Solvability of p-Laplacian Parabolic Equations with Constraints Coupled with Navier-Stokes ...

20 Pages·2017·0.16 MB·English

Funkcialaj Ekvacioj, 60 (2017) 1–20
Solvability of p-Laplacian Parabolic Equations with Constraints
Coupled with Navier-Stokes Equations in 3D Domains
by Using Largeness of p
By
Takeshi Fukao*, Yutaka Tsuzuki and Tomomi Yokota†
(Kyoto University of Education, Hiroshima Shudo University
and Tokyo University of Science, Japan)
Abstract. This paper is concerned with a system of p-Laplace heat equations with
constraints and Navier-Stokes equations. The existence and uniqueness of solutions
have been already proved for several types of the system in 2-dimensional domains.
This paper gives the existence result in 3-dimensional domains, where the di¤usion
term on heat equations is the p-Laplacian with pb3. This work provides a first
insight towards the full case pb2.
Key Words and Phrases. Heat equations, p-Laplacian, Time-dependent con-
straints, 3D Navier-Stokes equations, Subdi¤erential operators.
2010 Mathematics Subject Classification Numbers. Primary 35Q35, Secondary
47H05.
1. Introduction
1.1. Problem and related works
Let 0<T <y. Let W(cid:1)R3 be a bounded domain with smooth boundary
G. Then we consider the following problem (P):
8
>>>>>>>>>>>>><cqqqyyy1===aqqqttty(cid:3)(cid:3)(cid:3)aDDDcppyyy2þþþvvv(cid:4)(cid:4)(cid:4)‘‘‘yyyba¼ fff iiiinnnn QQQQð1:yðð¼yyÞÞÞ:ð¼::0¼¼;fTffðÞtðð;tt(cid:2);;xxxÞWÞÞA;AAQQQjjjcyy1¼¼<ccy1<gg;;c2g;
ðPÞ p 2 2
>>>>>>>>>>>>>:qdyyvði¼v0=;qv0(cid:4)tÞ¼;(cid:3)¼v0D¼yv0ðþ(cid:4)Þ;ðvvð(cid:4)0‘;Þ(cid:4)Þv¼¼vgððy(cid:4)ÞÞ(cid:3)‘p iiionnnnWQQS;;;:¼ð0;TÞ(cid:2)G;
0 0
* Supported by a Grant-in-Aid for Scientific Research (C) (No. 26400164), JSPS.
† Supported by a Grant-in-Aid for Scientific Research (C) (No. 25400119), JSPS.
2 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota
where y:Q!R, v¼ðv ;v ;v Þ:Q!R3 and p:Q!R represent the temper-
1 2 3
ature, the velocity and the pressure, respectively, and are unknown functions;
c ;c :Q!R are given obstacle functions with c ac , and f :Q!R,
1 2 1 2
g¼ðg ;g ;g Þ:R!R3, y :W!R, v :W!R3, and pb3 are also given,
1 2 3 0 0
and D is the so-called p-Laplace operator defined as D y:¼divðj‘yjp(cid:3)2‘yÞ.
p p
The problem including the fifth equation is the so-called ‘‘Boussinesq system’’.
From a viewpoint of physics, the problem (P) (practically in the case
p¼2) represents the model describing temperature y¼yðt;xÞ and velocity
v¼vðt;xÞ of incompressible fluid at each place x on the domain W at each time
tA½0;T(cid:5). Artificially, we impose constraint on temperature. In more detail,
the equation
qy=qt(cid:3)D yþv(cid:4)‘y¼ f
p
holds when c <y<c . On another hand if y¼c , then the left-hand side of
1 2 1
the above equation including the time derivative qy=qt bigger than the right-
hand side so that the constraint keeps. On the other hand if y¼c , similarly,
2
then the left-hand side is smaller than the right-hand side. Moreover, the
p-Laplacian D y implies the e¤ect of highlighting extreme values in the
p
temperature distribution, and such e¤ect becomes stronger depending on
largeness of p. Accordingly, convective e¤ect of the term v(cid:4)‘y also becomes
stronger. (P) is an artificial problem by taking such e¤ect into account.
