that
ì
ü
lim í A1/ 2 ( u1 ( t ) - u2 ( t ) ) 2 + u· 1 ( t ) - u· 2 ( t ) 2 ý = 0 .
þ
t®¥î
(7.8)
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Proof.
We rewrite problem (7.1) in the form
dy
------ + A y = B ( y , t ) ,
dt
y t = s = y0 ,
(7.9)
where
y = ( u ; u· ) ,
A y = ( -u· ; g u· + A u ) ,
B (y, t) = (0 ; B (u , t) ) .
Lemma 7.1.
There exists an exponential operator exp { - t A } in the space H =
= D ( A1/ 2 ) ´ H and
l1 g t ü
ì
g2
£ 2 1 + ------ × exp í - -------------------------y
2ý
l1
î 4 l1 + 2 g þ
where l 1 is the first eigenvalue of the operator A .
exp { -t A } y
H
H
(7.10)
,
Proof.
Let y 0 = ( u 0 ; u1 ) . Then it is evident that y ( t ) = e -tA y0 = ( u ( t ) ; u· ( t ) ) ,
where u ( t ) is a solution to the problem
(7.11)
u·· + g u· + A u = 0 ,
u t = 0 = u0 ,
u· t = 0 = u1 ,
(see Section 3.7 for the solvability of this problem and the properties of solutions). Let us consider the functional
g
--- æ u 2 + A1/ 2 u 2ö + n æ u , u· + --- uö ,
V(u) = 1
è
2 ø
2è
ø
0 < n < g,
on the space H = D ( A1/ 2 ) ´ H . It is clear that
1
--g- ) u· 2 + 1
--- ( 1 - n
--- A1/ 2 u 2 £ V ( u ) £
2
2
n- ö u·
--- + ----£ æ1
è 2 2g ø
2
--- + l-1 1 n g ) A1/ 2 u
+ (1
2
2
.
(7.12)
Moreover, for a solution u ( t ) to problem (7.11) from the class (7.3) with s = 0
we have that
d
----- V ( u ( t ) ) = - ( g - n ) u· 2 - n A1/ 2 u 2 .
dt
(7.13)
Therefore, it follows from (7.12) and (7.13) that
d
----- V ( u ( t ) ) + b V ( u ( t ) ) £
dt
n- ö bö u·
--- + ----£ - æg - n - æ 1
è
è 2 2g ø ø
2
--- + l-1 1 n gö bö A1/ 2 u
- æn - æ 1
è
è2
ø ø
Hence, for n = ( 1 ¤ 2 ) g and b = ( 2 + l-11 g 2 ) -1 × g we obtain that
2
.
Determining Functionals for Second Order in Time Equations
d
----- V ( u ( t ) ) + b V ( u ( t ) ) £ 0 ,
dt
1
--- æ u· 2 + A1/ 2 u 2ö £ V ( u ) £ æ 3
--- + l-11 g 2ö æ u· 2 + A1/ 2 u 2ö .
ø
è 4
øè
ø
4è
This implies estimate (7.10). Lemma 7.1 is proved.
It follows from (7.9) that
t
y (t) = e
-(t - s ) A
y (s) +
òe
- ( t - t )A
B ( y ( t ) , t ) dt .
s
Therefore, with the help of Lemma 7.1 for the difference of two solutions yj ( t ) =
= ( uj ( t ) ; u· j ( t ) ) , j = 1 , 2 , we obtain the estimate
t
y(t)
H
£ De
-b ( t - s)
y (s)
H
+D
òe
-b (t - t )
B ( u1 ( t ) ) - B ( u2 ( t ) ) dt , (7.14)
s
where y ( t ) = y1 ( t ) - y2 ( t ) and the constants D and b have the form
g l1
-.
b = -------------------------D = 2 1 + l-11 g 2 ,
4 l1 + 2 g 2
By virtue of the dissipativity (7.5) we can assume that yj ( t )
Therefore, equations (7.14) and (7.2) imply that
H
£ R for all t ³ s ³ s0 .
t
y (t)
H
£ De
-b ( t - s )
y (s)
H
+ D M(R)
òe
-b ( t - t )
Aq ( u1 ( t ) - u2 ( t ) ) dt .
