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A study of Galerkin method for the heat convection equations
Applied Mathematics and Computation 218 (2011) 520–531
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A study of Galerkin method for the heat convection equations Polina Vinogradova ⇑, Anatoli Zarubin Department of Natural Sciences, Far Eastern State Transport University, 680021, Khabarovsk, Serisheva 47, Russia
a r t i c l e
i n f o
Keywords: Approximate solution Error estimate Galerkin method Heat convection equation Orthogonal projection
a b s t r a c t This article investigates the Galerkin method for an initial boundary value problem for heat convection equations. The new error estimates for the approximate solutions and their derivatives in strong norm are obtained. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The large number of works is devoted to the study of various convection problems. It is possible to recognize two important directions of the study of convection phenomena. The first one is experimental and theoretical study of the convective stability. In detail these questions are considered, for example, in the monograph [1]. The other important direction is the numerical modeling of convection processes (see, for example, [2–5]). It allows to calculate the modes of convection at various meanings of Rayleigh, Reynolds numbers and at other parameters of the model. It is known that the main theoretical basis of numerical methods is the proof of convergence of the approximate solution to the exact one of the corresponding differential problem. The order of the convergence speed of approximate solutions of a nonlinear problem much depends on a kind of the nonlinear terms. It is often difficult to establish the convergence. In this case the basic information on the convergence of the computing procedure is found out by numerical experiments. In the present paper we study the Galerkin method for the approximate solving of an initial boundary value problem for a non-stationary quasi-linear system which describes the motion of the non-uniformly heated viscous incompressible fluid. The convergence of the Galerkin approximations in a strong norm is established, and also the asymptotic error estimates for the solutions and their derivatives in the uniform norm are obtained. 2. Statement of the problem and auxiliary assertions Let X be a bounded domain in R2with the smooth boundary oX, Q = X (0, T), S = oX (0, T], where T < 1. The initial boundary value problem for the heat convection in Boussinesq approximation is formulated in the following way ([1, 6, 7]): we seek a vector-function u(x, t):X [0, T] ? R2 and scalar functions p(x, t), h (x, t):X [0, T] ? R such that
@u mDu þ q1 0 rp þ ðu rÞu gbk3 h ¼ f ðx; tÞ; ðx; tÞ 2 Q; @t @h jDh þ u rh ¼ uðx; tÞ; ðx; tÞ 2 Q ; @t div uðx; tÞ ¼ 0; ðx; tÞ 2 Q ;
ð1Þ ð2Þ ð3Þ
uðx; tÞ ¼ 0; hðx; tÞ ¼ 0; ðx; tÞ 2 S;
ð4Þ
uðx; 0Þ ¼ 0; hðx; 0Þ ¼ 0; x 2 X;
ð5Þ
⇑ Corresponding author. E-mail addresses: [email protected] (P. Vinogradova), [email protected] (A. Zarubin). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.095
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
521
where m, j, q0, b > 0 are some phisical constants; k3 is a vertically directed up unit vector. Let Lp(X), 1 < p < +1, (respectively L1 (X)) be a space of real functions are absolutely integrable on X with the power of p according to Lebesque measure dx = dx1dx2 (respectively essentially bounded). These spaces with the norms
kukLp ðXÞ ¼
Z
1=p juðxÞjp dx X
respectively
kukL1 ðXÞ ¼ ess sup juðxÞj X
are Banach spaces. The space Lp(Q) is defined similarly. The Sobolev space W m p ðXÞ is a space of functions from Lp(X) whose all generalized partial derivatives up to order m inclusively belong to Lp(X) (m is a nonnegative integer). It is a Banach space with the norm
X
kukW mp ðXÞ ¼
!1=p j
kD
ukpLp ðXÞ
:
jjj6m
The space W 2m;m ðQ Þ (see [8]) with mbeing a nonnegative integer is a Banach space of functions from Lp(Q), which have genp eralized derivatives Drt Dsx with arbitrary nonnegative integers r and s satisfying inequality 2r + s 6 2m. The norm in W 2m;m ðQ Þ p is defined as
kukW 2m;m ðQÞ ¼ p
2m X X
kDrt Dsx ukLp ðQÞ :
j¼0 2rþs¼j
We put
W 1;0 p ðQ Þ ¼ fu 2 Lp ðQ Þ : Dx u 2 Lp ðQ Þg; n o W 12 ðXÞ ¼ u 2 W 12 ðXÞ : u ¼ 0 on @ X in the sense of traces :
2;1 The symbol W 2;1 2 ðQ Þ denotes the set of functions belonging W 2 ðQ Þ satisfying zero initial conditions and vanishing on S. We shall deal with two-dimensional vector-functions, each component of which belongs to one of the above defined spaces. We denote [Lp(X)]2 = Lp(X) Lp(X), [Lp(Q)]2 = Lp(Q) Lp(Q), etc. The norm, for example, in [Lp(X)]2 (p > 2) we shall h i2 h i2 denote by ½Lp ðXÞ . Let us denote similarly the norms in the spaces W 22 ðXÞ ; ½Lp ðQ Þ2 ; W 2;1 2 ðQ Þ .
The norm in L2(X) and in [L2(X)]2 will be denoted by kk and [], respectively. The inner product in L2(X) and in [L2(X)]2 will be denoted by (, ). Let J(X) be a set of solenoidal infinitely differentiable and finite on X vectors v(x) = (v1(x), v2(x)), J ðXÞ be the closure with h i2 1 respect to the norm of space W 2 ðXÞ .The elements of J ðQ Þ are the vectors v(x, t) that belong to J ðXÞ for almost all t. Let PJ be the orthogonal projection in [L2(X)]2 onto J ðXÞ. Using the operator PJ, the problem (1)–(5) can be written as
@u mPJ Du þ P J ððu rÞuÞ gbPJ ðk3 hÞ ¼ P J f ðx; tÞ; @t @h jDh þ u rh ¼ uðx; tÞ; ðx; tÞ 2 Q ; @t uðx; tÞ ¼ 0; hðx; tÞ ¼ 0; ðx; tÞ 2 S; uðx; 0Þ ¼ 0; hðx; 0Þ ¼ 0;
x 2 X:
ðx; tÞ 2 Q ;
ð6Þ ð7Þ ð8Þ ð9Þ
We consider the spectral problems
mP J De ¼ ke; eðxÞ ¼ 0;
e 2 J ðXÞ;
x 2 @X
and
jDm ¼ lm; mðxÞ ¼ 0;
x 2 @ X:
By ki we denote an eigenvalue, corresponding to the eigenvector ei(x), by li we denote an eigenvalue, corresponding to the h i2 eigenvector mi(x). The existence and the completeness of the eigenfunctions ei ðxÞ 2 W 22 ðXÞ \ J ðXÞ; mi ðxÞ 2 W 22 ðXÞ \ W 12 ðXÞ in the spaces [L2(X)]2 and L2(X) are proved in [9, 10]. Let Pn1 be the orthogonal projection in [L2(X)]2 onto the linear span of the vector-functions fei ðxÞgni¼1 ; P n2 be the orthogonal projection in L2(X) onto the linear span of the functions fmi ðxÞgni¼1 . The approximate solutions for the problem(6)–(9) are defined as
522