Concerning problems with obstacle functions such as c , c in (P), we can
1 2
find some mathematical studies. Brezis-Crandall-Pazy [3] originally dealt with
an obstacle problem in the framework of evolution equations in the case where
obstacle functions are independent of tA½0;T(cid:5). In the case where obstacle
functions depend on tA½0;T(cid:5), Yamazaki-Ito-Kenmochi [26] first studied such
obstacle problem under strict conditions on obstacle functions. After that,
Fukao-Kubo [9] dealt with such time-dependent obstacle problem without strict
conditions by employing a similar argument in [3].
There are some works on the solvability of (P) and related problems.
Fukao-Kubo [9] and Sobajima-Tsuzuki-Yokota [22] established the solvability
of the problem (P) with the di¤usion term D y altered into Dy in 2-dimensional
p
domains. Tsuzuki [24] also solved the problem (P) with the di¤usion term
D y in 2-dimensional domains. Other problems associated with the Boussinesq
p
system are dealt with in many papers (see e.g., Morimoto [15], Kubo [13],
Fukao-Kenmochi [6] and Fukao-Kubo [8]). The solvability of the model with
Dy and time-dependent constraint on velocity v in 2- and 3-dimensional domains
was recently studied by Fukao [5] and Fukao-Kenmochi [7]. On the other
hand, obstacle problems for elliptic equations of p-Laplacian type are dealt
with in Choe-Lewis [4], Mu [16], Rodrigues [19], and Rodrigues-Sancho´n-
Urbano [20].
Systems of p-Laplacian Parabolic Equations and Navier-Stokes Equations 3
However the works adopting the operator theory with the solvability of (P),
namely [9], [22] and [24], are done only in 2-dimensional domains. It should be
appreciated that we discuss the solvability in 3-dimensional domains even if we
consider an artificial setting with p-Laplacian. In this paper, we try to answer
the following question:
Can the solvability of (P) be extended for the 3D case along the operator
theory?
Actually, we can prove the existence of solutions to (P) with p-Laplacian D y in
p
the case pb3. That is a first step towards the full case pb2. The proof
is based on the theory for subdi¤erential operators of convex and indicator
functions, which is used in various problems (see e.g., Ito-Kenmochi [10],
Kenmochi-Shirakawa [11], Aiki [1], Kumazaki [14] and Yamazaki [25]).
In this section on later, we formulate the problem (P) and define solutions.
After that, we state the main results. The outline of our proof is as follows.
In Section 2 as preliminaries, we give the solvability of the approximate
heat equation with estimates and the Navier-Stokes equation, respectively. In
Section 3, we construct the approximate solution to the problem (P) by using
the solvability result obtained in the previous section, and then we estimate for
it and take the limit.
1.2. Previous works and our purposes
Let us recall the previous works [9], [22] and [24] in 2-dimensional domains.
In [9] and [22] it is shown that there exists a unique solution ðy;vÞ to the
problem (P) with p¼2 such that
ðC1Þ yAH1ð0;T;L2ðWÞÞ\Lyð0;T;H1ðWÞÞ\L2ð0;T;H2ðWÞÞ;
2 0
ðC2Þ vAH1ð0;T;ðH1ðWÞÞ(cid:6)Þ\Lyð0;T;L2ðWÞÞ\L2ð0;T;H1ðWÞÞ;
s s s
where L2ðWÞ and H1ðWÞ are the Lebesgue and Sobolev spaces of functions v
s s
such that divv¼0 (see Section 1.3). In [24] it is proved that the problem (P)
with p>2 admits a unique solution in the classes (C2) and
ðC1Þ yAH1ð0;T;L2ðWÞÞ\Lyð0;T;W1;pðWÞÞ with D yAL2ð0;T;L2ðWÞÞ:
p 0 p
There are two important parts to solve the problem (P) with pb2. One is
to assert that (C1) and (C2) imply
p
ð1:1Þ v(cid:4)‘yAL2ð0;T;L2ðWÞÞ:
Indeed, in the case p¼2 as in [9] and [22], noting that
4 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota
ðC1Þ0 ‘yALyð0;T;L2ðWÞÞ\L2ð0;T;H1ðWÞÞ;
2
ðC2Þ0 vALyð0;T;L2ðWÞÞ\L2ð0;T;H1ðWÞÞ;
s s
we see from the Gagliardo-Nirenberg interpolation that vAL4ð0;T;L4ðWÞÞ and
‘yAL4ð0;T;L4ðWÞÞ, and hence (1.1) holds. On the other hand, in the case
p>2 as in [24], the Gagliardo-Nirenberg interpolation with (C2)0 implies that
vALpð0;T;Lp(cid:6)ðWÞÞ, where p(cid:6) is defined as
1 1 1
ð1:2Þ þ ¼ ;
p p(cid:6) 2
moreover, it follows from (C1) that
p
ðC1Þ0 ‘yALyð0;T;LpðWÞÞ;
p
and hence (1.1) holds. The other of the two important parts is to introduce
the two mappings, which are for example the two solution operators of the
heat equation with fixed velocity and the Navier-Stokes equation with fixed
temperature, and to construct solutions to (P) by combining the two mappings.
