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Blowup analysis for two-dimensional viscous compressible, heat-conductive Navier–Stokes equations
Applied Mathematics and Computation 232 (2014) 719–726
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Blowup analysis for two-dimensional viscous compressible, heat-conductive Navier–Stokes equations q Yongfu Wang, Qin Zhang ⇑ Department of Mathematics, Sichuan University, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper, we study the finite time blow-up mechanism of the strong solution to the two-dimensional (2D) viscous, compressible, heat-conductive Navier–Stokes system and initial vacuum states are allowed. We show that if the solution to 2D viscous compressible flows blows up in finite time, then either the mass of the compressible fluid will concentrate in some points or the compressible gas will combust in the finite time. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Compressible Navier–Stokes equations Heat-conductive Blowup criterion Vacuum
1. Introduction and main results In this paper, we consider the 2D viscous compressible and heat-conductive Navier–Stokes equations
8 > < @ t q þ divðquÞ ¼ 0; @ t ðquÞ þ divðqu uÞ lDu ðl þ kÞrdivu þ rP ¼ 0; > : 2 ct ½@ t ðqhÞ þ divðquhÞ jDh þ Pdivu ¼ 2ljDðuÞj2 þ kðdivuÞ ;
ð1Þ
with the initial-boundary values,
ðqðx; tÞ; uðx; tÞÞjt¼0 ¼ ðq0 ðxÞ; u0 ðxÞÞ; u ¼ 0;
@h ¼ 0; @~ n
in X;
on @ X;
ð2Þ ð3Þ
where X is a bounded smooth domain in R2 , ~ n ¼ ðn1 ; n2 Þ is the unit outward normal to @ X. Here u, q, h and P ¼ Rqh ðR > 0Þ denote the velocity, density, absolute temperature and pressure, respectively. DðuÞ is the deformation tensor, which is described as
DðuÞ ¼
1 ru þ ðruÞtr : 2
The constant viscosity coefficient
l and k satisfy the physical restrictions
l > 0; l þ k P 0; q This work is supported in part by NSFC Grant 11171236, PCSIRT (IRT1273), Sichuan Youth Science & Technology Foundation and Fundamental Research Funds for the Central Universities. ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Wang), zqcfl[email protected] (Q. Zhang).
http://dx.doi.org/10.1016/j.amc.2014.01.103 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
in 2D space. In addition, we denote ct and j as the specific heat at constant volume and the coefficient of heat conduction, respectively. The existence and behaviors of solution to compressible Navier–Stokes equations have been studied by a large number of literatures (see [3,4,8,9,14,17–19,24,25]). In the absence of vacuum, the local existence and uniqueness of classical solutions were proved in [25,27]. Matsumura–Nishida [24] first investigated the global classical solution for initial data close to a nonvacuum equilibrium in some Sobolev space Hs . Later, Hoff [9] obtained a global weak solution for the discontinuous initial data, when the initial density is strictly positive. However, the regularity and uniqueness of such weak solutions remain still open, even in two dimensional space. But, the global existence of strong solution has been obtained in the case of some small initial data or small energy. Especially, for 3D compressible Navier–Stokes equations, Lions [19] obtained a global weak solution as long as solutions have finite energy and the pressure PðqÞ ¼ aqc with c P 95, and Feireisl extended the results with c > 32 in [8]. Jiang–Zhang [17] constructed the global existence of weak solution for c > 1 with the symmetric initial data. On the other hand, there are many literatures [2,5,6, 10–12,15,16,26] concerning the blowup mechanism for the compressible viscous flows, since the significant work [30] on blowup of the smooth solutions to the Navier–Stokes equations provided that the initial density has compact support. It is worth mentioning here that there are two kinds of known blowup criteria, namely Beale–Kato–Majda type (BKM) [1] for 3D Euler equations, more precisely, that if T < 1 is the maximal time for the existence of a strong (or classical) solution, then
lim kr ukL1 ð0;T;L1 Þ ¼ 1;
T!T
ðBeale-Kato-Majda typeÞ
ð4Þ
and Serrin-type [18] for 3D incompressible Navier–Stokes equations, then
lim kukLs ð0;T;Lr Þ ¼ 1;
T!T
ðSerrin typeÞ;
where
2 3 þ 6 1; s r
3 < r 6 1:
For the 2D case, Jiang–Ou [16] constructed a BKM-type blowup criterion for full Navier–Stokes equation (1) over a unit square domain or periodic domain of R2 , and Huang–Li–Wang [13] obtained a blowup criterion that Serrin-type for (1) in RN , as
lim kdivukL1 ð0;T;L1 Þ þ kukLs ð0;T;Lr Þ ¼ 1;
T!T
2 N þ ¼ 1; s r
N < r < 1:
ð5Þ
Recently, these results were improved only involving the divergence of the velocity field in two dimension by Wang [29], namely
lim kdivukL1 ð0;T;L1 Þ ¼ 1:
T!T
ð6Þ
The results mentioned in (4)–(6) are only in terms of velocity field. Additionally, there are lots of criteria with respect to the density q and the temperature h. For instance, Sun–Wang–Zhang [28] obtained the following criterion in 3D,
lim kqkL1 ð0;T;L1 Þ þ kq1 kL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ ¼ 1:
T!T
Especially, Fang–Zi–Zhang [6] obtained a similar result in 2D, where initial vacuum states are allowed. The aim of this paper is to investigate the further blowup mechanism to the solution of the 2D full compressible Navier– Stokes equations with initial vacuum state. The main results in this paper indicate that either the mass of the compressible viscous flows will concentrate in some space points or the gas will combust in the blowup time, provided that the strong solution blows up in finite time. More precisely, we show that
lim kqkL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ ¼ 1:
T!T
Throughout this paper, we denote the simple notation as
Z
fdx ¼
Z
fdx:
X
For 1 6 p 6 1 and integer k P 0, we denote the standard Lebesgue and Sobolev spaces as follows,
8 < Lp ¼ Lp ðXÞ; W k;p ¼ W k;p ðXÞ; Hk ¼ W k;2 ; n o : W 01;p ¼ u 2 W k;p ju ¼ 0 on @ X ; H10 ¼ W 1;2 0 : Set
f_ :¼ ft þ u rf ; denotes the material derivative of f.
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The local existence of the strong solution to the full Navier–Stokes equation (1) has been established in three dimension in [3]. The results and the method are adapted into the two-dimension case, which can be stated as follows. Proposition 1 (Local existence). Let q 2 ð2; 1Þ be a fixed constant. Assume that the initial data (q0 P 0; u0 ; h0 P 0) satisfy
q0 P 0; q0 2 W 1;q ; u0 2 H10 \ H2 ; h0 2 H2 ;
ð7Þ
with the compatibility conditions
(
pffiffiffiffiffiffi
lDu0 þ ðk þ lÞrdivu0 Rrðq0 h0 Þ ¼ q0 g 1 ; pffiffiffiffiffiffi jDh0 þ l2 jru0 þ ðru0 Þtr j2 þ kðdivu0 Þ2 ¼ q0 g 2
ð8Þ
for some g 1 ; g 2 2 L2 . Then there exist a positive constant T 0 and a unique strong solution ðq; u; hÞ to the system (1)–(3) such that
q P 0; q 2 Cð½0; T 0 ; W 1;q Þ; u 2 Cð½0; T 0 ; H10 \ H2 Þ; h 2 Cð½0; T 0 ; H2 Þ;
ðu; hÞ 2 L2 ð½0; T 0 ; W 2;q Þ;
ðut ; ht Þ 2 L2 ð½0; T 0 ; H1 Þ;
pffiffiffiffi pffiffiffiffi ð qut ; qht Þ 2 L1 ð½0; T 0 ; L2 Þ:
ð9Þ ð10Þ ð11Þ
Thus, our main theorem concerning the blowup criterion for the strong solution is stated as follows: Theorem 1.1. Suppose the assumptions in Proposition 1 are satisfied and ðq; u; hÞ is the strong solution to the problem (1)–(3). If T < 1 is the maximal existence time of the existence of the strong solution, then
lim kqkL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ ¼ 1:
T!T
ð12Þ
Now, we would like to give several comments on our results as follows. Remark 1. Theorem 1.1 implies that if the condition (12) is false for any T > 0, then the strong solution to the 2D full compressible Navier–Stokes equations exists global in time. Remark 2. The blowup criterion mentioned in Theorem 1.1 shows that the estimates that kqkL1 ð0;T;L1 Þ and khkL2 ð0;T;L1 Þ illustrate that the compressible fluid will either accumulate into some points or combust in the finite blowup time. Remark 3. We would like to mention the blowup criterion on compressible Magnethydrodynamics (MHD) flows, see [20–23,31,32] and the references therein. The analysis in this paper can be adapted into MHD flows in our forthcoming work. This paper is organized as follows. In Section 2, we established some useful estimates under the contrary of (12). The key estimate is the uniform bound of kukL2 ð0;T;L1 Þ . In Appendix A, we will mention not only the Lamé system and some related regularity estimates, but also the logarithmic Sobolev inequality. 2. Proof of the main results Let ðq; u; hÞ be a strong solution discussed in Theorem 1.1, we will verify it by using the contradiction argument that suppose that (12) is false, i.e.