s
The interpolation inequality (see Exercise 2.1.12)
Aq u £ u 1 - 2 q A1/ 2 u 2 q ,
--- ,
0 £ q £ 1
2
Theorem 2.1, and the result of Exercise 2.12 give us that
Aq u
1 - 2 q 1/ 2
£ CL max lj ( u ) + eL
A u ,
j
where eL = eL
( D ( A1/ 2 ) ,
H ) . Therefore,
t
y(t)
H
£ De
-b ( t - s )
y(s)
H
+ D M ( R ) × eL
òe
òe
-b ( t - t )
y(t)
H
dt +
s
t
+ C (L , D , R) ×
1 -2 q
-b ( t - t )
NL ( u ( t ) ) dt ,
s
where NL (u) = max { lj ( u ) : j = 1 , 2 , ¼ , N } . If we introduce a new unknown
function y ( t ) = e b t y ( t ) H in this inequality, then we obtain the equation
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t
y(t) £ Dy(s) + a
1-2q
where a = DM ( R ) e L
t
ò y(t) dt + C ò e
s
L ( u ( t ) ) dt
,
s
. We apply Gronwall’s lemma to obtain
5
t
y(t) £ D y(s) e
btN
a (t - s)
+
ò
C ea t
e -a t
s
ì
ï
í
ï
î
t
òe
bx
s
ü
ï
NL ( u ( x ) ) dx ý dt .
ï
þ
After integration by parts we get
t
y(t)
H
£ D y(s)
H
exp { -( b - a ) ( t - s ) } + C
òe
-( b - a ) ( t - t )
NL ( u ( t ) ) dt .
s
If equation (7.6) holds, then w = b - a > 0 . Therewith it is evident that for
0 < a < t - s we have the estimate
t
y(t)
H
£ D y (s)
H
e
-w ( t - s)
+C
-w ( t - t )
NL ( u ( t ) ) dt +
t -a
t-a
+ C
òe
òe
-w ( t - t )
NL ( u ( t ) ) dt .
s
Therefore, using the dissipativity property we obtain that
t
y (t)
H
£ DR × e
-w ( t - s)
+C
ò N ( u ( t ) ) dt + C ( R , L ) e
L
-w a
t-a
for t ³ s and 0 < a < t - s . If we fix a and tend the parameter t to infinity, then
with the help of (7.7) we find that
lim y ( t )
t®¥
H
£ C ( R , L ) e -w a
for any a > 0 . This implies (7.8). Theorem 7.1 is proved.
Unfortunately, because of the fact that condition (7.2) is assumed to hold only for
0 £ q < 1 ¤ 2 , Theorem 7.1 cannot be applied to the problem on nonlinear plate
oscillations considered in Chapter 4. However, the arguments in the proof of Theorem 7.1 can be slightly modified and the theorem can still be proved for this case
using the properties of solutions to linear nonautonomous problems (see Section 4.2).
However, instead of a modification we suggest another approach (see also [3]) which
helps us to prove the assertions on the existence of sets of determining functionals
for second order in time equations. As an example, let us consider a problem of plate
oscillations .
Determining Functionals for Second Order in Time Equations
Thus, in a separable Hilbert space H we consider the equation
ì ··
2
ï u + g u· + A2 u + M æè A1/ 2 u öø Au + L u = p ( t ) ,
í
ïu
= u0 , u· t = 0 = u1 .
î t=0
(7.15)
(7.16)
We assume that A is an operator with discrete spectrum and the function M ( z) lies
in C 1 ( R + ) and possesses the properties:
z
M (z) º
a)
ò M(x) dx
³ -az -b ,
(7.17)
0
where 0 £ a < l1 , b Î R , and l 1 is the first eigenvalue of the operator A ;
b) there exist numbers aj > 0 such that
z M ( z ) - a1 M ( z ) ³ a2 z1 + a - a3 ,
z ³ 0
with some constant a > 0 .
We also require the existence of 0 £ q < 1 and C > 0 such that
Lu £ C Aq u ,
u Î D ( Aq ) .
(7.18)
(7.19)
These assumptions enable us to state (see Sections 4.3 and 4.5) that if
u0 Î D ( A) ,
u1 Î H ,
p ( t ) Î L ¥ ( R+ , H ) ,
g > 0,
(7.20)
then problem (7.15) and (7.16) is uniquely solvable in the class of functions
W = C ( R+ ; D ( A) ) Ç C 1( R+ ; H ) .