Indeed in the both case p¼2 and p>2, they apply the Schauder fixed point
theorem (as in [9]) or the contraction mapping principle (as in [22] and [24]) to
the composition of the two mappings mentioned above.
The purpose of this paper is to establish the existence of solutions to
the problem (P) in 3-dimensional domains. We should pay more attention to
verifying (1.1) because of loss of regularity for solutions to (P). Indeed, (C2)0
implies that vALqð0;T;LrðWÞÞ with 4=ð3qÞþ2=r¼1 for all qA½2;y(cid:5) and
rA½2;6(cid:5). Therefore, to assert (1.1), it is required that
ð1:1Þ0 ‘yALqq~ð0;T;Lrr~ðWÞÞ with 2=qq~þ3=~rra1
for some qq~A½2;y(cid:5) and ~rrA½3;y(cid:5). If we consider the case p¼2, then (C1)0
2
implies that
ðC1Þ00 ‘yALqq~ð0;T;Lrr~ðWÞÞ with 4=ð3qq~Þþ2=~rr¼1
2
for all qq~A½2;y(cid:5) and ~rrA½2;6(cid:5). Since qq~ and ~rr in ðC1Þ00 satisfy
2
2=qq~þ3=~rr¼3=2 ð>1Þ;
ðC1Þ00 does not imply ð1:1Þ0. Therefore, in this paper, we assume pb3 so
2
that ðC1Þ0 holds and it implies ð1:1Þ0. Since we do not know the uniqueness
p
of solutions to the Navier-Stokes equation in 3-dimensional domains, we can
Systems of p-Laplacian Parabolic Equations and Navier-Stokes Equations 5
neither use the contraction mapping principle nor the Schauder fixed point
theorem as in [9], [22] and [24]. Indeed, the application of the contraction
mapping principle breaks down on a similar situation when we calculate
to prove the uniqueness of solutions to the Navier-Stokes equation in
3-dimensional domains. The application of the Schauder fixed point theorem
also breaks down because the solution operator mentioned above is multi-
valued. Therefore, it is necessary to introduce another method to construct
solutions to (P). In this paper, we use the method in [6] (see Section 3 in more
detail).
1.3. Formulation of the problem and main results
We use the following notation:
H :¼L2ðWÞ; V :¼W1;pðWÞ and H :¼L2ðWÞ; V :¼H1ðWÞ:
p 0 s s
Here H and V are equipped with standard inner product and k(cid:4)k :¼
p Vp
k‘(cid:4)k , respectively. Also, L2ðWÞ and H1ðWÞ are the closures of D ðWÞ
LpðWÞ s s s
:¼fvACyðWÞjdivv¼0 in Wg in L2ðWÞ and H1ðWÞ, respectively. Then we
0
have the dense and compact imbeddings V ,!H and V ,!H ,!V(cid:6). Under
p
these spaces, we define the closed and convex set KðtÞ as
KðtÞ:¼fzAHjc ðtÞazac ðtÞ a:e: on Wg; tA½0;T(cid:5);
1 2
and we define að(cid:4);(cid:4)Þ:V (cid:2)V !R and bð(cid:4);(cid:4);(cid:4)Þ:V (cid:2)V (cid:2)V !R respectively as
aðu;zÞ:¼ X3 ð quj qzj dx; u;zAV;
qx qx
i;j¼1 W i i
bðu;v;zÞ:¼ X3 ð u qvjz dx; u;v;zAV;
iqx j
i;j¼1 W i
and we also define A:V !V(cid:6) and B:V (cid:2)V !V(cid:6) respectively as
hAu;zi :¼aðu;zÞ; u;zAV;
V(cid:6);V
hBðu;vÞ;zi :¼bðu;v;zÞ; u;v;zAV;
V(cid:6);V
where A is the so-called Stokes operator and represented as (cid:3)PD, where
P:L2ðWÞ!H is the Helmholtz projection.