lim kqkL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ 6 M < þ1;
T!T
ð13Þ
where M is independent of T. First, we will establish the fundamental energy estimates. Lemma 2.1. Under the condition (13), it holds that for any 0 6 T < T ,
sup
Z
Z ðqh þ qjuj2 Þdx þ
06t6T
T
jruj2 dx dt 6 C:
0
Proof. Applying standard maximum principle in [5,7] to (1)3 together with h0 P 0 yields to
inf hðx; tÞ P 0:
R2 ½0;T
ð14Þ
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
First, the specific energy E , ct h þ 12 juj2 satisfies the following energy equation
@ t ðqEÞ þ divðqEu þ PuÞ ¼ divðuTÞ þ jDh;
ð15Þ
tr
where T ¼ lðru þ ðruÞ Þ þ ðl þ kÞdivuI is the stress tensor, I is a 3 3 unit matrix. Integrating (15) over X ½0; T, and according to the boundary condition (3), we obtain that
sup
Z
ðqh þ qjuj2 Þdx 6 C:
ð16Þ
06t6T
On the other hand, multiplying (1)2 by u and integrating the resulting equation over X, then by the Young’s inequality and (16), we obtain that
1 d 2 dt
Z
qjuj2 dx þ
Z
ljruj2 þ ðl þ kÞðdivuÞ2 dx 6 C
Z
1
qhjdivujdx 6 Ckhk2L1
6 CkhkL1
Z
qhdx þ ðl þ kÞ
6 CkhkL1 þ ðl þ kÞ
Z
Z Z
1
qh2 jdivujdx jdivuj2 dx
jdivuj2 dx;
which implies
d dt
Z
qjuj2 dx þ 2l
Z
jruj2 dx 6 CkhkL1 ;
ð17Þ
thus, (17) together with (13) and Gronwall’s inequality give that
Z
T
Z
jruj2 dx dt 6 C;
0
which together with (16) gives (14). h Next, the key estimate on ru will be given in the following Lemma 2.2. Lemma 2.2. Under the condition (13), it holds that for 0 6 T < T ,
sup
Z
Z ðqh2 þ jruj2 Þdx þ
06t6T
T
0
krhk2L2 dt 6 C:
ð18Þ
Proof. First, multiplying (1)2 by ut and integrating the resulting equation over X, after integrating by parts, give that
1 d 2 dt
Z
ljruj2 þ ðl þ kÞðdivuÞ2 dx þ
Z
_ 2 dx ¼ qjuj
Z
qu_ ðu rÞudx þ
Z
Pdivut dx;
ð19Þ
The following, we will estimate the two terms of right-hand of (19). On the one hand, with aid of Young’s inequality, we obtain that
Z
qu_ ðu rÞudx 6
1 4
Z
_ 2 dx þ C qjuj
Z
juj2 jruj2 dx;
ð20Þ
on the other hand, a series of directly computation show that
Z
Pdivut dx ¼
d dt
Z
Pdivudx
Z
Pt divudx:
R In order to estimate Pt divudx, we split u into two parts, namely, u ¼ v þ w; null boundary condition, respectively,
ð21Þ
v and w satisfy the following equations with
lDv þ ðl þ kÞrdivv ¼ rP
ð22Þ
_ lDw þ ðl þ kÞrdivw ¼ qu;
ð23Þ
and
then, by virtue of Proposition 3, we obtain that
and
krv kL3 6 CkqhkL3
ð24Þ
pffiffiffiffi _ L2 : kr2 wkL2 6 Ck quk
ð25Þ
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
Now
Z
Pt divudx ¼
Z
Pt divv dx þ
Z
Pt divwdx:
Herein, using (15) and (22), after integrating by parts, we obtain that
Z
Pt divv dx ¼
Z
rPt v dx ¼
Z
ðlDv t þ ðl þ kÞrdivv t Þv dx ¼
1 d 2 dt
Z
ljrv j2 þ ðl þ kÞðdivv Þ2 dx
ð26Þ
and
Z
Z
Z R 1 RðqhÞt divwdx ¼ ðqEÞt divwdx ðqjuj2 Þt divwdx ct 2 Z R ðqEu þ Pu jrh lru u lu ru kudivuÞ rdivwdx ¼ ct Z R 1 divðquÞjuj2 qut u divwdx þ ct 2 Z R 1 qjuj2 u þ ðct þ RÞqhu jrh lru u lu ru kudivu rdivwdx ¼ ct 2 Z R 1 ðqu_ udivw þ qujuj2 rdivwÞdx ct 2 Z R ððct þ RÞqhu jrh lru u lu ru kudivuÞ rdivwdx ¼ ct Z Z Z R _ ¼ I1 þ I2 : qu_ udivwdx 6 C ðqhjuj þ jrhj þ jujjrujÞjrdivwjdx þ C qjujjujjdivwjdx ct
Pt divwdx ¼
Z
For the first term I1 , we get the following estimate by Young’s inequality, (13) and (14),
I1 ¼ C
Z