(7.21)
Therewith there exists R > 0 such that
Au ( t ) 2 + u· ( t ) 2 £ R 2 ,
t ³ t0 ( u0 , u1 )
for any solution u ( t ) Î W to problem (7.15) and (7.16).
(7.22)
Theorem 7.2.
Assume that conditions (7.17)–(7.20) hold. Let L = { lj : j = 1 , 2 , ¼ , N }
be a set of continuous linear functionals on D ( A) . Then there exists e0 > 0
depending both on R and the parameter of equation (7.15) such that the
condition e º eL ( D ( A) ; H ) < e0 implies that L is an asymptotically determining set of functionals with respect to D ( A) ´ H for problem (7.15) and
(7.16) in the class of solutions W , i.e. for two solutions u1 ( t ) and u2 ( t )
from W the condition
t +1
lim
t®¥
implies that
ò
t
lj ( u1 ( t ) - u2 ( t ) ) 2 dt = 0 ,
lj Î L
(7.23)
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ì
ü
lim í u· 1 ( t ) - u· 2 ( t ) 2 + A ( u1 ( t ) - u2 ( t ) ) 2 ý = 0 .
t®¥î
þ
(7.24)
Proof.
Let u1 ( t ) and u2 ( t ) be solutions to problem (7.15) and (7.16) lying in W . Due
to equation (7.22) we can assume that these solutions possess the property
Auj ( t ) 2 + u· j ( t ) 2 £ R 2 ,
t ³ 0,
j = 1, 2 .
(7.25)
Let us consider the function u ( t ) = u1 ( t ) - u 2 ( t ) as a solution to equation
u·· + g u· + A2 u + M æ A1/ 2 u1 ( t ) 2ö Au = F ( u1 ( t ) , u2 ( t ) ) ,
è
ø
(7.26)
where
F ( u1 , u2 ) = M æ A1/ 2 u2 2ö - M æ A1/ 2 u1 2ö Au2 - L ( u1 - u2 ) .
è
ø
è
ø
It follows from (7.19) and (7.25) that
F ( u1 ( t ) , u 2 ( t ) )
£ C R æ A1/ 2 u + Aq u ö .
è
ø
(7.27)
Let us consider the functional
ì
ü
g
--- E ( u , u· ; t ) + n í ( u , u· ) + --- u 2 ý
V ( u , u· ; t ) = 1
2
2
î
þ
on the space H = D ( A) ´ H , where
(7.28)
E ( u , u· ; t ) = u· 2 + Au 2 + M æ A1/ 2 u1 ( t ) 2ö A1/ 2 u 2 + m u 2
è
ø
and the positive parameters m and n will be chosen below. It is clear that for
( u ; u· ) Î H we have
E ( u , u· ; t ) ³
u· 2 + Au 2 + m R A1/ 2 u 2 + m u 2 ,
where mR = min { M ( z ) : 0 £ z £ l-1 1 R 2 } . Moreover,
1- u· 2 £ ( u , u· ) + --g- u· 2 £ ----1- u·
- ----2
2g
2g
Therefore, the value m can be chosen such that
a1 æ Au
è
2
2
+ u· 2ö £ V ( u , t ) £ a 2 æ Au
ø
è
+g u
2
2
.
+ u· 2ö
ø
(7.29)
for all 0 < n < g , where a 1 and a2 are positive numbers depending on R . Let us
now estimate the value ( d ¤ dt ) V ( u ( t ) , u· ( t ) ; t ) . Due to (7.26) we have that
d
1
--- ----- E ( u ( t ) , u· ( t ) ; t ) = - g u· ( t ) 2 + m ( u ( t ) , u· ( t ) ) +
2 dt
+ M ¢ æ A1/ 2 u1 ( t ) 2ö ( A u1 ( t ) , u· 1 ( t ) ) A1/ 2 u ( t )
è
ø
2
+ ( F ( u1 ( t ) , u 2 ( t ) ) , u· ( t ) ) .
Determining Functionals for Second Order in Time Equations
With the help of (7.25) and (7.27) we obtain that
g
d
1
--- ----- E ( u , u· ; t ) £ - --- u· ( t ) 2 + C R æ A1/ 2 u 2 + Aq u 2ö .
