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130
a unique strong solution
on
(0,
T
t
)
provided that
a € L
n
{Q)
satisfies
the
compatibility
conditions
V
•
a =
0,
v
•
a\gn —
0 and
(j){a)
= F(0).
This improves
a
related result
of
Pranzke [17]
and can be
proved
in the
same manner
as the
proof
of
Theorem
2
with
the
aid of the auxiliary function of Heywood
([20],
Lemma 11), which
is
used
for
the reduction
of the problem
to an
equivalent one with
the
vanishing flux condition
(5).
Up
to
now we have
the
same global existence result
as
Theorem
2 for
the whole space
(Kato [24]),
the
half space (Ukai [32]), bounded domains (Giga
and
Miyakawa [18])
and
exterior domains (Iwashita [23]) since the
L
q
-U
estimates
(3) and (4)
are well established
for these four types
of
domains. Especially
for
exterior domains with
n > 3,
based
on (3)
for
q = r due to
Borchers
and
Sohr [5], some partial results were given
by
Iwashita [23],
Giga and Sohr [19]
and
Borchers
and
Miyakawa [4];
in
particular, Iwashita proved
(3) for
1
< q < r < oo and (4) for 1 < q < r < n,
which made
it
possible
to
construct
a
global
solution. Later
on, due to the
following authors,
(3) for n > 2,1 < q < r < co
(q
^ oo)
and
(4) for 1 < q < r < n = 2
were also derived: Chen [8]
(n =
3), Shibata [31]
(n = 3),
Borchers
and
Varnhorn
[6] (n = 2, (3) for q = r), Dan and
Shibata [10],
[11] (n = 2),
Dan, Kobayashi
and
Shibata [9]
(n = 2,3) and
Maremonti
and
Solonnikov [28]
(n > 2).
In
the
proof
of
the
L
q
-U
estimates,
it
seems
to be
heuristically reasonable
to
combine
some local decay properties near
the
aperture with
the
L
q
-L
r
estimates
of the
Stokes
semigroup
for the
half space (Ukai [32])
by
means
of a
localization procedure since
the
aperture domain
n is
obtained from
H
+
U
H_
by a
perturbation within
a
compact region.
Indeed, Abels
[1]
used this idea that
was
well developed
by
Iwashita
[23] and,
later,
Kobayashi
and
Shibata [25]
in
the case of exterior domains. We should however note that
the boundary
dQ is
noncompact; thus,
a
difficulty
is to
deduce
the
sharp local energy
decay estimate
\\e'
tA
f\\w^(n
R)
< Ct-
n
'"\\f\\
ma)
, t > 1, (7)
for
/ e
Ll(Q), 1
< q < co,
where Q,
R
= {x
G
CI;
\x\
< R}, but
this
is the
essential part
of
our
proof.
The
estimate
(7)
improves
the
local energy decay given
by
Abels
[1], in
which
a
little slower rate t~
n
/
2l
i+
e
was shown.
In
[1], similarly
to
Iwashita [23],
a
resolvent
expansion around
the
origin A
= 0 was
derived
in
some weighted function spaces.
To
this end, Abels made
use of
the Ukai formula
of
the Stokes semigroup
for the
half space
([32])
and, in
order
to
estimate
the
Riesz operator appearing
in
this formula,
he had
to introduce Muckenhoupt weights, which caused some restrictions although
his
analysis
itself
is of
interest.
On the
other hand, Kobayashi
and
Shibata
[25]
refined
the
proof
of Iwashita
in
some sense
and
obtained
the L
q
-U
estimates
of
the Oseen semigroup
for
the 3-dimensional exterior domain.
As a
particular case,
the
result
of
[25] includes
the
estimates
of the
Stokes semigroup
as
well.
To
prove Theorem
1 we
employ
in
principle
the strategy developed
by
[25] (without using
any
weighted function space)
and
extend
the method
to
general
n > 3. The
proof is divided into three steps
and the
outline
is as
follows.
The first step
is the
investigation
of
the Stokes resolvent
for the
half space
H = H
+
or
H-. We
derive some regularity estimates near
the
origin A
= 0 of
(A
+
A
ff
)
_1
P
ff
/
when
/ e L
q
(H) has a
bounded support, where
A
H
=
— P#A
is the
Stokes operator
for
the half space
H.