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
un ðx; tÞ ¼ hn ðx; tÞ ¼
n X i¼1 n X
ai ðtÞei ðxÞ; ci ðtÞmi ðxÞ;
i¼1
where unknown functions ai(t) and ci(t) (i = 1, 2, . . . , n) are the exact solution of the following problem:
@un mPJ Dun þ Pn1 PJ ððun rÞun Þ gbPn1 PJ ðk3 hn Þ ¼ Pn1 PJ f ðx; tÞ; @t @hn jDhn þ Pn2 ðun rhn Þ ¼ Pn2 uðx; tÞ; ðx; tÞ 2 Q ; @t un ðx; 0Þ ¼ 0; hn ðx; 0Þ ¼ 0; x 2 X:
ðx; tÞ 2 Q ;
From now on, by C we denote the difference positive constants independent of n. Later the following multiplicative inequalities will be used very often (see, for example, [11, 12]). Let
m1 ¼
ð10Þ ð11Þ ð12Þ
v 2 W lp
1 1
ðXÞ,
j3 j2 2 ; ji ¼ li ; j1 < j2 < j3 ; l3 6 l2 < l1 : pi j3 j1
Then
kv kW l2 ðXÞ 6 Ckv k1l3m1 kv km1 l1 W p ðXÞ
p2
3
W p ðXÞ
ð13Þ
:
1
This inequality also holds with l2 = l3 = 0 and p2 = 1. Let u 2 W 2m;m ðQ Þ and p(2m 2h s) > 4. Then any derivative Dht Dax u with —a— = s belongs to the space Lr(Q) with any p r P p including r = 1 and the inequality
kDht Dax ukLr ðQ Þ 6 e2m2hs4ðp r Þ kukW 2m;m ðQÞ þ e2hs4ðp r Þ kukLp ðQ Þ 1 1
1 1
p
holds for any e > 0. In the multiplicative form this inequality can be written as
kDht Dax ukLr ðQ Þ 6 CkukbW 2m;m ðQÞ kuk1b Lp ðQ Þ ;
ð14Þ
p
where b ¼ 2h þ s þ 4 1p 1r =ð2mÞ. h i2 Lemma 2.1. Let f(x, t) 2 [L2(Q)]2, u(x, t) 2 L2(Q). Then problem (10)–(12) has a unique solution un ðx; tÞ 2 W 2;1 \J 2 ðQ Þ ðQ Þ; hn ðx; tÞ 2 W 2;1 2 ðQ Þ for each n. The inequalities
½un ðx; tÞW 2;1 ðQÞ 6 C;
ð15Þ
khn ðx; tÞkW 2;1 ðQ Þ 6 C
ð16Þ
2
2
hold. Proof. We take the inner product of (11) in L2(X) by hn(x, s) and integrate the resulting relation over the interval [0, t], t 6 T. Then, using the equality
ðPn2 ðun rhn Þ; hn Þ ¼ 0; we obtain
sup khn ðx; tÞk2 þ
Z
06t6T
t
krhn k2 ds 6 C:
ð17Þ
0
Similarly, using the equality
ðPn1 PJ ððun rÞun Þ; un Þ ¼ 0 and (17), we have
sup ½un ðx; tÞ2 þ 06t6T
Z
t
½run 2 ds 6 C:
0
We multiply the Eq. (10) in [L2(X)]2 by PJDun and integrate the resulting relation over the interval [0, t], t 6 T. Then
ð18Þ
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
1 ½run ðx; tÞ2 þ m 2
Z
t
2 PJ Dun ðx; sÞ ds 6
Z
0
t
0
½f ðx; sÞ P J Dun ðx; sÞ ds þ gb
t
khn ðx; sÞk PJ Dun ðx; sÞ ds
0
Z
þ
Z
523
t
½ðun ðx; sÞ rÞun ðx; sÞ PJ Dun ðx; sÞ ds:
0
Now, using Cauchy inequality jakbj 6 2e jaj2 þ 21e jbj2 , for sufficiently small e > 0 we obtain
½run ðx; tÞ2 þ
Z
t
0
Z t ½PJ Dun ðx; sÞ2 ds 6 C ½f 2L2 ðQ Þ þ khn k2L2 ðQ Þ þ ½ðun rÞun 2 ds : 0
From this and (17) it follows that
½run ðx; tÞ2 þ
Z
t
Z t ½PJ Dun ðx; sÞ2 ds 6 C 1 þ ½ðun ðx; sÞ rÞun ðx; sÞ2 ds :
0
ð19Þ
0
Let us estimate the integral of the right-hand side of (19). We apply the Hölder inequality, then
J
Z
t
½ðun rÞun 2 ds 6
0
Z
t
0
½un 2L4 ðXÞ ½run 2L4 ðXÞ ds:
By using (13) for the spaces W 14 ðXÞ, W 22 ðXÞ and W 12 ðXÞ, we get
J6C
Z
t
½un 2L4 ðXÞ ½run ½un W 2 ðXÞ ds: 2
0
From this and coercive inequality (see, for example, [9]):
½zðxÞW 2 ðXÞ 6 C½PJ DzðxÞ; 8zðxÞ 2 ½W 22 ðXÞ2 \ J ðXÞ;
ð20Þ
2
we obtain
J6C
Z 0
t
½un 2L4 ðXÞ ½P J Dun ½run ds:
Therefore, from (19) it follows that
½run ðx; tÞ2 þ
Z
t
0
Z t Z 1 t ½PJ Dun ðx; sÞ2 ds 6 C 1 þ e ½PJ Dun 2 ds þ ½un 4L4 ðXÞ ½run 2 ds :
e
0
0
Choosing sufficiently small e > 0, we have
Z t ½run ðx; tÞ2 6 C 1 þ ½un 4L4 ðXÞ ½run 2 ds : 0
Now, applying the Gronwall inequality (see [13]), we come to an estimate
Z t ½run ðx; tÞ2 6 C exp C ½un 4L4 ðXÞ ds : 0
We use the inequality (13) for the spaces L4 ðXÞ; W 12 ðXÞ; L2 ðXÞ and also inequality (18), then
Z 0
T
½un 4L4 ðXÞ ds 6 C
Z
T
½un 2 ½run 2 ds 6 C
0
Z
T
½run 2 ds:
0
Thus,
sup ½run ðx; tÞ 6 C:
ð21Þ
06t6T
By applying the estimate (21), from the Eq. (11) it follows that
sup krhn ðx; tÞk 6 C:
ð22Þ
06t6T
From coercive inequality (see [9]):
½zW 2;1 ðQÞ 6 C 2
@ mPJ D z ; @t L2 ðQÞ
h
i2
8z 2 W 2;1 2 ðQ Þ
we have
½un W 2;1 ðQÞ 6 C ½f L2 ðQ Þ þ khn kL2 ðQ Þ þ ½ðun rÞun L2 ðQÞ : 2
\ J ðQ Þ;
ð23Þ
524
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
Then
½un W 2;1 ðQÞ 6 C 1 þ ½un L6 ðQ Þ ½run L3 ðQÞ : 2
2;1 By using the inequality (14) for the spaces W 1;0 3 ðQ Þ; W 2 ðQ Þ; L2 ðQ Þ, we get
5=6 1=6 ½un W 2;1 ðQÞ 6 C 1 þ ½un L6 ðQ Þ ½un W 2;1 ðQ Þ ½un L2 ðQ Þ : 2
Since the space
½un W 2;1 ðQÞ 2
2
W 12 ð
XÞ is embedded in L6(X) and the inequalities (18), (21) hold; it follows that : 6 C 1 þ ½un 5=62;1 W 2 ðQÞ
Thus, the estimate (15) holds. By analogy we obtain (16). h
3. Error estimates for Galerkin method In this section we establish the error estimates for the approximate solutions, for the gradient of the approximate solutions and for the derivative with respect to t. Theorem 3.1. Suppose that f(x, t) 2 [L2(Q)]2 and u(x, t) 2 L2(Q). Then 1=2 1=2 sup ½un ðx; tÞ uðx; tÞ 6 Cðknþ1 þ lnþ1 Þ;
ð24Þ
1=2 sup khn ðx; tÞ hðx; tÞk 6 Cðk1=2 nþ1 þ lnþ1 Þ;
ð25Þ
lim ½un ðx; tÞ uðx; tÞW 2;1 ðQ Þ ¼ 0;
ð26Þ
lim khn ðx; tÞ hðx; tÞkW 2;1 ðQÞ ¼ 0;
ð27Þ
06t6T
06t6T n!1
2
n!1
2
where u(x, t) and h(x, t) are the solution of the problem (6)–(9). Proof. For the differences un(x, t) u(x, t) and hn(x, t) h(x, t) we have
ðun uÞ0 mPJ Dðun uÞ ¼ ðI Pn1 ÞPJ ððun rÞun f Þ þ PJ ððu rÞu ðun rÞun Þ gbðI Pn1 ÞPJ ðk3 hn Þ þ gbPJ ðk3 hn k3 hÞ;
ð28Þ
ðhn hÞ0 jDðhn hÞ ¼ ðI Pn2 Þðrhn un uÞ rhn un þ rh u:
ð29Þ
2
We take the inner product of the Eq. (28) in [L2(X)] by un(x, t) u(x, t), multiply the Eq. (29) by hn(x, t) h(x, t) in L2(X) and integrate over the interval [0, s], s 6 T. Then
1 ½un u2 þ m 2
Z
s
½rðun uÞ2 dt 6
0
Z
s
jðP J ðf þ ðun rÞun gbk3 hn Þ; ðI Pn1 Þðun uÞÞjdt Z s Z s jððun rÞun þ ðu rÞu; un uÞjdt þ gb jðk3 hn k3 h; un uÞjdt; þ 0
0
1 khn hk2 þ j 2
Z
s
Z
krðhn hÞk2 dt 6
0
s
jðrhn un u; ðI Pn2 Þðhn hÞÞjdt þ
Z
0
s
jðrh u rhn un ; hn hÞjdt:
0
Since hn hbelongs to the space W 2;1 2 ðQ Þ, from (31) we obtain
1 khn hk2 þ j 2
Z
s
0
1
2 krðhn hÞk2 dt 6 lnþ1
Z
s
krðhn hÞkkrhn un ukdt þ
0
Z
s
jðrhn ðun uÞ; hn hÞjdt:
0
1;0 Because the space W 2;1 2 ðQ Þ is embedded in L4(Q) and in W 4 ðQ Þ, by using (15), (16), we get
1 khn hk2 þ j 2
Z
s
C
krðhn hÞk2 dt 6
e
0
l1 nþ1 þ
e
Z
2
s
krðhn hÞk2 dt þ
0
Z
0
s
krhn k½u un 2L4 ðXÞ khn hk2L4 ðXÞ dt:
From this and (22) for sufficiently small e > 0 we have
khn hk2 þ
Z 0
s
krðhn hÞk2 dt 6 C
ð30Þ
0
l1 nþ1 þ
Z 0
s
½u un 2L4 ðXÞ khn hk2L4 ðXÞ dt :
We use the inequality (13) for the spaces L4 ðXÞ; W 12 ðXÞ and L2(X), then
ð31Þ
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
khn hk2 þ
Z
s
krðhn hÞk2 dt 6 C
0
l1 nþ1 þ
Z
s
0