Under these settings we define solutions to the problem (P).
Definition 1.1. A pair ðy;vÞAL2ðQÞ(cid:2)L2ðQÞ is called a solution to the
problem (P) if the following conditions are satisfied:
6 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota
(D1) y and v belong to the following classes, respectively:
ðD1aÞ yAH1ð0;T;HÞ\Lyð0;T;V Þ; D yAL2ð0;T;HÞ;
p p
ðD1bÞ yðtÞAKðtÞ for all tA½0;T(cid:5);
ðD1cÞ vAW1;4=3ð0;T;V(cid:6)Þ\Lyð0;T;HÞ\L2ð0;T;VÞ;
(D2) y and v satisfy (D2a) for a.a. tAð0;TÞ and all zAKðtÞ and (D2b)
for all zAV respectively:
ðD2aÞ ððdy=dtÞðtÞ(cid:3)D yðtÞþvðtÞ(cid:4)‘yðtÞ(cid:3) fðtÞ;yðtÞ(cid:3)zÞ a0;
p H
ðD2bÞ hdv=dt;zi þaðv;zÞþbðv;v;zÞ¼ðPgðyÞ;zÞ a:e: on ð0;TÞ
V(cid:6);V H
ð,dv=dtþAvþBðv;vÞ¼PgðyÞ in V(cid:6) a:e: on ð0;TÞÞ;
(D3) ðyð0Þ;vð0ÞÞ¼ðy ;v Þ in H(cid:2)H.
0 0
In this paper, especially in the two main theorems, we assume the following
conditions:
(A1) c ;c AH1ð0;T;HÞ, D c AL2ð0;T;HÞ, cðtÞAW1;pðWÞ a.a. tA
1 2 p i i
ð0;TÞ for i¼1;2, c ðtÞac ðtÞ a.e. on W, c ðtÞa0ac ðtÞ a.e. on G for
1 2 1 2
all tA½0;T(cid:5);
(A2) f AL2ð0;T;HÞ and g:R!R3 is Lipschitz continuous;
(A3) ðy ;v ÞAðV \Kð0ÞÞ(cid:2)H.
0 0 p
Now we are in a position to state the main results.
Theorem 1.1. Assume (A1), (A2), (A3) and
p>4:
Then there exists at least one solution ðy;vÞ to the problem (P).
Theorem 1.2. Let pb3. Assume (A1), (A2), (A3) and
c ;c ALyð0;T;W1;pðWÞÞ:
1 2
Then there exists at least one solution ðy;vÞ to the problem (P).
Remark 1.1. The assumption pb3 assures that V ,!Lp(cid:6)ðWÞ with p(cid:6)
defined as (1.2) and kzkLp(cid:6)ðWÞackzkV for some positive constant c. Therefore,
yALyð0;T;V Þ and vAL2ð0;T;VÞ imply that
p
v(cid:4)‘yAL2ð0;T;HÞ
with estimate
ð1:3Þ kv(cid:4)‘yk ackvk kyk :
L2ð0;T;HÞ L2ð0;T;VÞ Lyð0;T;VpÞ
This fact is fundamental in dealing with the heat equation in L2ð0;T;HÞ.