è
ø
2
2 dt
Using (7.26) and (7.27) it is also easy to find that
d
----dt
ì
ü
g
·
í ( u , u ) + --2- ( u , u ) ý = u·
î
þ
2
+ ( u , u·· + g u· ) £
u· 2 - Au 2 + MR A1/ 2 u 2 + CR æ A1/ 2 u 2 + Aq u 2ö u ,
è
ø
£
where
ì
ü
MR = max í M ( z ) : 0 £ z £ l-11 R 2 ý .
î
þ
We choose n = g ¤ 4 and use the estimate of the form
Ab u
£
Au b × u 1 - b £ e Au + Ce u ,
0 < b < 1,
e > 0,
to obtain that
g ü
d
d ì
----- V ( u , u· ; t ) = ----- í 1
--- E ( u , u· ; t ) + n æ u , u· + --- uö ý £
è
ø
2
2
dt
dt î
þ
g
£ - --- æ Au
8è
2
+ u· 2ö + C R u ( t )
ø
2
.
Therefore, using the estimate
u ( t ) 2 £ CL max lj ( u ) 2 + 2 eL ( D ( A) , H ) Au 2
j
and equation (7.29) we obtain the inequality
d
----- V ( u ( t ) , u· ( t ) ; t ) + w V ( u ( t ) , u· ( t ) ; t ) £ C max lj ( u ( t ) ) 2 ,
dt
j
g -1
provided eL ( D ( A) , H ) < e 0 = ----- C R . Here w is a positive constant. As above, this
16
easily implies (7.24), provided (7.23) holds. Theorem 7.2 is proved.
E x e r c i s e 7.1 Show that the method used in the proof of Theorem 7.2 also
enables us to obtain the assertion of Theorem 7.1 for problem (7.1).
E x e r c i s e 7.2 Using the results of Section 4.2 related to the linear variant of
equation (7.15), prove that the method of the proof of Theorem 7.1
can also be applied in the proof of Theorem 7.2.
Thus, the methods presented in the proofs of Theorems 7.1 and 7.2 are close to each
other. The same methods with slight modifications can also be used in the study
of problems like (7.1) with additional retarded terms (see [3]).
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E x e r c i s e 7.3 Using the estimates for the difference of two solutions to equation (7.15) proved in Lemmata 4.6.1 and 4.6.2, find an analogue
of Theorems 1.3 and 4.4 for the problem (7.15) and (7.16).
5
§ 8
On Boundary Determining Functionals
The fact (see Sections 5–7 as well as paper [3]) that in some cases determining functionals can be defined on some auxiliary space admits in our opinion an interesting
generalization which leads to the concept of boundary determining functionals.
We now clarify this by giving the following simple example.
In a smooth bounded domain W Ì R d we consider a parabolic equation with
the nonlinear boundary condition
ì
ï
ï
í
ï
ï
î
¶-----u
= n Du - f ( u ) ,
¶t
¶u-----¶n
¶W
+ h (u)
¶W
x ÎW, t > 0 ,
= 0,
(8.1)
u t = 0 = u0 ( x ) .
Assume that n is a positive parameter, f ( z ) and h ( z ) are continuously differentiable functions on R 1 such that
f ¢( z ) ³ - a ,
h ¢(z) £ b ,
(8.2)
where a ³ 0 and b > 0 are constants. Let
W = C 2 , 1 ( W ´ R+ ) Ç C 1 , 0 ( W ´ R+ ) .
(8.3)
Here C 2 , 1 ( W ´ R + ) is a set of functions u ( x , t ) on W ´ R+ that are twice continuously differentiable with respect to x and continuously differentiable with respect
to t . The notation C 1 , 0 ( W ´ R+ ) has a similar meaning, the bar denotes the closure
of a set.
Let u1 ( x , t ) and u2 ( x , t ) be two solutions to problem (8.1) lying in the class
W (we do not discuss the existence of such solutions here and refer the reader to
the book [7]). We consider the difference u ( t ) = u1 ( t ) - u 2 ( t ) . Then (8.1) evidently
implies that
d
1
--- ----- u ( t ) 2 2
+ n Ñu ( t ) 2 2
+ ( f ( u1 ( t ) ) - f ( u2 ( t ) ) , u ( t ) ) 2
=
L (W)
L (W)
2 dt
L (W)
= -n
ò u (t ) (h(u (t) ) - h (u (t) ) ) d s .