525
½u un 2L4 ðXÞ krðhn hÞkkhn hkdt :
Now, by applying Cauchy inequality, we obtain
khn hk2 þ
Z
s
krðhn hÞk2 dt 6 C
0
l1 nþ1 þ
Z
e 2
s
krðhn hÞk2 dt þ
0
1 2e
Z 0
s
½u un 4L4 ðXÞ khn hk2 dt :
From this for sufficiently small e > 0 it follows that
khn hk2 þ
Z
s
krðhn hÞk2 dt 6 C
0
l1 nþ1 þ
Z 0
s
½u un 4L4 ðXÞ khn hk2 dt :
ð32Þ
By analogy, from the Eq. (30) we have
½un u2 þ
Z
s
0
Z s Z s ½rðun uÞ2 dt 6 C k1 ½u un 4L4 ðXÞ ½un u2 dt þ khn hk2 dt : nþ1 þ 0
ð33Þ
0
Adding (32) and (33), we come to the inequality
½un u2 þ khn hk2 6 C
1 l1 nþ1 þ knþ1 þ
Z 0
s
ð½u un 4L4 ðXÞ þ 1Þðkh hn k2 þ ½u un 2 Þdt :
We use the Gronwall inequality, then
Z T 4 4 1 ½un u2 þ khn hk2 6 Cðl1 þ k Þ exp C ð½u u þ kh h k þ 1Þdt : n L4 ðXÞ n L4 ðXÞ nþ1 nþ1
ð34Þ
0
Since
Z
T
0
½u un 4L4 ðXÞ dt 6 C
Z
T
½rðu un Þ2 ½un u2 dt;
0
from (18) and (21) it follows that
Z
T
0
½u un 4L4 ðXÞ dt 6 C:
In the similar way, it is possible to obtain
Z 0
T
½kh hn k4L4 ðXÞ dt 6 C:
From (34) and from last two inequalities we obtain the estimates (24), (25). Now we shall prove the relations (26) and (27). We put
@un mPJ Dun þ P J ððun rÞun Þ gbPJ ðk3 hn Þ PJ f ; @t @hn d2n ¼ jDhn þ rhn un u: @t
d1n ¼
Since un and hn are the solution of the problem (10)–(12), then
d1n ¼ ðI Pn1 ÞPJ f þ ðI Pn1 ÞPJ ððun rÞun Þ gbðI Pn1 ÞPJ ðk3 hn Þ: Hence,
1 dn L
2 ðQ Þ
6 ½ðI Pn1 ÞPJ f L2 ðQ Þ þ ½ðI Pn1 ÞPJ ððun rÞun ÞL2 ðQÞ þ ½gbðI Pn1 ÞPJ ðk3 hn ÞL2 ðQ Þ :
From (15), (16) and from the embedding theorems (see [12]) it follows that the sets {PJ(unr)un} and {hn} are compact in L2(Q). It is known that the sequence of the bounded operators converges uniformly on a compact set, therefore
1 dn L
2 ðQ Þ
! 0;
n ! 1:
By analogy, we establish that
kd2n kL2 ðQ Þ ! 0;
n ! 1:
For difference un u we have the identity
@ðun uÞ mPJ Dðun uÞ ¼ d1n þ PJ ððu rÞu ðun rÞun Þ þ gbPJ ðk3 ðhn hÞÞ: @t
ð35Þ
526
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
From this and from (23) it follows that
½un uW 2;1 ðQ Þ 6 C 2
d1n L
þ ½ðu rÞu ðun rÞun L2 ðQ Þ þ ½gbk3 ðhn hÞL2 ðQ Þ :
2 ðQ Þ
ð36Þ
From (25) we obtain
½gbk3 ðhn hÞL2 ðQ Þ ! 0;
n ! 1:
ð37Þ
Now we estimate the second addend of the right-hand side of (36)
½ðu rÞu ðun rÞun L2 ðQ Þ 6 ½un uL4 ðQ Þ ½ruL4 ðQ Þ þ ½un L6 ðQÞ ½rðun uÞL3 ðQ Þ : 1;0 By using the inequality (15) and the embedding of the space W 2;1 2 ðQ Þ in L6(Q) and in W 4 ðQ Þ, we come to the estimate
½ðu rÞu ðun rÞun L2 ðQ Þ 6 Cð½un uL4 ðQÞ þ ½rðun uÞL3 ðQ Þ Þ: 1;0 2;1 Further, applying the inequality (14) for spaces L4 ðQ Þ; W 2;1 2 ðQ Þ and L2(Q), and also for W 3 ðQ Þ; W 2 ðQ Þ and L2(Q),we have
½ðu rÞu ðun rÞun L2 ðQ Þ 6 Cð½un u1=22;1
W 2 ðQ Þ
5=6 ½un u1=2 L2 ðQ Þ þ ½un u 2;1
W 2 ðQÞ
½un u1=6 L2 ðQ Þ Þ:
From this and (15), (24) it follows that
½ðu rÞu ðun rÞun L2 ðQ Þ ! 0;
n ! 1:
ð38Þ
Using (35)–(38), we obtain (26). By analogy, we come to the estimate (27). h 2 2;1 Let the vector-function z (x, t) belongs to the space ½W ðQ Þ \ J ðQ Þ and let the function z2(x,t) belongs to the space 1 2 W 2;1 2 ðQ Þ. We consider the following problem
@v mPJ Dv þ PJ ððz1 rÞv þ ðv rÞz1 Þ gbPJ ðk3 wÞ ¼ PJ hðx; tÞ; @t @w jDw þ rw z1 þ rz2 v ¼ h1 ðx; tÞ; ðx; tÞ 2 Q ; @t div v ¼ 0;
v ðx; tÞ ¼ 0; v ðx; 0Þ ¼ 0;
ðx; tÞ 2 Q;
ð39Þ ð40Þ ð41Þ
wðx; tÞ ¼ 0;
ðx; tÞ 2 S;
ð42Þ
wðx; 0Þ ¼ 0;
x 2 X:
ð43Þ
Lemma 3.1. Suppose that h(x, t) 2 [L2(Q)]2, h1(x, t) 2 L2(Q). Then problem (39)–(43) has a unique solution h i2 2;1 \ J ðQ Þ; w 2 W 2;1 \ J ðQ Þ and z2 2 W 2;1 2 ðQ Þ for any functions z1 2 W 2 ðQ Þ 2 ðQ Þ such that
½z1 W 2;1 ðQ Þ 6 R; 2
v2
h
W 2;1 2 ðQ Þ
i2
kz2 kW 2;1 ðQ Þ 6 R; 2
where R is a positive constant. Proof. We put
Kðz1 ; z2 Þw ¼
PJ ððz1 rÞI þ ðI rÞz1 Þ gbP J ðk3 IÞ
rz2 I
r I z1
v : w
Then
kKðz1 ; z2 Þwk½L2 ðXÞ3 6 ½ðz1 rÞv þ krw z1 k þ ½ðv rÞz1 þ krz2 v k þ gb½k3 w:
ð44Þ
By using Hölder inequality and (13), we obtain 3
1
I1 ¼ ½ðz1 rÞv 6 ½z1 L4 ðXÞ ½rv L4 ðXÞ 6 C½z1 L4 ðXÞ ½v 4W 2 ðXÞ ½v 4 : 2
From (20) we have 3
1
I1 6 C½z1 L4 ðXÞ ½PJ Dv 4 ½v 4 :
ð45Þ
Because z1(x, 0) = 0, we see that
Z 0
s
Z X
@z1 3 1 z dxdt ¼ 4 @t 1
Z Z X
0
s
@ 4 1 z dtdx ¼ @t 1 4
Z X
z41 dx:
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
527
From this it follows that
Z X
z41 dx 6 C½z1 W 2;1 ðQ Þ ½z1 3L6 ðQÞ : 2
W 2;1 2 ðQ Þ
The space
Z X
is embedded in L6(Q) (see [12]). Therefore,
z41 dx 6 C½z1 4W 2;1 ðQ Þ : 2
Hence, from (45) we have 3
1
I1 6 CR½PJ Dv 4 ½v 4 :
ð46Þ
By analogy, we come to estimate 3
1
I2 ¼ krw z1 k 6 CRkDwk4 kwk4 : W 22 ð
The space
ð47Þ
XÞ is embedded in L1(X) (see [12]). Therefore,
I3 ¼ ½ðv rÞz1 6 C½v L1 ðXÞ ½rz1 : By using the inequality (13) for the spaces L1 ðXÞ; W 22 ðXÞand L2(X), we get 1
1
I3 6 C½rz1 ½PJ Dv 2 ½v 2 :
ð48Þ
It is obvious that
Z
s
Z
0
X
@z1 Dz1 dxdt ¼ @t
Z
s
Z
0
X
@ rz1 1 rz1 dxdt ¼ 2 @t
Z Z X
s 0
@jrz1 j2 1 dtdx ¼ ½rz1 2 : 2 @t
From this and from (48) it follows that 1
1
I3 6 CR½PJ Dv 2 ½v 2 :
ð49Þ
Likewise, we come to estimate 1
1
I4 ¼ krz2 v k 6 CR½PJ Dv 2 ½v 2 :
ð50Þ
From 44, 46, 47, 49 and 50 we obtain
3 1 3 1 1 1 kKðz1 ; z2 Þwk½L2 ðXÞ3 6 CR ½PJ Dv 4 ½v 4 þ kDwk4 kwk4 þ ½PJ Dv 2 ½v 2 þ Ckwk:
ð51Þ
We denote
A¼
mPJ D
0
0
jD
;
then
kAwk½L2 ðXÞ3 ¼ ½mPJ Dv þ k jDwk: From this and (51), by using the positive definiteness of the operators (PJD) and (D), it follows that 3
kKðz1 ; z2 Þwk½L2 ðXÞ3 6 CkAwk4½L
1
2 ðXÞ
3
kwk4½L
2 ðXÞ
3
:
ð52Þ
Thus, the operator K(z1, z2) is subordinated to the operator A with order 34. Therefore, the statement of this lemma follows from Theorem 2.1 of the paper [14]. h Theorem 3.2. Let f(x, t) 2 [C1([0, T];L2(X))]2,f(x, 0) = 0, u(x, t) 2 C1([0, T];L2(X)),u(x,0) = 0. Then
1 @ðhn hÞ 1 @ðun uÞ 6 C k8 þ l8 ; þ sup nþ1 nþ1 @t @t 06t6T 06t6T 1 14 4 þ lnþ1 : sup ½rðun uÞ þ sup krðhn hÞk 6 C knþ1 sup
06t6T
ð53Þ ð54Þ
06t6T
Proof. In the problem (39)–(43) we put z1 = u, z2 = h, where u and hare the solution of the problem (6)–(9), h ¼ @f , h1 ¼ @@tu. @t 2 2;1 Then, according to Lemma 3.1, the problem (39)–(43) has a unique solution v 2 ½W 2;1 ðQ Þ \ J ðQ Þ, w 2 W ðQ Þ. Now, we 2 2 set z1 = un, z2 = hn, where un and hnare the solution of the problem (10)–(12). Again, using Lemma 3.1, we obtain that the 2 2;1 problem (39)–(43) has a unique solution v ðnÞ ¼ v ðn; x; tÞ 2 ½W 2;1 2 ðQ Þ \ J ðQ Þ; wðnÞ ¼ wðn; x; tÞ 2 W 2 ðQ Þ for each n.
528
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
Let w ¼
v ; wðnÞ ¼ wðn; x; tÞ ¼ v ðnÞ ; F ¼ wðnÞ w
PJ @f @t
!