Systems of p-Laplacian Parabolic Equations and Navier-Stokes Equations 7
2. Preliminaries
2.1. Solvability of the approximate heat equation
To construct solutions to the heat equation, we regard the variational
inequality (D2a) as the following evolution equation in H
dy
ðD2aÞ0 ðtÞþqjðyðtÞÞþqI ðyðtÞÞC fðtÞ(cid:3)vðtÞ(cid:4)‘yðtÞ a:a: tAð0;TÞ
dt KðtÞ
by using subdi¤erential operators, and we also introduce the Yosida approx-
imation and the Moreau-Yosida approximation.
We define two proper, lower semi-continuous and convex functions as
follows:
81ð
>< j‘yjpdx; yAV ;
ð2:1Þ jðyÞ:¼ p p
W
>:y; yAHnV ;
p
(cid:1)0; yAKðtÞ;
ð2:2Þ I ðyÞ:¼ tA½0;T(cid:5):
KðtÞ y; yAHnKðtÞ;
We introduce the subdi¤erential operator qf:DðqfÞ(cid:1)H !2H, for a proper,
lower semi-continuous and convex function f:H !R[fyg with DðfÞ:¼
fuAHjfðuÞ<yg0q, defined as u(cid:6) AqfðuÞ if
ðu(cid:6);z(cid:3)uÞ afðzÞ(cid:3)fðuÞ for all zAH;
H
and uADðqfÞ if qfðuÞ0q. It is well-known that
qjðyÞ¼(cid:3)D y for all yADðqjÞ¼fzAV jD zAHg;
p p p
xAqI ðyÞ,ð(cid:3)x;y(cid:3)zÞ a0 ðzAKðtÞÞ for all yADðqI Þ¼KðtÞ;
KðtÞ H KðtÞ
and hence (D2a) is equivalent to the evolution equation (D2a)0.
To consider approximate heat equations, we also introduce the Yosida
approximation ðqfÞ of qf and the Moreau-Yosida approximation f of f,
n n
which are defined as
(cid:2) 1 (cid:3)(cid:3)1
Jqf :¼ I þ qf ; ðqfÞ :¼nðI (cid:3)JqfÞ;
n n n n
(cid:2) (cid:3)
n 1
f ðzÞ:¼ inf kz(cid:3)yk2 þfðyÞ ¼ kðqfÞ ðzÞk2 þfðJqfðzÞÞ; zAH;
n yAH 2 H 2n n H n
8 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota
where I is the identity, and ðqfÞ ¼qðf Þ¼:qf holds. In particular, the
n n n
following calculation of I (defined in (2.2)) is obtained for all tA½0;T(cid:5):
KðtÞ
ð2:3Þ JqIKðtÞðzÞ¼P ðzÞ¼zþ½z(cid:3)c ðtÞ(cid:5)(cid:3)(cid:3)½z(cid:3)c ðtÞ(cid:5)þ;
n KðtÞ 1 2
ð2:4Þ qIn ðzÞ¼nð(cid:3)½z(cid:3)c ðtÞ(cid:5)(cid:3)þ½z(cid:3)c ðtÞ(cid:5)þÞ;
KðtÞ 1 2
ð2:5Þ kqIn ðzÞk2 ¼n2ðk½z(cid:3)c ðtÞ(cid:5)(cid:3)k2 þk½z(cid:3)c ðtÞ(cid:5)þk2Þ
KðtÞ H 1 H 2 H
for all nAN and zAH, where P is the projection onto KðtÞ and xG:¼
KðtÞ
maxfGx;0g.
The proof of the main theorem proceeds as follows. First, we solve the
approximate heat equation (NH) (where qI is approximated by qIn ) for
n KðtÞ KðtÞ
given v. Next, we solve the Navier-Stokes equation (NS) for given y. Finally,
we construct the approximate solution ðy ;v Þ and take the limit.
n n
To obtain the estimate for solutions to the approximate heat equation, we
introduce the following two lemmas.