ðqhjuj þ jrhj þ jujjrujÞjrdivwjdx 6 ekr2 wk2L2 þ C
Z
qh2 juj2 dx þ Ckrhk2L2 þ C
Z
juj2 jruj2 dx:
ð27Þ
ð28Þ
For the second one, a series of directly computations and Hölder inequality yield to
I2 ¼ C
Z
pffiffiffiffi
_ L2 kukL1 kdivwkL2 6 e _ qjujjujjdivwjdx 6 Ck quk
Z
_ 2 dx þ Ckuk2L1 kruk2L2 þ krv k2L2 ; qjuj
ð29Þ
thus, substituting (20) and (24)–(29) into (19), we obtain that
d dt
Z
ljruj2 þ ðl þ kÞðdivuÞ2 2Pdivu þ ljrv j2 þ ðl þ kÞðdivv Þ2 dx þ
Z
_ 2 dx qjuj
Z 6 C 1 krhk2L2 þ Ckuk2L1 kruk2L2 þ krv k2L2 þ qh2 dx :
ð30Þ
Next, multiplying (1)3 by h and integrating the resulting equation over X lead to
d dt
Z
qh2 dx þ 2jkrhk2L2 ¼ 2
Z
qhdivudx þ 4l
6 CkhkL1
Z
Z
jDðuÞj2 hdx þ 2k
Z
2
ðdivuÞ hdx
jdivujdx þ CkhkL1 kruk2L2 6 CkhkL1 kruk2L2 þ 1 ;
ð31Þ
due to interpolation inequality. Choose a constant C 2 suitable large that satisfies the following conditions,
(
C 2 P C12þ1 j ; l 2
kruk2L2 2Pdivu þ C 2 qh2 P qh2 :
Thus, adding (31) multiplied by the constant C 2 to (30), we obtain that
d dt
Z
ljruj2 þ ðl þ kÞðdivuÞ2 2Pdivu þ ljrv j2 þ ðl þ kÞðdivv Þ2 þ C 2 qh2 dx þ
Z 6 CkhkL1 kruk2L2 þ 1 þ Ckuk2L1 kruk2L2 þ krv k2L2 þ qh2 dx :
Z
_ 2 dx þ qjuj
Z
jrhj2 dx ð32Þ
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
Let
Z t pffiffiffiffi pffiffiffiffi _ 2L2 þ krhk2L2 ds: wðtÞ ¼ e þ sup kruðsÞk2L2 þ k qhðsÞk2L2 þ krv ðsÞk2L2 þ k quk s2½0;t
ð33Þ
0
It follows from (32) and (33) and Gronwall’s inequality that
Z
wðTÞ 6 CwðsÞ exp
T
CkuðsÞk2L1 ds
s
ð34Þ
for any 0 6 s 6 T < T . Now, we have to establish the estimate of kukL2 ðs;T;L1 Þ . In fact, using of Lemma 2.3 shows that
kuk2L2 ðs;T;L1 Þ 6 C 1 þ kuk2L2 ðs;T;H1 Þ lnðe þ kukL2 ðs;T;W 1;3 Þ Þ 6 C 1 þ kruk2L2 ðs;T;L2 Þ lnðe þ kukL2 ðs;T;W 1;3 Þ Þ ;
ð35Þ
due to the Poincaré inequality. By virtue of (22)–(25), we obtain that
kukW 1;3 6 CkukL3 þ CkrukL3 6 CkukH1 þ Ckrv kL3 þ CkrwkL3 6 CkrukL2 þ Ckwk 2;6 þ CkPkL3 W 5 pffiffiffiffi pffiffiffiffi _ 6 þ CkrukL2 þ CkhkL3 6 Ck quk _ L2 þ CkrukL2 þ CkrhkL2 þ C; 6 Ck quk L5
ð36Þ
where we have used the Poincaré inequality, interpolation inequality and Sobolev embedding theorem. The estimate of khkL3 is stated as follows,
khkL3 6 CkrhkL2 þ C: due to
kh hkL2 ðXÞ 6 CkrhkL2 ðXÞ and
khkL3 6 CkhkH1 6 CkrhkL2 þ CkhkL2 ; R h ¼ jX1 j hdx. where Substituting (35) and (36) into (34), we show that
wðTÞ 6 CwðsÞðCwðTÞÞ
C 3 kruk22
L ðs;T;L2 Þ
ð37Þ
:
With aid of (14), (37) and choosing some s which is close enough to T , we obtain
lim C 3 kruk2L2 ðs;T;L2 Þ 6
T!T
1 ; 2
then
wðTÞ 6 Cw2 ðsÞ < 1: Thus, we finish the proof of Lemma 2.2. h Next, we would like to improve the higher regularity estimates on q; u and h, which play the important role to extend the local strong solution to a global one. The proof is standard, and we would like to point it out to the readers in [29]. Proposition 2. Under the condition (13), it holds that for 0 6 T < T , then
sup
Z Z _ 2 dx þ jrhj2 þ qjuj
06t6T
sup 06t6T
0
Z
qh_ 2 dx þ
Z
T
0
T
pffiffiffiffi _ 22 þ kruk _ 2L2 dt 6 C; k qhk L
_ 22 dt 6 C; krhk L
sup kqkH1 \W 1;q þ krukH1 þ krhkH1 6 C:
ð38Þ
ð39Þ ð40Þ
06t6T
Proof of Theorem 1.1. Suppose that (12) was false, that is, (13) holds. In fact, due to Lemmas 2.1–2.3 and Proposition 3, the functions ðq; u; hÞðx; T Þ , limt!T ðq; u; hÞðx; tÞ satisfy the conditions imposed on the initial data (7) at the time t ¼ T . Fur_ qh_ 2 Cð½0; T; L2 Þ, which imply thermore, we have that qu;