@u @t
, then
@w þ Aw þ Kðu; hÞw ¼ F; wðx; 0Þ ¼ 0; @t @wðnÞ þ AwðnÞ þ Kðun ; hn ÞwðnÞ ¼ F; wðn; x; 0Þ ¼ 0: @t
Pn1 Let P n ¼ 0 (56) we have
0 P n2
ð55Þ ð56Þ
be the orthogonal projection in [L2(X)]3 onto the linear span of the elements fei ; mi gni¼1 . Then from (55),
@ðw wðnÞÞ þ Aðw wðnÞÞ þ Pn Kðu; hÞðw wðnÞÞ ¼ ðPn IÞKðu; hÞðw wðnÞÞ þ ðKðun ; hn Þ Kðu; hÞÞwðnÞ: @t
ð57Þ
According to the inequality (52), the operator K(u, h) is subordinated to the operator A with order 34. Hence, from [14] it fol
lows that the operator @t@ þ A þ Pn Kðu; hÞ is invertible and
@ þ A þ Pn Kðu; hÞ @t
0 1 1 1 !1 1 @ @ @ @ A: ¼ I P n I þ Kðu; hÞ Pn Kðu; hÞ þA þA þA @t @t @t
1
Therefore, from (57) we have
@ þA w wðnÞ ¼ @t
1
0
1 1 !1 1 @ @I Pn I þ Kðu; hÞ @ þ A A þA Pn Kðu; hÞ @t @t
ððPn IÞKðu; hÞðw wðnÞÞ þ ðKðun ; hn Þ Kðu; hÞÞwðnÞÞ:
ð58Þ
We estimate each addend of the right-hand side of (58)
kJ 1 k½L2 ðXÞ3
0 1 1 1 !1 1 @ @ @ @I Pn I þ Kðu; hÞ AðPn IÞKðu; hÞðw wðnÞÞk þA þA ¼ Pn Kðu; hÞ ½L2 ðXÞ3 @t þ A @t @t 1 1 1 !1 @ @ @ þA þA þA 6 ðPn IÞKðu; hÞðw wðnÞÞ þ Pn I þ Kðu; hÞ Pn @t @t @t 3 ½L2 ðXÞ
1 @ ðu; hÞ þA ðPn IÞKðu; hÞðw wðnÞÞk @t
In [14] the estimates were established, which with reference to the given problem can be written as
1 @ 12 12 þA ðPn IÞg 6 Cðknþ1 þ lnþ1 Þkgk½L2 ðQ Þ3 ; 8g 2 ½L2 ðQÞ3 ; @t ½L2 ðXÞ3 1 !1 I þ Kðu; hÞ @ þ A Pn 6 C; @t ½L2 ðQÞ3 !½L2 ðQ Þ3 1 @ þA g 6 Ckgk½L2 ðQ Þ3 ; 8g 2 ½L2 ðQ Þ3 : @t 3
ð59Þ
ð60Þ
ð61Þ
½L2 ðXÞ
Taking into account (59)–(61), we come to the estimate
1 @ 12 12 þA kJ 1 k½L2 ðXÞ3 6 Cðknþ1 þ lnþ1 ÞkKðu; hÞðw wðnÞÞk½L2 ðQ Þ3 þ C Kðu; hÞ ðPn IÞKðu; hÞðw wðnÞÞ @t
:
ð62Þ
½L2 ðQÞ3
From (52) we have 3
kKðu; hÞðw wðnÞÞk½L2 ðQ Þ3 6 CkAðw wðnÞÞk4½L
2 ðQÞ
1
3
kw wðnÞk4½L
2 ðQÞ
3
:
From Lemma 3.1 it follows that
sup kwðnÞk½L2 ðXÞ3 6 C;
06t6T
kAwðnÞk½L2 ðQ Þ3 6 C:
ð63Þ
Therefore, from the last three inequalities we obtain
kKðu; hÞðw wðnÞÞk½L2 ðQ Þ3 6 C:
ð64Þ
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P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
Now we shall consider the second addend of (62)
1 @ þA ðPn IÞKðu; hÞðw wðnÞÞ Kðu; hÞ @t
½L2 ðQ Þ3
3 1 4 @ þA 6 C A ðP n IÞKðu; hÞðw wðnÞÞ @t
½L2 ðQÞ3
1 1 4 @ þA ðPn IÞKðu; hÞðw wðnÞÞ @t
:
½L2 ðQ Þ3
Thus, taking into account the inequalities (59) and (64), we derive
1 @ ðPn IÞKðu; hÞðw wðnÞÞ þA Kðu; hÞ @t
1 18 8 6 C knþ1 þ lnþ1 :
½L2 ðQ Þ3
This, together with (62), (64), leads to the estimate
1 18 8 kJ 1 k½L2 ðXÞ3 6 C knþ1 þ lnþ1 :
ð65Þ
We estimate the second addend of the right-hand side of (58)
kJ 2 k½L2 ðXÞ3
0 1 1 1 !1 1 @ @ @ @I Pn I þ Kðu; hÞ AððKðun ; hn Þ Kðu; hÞÞwðnÞÞk þA þA ¼ Pn Kðu; hÞ ½L2 ðXÞ3 @t þ A @t @t 6 CkðKðun ; hn Þ Kðu; hÞÞwðnÞk½L2 ðQÞ3 : ð66Þ
It is obvious that
kðKðun ; hn Þ Kðu; hÞÞwðnÞk½L2 ðQÞ3 6 ½PJ ðððun uÞ rÞv ðnÞÞL2 ðQ Þ þ krwðnÞðun uÞkL2 ðQ Þ þ ½PJ ððv ðnÞ rÞ ðun uÞÞL2 ðQ Þ þ krðhn hÞ v ðnÞkL2 ðQÞ :
ð67Þ
Now we shall consider each addend of the right-hand side of (67). We apply Hölder inequality, then
I1 ¼ ½PJ ðððun uÞ rÞv ðnÞÞL2 ðQÞ 6 ½un uL4 ðQ Þ ½rv ðnÞL4 ðQÞ : Next, we use the estimate (63) and inequality (13) for the spaces L4 ðXÞ; W 22 ðXÞ; L2 ðXÞ. Therefore,
I1 6 C
Z
T
½un u2W 2 ðXÞ ½un u3 dt
14 :
2
0
From this and from Theorem 3.1 we obtain 3
8 I1 6 Cknþ1 :
ð68Þ
By analogy, we obtain 3
8 I2 ¼ krwðnÞðun uÞkL2 ðQÞ 6 Cknþ1 ; I3 ¼ ½PJ ððv ðnÞ rÞðun uÞÞL2 ðQ Þ 6 ½v ðnÞL6 ðQ Þ ½rðun uÞL3 ðQ Þ :
ð69Þ
Now we apply the estimate (63) and the inequality (13) for the spaces W 13 ðXÞ; W 22 ðXÞ; L2 ðXÞ. Then
I3 6 C
Z
T
½un
0
u2W 2 ðXÞ ½un 2
udt
13 :
From preceding relation and also from Lemma 2.1 and Theorem 3.1 it follows that 1
6 I3 6 Cknþ1 :
ð70Þ
Likewise, we obtain 1
6 I4 ¼ krðhn hÞv ðnÞkL2 ðQ Þ 6 C lnþ1 :
ð71Þ
According to the inequalities (66)–(71), we have 1
1
6 6 kJ 2 k½L2 ðXÞ3 6 Cðknþ1 þ lnþ1 Þ:
ð72Þ
530
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
From 58, 65 and 72 it follows
1 18 8 : kw wðnÞk½L2 ðXÞ3 6 C knþ1 þ lnþ1 We put wn ¼
ð73Þ
v n ; z ¼ un ; z ¼ u , then n wn hn h
@wn þ Awn þ Pn Kðun ; hn Þwn ¼ P n F; @t
wn ðx; 0Þ ¼ 0:
ð74Þ
The Eqs. (74) and (56) leads to the relation
@ðwn wðnÞÞ þ Aðwn wðnÞÞ þ Pn Kðun ; hn Þðwn wðnÞÞ ¼ ðI Pn ÞðKðun ; hn ÞwðnÞ FÞ: @t By analogy with the proof of (73), it is easy to establish that
1 18 8 kwn wðnÞk½L2 ðXÞ3 6 C knþ1 þ lnþ1 :
ð75Þ
From (26) and (27) it follows that
@z lim w ¼ 0: n n!1 @t ½L2 ðQ Þ3
ð76Þ
It is obvious that
w @z wn @z 6 kw wðnÞk : 3 þ kwn wðnÞk 3 þ ½L2 ðQ Þ ½L2 ðQ Þ @t ½L2 ðQÞ3 @t ½L2 ðQ Þ3 The last inequality, together with 73, 75, 76 leads to the equality w ¼ @z almost everywhere in Q. Since @t
@ðzn zÞ @t
½L2 ðXÞ3
@z wðnÞ 6 þ kwðnÞ wn k½L2 ðXÞ3 ; @t ½L2 ðXÞ3
from (73) and (75) we obtain the estimate (53). Because zn is the solution of the problem (10)–(12), we see that
@zn kAzn k½L2 ðXÞ3 6 þ ½Pn1 PJ ððun rÞun Þ þ kPn2 ðrhn un Þk þ gb½Pn1 P J ðk3 hn Þ þ ½Pn1 PJ f þ kPn2 uk: @t ½L2 ðXÞ3 Hence, from (53) and (17) we have
kAzn k½L2 ðXÞ3 6 C þ ½ðun rÞun þ krhn un k: Let us apply the Hölder inequality, then
kAzn k½L2 ðXÞ3 6 C þ ½un L6 ðXÞ ½run L3 ðXÞ þ ½un L6 ðXÞ krhn kL3 ðXÞ : Since the space W 12 ðXÞ is embedded in L6(X), from (21) and (22) it follows that
kAzn k½L2 ðXÞ3 6 Cð1 þ ½run L3 ðXÞ þ krhn kL3 ðXÞ Þ: Now we use the inequality (13) for the spaces W 13 ðXÞ; W 22 ðXÞ; L2 ðXÞ and the inequalities (17), (18). Then
2 2 kAzn k½L2 ðXÞ3 6 C 1 þ ½un 3W 2 ðXÞ þ khn k3W 2 ðXÞ : 2
2
From it and coercive inequality (20) we obtain
2 kAzn k½L2 ðXÞ3 6 C 1 þ kAzn k3½L
3 2 ðXÞ
:
Therefore,
sup kAzn k½L2 ðXÞ3 6 C:
06t6T
We use the moment inequality (see [15]), then 1
½rðun uÞ þ krðhn hÞk 6 kAðzn zÞk2½L
1
2 ðXÞ
3
kzn zk2½L
3 2 ðXÞ
:
From this the desired estimate (54) follows. The proof of the theorem is complete.