Lemma 2.1 ([22, Lemma 3.1]). Let nAN and uAW1;1ð0;T;HÞ. Assume
that c , c satisfy (A1). Then the following inequality holds:
1 2
d
ðu0ðtÞ;qIn ðuðtÞÞÞ b In ðuðtÞÞ(cid:3)C(cid:6)ðtÞkqIn ðuðtÞÞk a:a: tAð0;TÞ;
KðtÞ H dt KðtÞ KðtÞ H
where C(cid:6) AL2ð0;TÞ is defined as
C(cid:6)ðtÞ:¼ðkc0ðtÞk2 þkc0ðtÞk2Þ1=2; tA½0;T(cid:5):
1 H 2 H
Lemma 2.2 ([24, Lemma 2.2]). Let nAN and zADðqjÞ. Assume that c ,
1
c satisfy (A1). Then the following estimate holds:
2
ðqjðzÞ;qIn ðzÞÞ b(cid:3)CDpðtÞkqIn ðzÞk a:a: tAð0;TÞ;
KðtÞ H KðtÞ H
where CDp AL2ð0;TÞ is defined as
CDpðtÞ:¼ðkDpc1ðtÞkH2 þkDpc2ðtÞkH2Þ1=2; tA½0;T(cid:5):
The next lemma gives the estimates for the approximate heat equation.
Lemma 2.3. Assume (A1), (A2) and y AV . Let nAN, vAL2ð0;T;VÞ
0 p
and yALyð0;T;V Þ. Suppose yy~ belonging to the class (D1a) is a solution to
p
the following problem:
8
>><ðdyy~=dtÞðtÞþqjðyy~ðtÞÞþqIKnðtÞðyy~ðtÞÞ
ðNNgHHÞn ¼ fðtÞ(cid:3)vðtÞ(cid:4)‘yðtÞ a:a: tAð0;TÞ;
>>:yy~ð0Þ¼y :
0
Systems of p-Laplacian Parabolic Equations and Navier-Stokes Equations 9
Then for all tA½0;T(cid:5),
ð2:6Þ kdyy~=dtk2 þjðyy~ðtÞÞþIn ðyy~ðtÞÞ
L2ð0;T;HÞ KðtÞ
þkqjðyy~Þk2 þkqIn ðyy~Þk2
L2ð0;T;HÞ Kð(cid:4)Þ L2ð0;T;HÞ
aC ðkvk2 kyk2 þIn ðy Þþ1Þ;
1 L2ð0;T;VÞ Lyð0;T;VpÞ Kð0Þ 0
and moreover, if y¼yy~¼:y(cid:6) and y AKð0Þ, then for all tA½0;T(cid:5),
0
ð2:7Þ kdy(cid:6)=dtk2 þjðy(cid:6)ðtÞÞþIn ðy(cid:6)ðtÞÞ
L2ð0;T;HÞ KðtÞ
þkqjðy(cid:6)Þk2 þkqIn ðy(cid:6)Þk2
L2ð0;T;HÞ Kð(cid:4)Þ L2ð0;T;HÞ
aC0ðkvkp(cid:6) þ1Þ:
1 L2ð0;T;VÞ
Here C and C0 are positive constants depending only on T, jðy Þ, f, c and c
1 1 0 1 2
(independent of n) and p(cid:6) is the constant defined as (1.2).
Proof. We multiply the equation in ðNNgHHÞ by qjðyy~ðtÞÞþqIn ðyy~ðtÞÞ.