_ _ _ qhÞðx; _ qhÞðx; T Þ ¼ lim ðqu; tÞ 2 L2 : ðqu; t!T
Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
725
Hence, we have
ðlDu ðl þ kÞrdivu þ RrðqhÞÞjt¼T ¼
pffiffiffiffi
qðx; T Þg 1 ðx; T Þ;
pffiffiffiffi
jDh þ 2ljDðuÞj2 þ kðdivuÞ2 jt¼T ¼ qðx; T Þg 2 ðx; T Þ;
where
( g 1 ðxÞ ,
1
_ q2 ðx; T ÞðquÞðx; T Þ; for x 2 fxjqðx; T Þ > 0g; 0; for x 2 fxjqðx; T Þ ¼ 0g
and
( g 2 ðxÞ ,
1
q2 ðx; T Þ½qh_ þ Rqhdivuðx; T Þ; for x 2 fxjqðx; T Þ > 0g; 0; for x 2 fxjqðx; T Þ ¼ 0g:
It is clear that g 1 ; g 2 2 L2 are due to the estimates (38)–(40). Thus, ðq; u; hÞðx; T Þ satisfy compatibility conditions (8). Therefore, the local strong solution beyond T can be extended by taking ðq; u; hÞðx; T Þ as the initial data and using Proposition 1, which contradicts the maximality of T . Then the assumption (13) does not hold, and we complete the proof of Theorem 1.1. h Acknowledgments The authors would like to thank the referees to giving many helpful suggestions to improve our paper. Appendix A In the appendix, we introduce the following Lamé system, which has been used in the a priori estimates,
lDU þ ðl þ kÞrdivU ¼ F; in X; UðxÞ ¼ 0;
ð41Þ
on @ X:
Suppose U 2 H10 is a weak solution to the Lamé system, we have some regularity estimates for system (41), the proof has been shown in [28]. Proposition 3. Let q 2 ð1; 1Þ, then there exists some constant C depending only on k;
l; p and X such that
(1) if F 2 Lp , then
kUkW 2;p 6 CkFkLp ; (2) if F 2 W
1;p
ð42Þ p
(i.e, F ¼ div f with f ¼ ðfij Þ; ðfij 2 L ðXÞÞ, then
kUkW 1;p 6 Ckf kLp :
ð43Þ
Finally, in order to acquire the bound of kukL2 ð0;T;L1 Þ in Section 2, we introduce the logarithmic Sobolev inequality, and we would like to refer its proof in [14]. Lemma 2.3. Let X be a bounded domain in R2 , and f 2 L2 ð0; T; H10 \ W 1;p Þ with p > 2. Then there exists a constant C depending only on p such that
kf k2L2 ð0;T;L1 Þ 6 C 1 þ kf k2L2 ð0;T;H1 Þ lnðe þ kf k2L2 ð0;T;W 1;p Þ Þ : References [1] [2] [3] [4] [5] [6] [7] [8]
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[9] D. Hoff, Global solution of the Navier–Stokes equations for multidimensional compressible flow with discontinous initial data, J. Differ. Equ. 120 (1995) 215–254. [10] X.D. Huang, Some results on blowup of solutions to the compressible Navier–Stokes equations, Methods Appl. Anal. 16 (2009) 479–490. [11] X.D. Huang, J. Li, Z.P. Xin, Serrin-type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal. 43 (2011) 1872–1886. [12] X.D. Huang, J. Li, Z.P. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Commun. Math. Phys. 301 (2011) 23–35. [13] X.D. Huang, J. Li, Y. Wang, Serrin-type blowup criterion for the full compressible Navier–Stokes system, Arch. Ration. Mech. Anal. 207 (2013) 303–316. [14] X.D. Huang, Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differ. Equ. 254 (2013) 511–527. [15] X.D. Huang, Z.P. Xin, A blow-up criterion for classical solutions to the compressible Navier–Stokes equations, Sci. China Math. 53 (3) (2010) 671–686. [16] S. Jiang, Y.B. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Acta Math. Sci. Ser. B Engl. Ed. 30 (6) (2010) 1851–1864. [17] S. Jiang, P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier–Stokes equations, Commun. Math. Phys. 215 (2001) 559– 581. [18] J. Serrin, On the interior regularity of weak solutions of Navier–Stokes equation, Arch. Ration. Mech. Anal. 9 (1962) 187–195. [19] P.L. Lions, Mathematical Topics in Fluid Mechanics, Compressible Models, vol. 2, Oxford University Press, New York, 1998. [20] S.Q. Liu, H.B. Yu, J.W. Zhang, Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum, J. Differ. Equ. 254 (2013) 229– 255. [21] M. Lu, Y. Du, Z.A. Yao, Blow-up phenomena for the 3D compressible MHD equations, Discrete Contin. Dyn. Syst. 32 (2012) 1835–1855. [22] M. Lu, Y. Du, Z.A. Yao, Blow-up criterion for compressible MHD equations, J. Math. Anal. Appl. 379 (2012) 425–438. [23] M. Lu, Y. Du, Z.A. Yao, Z.J. Zhang, A blow-up criterion for the 3D compressible MHD equations, Commun. Pure Appl. Anal. 11 (2012) 1167–1183. [24] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980) 67–104. [25] J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général, Bull. Soc. Math. France 90 (1962) 487–497. [26] O. Rozanova, Blow up of smooth solutions to the compressible Navier–Stokes equations with the data highly decreasing at infinity, J. Differ. Equ. 245 (2008) 1762–1774. [27] J. Seerin, On the uniqueness of compressible fluid motion, Arch. Ration. Math. Anal. 3 (1959) 271–288. [28] Y.Z. Sun, C. Wang, Z.F. Zhang, A Beal–Kate–Majda blow-up criterion for the 3D compressible Navier–Stokes equations, J. Math. Pure Appl. 95 (2011) 36– 47. [29] Y. Wang, One new blowup criterion for the 2D full compressible Navier–Stokes system, Nonlinear Anal. Real World Appl. 16 (2014) 214–226. [30] Z.P. Xin, Blowup of smooth solutions to the compressible Navier–Stokes equation with compact density, Commun. Pure Appl. Math. 51 (1998) 229– 240. [31] X.Y. Xu, J.W. Zhang, A blow-up criterion for 3D non-resistive compressible magnetohydrodynamic equations with initial vacuum, Nonlinear Anal. Real World Appl. 12 (2011) 3442–3451. [32] X.Y. Xu, J.W. Zhang, A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Methods Appl. Sci. 22 (2) (2012) 1150010. 23pp.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Blowup analysis for two-dimensional viscous compressible, heat-conductive Navier–Stokes equations q Yongfu Wang, Qin Zhang ⇑ Department of Mathematics, Sichuan University, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper, we study the finite time blow-up mechanism of the strong solution to the two-dimensional (2D) viscous, compressible, heat-conductive Navier–Stokes system and initial vacuum states are allowed. We show that if the solution to 2D viscous compressible flows blows up in finite time, then either the mass of the compressible fluid will concentrate in some points or the compressible gas will combust in the finite time. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Compressible Navier–Stokes equations Heat-conductive Blowup criterion Vacuum
1. Introduction and main results In this paper, we consider the 2D viscous compressible and heat-conductive Navier–Stokes equations