h
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
531
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G.Z. Gershuni, E.M. Zhuhovitsky, Convective Stability of Incompressible Fluid, Nauka, Moscow, 1972 (in Russian). B.J. Daly, A numerical study of turbulent transitions in convective flow, J. Fluid Mech. 64 (1974) 129–165. J.H. Curry, J.R. Herring, J. Loncatic, S.A. Orszag, Order and disorder in two- and three-dimensional Benard convection, J. Fluid Mech. 147 (1984) 1–38. J. Werne, E.E. DeLuca, R. Rosner, F. Cattaneo, Numerical simulation of soft and hard turbulence: preliminary results for two-dimensional convection, Phys. Rev. Lett. 64 (20) (1990) 2370–2373. V. Ambethkar, Numerical solutions of heat and mass transfer effects of an unsteady MHD free convective flow past an infinite vertical plate with constant suction, J. Naval Arch. Marine Eng. 5 (2008) 28–36. M. Shihbot, W.P. Kotorynski, The initial value problem for a viscous heat-conducting equations, J. Math. Anal. Appl. 45 (1) (1974) 1–22. V.G. Babskii, N.D. Kopachevskii, A.D. Myshkis, L.A. Slobozhanin, A.D. Tyuptsov, Fluid Mechanics of Weightlessness, Nauka, Moscow, 1976 (in Russian). O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967. pp. 736; O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, English Translation: Transactions on Mathematics Monograph, vol. 23, AMS, Rhode Island, 1968. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, Gordon and Breach, New York, 1969. S. Agmon, On the eigenfunctions and on the eigenvalue of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962) 119–147. V.P. Glushko, S.G. Krein, Inequalities for the norms of derivative in spaces Lp with weight, Sib. Mat. J. 1 I.3 (1960) 343–382 (in Russian). V.A. Solonnikov, On estimates of the solutions of elliptic and parabolic systems in Lp, Tr. MIAN SSSR 102 (1967) 137–160 (in Russian). F. Beckenbach, R. Bellman, Inequalities, Springer-Verlag, Berlin, 1961. P. Vinogradova, A. Zarubin, Projection method for Cauchy problem for operator-differential equation, Numer. Funct. Anal. Optim. 30 (1-2) (2009) 148–167. S.G. Krein, Linear Differential Equations in Banach Spaces, Transactions on Mathematics Monograph, vol. 29, AMS, Providence, 1971.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A study of Galerkin method for the heat convection equations Polina Vinogradova ⇑, Anatoli Zarubin Department of Natural Sciences, Far Eastern State Transport University, 680021, Khabarovsk, Serisheva 47, Russia
a r t i c l e
i n f o
Keywords: Approximate solution Error estimate Galerkin method Heat convection equation Orthogonal projection
a b s t r a c t This article investigates the Galerkin method for an initial boundary value problem for heat convection equations. The new error estimates for the approximate solutions and their derivatives in strong norm are obtained. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The large number of works is devoted to the study of various convection problems. It is possible to recognize two important directions of the study of convection phenomena. The first one is experimental and theoretical study of the convective stability. In detail these questions are considered, for example, in the monograph [1]. The other important direction is the numerical modeling of convection processes (see, for example, [2–5]). It allows to calculate the modes of convection at various meanings of Rayleigh, Reynolds numbers and at other parameters of the model. It is known that the main theoretical basis of numerical methods is the proof of convergence of the approximate solution to the exact one of the corresponding differential problem. The order of the convergence speed of approximate solutions of a nonlinear problem much depends on a kind of the nonlinear terms. It is often difficult to establish the convergence. In this case the basic information on the convergence of the computing procedure is found out by numerical experiments. In the present paper we study the Galerkin method for the approximate solving of an initial boundary value problem for a non-stationary quasi-linear system which describes the motion of the non-uniformly heated viscous incompressible fluid. The convergence of the Galerkin approximations in a strong norm is established, and also the asymptotic error estimates for the solutions and their derivatives in the uniform norm are obtained. 2. Statement of the problem and auxiliary assertions Let X be a bounded domain in R2with the smooth boundary oX, Q = X (0, T), S = oX (0, T], where T < 1. The initial boundary value problem for the heat convection in Boussinesq approximation is formulated in the following way ([1, 6, 7]): we seek a vector-function u(x, t):X [0, T] ? R2 and scalar functions p(x, t), h (x, t):X [0, T] ? R such that
@u mDu þ q1 0 rp þ ðu rÞu gbk3 h ¼ f ðx; tÞ; ðx; tÞ 2 Q; @t @h jDh þ u rh ¼ uðx; tÞ; ðx; tÞ 2 Q ; @t div uðx; tÞ ¼ 0; ðx; tÞ 2 Q ;
ð1Þ ð2Þ ð3Þ
uðx; tÞ ¼ 0; hðx; tÞ ¼ 0; ðx; tÞ 2 S;
ð4Þ
uðx; 0Þ ¼ 0; hðx; 0Þ ¼ 0; x 2 X;
ð5Þ
⇑ Corresponding author. E-mail addresses: [email protected] (P. Vinogradova), [email protected] (A. Zarubin). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.095
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
521
where m, j, q0, b > 0 are some phisical constants; k3 is a vertically directed up unit vector. Let Lp(X), 1 < p < +1, (respectively L1 (X)) be a space of real functions are absolutely integrable on X with the power of p according to Lebesque measure dx = dx1dx2 (respectively essentially bounded). These spaces with the norms
kukLp ðXÞ ¼
Z
1=p juðxÞjp dx X
respectively
kukL1 ðXÞ ¼ ess sup juðxÞj X
are Banach spaces. The space Lp(Q) is defined similarly. The Sobolev space W m p ðXÞ is a space of functions from Lp(X) whose all generalized partial derivatives up to order m inclusively belong to Lp(X) (m is a nonnegative integer). It is a Banach space with the norm
X
kukW mp ðXÞ ¼
!1=p j
kD
ukpLp ðXÞ
:
jjj6m
The space W 2m;m ðQ Þ (see [8]) with mbeing a nonnegative integer is a Banach space of functions from Lp(Q), which have genp eralized derivatives Drt Dsx with arbitrary nonnegative integers r and s satisfying inequality 2r + s 6 2m. The norm in W 2m;m ðQ Þ p is defined as
kukW 2m;m ðQÞ ¼ p
2m X X
kDrt Dsx ukLp ðQÞ :
j¼0 2rþs¼j
We put
W 1;0 p ðQ Þ ¼ fu 2 Lp ðQ Þ : Dx u 2 Lp ðQ Þg; n o W 12 ðXÞ ¼ u 2 W 12 ðXÞ : u ¼ 0 on @ X in the sense of traces :
2;1 The symbol W 2;1 2 ðQ Þ denotes the set of functions belonging W 2 ðQ Þ satisfying zero initial conditions and vanishing on S. We shall deal with two-dimensional vector-functions, each component of which belongs to one of the above defined spaces. We denote [Lp(X)]2 = Lp(X) Lp(X), [Lp(Q)]2 = Lp(Q) Lp(Q), etc. The norm, for example, in [Lp(X)]2 (p > 2) we shall h i2 h i2 denote by ½Lp ðXÞ . Let us denote similarly the norms in the spaces W 22 ðXÞ ; ½Lp ðQ Þ2 ; W 2;1 2 ðQ Þ .