n KðtÞ
Using Lemmas 2.1 and 2.2, we see that for a.a. tAð0;TÞ,
d
½jðyy~ðtÞÞþIn ðyy~ðtÞÞ(cid:5)þkqjðyy~ðtÞÞk2 þkqIn ðyy~ðtÞÞk2
dt KðtÞ H KðtÞ H
aðC(cid:6)ðtÞþ2CDpðtÞÞkqIn ðyy~ðtÞÞk
KðtÞ H
þðkfðtÞk þkvðtÞ(cid:4)‘yðtÞk Þðkqjðyy~ðtÞÞk þkqIn ðyy~ðtÞÞk Þ
H H H KðtÞ H
aðC(cid:6)ðtÞþ2CDpðtÞþkfðtÞk þckvðtÞk kyðtÞk Þ
H V Vp
(cid:2)ðkqjðyy~ðtÞÞk þkqIn ðyy~ðtÞÞk Þ
H KðtÞ H
a4ðjC(cid:6)ðtÞj2þ4jCDpðtÞj2þkfðtÞk2 þc2kvðtÞk2kyðtÞk2 Þ
H V Vp
1
þ ðkqjðyy~ðtÞÞk2 þkqIn ðyy~ðtÞÞk2Þ:
2 H KðtÞ H
Integrate it over ½0;t0(cid:5) for all t0 A½0;T(cid:5) (denoted by t again). Then we see that
for all tA½0;T(cid:5),
2jðyy~ðtÞÞþ2In ðyy~ðtÞÞþkqjðyy~Þk2 þkqIn ðyy~Þk2
KðtÞ L2ð0;t;HÞ Kð(cid:4)Þ L2ð0;t;HÞ
a2jðy Þþ2In ðy Þþ8ðkC(cid:6)k2 þ4kCDpk2 þkfk2
0 Kð0Þ 0 L2ð0;TÞ L2ð0;TÞ L2ð0;T;HÞ
þc2kvk2 kyk2 Þ:
L2ð0;T;VÞ Lyð0;T;VpÞ
10 Takeshi Fukao, Yutaka Tsuzuki and Tomomi Yokota
This inequality together with the equation in ðNNgHHÞ gives the estimate for
n
kdyy~=dtk2 and hence we obtain (2.6). Furthermore, if y¼yy~¼:y(cid:6) and
L2ð0;T;HÞ
y AKð0Þ, then (2.6) implies (2.7). To see it, note In ðy Þ¼0 and use the
0 Kð0Þ 0
Young inequality with (1.2). r
Now we state the solvability of the approximate heat equation in ðNNgHHÞ
n
with y¼yy~¼:y(cid:6) for given velocity.
n
Proposition 2.4. Assume (A1), (A2) and y AV . Let nAN. Then for all
0 p
vAL2ð0;T;VÞ, there exists a unique solution y(cid:6) belonging to the class (D1a)
n
such that
8ðdy(cid:6)=dtÞðtÞþqjðy(cid:6)ðtÞÞþqIn ðy(cid:6)ðtÞÞ
>< n n KðtÞ n
ðNHÞ ¼ fðtÞ(cid:3)vðtÞ(cid:4)‘y(cid:6)ðtÞ a:a: tAð0;TÞ;
n n
>:y(cid:6)ð0Þ¼y :
n 0
Proof. Since j is proper, lower-semi continuous and convex and qIn is
KðtÞ
Lipschitz continuous, we see that for all yALyð0;T;V Þ there exists a unique
p
solution yy~belonging to the class (D1a) to the problem ðNNgHHÞ as in Lemma 2.3,
n
and hence we obtain the estimate (2.6). Here we introduce the mapping
S :yAX !yy~AX
n M M
with the compact and convex set X defined as
M
X :¼fyACð½0;T(cid:5);HÞjkdy=dtk2 þkykp aMg
M L2ð0;T;HÞ Lyð0;T;VpÞ
with the topology on Cð½0;T(cid:5);HÞ, where M >0 is large enough. Using the
Schauder fixed point theorem, we obtain a solution y(cid:6) to (NH) in the same
n n
way as in [24, Proof of Proposition 3.2]. The uniqueness is obtained by the
accretivity of qj and qIn and the divergence freeness of v. r
KðtÞ
2.2. Further estimate toward Theorem 1.2
In this section we suppose c ;c ALyð0;T;W1;pðWÞÞ, which is the
1 2
assumption of Theorem 1.2.
Lemma 2.5. Let nAN, vAV and zAV . Assume that c , c satisfy (A1)
p 1 2
and c ALyð0;T;W1;pðWÞÞ for i ¼1;2. Then the following estimate holds:
i
ðv(cid:4)‘z;qIn ðzÞÞ b(cid:3)C‘ðtÞkvk kqIn ðzÞk a:a: tAð0;TÞ;
KðtÞ H V KðtÞ H
where C‘ ALyð0;TÞ is defined as
C‘ðtÞ:¼cðk‘c ðtÞk2 þk‘c ðtÞk2 Þ1=2; tA½0;T(cid:5):
1 LpðWÞ 2 LpðWÞ