Although
the
obtained estimates
do not
seem
to be
optimal compared
with those shown
by
[25]
for
the whole space,
the
results are sufficient
for
our aim and
the
131
proof is rather elementary: in fact, we represent the resolvent
(A
+ A#)
_1
in terms of the
semigroup e~
tA
" and, with the aid of local energy decay properties of this semigroup, we
have only to perform several integrations by parts and to estimate the resulting formulae.
One needs neither Fourier analysis nor resolvent expansions.
In the next step, based on the results for the half space, we proceed to the analysis
of the Stokes resolvent for the aperture domain fi. To do so, in an analogous way to
[23],
[25] and [1], we first construct the resolvent (A + A)~
1
Pf near the origin A = 0 for
/ € L
q
(Q) with bounded support by use of the operator (A +
AH)~
1
PH,
the Stokes flow
in a bounded domain and a cut-off function together with the result of Bogovskii [2] on
the boundary value problem for the equation of continuity. And then, for the same /
as above, we deduce essentially the same regularity estimates near the origin A = 0 of
(A + A)~
l
Pf as shown in the previous step.
In the final step we prove (7) and thereby (4) for q = r G (l,n] as well as (3) for
r = oo, from which the other cases follow. Some of the estimates obtained in the previous
step enable us to justify a representation formula of the semigroup e~
tA
Pf in
W
l
'
q
(Q
R
)
in
terms of the Fourier inverse transform of 9™(is +
A)
-1
Pf when / G L
q
(Q) has a bounded
support, where n = 2m +
1
or n = 2m + 2. We then appeal to the lemma due to Shibata
([30],
[25]), which tells us a relation between the regularity of a function at the origin and
the decay property of its Fourier inverse image, so that we obtain another local energy
decay estimate
\\e~
tA
Pf\\
wl
„
{nR)
< Cr"/
2
+l/||
L
,
(n)
, t > 1, (8)
for / G
L
q
(Q,),
1 < q < oo, with bounded support, where e > 0 is arbitrary. Estimate (8)
was shown in [1] only for solenoidal data / G Z/'(0) with bounded support, from which
(7) with the rate replaced by
t~
n
/
2q+s
follows through an interpolation argument. But it
is crucial for the proof of (7) to use (8) even for data which are not solenoidal (so that the
support of Pf is unbounded). In order to deduce (7) from (8), we develop the method in
[23] and [25] based on a localization argument using a cut-off function. In fact, we regard
the Stokes flow for the aperture domain Q as the sum of the Stokes flows for the half spaces
H± and a certain perturbed flow. Since the Stokes flow for the half space enjoys the L'-L°°
decay estimate with the rate
i~"/
2
«
(Borchers and Miyakawa [3]), our main task is to show
(7) for the perturbation part. In contrast to the case of exterior domains, the support
of the derivative of the cut-off function touches the boundary dil; indeed, this difficulty
occurs in all stages of localization procedures in the course of the proof and thus we have
to carry out such procedures carefully. Furthermore, the remainder term arising from the
above-mentioned localization argument involves the pressure of the nonstationary Stokes
system in the half space and, therefore, does not belong to any solenoidal function space.
Hence, in order to treat this term, (8) is necessary for non-solenoidal data, while that is
not the case for the exterior problem.
Acknowledgment. The present work was partially supported by the Alexander von Hum-
boldt research fellowship.
132
References
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L
q
-L
r
estimates for the non-stationary Stokes equations in an aperture
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2
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2
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L
q
-L
T
estimates of the Stokes semigroup in a 2-
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?
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Univ. Ferrara Sez. VII. Sc. Mat. 46, 161-173 (2000).
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r
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L
q
-L
r
estimates for solutions of the nonstationary Stokes equations in
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q
spaces, Math.
Ann.
285, 265-288 (1989).