The norm in L2(X) and in [L2(X)]2 will be denoted by kk and [], respectively. The inner product in L2(X) and in [L2(X)]2 will be denoted by (, ). Let J(X) be a set of solenoidal infinitely differentiable and finite on X vectors v(x) = (v1(x), v2(x)), J ðXÞ be the closure with h i2 1 respect to the norm of space W 2 ðXÞ .The elements of J ðQ Þ are the vectors v(x, t) that belong to J ðXÞ for almost all t. Let PJ be the orthogonal projection in [L2(X)]2 onto J ðXÞ. Using the operator PJ, the problem (1)–(5) can be written as
@u mPJ Du þ P J ððu rÞuÞ gbPJ ðk3 hÞ ¼ P J f ðx; tÞ; @t @h jDh þ u rh ¼ uðx; tÞ; ðx; tÞ 2 Q ; @t uðx; tÞ ¼ 0; hðx; tÞ ¼ 0; ðx; tÞ 2 S; uðx; 0Þ ¼ 0; hðx; 0Þ ¼ 0;
x 2 X:
ðx; tÞ 2 Q ;
ð6Þ ð7Þ ð8Þ ð9Þ
We consider the spectral problems
mP J De ¼ ke; eðxÞ ¼ 0;
e 2 J ðXÞ;
x 2 @X
and
jDm ¼ lm; mðxÞ ¼ 0;
x 2 @ X:
By ki we denote an eigenvalue, corresponding to the eigenvector ei(x), by li we denote an eigenvalue, corresponding to the h i2 eigenvector mi(x). The existence and the completeness of the eigenfunctions ei ðxÞ 2 W 22 ðXÞ \ J ðXÞ; mi ðxÞ 2 W 22 ðXÞ \ W 12 ðXÞ in the spaces [L2(X)]2 and L2(X) are proved in [9, 10]. Let Pn1 be the orthogonal projection in [L2(X)]2 onto the linear span of the vector-functions fei ðxÞgni¼1 ; P n2 be the orthogonal projection in L2(X) onto the linear span of the functions fmi ðxÞgni¼1 . The approximate solutions for the problem(6)–(9) are defined as
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P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
un ðx; tÞ ¼ hn ðx; tÞ ¼
n X i¼1 n X
ai ðtÞei ðxÞ; ci ðtÞmi ðxÞ;
i¼1
where unknown functions ai(t) and ci(t) (i = 1, 2, . . . , n) are the exact solution of the following problem:
@un mPJ Dun þ Pn1 PJ ððun rÞun Þ gbPn1 PJ ðk3 hn Þ ¼ Pn1 PJ f ðx; tÞ; @t @hn jDhn þ Pn2 ðun rhn Þ ¼ Pn2 uðx; tÞ; ðx; tÞ 2 Q ; @t un ðx; 0Þ ¼ 0; hn ðx; 0Þ ¼ 0; x 2 X:
ðx; tÞ 2 Q ;
From now on, by C we denote the difference positive constants independent of n. Later the following multiplicative inequalities will be used very often (see, for example, [11, 12]). Let
m1 ¼
ð10Þ ð11Þ ð12Þ
v 2 W lp
1 1
ðXÞ,
j3 j2 2 ; ji ¼ li ; j1 < j2 < j3 ; l3 6 l2 < l1 : pi j3 j1
Then
kv kW l2 ðXÞ 6 Ckv k1l3m1 kv km1 l1 W p ðXÞ
p2
3
W p ðXÞ
ð13Þ
:
1
This inequality also holds with l2 = l3 = 0 and p2 = 1. Let u 2 W 2m;m ðQ Þ and p(2m 2h s) > 4. Then any derivative Dht Dax u with —a— = s belongs to the space Lr(Q) with any p r P p including r = 1 and the inequality
kDht Dax ukLr ðQ Þ 6 e2m2hs4ðp r Þ kukW 2m;m ðQÞ þ e2hs4ðp r Þ kukLp ðQ Þ 1 1
1 1
p
holds for any e > 0. In the multiplicative form this inequality can be written as
kDht Dax ukLr ðQ Þ 6 CkukbW 2m;m ðQÞ kuk1b Lp ðQ Þ ;
ð14Þ
p
where b ¼ 2h þ s þ 4 1p 1r =ð2mÞ. h i2 Lemma 2.1. Let f(x, t) 2 [L2(Q)]2, u(x, t) 2 L2(Q). Then problem (10)–(12) has a unique solution un ðx; tÞ 2 W 2;1 \J 2 ðQ Þ ðQ Þ; hn ðx; tÞ 2 W 2;1 2 ðQ Þ for each n. The inequalities
½un ðx; tÞW 2;1 ðQÞ 6 C;
ð15Þ
khn ðx; tÞkW 2;1 ðQ Þ 6 C
ð16Þ
2
2
hold. Proof. We take the inner product of (11) in L2(X) by hn(x, s) and integrate the resulting relation over the interval [0, t], t 6 T. Then, using the equality
ðPn2 ðun rhn Þ; hn Þ ¼ 0; we obtain
sup khn ðx; tÞk2 þ
Z
06t6T
t
krhn k2 ds 6 C:
ð17Þ
0
Similarly, using the equality
ðPn1 PJ ððun rÞun Þ; un Þ ¼ 0 and (17), we have
sup ½un ðx; tÞ2 þ 06t6T
Z
t
½run 2 ds 6 C:
0
We multiply the Eq. (10) in [L2(X)]2 by PJDun and integrate the resulting relation over the interval [0, t], t 6 T. Then
ð18Þ
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
1 ½run ðx; tÞ2 þ m 2
Z
t
2 PJ Dun ðx; sÞ ds 6
Z
0
t
0
½f ðx; sÞ P J Dun ðx; sÞ ds þ gb
t
khn ðx; sÞk PJ Dun ðx; sÞ ds
0
Z
þ
Z
523
t
½ðun ðx; sÞ rÞun ðx; sÞ PJ Dun ðx; sÞ ds:
0
Now, using Cauchy inequality jakbj 6 2e jaj2 þ 21e jbj2 , for sufficiently small e > 0 we obtain
½run ðx; tÞ2 þ
Z
t
0
Z t ½PJ Dun ðx; sÞ2 ds 6 C ½f 2L2 ðQ Þ þ khn k2L2 ðQ Þ þ ½ðun rÞun 2 ds : 0
From this and (17) it follows that
½run ðx; tÞ2 þ
Z
t
Z t ½PJ Dun ðx; sÞ2 ds 6 C 1 þ ½ðun ðx; sÞ rÞun ðx; sÞ2 ds :
0
ð19Þ
0
Let us estimate the integral of the right-hand side of (19). We apply the Hölder inequality, then
J
Z
t
½ðun rÞun 2 ds 6
0
Z
t
0
½un 2L4 ðXÞ ½run 2L4 ðXÞ ds:
By using (13) for the spaces W 14 ðXÞ, W 22 ðXÞ and W 12 ðXÞ, we get
J6C
Z
t
½un 2L4 ðXÞ ½run ½un W 2 ðXÞ ds: 2
0
From this and coercive inequality (see, for example, [9]):
½zðxÞW 2 ðXÞ 6 C½PJ DzðxÞ; 8zðxÞ 2 ½W 22 ðXÞ2 \ J ðXÞ;
ð20Þ
2
we obtain
J6C
Z 0
t
½un 2L4 ðXÞ ½P J Dun ½run ds:
Therefore, from (19) it follows that
½run ðx; tÞ2 þ
Z
t
0
Z t Z 1 t ½PJ Dun ðx; sÞ2 ds 6 C 1 þ e ½PJ Dun 2 ds þ ½un 4L4 ðXÞ ½run 2 ds :
e
0
0
Choosing sufficiently small e > 0, we have
Z t ½run ðx; tÞ2 6 C 1 þ ½un 4L4 ðXÞ ½run 2 ds : 0
Now, applying the Gronwall inequality (see [13]), we come to an estimate
Z t ½run ðx; tÞ2 6 C exp C ½un 4L4 ðXÞ ds : 0
We use the inequality (13) for the spaces L4 ðXÞ; W 12 ðXÞ; L2 ðXÞ and also inequality (18), then
Z 0
T
½un 4L4 ðXÞ ds 6 C
Z
T
½un 2 ½run 2 ds 6 C
0
Z
T
½run 2 ds:
0
Thus,
sup ½run ðx; tÞ 6 C:
ð21Þ
06t6T
By applying the estimate (21), from the Eq. (11) it follows that
sup krhn ðx; tÞk 6 C:
ð22Þ
06t6T
From coercive inequality (see [9]):
½zW 2;1 ðQÞ 6 C 2
@ mPJ D z ; @t L2 ðQÞ
h
i2
8z 2 W 2;1 2 ðQ Þ
we have
½un W 2;1 ðQÞ 6 C ½f L2 ðQ Þ þ khn kL2 ðQ Þ þ ½ðun rÞun L2 ðQÞ : 2
\ J ðQ Þ;
ð23Þ
524
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
Then
½un W 2;1 ðQÞ 6 C 1 þ ½un L6 ðQ Þ ½run L3 ðQÞ : 2
2;1 By using the inequality (14) for the spaces W 1;0 3 ðQ Þ; W 2 ðQ Þ; L2 ðQ Þ, we get
5=6 1=6 ½un W 2;1 ðQÞ 6 C 1 þ ½un L6 ðQ Þ ½un W 2;1 ðQ Þ ½un L2 ðQ Þ : 2
Since the space
½un W 2;1 ðQÞ 2
2
W 12 ð
XÞ is embedded in L6(X) and the inequalities (18), (21) hold; it follows that : 6 C 1 þ ½un 5=62;1 W 2 ðQÞ
Thus, the estimate (15) holds. By analogy we obtain (16). h
3. Error estimates for Galerkin method In this section we establish the error estimates for the approximate solutions, for the gradient of the approximate solutions and for the derivative with respect to t. Theorem 3.1. Suppose that f(x, t) 2 [L2(Q)]2 and u(x, t) 2 L2(Q). Then 1=2 1=2 sup ½un ðx; tÞ uðx; tÞ 6 Cðknþ1 þ lnþ1 Þ;
ð24Þ
1=2 sup khn ðx; tÞ hðx; tÞk 6 Cðk1=2 nþ1 þ lnþ1 Þ;
ð25Þ
lim ½un ðx; tÞ uðx; tÞW 2;1 ðQ Þ ¼ 0;
ð26Þ
lim khn ðx; tÞ hðx; tÞkW 2;1 ðQÞ ¼ 0;
ð27Þ
06t6T
06t6T n!1
2
n!1
2
where u(x, t) and h(x, t) are the solution of the problem (6)–(9). Proof. For the differences un(x, t) u(x, t) and hn(x, t) h(x, t) we have
ðun uÞ0 mPJ Dðun uÞ ¼ ðI Pn1 ÞPJ ððun rÞun f Þ þ PJ ððu rÞu ðun rÞun Þ gbðI Pn1 ÞPJ ðk3 hn Þ þ gbPJ ðk3 hn k3 hÞ;
ð28Þ
ðhn hÞ0 jDðhn hÞ ¼ ðI Pn2 Þðrhn un uÞ rhn un þ rh u:
ð29Þ
2
We take the inner product of the Eq. (28) in [L2(X)] by un(x, t) u(x, t), multiply the Eq. (29) by hn(x, t) h(x, t) in L2(X) and integrate over the interval [0, s], s 6 T. Then
1 ½un u2 þ m 2
Z
s
½rðun uÞ2 dt 6
0
Z
s
jðP J ðf þ ðun rÞun gbk3 hn Þ; ðI Pn1 Þðun uÞÞjdt Z s Z s jððun rÞun þ ðu rÞu; un uÞjdt þ gb jðk3 hn k3 h; un uÞjdt; þ 0
0
1 khn hk2 þ j 2
Z
s
Z
krðhn hÞk2 dt 6
0
s
jðrhn un u; ðI Pn2 Þðhn hÞÞjdt þ
Z
0
s
jðrh u rhn un ; hn hÞjdt:
0