8 > < @ t q þ divðquÞ ¼ 0; @ t ðquÞ þ divðqu uÞ lDu ðl þ kÞrdivu þ rP ¼ 0; > : 2 ct ½@ t ðqhÞ þ divðquhÞ jDh þ Pdivu ¼ 2ljDðuÞj2 þ kðdivuÞ ;
ð1Þ
with the initial-boundary values,
ðqðx; tÞ; uðx; tÞÞjt¼0 ¼ ðq0 ðxÞ; u0 ðxÞÞ; u ¼ 0;
@h ¼ 0; @~ n
in X;
on @ X;
ð2Þ ð3Þ
where X is a bounded smooth domain in R2 , ~ n ¼ ðn1 ; n2 Þ is the unit outward normal to @ X. Here u, q, h and P ¼ Rqh ðR > 0Þ denote the velocity, density, absolute temperature and pressure, respectively. DðuÞ is the deformation tensor, which is described as
DðuÞ ¼
1 ru þ ðruÞtr : 2
The constant viscosity coefficient
l and k satisfy the physical restrictions
l > 0; l þ k P 0; q This work is supported in part by NSFC Grant 11171236, PCSIRT (IRT1273), Sichuan Youth Science & Technology Foundation and Fundamental Research Funds for the Central Universities. ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Wang), zqcfl[email protected] (Q. Zhang).
http://dx.doi.org/10.1016/j.amc.2014.01.103 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
in 2D space. In addition, we denote ct and j as the specific heat at constant volume and the coefficient of heat conduction, respectively. The existence and behaviors of solution to compressible Navier–Stokes equations have been studied by a large number of literatures (see [3,4,8,9,14,17–19,24,25]). In the absence of vacuum, the local existence and uniqueness of classical solutions were proved in [25,27]. Matsumura–Nishida [24] first investigated the global classical solution for initial data close to a nonvacuum equilibrium in some Sobolev space Hs . Later, Hoff [9] obtained a global weak solution for the discontinuous initial data, when the initial density is strictly positive. However, the regularity and uniqueness of such weak solutions remain still open, even in two dimensional space. But, the global existence of strong solution has been obtained in the case of some small initial data or small energy. Especially, for 3D compressible Navier–Stokes equations, Lions [19] obtained a global weak solution as long as solutions have finite energy and the pressure PðqÞ ¼ aqc with c P 95, and Feireisl extended the results with c > 32 in [8]. Jiang–Zhang [17] constructed the global existence of weak solution for c > 1 with the symmetric initial data. On the other hand, there are many literatures [2,5,6, 10–12,15,16,26] concerning the blowup mechanism for the compressible viscous flows, since the significant work [30] on blowup of the smooth solutions to the Navier–Stokes equations provided that the initial density has compact support. It is worth mentioning here that there are two kinds of known blowup criteria, namely Beale–Kato–Majda type (BKM) [1] for 3D Euler equations, more precisely, that if T < 1 is the maximal time for the existence of a strong (or classical) solution, then
lim kr ukL1 ð0;T;L1 Þ ¼ 1;
T!T
ðBeale-Kato-Majda typeÞ
ð4Þ
and Serrin-type [18] for 3D incompressible Navier–Stokes equations, then
lim kukLs ð0;T;Lr Þ ¼ 1;
T!T
ðSerrin typeÞ;
where
2 3 þ 6 1; s r
3 < r 6 1:
For the 2D case, Jiang–Ou [16] constructed a BKM-type blowup criterion for full Navier–Stokes equation (1) over a unit square domain or periodic domain of R2 , and Huang–Li–Wang [13] obtained a blowup criterion that Serrin-type for (1) in RN , as