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LP
solutions of the Navier-Stokes equation in R
m
, with applications
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Global attractors for multivalued flows associated with
sub differentials
Noboyuki KENMOCHI
Department of Mathematics
Faculty of Education, Chiba University,
1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan
Email: kenmochi@math.e.chiba-u.ac.jp
Noriaki YAMAZAKI
Department of Mathematical Science
Common Subject Division
Muroran Institute of Technology
1-27 Mizumoto-cho, Muroran, 050-8585 Japan
Email: noriaki@mmm.muroran-it.ac.jp
Abstract
In a real Hilbert space we consider a class of non-autonomous evolution equa-
tions including time-dependent subdifferentials and their perturbations. We are
interested in the case when the Cauchy problems have in general multiple solutions,
namely they may lose the uniqueness of solutions. In such a situation the associ-
ated flows (or dynamical systems) are multivalued, and there have not obtained so
many theoretical results especially in the non-autonomous case. In this paper we
investigate the large time behaviour of the multivalued flows from the view-point
of attractors. In fact, we construct a global attractor for our multivalued flows,
and discuss the relationship to the one for the flows associated with the limiting
autonomous system.
1 Introduction
In this paper let us consider a nonlinear evolution equation in a real Hilbert space H of
the form:
u'(i) + dip*{u{t)) + g(t, u{t)) 3 f(t) in if, t > s (> 0), (1.1)
135
136
where
d(p*
is the subdifferential of time-dependent proper l.s.c. and convex function
tp*
on H, g(t, •) is a single-valued operator in H which is rather small relative to
ip*
and / is
a forcing term given in Lf
oc
([0, +00); H).
We assume that
ip*,
g(i, •) and f(t) respectively converge to a convex function
ip°°
on
H, an operator g°°(-) in H and an element /°° in H in appropriate senses as t
—>•
+00.
Then, we have the limiting autonomous system (as s
—>•
+00):
v'{t) +
d<p'
x
{v{t))
+ g°
0
{v{t))3 J™
intf,
t > 0. (1.2)
In this paper, assuming that the Cauchy problems for (1.1) and (1.2) have in general
multiple solutions, we shall discuss the large-time behaviour of solutions of (1.1). In such
a situation the solution operator E(t,s) (0 < s < t < +00) for (1.1) is multivalued.
Namely, E(t,s) (0 < s < t < +00) is the multivalued operator from D(ip
s
) into D{ip
l
)
which assigns to each u
0
G
D(<p
s
)
the set
_,, . f „ There is a solution u of (1.1) on [s, +00) 1 ., „.
E(t, s)u
0
:= |z
G
H ^ ^
u{s) =
^ ^ ^
(t)
^ j . (1.3)
Of course, the solution operator 5(i) (i > 0) for the limiting autonomous system (1.2)
is similarly defined as a multivalued operator in Z?(<y9°°), and the family {S(t);t £ R
+
}
forms a multivalued semigroup, where R
+
:= [0,+00).
The main object of this paper is to show not only that there exists a global attractor
for non-autonomous systems (1.1), but also that it is semi-invariant under the multivalued
semigroup S(t) associated with the limiting autonomous system (1.2). Moreover we shall
give a sufficient condition in order that the global attractor for (1.1) is invariant under
S(t).
Notation. Throughout this paper, let H be a real Hilbert space with norm | • \
H
and
inner product (-,•)»• F°
r a
P
ro
P
er
l.s.c. convex function ip on H we use the notation
D(<p),
dip and
D(d<p)
to indicate the effective domain of
ip,
the subdifferential of
<p
and
the domain of
dip,
respectively; for their precise definitions and basic properties see [1].
For two sets A and B in H, we define the so-called Hausdorff semi-distance
dist
H
{A,B) := sup inf \x
—
y\
H
-
2 Global boundedness of solutions
In what follows, system (1.1) is referred as (E
s
) and its Cauchy problem associated with
initial value u
0
as
(E
S
;K
0
),
namely
(E
•
u ) I
U
'
W +
^'M*)) + 9(t, u(*)) 3 f(t) in H, t > s (> 0),
s
'
0>
\ u(s) =
Mo-
Definition 2.1. Let J be any interval in R
+
with initial time s. Then
(i) A function u : J -> H is called a solution of (E,) on J, if u E C(J; H) n W^(J; H),
<^
()
(u(-)) e L}
0C
(J), u{t) e Didip*) for a.e. t e J and
f(t) - u'(t) - g(t,u(t)) e V(w(t)), a.e. t G J.
137
(ii) A function u
:
J —> H is called a solution of the Cauchy problem (E
s
; w
0
) on J with
given initial value u
0
G H, if it is a solution of (E
s
) on J satisfying u(s) = u
0
.