Since hn hbelongs to the space W 2;1 2 ðQ Þ, from (31) we obtain
1 khn hk2 þ j 2
Z
s
0
1
2 krðhn hÞk2 dt 6 lnþ1
Z
s
krðhn hÞkkrhn un ukdt þ
0
Z
s
jðrhn ðun uÞ; hn hÞjdt:
0
1;0 Because the space W 2;1 2 ðQ Þ is embedded in L4(Q) and in W 4 ðQ Þ, by using (15), (16), we get
1 khn hk2 þ j 2
Z
s
C
krðhn hÞk2 dt 6
e
0
l1 nþ1 þ
e
Z
2
s
krðhn hÞk2 dt þ
0
Z
0
s
krhn k½u un 2L4 ðXÞ khn hk2L4 ðXÞ dt:
From this and (22) for sufficiently small e > 0 we have
khn hk2 þ
Z 0
s
krðhn hÞk2 dt 6 C
ð30Þ
0
l1 nþ1 þ
Z 0
s
½u un 2L4 ðXÞ khn hk2L4 ðXÞ dt :
We use the inequality (13) for the spaces L4 ðXÞ; W 12 ðXÞ and L2(X), then
ð31Þ
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
khn hk2 þ
Z
s
krðhn hÞk2 dt 6 C
0
l1 nþ1 þ
Z
s
0
525
½u un 2L4 ðXÞ krðhn hÞkkhn hkdt :
Now, by applying Cauchy inequality, we obtain
khn hk2 þ
Z
s
krðhn hÞk2 dt 6 C
0
l1 nþ1 þ
Z
e 2
s
krðhn hÞk2 dt þ
0
1 2e
Z 0
s
½u un 4L4 ðXÞ khn hk2 dt :
From this for sufficiently small e > 0 it follows that
khn hk2 þ
Z
s
krðhn hÞk2 dt 6 C
0
l1 nþ1 þ
Z 0
s
½u un 4L4 ðXÞ khn hk2 dt :
ð32Þ
By analogy, from the Eq. (30) we have
½un u2 þ
Z
s
0
Z s Z s ½rðun uÞ2 dt 6 C k1 ½u un 4L4 ðXÞ ½un u2 dt þ khn hk2 dt : nþ1 þ 0
ð33Þ
0
Adding (32) and (33), we come to the inequality
½un u2 þ khn hk2 6 C
1 l1 nþ1 þ knþ1 þ
Z 0
s
ð½u un 4L4 ðXÞ þ 1Þðkh hn k2 þ ½u un 2 Þdt :
We use the Gronwall inequality, then
Z T 4 4 1 ½un u2 þ khn hk2 6 Cðl1 þ k Þ exp C ð½u u þ kh h k þ 1Þdt : n L4 ðXÞ n L4 ðXÞ nþ1 nþ1
ð34Þ
0
Since
Z
T
0
½u un 4L4 ðXÞ dt 6 C
Z
T
½rðu un Þ2 ½un u2 dt;
0
from (18) and (21) it follows that
Z
T
0
½u un 4L4 ðXÞ dt 6 C:
In the similar way, it is possible to obtain
Z 0
T
½kh hn k4L4 ðXÞ dt 6 C:
From (34) and from last two inequalities we obtain the estimates (24), (25). Now we shall prove the relations (26) and (27). We put
@un mPJ Dun þ P J ððun rÞun Þ gbPJ ðk3 hn Þ PJ f ; @t @hn d2n ¼ jDhn þ rhn un u: @t
d1n ¼
Since un and hn are the solution of the problem (10)–(12), then
d1n ¼ ðI Pn1 ÞPJ f þ ðI Pn1 ÞPJ ððun rÞun Þ gbðI Pn1 ÞPJ ðk3 hn Þ: Hence,
1 dn L
2 ðQ Þ
6 ½ðI Pn1 ÞPJ f L2 ðQ Þ þ ½ðI Pn1 ÞPJ ððun rÞun ÞL2 ðQÞ þ ½gbðI Pn1 ÞPJ ðk3 hn ÞL2 ðQ Þ :
From (15), (16) and from the embedding theorems (see [12]) it follows that the sets {PJ(unr)un} and {hn} are compact in L2(Q). It is known that the sequence of the bounded operators converges uniformly on a compact set, therefore
1 dn L
2 ðQ Þ
! 0;
n ! 1:
By analogy, we establish that
kd2n kL2 ðQ Þ ! 0;
n ! 1:
For difference un u we have the identity
@ðun uÞ mPJ Dðun uÞ ¼ d1n þ PJ ððu rÞu ðun rÞun Þ þ gbPJ ðk3 ðhn hÞÞ: @t
ð35Þ
526
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
From this and from (23) it follows that
½un uW 2;1 ðQ Þ 6 C 2
d1n L
þ ½ðu rÞu ðun rÞun L2 ðQ Þ þ ½gbk3 ðhn hÞL2 ðQ Þ :
2 ðQ Þ
ð36Þ
From (25) we obtain
½gbk3 ðhn hÞL2 ðQ Þ ! 0;
n ! 1:
ð37Þ
Now we estimate the second addend of the right-hand side of (36)
½ðu rÞu ðun rÞun L2 ðQ Þ 6 ½un uL4 ðQ Þ ½ruL4 ðQ Þ þ ½un L6 ðQÞ ½rðun uÞL3 ðQ Þ : 1;0 By using the inequality (15) and the embedding of the space W 2;1 2 ðQ Þ in L6(Q) and in W 4 ðQ Þ, we come to the estimate
½ðu rÞu ðun rÞun L2 ðQ Þ 6 Cð½un uL4 ðQÞ þ ½rðun uÞL3 ðQ Þ Þ: 1;0 2;1 Further, applying the inequality (14) for spaces L4 ðQ Þ; W 2;1 2 ðQ Þ and L2(Q), and also for W 3 ðQ Þ; W 2 ðQ Þ and L2(Q),we have
½ðu rÞu ðun rÞun L2 ðQ Þ 6 Cð½un u1=22;1
W 2 ðQ Þ
5=6 ½un u1=2 L2 ðQ Þ þ ½un u 2;1
W 2 ðQÞ
½un u1=6 L2 ðQ Þ Þ:
From this and (15), (24) it follows that
½ðu rÞu ðun rÞun L2 ðQ Þ ! 0;
n ! 1:
ð38Þ
Using (35)–(38), we obtain (26). By analogy, we come to the estimate (27). h 2 2;1 Let the vector-function z (x, t) belongs to the space ½W ðQ Þ \ J ðQ Þ and let the function z2(x,t) belongs to the space 1 2 W 2;1 2 ðQ Þ. We consider the following problem
@v mPJ Dv þ PJ ððz1 rÞv þ ðv rÞz1 Þ gbPJ ðk3 wÞ ¼ PJ hðx; tÞ; @t @w jDw þ rw z1 þ rz2 v ¼ h1 ðx; tÞ; ðx; tÞ 2 Q ; @t div v ¼ 0;
v ðx; tÞ ¼ 0; v ðx; 0Þ ¼ 0;
ðx; tÞ 2 Q;
ð39Þ ð40Þ ð41Þ
wðx; tÞ ¼ 0;
ðx; tÞ 2 S;
ð42Þ
wðx; 0Þ ¼ 0;
x 2 X:
ð43Þ
Lemma 3.1. Suppose that h(x, t) 2 [L2(Q)]2, h1(x, t) 2 L2(Q). Then problem (39)–(43) has a unique solution h i2 2;1 \ J ðQ Þ; w 2 W 2;1 \ J ðQ Þ and z2 2 W 2;1 2 ðQ Þ for any functions z1 2 W 2 ðQ Þ 2 ðQ Þ such that
½z1 W 2;1 ðQ Þ 6 R; 2
v2
h
W 2;1 2 ðQ Þ
i2
kz2 kW 2;1 ðQ Þ 6 R; 2
where R is a positive constant. Proof. We put
Kðz1 ; z2 Þw ¼
PJ ððz1 rÞI þ ðI rÞz1 Þ gbP J ðk3 IÞ
rz2 I
r I z1
v : w
Then
kKðz1 ; z2 Þwk½L2 ðXÞ3 6 ½ðz1 rÞv þ krw z1 k þ ½ðv rÞz1 þ krz2 v k þ gb½k3 w:
ð44Þ
By using Hölder inequality and (13), we obtain 3
1
I1 ¼ ½ðz1 rÞv 6 ½z1 L4 ðXÞ ½rv L4 ðXÞ 6 C½z1 L4 ðXÞ ½v 4W 2 ðXÞ ½v 4 : 2
From (20) we have 3
1
I1 6 C½z1 L4 ðXÞ ½PJ Dv 4 ½v 4 :
ð45Þ
Because z1(x, 0) = 0, we see that
Z 0
s
Z X
@z1 3 1 z dxdt ¼ 4 @t 1
Z Z X
0
s
@ 4 1 z dtdx ¼ @t 1 4
Z X
z41 dx:
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
527
From this it follows that
Z X
z41 dx 6 C½z1 W 2;1 ðQ Þ ½z1 3L6 ðQÞ : 2
W 2;1 2 ðQ Þ
The space
Z X
is embedded in L6(Q) (see [12]). Therefore,
z41 dx 6 C½z1 4W 2;1 ðQ Þ : 2
Hence, from (45) we have 3
1
I1 6 CR½PJ Dv 4 ½v 4 :
ð46Þ
By analogy, we come to estimate 3
1
I2 ¼ krw z1 k 6 CRkDwk4 kwk4 : W 22 ð
The space
ð47Þ
XÞ is embedded in L1(X) (see [12]). Therefore,
I3 ¼ ½ðv rÞz1 6 C½v L1 ðXÞ ½rz1 : By using the inequality (13) for the spaces L1 ðXÞ; W 22 ðXÞand L2(X), we get 1
1
I3 6 C½rz1 ½PJ Dv 2 ½v 2 :
ð48Þ
It is obvious that
Z
s
Z
0
X
@z1 Dz1 dxdt ¼ @t
Z
s
Z
0
X
@ rz1 1 rz1 dxdt ¼ 2 @t
Z Z X
s 0
@jrz1 j2 1 dtdx ¼ ½rz1 2 : 2 @t
From this and from (48) it follows that 1
1
I3 6 CR½PJ Dv 2 ½v 2 :
ð49Þ
Likewise, we come to estimate 1
1
I4 ¼ krz2 v k 6 CR½PJ Dv 2 ½v 2 :
ð50Þ
From 44, 46, 47, 49 and 50 we obtain
3 1 3 1 1 1 kKðz1 ; z2 Þwk½L2 ðXÞ3 6 CR ½PJ Dv 4 ½v 4 þ kDwk4 kwk4 þ ½PJ Dv 2 ½v 2 þ Ckwk:
ð51Þ
We denote
A¼
mPJ D
0
0
jD
;
then
kAwk½L2 ðXÞ3 ¼ ½mPJ Dv þ k jDwk: From this and (51), by using the positive definiteness of the operators (PJD) and (D), it follows that 3
kKðz1 ; z2 Þwk½L2 ðXÞ3 6 CkAwk4½L
1
2 ðXÞ
3
kwk4½L
2 ðXÞ
3
:
ð52Þ
Thus, the operator K(z1, z2) is subordinated to the operator A with order 34. Therefore, the statement of this lemma follows from Theorem 2.1 of the paper [14]. h Theorem 3.2. Let f(x, t) 2 [C1([0, T];L2(X))]2,f(x, 0) = 0, u(x, t) 2 C1([0, T];L2(X)),u(x,0) = 0. Then
1 @ðhn hÞ 1 @ðun uÞ 6 C k8 þ l8 ; þ sup nþ1 nþ1 @t @t 06t6T 06t6T 1 14 4 þ lnþ1 : sup ½rðun uÞ þ sup krðhn hÞk 6 C knþ1 sup
06t6T
ð53Þ ð54Þ
06t6T
Proof. In the problem (39)–(43) we put z1 = u, z2 = h, where u and hare the solution of the problem (6)–(9), h ¼ @f , h1 ¼ @@tu. @t 2 2;1 Then, according to Lemma 3.1, the problem (39)–(43) has a unique solution v 2 ½W 2;1 ðQ Þ \ J ðQ Þ, w 2 W ðQ Þ. Now, we 2 2 set z1 = un, z2 = hn, where un and hnare the solution of the problem (10)–(12). Again, using Lemma 3.1, we obtain that the 2 2;1 problem (39)–(43) has a unique solution v ðnÞ ¼ v ðn; x; tÞ 2 ½W 2;1 2 ðQ Þ \ J ðQ Þ; wðnÞ ¼ wðn; x; tÞ 2 W 2 ðQ Þ for each n.
528
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
Let w ¼
v ; wðnÞ ¼ wðn; x; tÞ ¼ v ðnÞ ; F ¼ wðnÞ w
PJ @f @t
!