1
¶W
2
On Boundary Determining Functionals
Using (8.2) we obtain that
d
1
--- ----- u ( t ) 2 2
+ n Ñu ( t ) 2 2
- a u ( t ) 22
£ n b × u (t) 2 2
. (8.4)
L (W)
L (W)
L (W)
L (¶ W)
2 dt
One can show that there exist constants c 1 and c2 depending on the domain W
only and such that
u 22
L (W)
u 2 1/ 2
H
£ c1 æ u
è
(¶ W)
2
L2( ¶ W )
£ c2 æ u
è
+ Ñu
2
L2( W )
ö
2
L2( W ) ø
+ Ñu
,
ö
2
L2( W ) ø
(8.5)
.
(8.6)
Here H s ( ¶ W ) is the Sobolev space of the order s on the boundary of the domain
W . Equations (8.4)–(8.6) enable us to obtain the following assertion.
Theorem 8.1.
Let L = { lj : j = 1 , ¼ , N } be a set of continuous linear functionals
on the space H 1/ 2 ( ¶ W ) . Assume that a c1 < n and
eL º eL ( H 1/ 2 ( ¶ W ) , L 2 ( ¶ W ) ) <
1/ 2
n - a c1
-----------------------------------------------º e0 ,
n ( 1 + c1 ) c2 ( 1 + b )
(8.7)
where the constants n , a , b , c 1 , and c2 are defined in equations (8.1),,
(8.2),, (8.5),, and (8.6).. Then L is an asymptotically determining set with
respect to L 2 ( W ) for problem (8.1) in the class of classical solutions W .
Proof.
Let u ( t ) = u1 ( t ) - u2 ( t ) , where uj ( t ) Î W are solutions to problem (8.1).
Theorem 2.1 implies that
u 22
L (¶ W)
2
£ CL , d max lj ( u ) 2 + ( 1 + d ) eL
u
j
2
H 1/ 2 ( W )
(8.8)
for any d > 0 . Equations (8.5) and (8.6) imply that
u 22
L (¶ W)
2 æ
£ CL , d max lj ( u ) 2 + ( 1 + c1 ) c 2 ( 1 + d ) eL
u
è
j
2
L2( ¶ W )
+ Ñu
2
ö
L2 ( W )ø
.
Therefore, equation (8.4) gives us that
d
1
--- ----- u 2 2
£
+ n u 22
+ Ñu 2 2
- a u (t) 2 2
L (W)
L (¶ W)
L (W)
L (W)
2 dt
£ n (1 + b) u(t)
L2 ( ¶ W )
2æ
£ n ( 1 + c 1 ) c 2 ( 1 + d ) ( 1 + b ) eL
u
è
2
L2( ¶ W )
+ Ñu
2
ö
L2( W )ø
+
+ C max lj ( u ) 2 .
j
Using estimate (8.5) once again we get
d
1
n- ì 1 - ( 1 + c ) c ( 1 + d) ( 1 + b ) e 2 - a c----1- ü u 2 £ C max l ( u ) 2 , (8.9)
--- ----- u 2 + --L
1 2
j
c1 í
n ýþ
2 dt
j
î
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Theory of Functionals that Uniquely Determine Long-Time Dynamics
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provided that
5
c
2
1 - ( 1 + c1 ) c2 ( 1 + d ) ( 1 + b ) eL - a ----1- > 0 .
(8.10)
n
It is evident that (8.10) with some d > 0 follows from (8.7). Therefore, inequality
(8.9) enables us to complete the proof of the theorem.
Thus, the analogue of Theorem 3.1 for smooth surfaces enables us to state that problem (6.1) has finite determining sets of boundary local surface averages.
An assertion similar to Theorem 8.1 can also be obtained (see [3]) for a nonlinear
wave equation of the form
¶u
¶2u
--------+ g ------ = Du - f ( u ) ,
¶t
¶t 2
¶u
-----¶n
G
¶u
= - a -----¶t
G
- j æ u Gö ,
è
ø
u ¶W\G = 0 ,
x Î W Ì Rd ,
u t = 0 = u0 ( x ) ,
t > 0,
¶u
------¶t
t=0
= u1 ( x ) .
Here G is a smooth open subset on the boundary of W , f ( u ) and j ( u ) are bounded continuously differentiable functions, and a and g are positive parameters.
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