Now, let {o
r
} := {a
r
; r G R
+
} and
{&,;
r G R
+
} be two families of absolutely continu-
ous (real) functions a
r
, b
r
on R
+
, with parameter r G R
+
, such that
a'
r
G L
l
{R+) n
L
2
{R
+
),
b'
r
G L^R+y, (2.1)
by (2.1), the limits a
r
(+oo) := lim aJt) and 6
r
(+oo) := lim b
r
(i) exist, so a
r
and b
r
t—>+oo £—y+oo
are considered as continuous functions on R
+
:= [0,+oo].
With these families {a
r
} and {b
r
}, we specify a class
$({a
r
},
{6
r
}) of families {</?'} :=
{<p';
i G -R
+
} as follows.
Definition 2.2. {ip
1
} G
<&({a
r
},
{6
r
}) if and only if
tp*
is a proper l.s.c. convex functions
on H satisfying the following property (*):
(*) For each r G R
+
, s,t G [0, +oo] and z G
D(<p
s
)
with |z|# < r, there exists z G £>(y
f
)
such that
|5-«U<K(t)-0,(8)1(1 +
1^(2)1*)
and
V
'(5)-^W<|6r(*)-6r(s)|(l + |^W|).
For
{<£*}
G
$({a
r
},
{&,•}),
d{-, •) and /(•), we further require the following conditions:
(Al) There exists a positive constant C\ > 0 such that
<p\z) >
d\z\
2
H
,
Vi G
R~;,
VZ G Z%').
(A2) For every finite r > 0 and t G i?
+
, the level set {z £ H ; tp
t
(z) < r} is compact in
H.
(A3) Diip
1
) C D(g(t, •)) C .ff for any t G R+, and </(-,u(')) is strongly measurable on J
for any interval J C -R+ and u G
Lf
oc
(J;
H) with <£>(''(«(•))
G
L}
0C
(J).
(A4) There are positive constants C
2
, C
3
such that
lff(*,2)l!r < <V(*) + Cs, Vi
G
S^, Vz G D(v»
t
)-
(A5) (demi-closedness) If z„ G D((p
tn
), {t
n
} C f?
+
, {<p
u
{z
n
)} is bounded, z„ -+ z in ff
and t
n
-> £ (as n -4 +oo), then
<;(£„,
z„) -> #(t, 2) weakly in H.
(A6) For each bounded subset B of if, there exist positive constants Ci{B) and C
5
(B)
such that
¥>*(*) + (g(t,z),z- b)
E
>
C
4
(B)\z\j,
- C
5
(B), Vt eR~;,\/ze Dtf), V6
G
B.
138
As
to the
existence
and
global boundedness
of a
solution
for
(E
s
;
u
0
) on
[s,
+00),
we
have
the
following theorem, which
can be
proved
in the
same
way as in [8].
Theorems
2.1. [8,
Theorems 2.1,
2.2] Let f £
Lf
0C
(R
+
;H)
and
assume that {</?'}€
$({a
r
},
{b
r
}) and
(A1)-(A6)
hold.
Then,
for
each
s > 0 and u
0
e
D(<p
s
),
(E
s
;
u
0
) has at
least
one
solution
u on
[s,
+00).
In
addition,
if
S
f
:=
sup
\f\mt,t+i;H)
<
+°°>
then,
the
solution
u of
(E
s
;
uo)
satisfies
the
following estimate:
rt+l
sup
\u(t)\
2
H
+
sup
/
ip
T
(u(r))dT
<
JVi(l
+ S) +
|u
0
|lf),
where
Ni is a
positive constant independent
of f, s > 0 and «o 6
D(ip
s
). Moreover,
for
each
6 > 0 and
each bounded
set B C H,
there
is a
constant Ni{5,
B) > 0,
depending only
on
5 > 0 and B,
such that
sup
\u'\W
t+1
.
H)
+ sup
f\u{t))
<
N
2
{5,B),
t>s+s t>
s
+e
as long
as s > 0 and «o €
D(tp
s
) D
B.
3 Global attractor
of the
non-autonomous system
By Theorem
2.1 we see
that
the
solution operator E(t,s)
(0 < s < t < +00) for (E
s
)
is well defined
as a
multivalued operator from D(tp
s
) into D(ip
l
) which assigns
to
each
Uo
6
D(tp
s
)
the set E(t,
s)uo given
by
(1.3). Also,
it is
easy
to
check
the
following evolu-
tion properties
(El) and (E2):
(El)
E(s, s) =1 (the
identity)
on
D(<p
s
)
for any s > 0.