@u @t
, then
@w þ Aw þ Kðu; hÞw ¼ F; wðx; 0Þ ¼ 0; @t @wðnÞ þ AwðnÞ þ Kðun ; hn ÞwðnÞ ¼ F; wðn; x; 0Þ ¼ 0: @t
Pn1 Let P n ¼ 0 (56) we have
0 P n2
ð55Þ ð56Þ
be the orthogonal projection in [L2(X)]3 onto the linear span of the elements fei ; mi gni¼1 . Then from (55),
@ðw wðnÞÞ þ Aðw wðnÞÞ þ Pn Kðu; hÞðw wðnÞÞ ¼ ðPn IÞKðu; hÞðw wðnÞÞ þ ðKðun ; hn Þ Kðu; hÞÞwðnÞ: @t
ð57Þ
According to the inequality (52), the operator K(u, h) is subordinated to the operator A with order 34. Hence, from [14] it fol
lows that the operator @t@ þ A þ Pn Kðu; hÞ is invertible and
@ þ A þ Pn Kðu; hÞ @t
0 1 1 1 !1 1 @ @ @ @ A: ¼ I P n I þ Kðu; hÞ Pn Kðu; hÞ þA þA þA @t @t @t
1
Therefore, from (57) we have
@ þA w wðnÞ ¼ @t
1
0
1 1 !1 1 @ @I Pn I þ Kðu; hÞ @ þ A A þA Pn Kðu; hÞ @t @t
ððPn IÞKðu; hÞðw wðnÞÞ þ ðKðun ; hn Þ Kðu; hÞÞwðnÞÞ:
ð58Þ
We estimate each addend of the right-hand side of (58)
kJ 1 k½L2 ðXÞ3
0 1 1 1 !1 1 @ @ @ @I Pn I þ Kðu; hÞ AðPn IÞKðu; hÞðw wðnÞÞk þA þA ¼ Pn Kðu; hÞ ½L2 ðXÞ3 @t þ A @t @t 1 1 1 !1 @ @ @ þA þA þA 6 ðPn IÞKðu; hÞðw wðnÞÞ þ Pn I þ Kðu; hÞ Pn @t @t @t 3 ½L2 ðXÞ
1 @ ðu; hÞ þA ðPn IÞKðu; hÞðw wðnÞÞk @t
In [14] the estimates were established, which with reference to the given problem can be written as
1 @ 12 12 þA ðPn IÞg 6 Cðknþ1 þ lnþ1 Þkgk½L2 ðQ Þ3 ; 8g 2 ½L2 ðQÞ3 ; @t ½L2 ðXÞ3 1 !1 I þ Kðu; hÞ @ þ A Pn 6 C; @t ½L2 ðQÞ3 !½L2 ðQ Þ3 1 @ þA g 6 Ckgk½L2 ðQ Þ3 ; 8g 2 ½L2 ðQ Þ3 : @t 3
ð59Þ
ð60Þ
ð61Þ
½L2 ðXÞ
Taking into account (59)–(61), we come to the estimate
1 @ 12 12 þA kJ 1 k½L2 ðXÞ3 6 Cðknþ1 þ lnþ1 ÞkKðu; hÞðw wðnÞÞk½L2 ðQ Þ3 þ C Kðu; hÞ ðPn IÞKðu; hÞðw wðnÞÞ @t
:
ð62Þ
½L2 ðQÞ3
From (52) we have 3
kKðu; hÞðw wðnÞÞk½L2 ðQ Þ3 6 CkAðw wðnÞÞk4½L
2 ðQÞ
1
3
kw wðnÞk4½L
2 ðQÞ
3
:
From Lemma 3.1 it follows that
sup kwðnÞk½L2 ðXÞ3 6 C;
06t6T
kAwðnÞk½L2 ðQ Þ3 6 C:
ð63Þ
Therefore, from the last three inequalities we obtain
kKðu; hÞðw wðnÞÞk½L2 ðQ Þ3 6 C:
ð64Þ
529
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
Now we shall consider the second addend of (62)
1 @ þA ðPn IÞKðu; hÞðw wðnÞÞ Kðu; hÞ @t
½L2 ðQ Þ3
3 1 4 @ þA 6 C A ðP n IÞKðu; hÞðw wðnÞÞ @t
½L2 ðQÞ3
1 1 4 @ þA ðPn IÞKðu; hÞðw wðnÞÞ @t
:
½L2 ðQ Þ3
Thus, taking into account the inequalities (59) and (64), we derive
1 @ ðPn IÞKðu; hÞðw wðnÞÞ þA Kðu; hÞ @t
1 18 8 6 C knþ1 þ lnþ1 :
½L2 ðQ Þ3
This, together with (62), (64), leads to the estimate
1 18 8 kJ 1 k½L2 ðXÞ3 6 C knþ1 þ lnþ1 :
ð65Þ
We estimate the second addend of the right-hand side of (58)
kJ 2 k½L2 ðXÞ3
0 1 1 1 !1 1 @ @ @ @I Pn I þ Kðu; hÞ AððKðun ; hn Þ Kðu; hÞÞwðnÞÞk þA þA ¼ Pn Kðu; hÞ ½L2 ðXÞ3 @t þ A @t @t 6 CkðKðun ; hn Þ Kðu; hÞÞwðnÞk½L2 ðQÞ3 : ð66Þ
It is obvious that
kðKðun ; hn Þ Kðu; hÞÞwðnÞk½L2 ðQÞ3 6 ½PJ ðððun uÞ rÞv ðnÞÞL2 ðQ Þ þ krwðnÞðun uÞkL2 ðQ Þ þ ½PJ ððv ðnÞ rÞ ðun uÞÞL2 ðQ Þ þ krðhn hÞ v ðnÞkL2 ðQÞ :
ð67Þ
Now we shall consider each addend of the right-hand side of (67). We apply Hölder inequality, then
I1 ¼ ½PJ ðððun uÞ rÞv ðnÞÞL2 ðQÞ 6 ½un uL4 ðQ Þ ½rv ðnÞL4 ðQÞ : Next, we use the estimate (63) and inequality (13) for the spaces L4 ðXÞ; W 22 ðXÞ; L2 ðXÞ. Therefore,
I1 6 C
Z
T
½un u2W 2 ðXÞ ½un u3 dt
14 :
2
0
From this and from Theorem 3.1 we obtain 3
8 I1 6 Cknþ1 :
ð68Þ
By analogy, we obtain 3
8 I2 ¼ krwðnÞðun uÞkL2 ðQÞ 6 Cknþ1 ; I3 ¼ ½PJ ððv ðnÞ rÞðun uÞÞL2 ðQ Þ 6 ½v ðnÞL6 ðQ Þ ½rðun uÞL3 ðQ Þ :
ð69Þ
Now we apply the estimate (63) and the inequality (13) for the spaces W 13 ðXÞ; W 22 ðXÞ; L2 ðXÞ. Then
I3 6 C
Z
T
½un
0
u2W 2 ðXÞ ½un 2
udt
13 :
From preceding relation and also from Lemma 2.1 and Theorem 3.1 it follows that 1
6 I3 6 Cknþ1 :
ð70Þ
Likewise, we obtain 1
6 I4 ¼ krðhn hÞv ðnÞkL2 ðQ Þ 6 C lnþ1 :
ð71Þ
According to the inequalities (66)–(71), we have 1
1
6 6 kJ 2 k½L2 ðXÞ3 6 Cðknþ1 þ lnþ1 Þ:
ð72Þ
530
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
From 58, 65 and 72 it follows
1 18 8 : kw wðnÞk½L2 ðXÞ3 6 C knþ1 þ lnþ1 We put wn ¼
ð73Þ
v n ; z ¼ un ; z ¼ u , then n wn hn h
@wn þ Awn þ Pn Kðun ; hn Þwn ¼ P n F; @t
wn ðx; 0Þ ¼ 0:
ð74Þ
The Eqs. (74) and (56) leads to the relation
@ðwn wðnÞÞ þ Aðwn wðnÞÞ þ Pn Kðun ; hn Þðwn wðnÞÞ ¼ ðI Pn ÞðKðun ; hn ÞwðnÞ FÞ: @t By analogy with the proof of (73), it is easy to establish that
1 18 8 kwn wðnÞk½L2 ðXÞ3 6 C knþ1 þ lnþ1 :
ð75Þ
From (26) and (27) it follows that
@z lim w ¼ 0: n n!1 @t ½L2 ðQ Þ3
ð76Þ
It is obvious that
w @z wn @z 6 kw wðnÞk : 3 þ kwn wðnÞk 3 þ ½L2 ðQ Þ ½L2 ðQ Þ @t ½L2 ðQÞ3 @t ½L2 ðQ Þ3 The last inequality, together with 73, 75, 76 leads to the equality w ¼ @z almost everywhere in Q. Since @t
@ðzn zÞ @t
½L2 ðXÞ3
@z wðnÞ 6 þ kwðnÞ wn k½L2 ðXÞ3 ; @t ½L2 ðXÞ3
from (73) and (75) we obtain the estimate (53). Because zn is the solution of the problem (10)–(12), we see that
@zn kAzn k½L2 ðXÞ3 6 þ ½Pn1 PJ ððun rÞun Þ þ kPn2 ðrhn un Þk þ gb½Pn1 P J ðk3 hn Þ þ ½Pn1 PJ f þ kPn2 uk: @t ½L2 ðXÞ3 Hence, from (53) and (17) we have
kAzn k½L2 ðXÞ3 6 C þ ½ðun rÞun þ krhn un k: Let us apply the Hölder inequality, then
kAzn k½L2 ðXÞ3 6 C þ ½un L6 ðXÞ ½run L3 ðXÞ þ ½un L6 ðXÞ krhn kL3 ðXÞ : Since the space W 12 ðXÞ is embedded in L6(X), from (21) and (22) it follows that
kAzn k½L2 ðXÞ3 6 Cð1 þ ½run L3 ðXÞ þ krhn kL3 ðXÞ Þ: Now we use the inequality (13) for the spaces W 13 ðXÞ; W 22 ðXÞ; L2 ðXÞ and the inequalities (17), (18). Then
2 2 kAzn k½L2 ðXÞ3 6 C 1 þ ½un 3W 2 ðXÞ þ khn k3W 2 ðXÞ : 2
2
From it and coercive inequality (20) we obtain
2 kAzn k½L2 ðXÞ3 6 C 1 þ kAzn k3½L
3 2 ðXÞ
:
Therefore,
sup kAzn k½L2 ðXÞ3 6 C:
06t6T
We use the moment inequality (see [15]), then 1
½rðun uÞ þ krðhn hÞk 6 kAðzn zÞk2½L
1
2 ðXÞ
3
kzn zk2½L
3 2 ðXÞ
:
From this the desired estimate (54) follows. The proof of the theorem is complete.
h
P. Vinogradova, A. Zarubin / Applied Mathematics and Computation 218 (2011) 520–531
531
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