(E2)
E(t
2
, s)z =
EfatjEtti,
s)z for any 0 < s < t
x
< t
2
< +00 and z
G D(<p
s
).
Now, suppose that
the
source term
/(<)
converges
to
some element
f°° € H in the
sense
of
Stepanov
as t
—»•
+00, i.e.
\f{t
+ •)-
/°°lL»(o,iiH)
—^0 as t -*
+00.
(3.1)
Moreover, note that
{<p
l
;
t e R+} e
$({o
r
},
{6
r
}) implies that
tp*
converges
to
(p°°
on H as t
—¥
+00 in the
sense
of
Mosco [6].
(3.2)
Then,
by
assumptions
(A5), (3.1) and
(3.2),
as s
—>•
+00 the
limiting system
of (E„) is of
the form:
(Eoo)
i/(t) +
0p
oo
(i;(t))+fl
o
>(t))9/"
in#, t > 0,
where
g°° :=
g(oo,-).
By
(Eoo^o)
we
denote
the
Cauchy problem
for (Eoo)
associated
with initial value
v
0
.
139
The limiting autonomous system (E
m
) can be considered as a special case of (E
s
)
with
<p<
=
<p°°,
g(t, •) = g°°(-) and f(t) = f°°. Therefore, for each v
0
6
D{<p°°)
there is at
least one solution of (E^iio) on R
+
which is globally bounded on R
+
in the same sense
as Theorem 2.1, and the family of solution operators {S(t);t 6 R
+
} on _D(^°°) is well
defined; namely, S(t) assigns to each v
0
€ D(tp°°) the set
s(t)v
0
:
=
L
e
Dip*)
Theie
!\i ftr"
of (E
r
;
S
on R+
w
^
v
' such that u(0) = v
0
and ti(i) = z.
Clearly, the following properties are satisfied:
(51) S(0) = 7on%°°).
(52) S{t + s)z = S{t)S{s)z, \/z €
D{<p°°),
Vs,
t G R
+
.
Moreover, we can easily show the closedness of S(-) and E
(•){•)
in the following sense.
Lemma 3.1. (i) Assume that t
n
,t G R
+
with t
n
—>
£, ^on, v
0
6 D{ip°°) with v
0n
—> D
0
in
i? and an element z
n
6 S(£„)i;on converges to some element z in H as n
—>
+oo. Then,
z G S(t)vo.
(M)
Assume £fta£ s
n
, s,
£„,
t G -R+ tuitfj s„
—>
s and i„
—>•
i, won £ D((p
s
"), u
0
G D(ip
s
) with
^on -^ u
0
in H and an element z
n
G S(i
n
+ s
n
, s
n
)«on converges to some element z in H
as n -¥ +oo. 27»en, z G £(£ + s, s)w
0
- J" particular, if s = +oo, tten z G S(£)ito-
Next, to construct a global attractor for the non-autonomous system (E
s
), we need
the concept of w-limit sets under E(t, s) or S(t).
Definition 3.1. Let B(H) be a family of bounded subsets of H. Then, for each B € B(H)
the sets
ws(B):=n U E(t + s,s)(D(^jnB)
T>Ot>T,!>0
and __
=
^___
us(B):=n\JS(t){D(<p>»)nB)
T>Ot>T
are called respectively the w-limit sets of B under E(t, s) and S(t).
Remark 3.1. (cf. [5, Lemma 1]) By the definition of the w-limit set
U>E(B),
it is easy
to see that x €
COE(B)
if and only if there exist sequences {£„} C R+ with t
n
—>
+oo,
{s
n
} C R+, {z
n
} C B with z
n
G £)(</3
s
«) and {z
n
} C # with i„ G U(t„ + s„, s„)z„ such
that
x
n
—>• a;
in if as n
—>
+oo.
So far as the construction of a global attractor for {S(t)} is concerned, we can do it
by modifying the standard technique established in [2] and [9]. The key is the following
lemma:
Lemma 3.2. (cf. [3, Lemma 4.1]) There exists a compact and convex subset B
Q
O/-D(</J°°)
such that
sup
<p°°{z)
< +oo
ZG-Bo