lim kdivukL1 ð0;T;L1 Þ þ kukLs ð0;T;Lr Þ ¼ 1;
T!T
2 N þ ¼ 1; s r
N < r < 1:
ð5Þ
Recently, these results were improved only involving the divergence of the velocity field in two dimension by Wang [29], namely
lim kdivukL1 ð0;T;L1 Þ ¼ 1:
T!T
ð6Þ
The results mentioned in (4)–(6) are only in terms of velocity field. Additionally, there are lots of criteria with respect to the density q and the temperature h. For instance, Sun–Wang–Zhang [28] obtained the following criterion in 3D,
lim kqkL1 ð0;T;L1 Þ þ kq1 kL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ ¼ 1:
T!T
Especially, Fang–Zi–Zhang [6] obtained a similar result in 2D, where initial vacuum states are allowed. The aim of this paper is to investigate the further blowup mechanism to the solution of the 2D full compressible Navier– Stokes equations with initial vacuum state. The main results in this paper indicate that either the mass of the compressible viscous flows will concentrate in some space points or the gas will combust in the blowup time, provided that the strong solution blows up in finite time. More precisely, we show that
lim kqkL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ ¼ 1:
T!T
Throughout this paper, we denote the simple notation as
Z
fdx ¼
Z
fdx:
X
For 1 6 p 6 1 and integer k P 0, we denote the standard Lebesgue and Sobolev spaces as follows,
8 < Lp ¼ Lp ðXÞ; W k;p ¼ W k;p ðXÞ; Hk ¼ W k;2 ; n o : W 01;p ¼ u 2 W k;p ju ¼ 0 on @ X ; H10 ¼ W 1;2 0 : Set
f_ :¼ ft þ u rf ; denotes the material derivative of f.
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The local existence of the strong solution to the full Navier–Stokes equation (1) has been established in three dimension in [3]. The results and the method are adapted into the two-dimension case, which can be stated as follows. Proposition 1 (Local existence). Let q 2 ð2; 1Þ be a fixed constant. Assume that the initial data (q0 P 0; u0 ; h0 P 0) satisfy
q0 P 0; q0 2 W 1;q ; u0 2 H10 \ H2 ; h0 2 H2 ;
ð7Þ
with the compatibility conditions
(
pffiffiffiffiffiffi
lDu0 þ ðk þ lÞrdivu0 Rrðq0 h0 Þ ¼ q0 g 1 ; pffiffiffiffiffiffi jDh0 þ l2 jru0 þ ðru0 Þtr j2 þ kðdivu0 Þ2 ¼ q0 g 2
ð8Þ
for some g 1 ; g 2 2 L2 . Then there exist a positive constant T 0 and a unique strong solution ðq; u; hÞ to the system (1)–(3) such that
q P 0; q 2 Cð½0; T 0 ; W 1;q Þ; u 2 Cð½0; T 0 ; H10 \ H2 Þ; h 2 Cð½0; T 0 ; H2 Þ;
ðu; hÞ 2 L2 ð½0; T 0 ; W 2;q Þ;
ðut ; ht Þ 2 L2 ð½0; T 0 ; H1 Þ;
pffiffiffiffi pffiffiffiffi ð qut ; qht Þ 2 L1 ð½0; T 0 ; L2 Þ:
ð9Þ ð10Þ ð11Þ
Thus, our main theorem concerning the blowup criterion for the strong solution is stated as follows: Theorem 1.1. Suppose the assumptions in Proposition 1 are satisfied and ðq; u; hÞ is the strong solution to the problem (1)–(3). If T < 1 is the maximal existence time of the existence of the strong solution, then
lim kqkL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ ¼ 1:
T!T
ð12Þ
Now, we would like to give several comments on our results as follows. Remark 1. Theorem 1.1 implies that if the condition (12) is false for any T > 0, then the strong solution to the 2D full compressible Navier–Stokes equations exists global in time. Remark 2. The blowup criterion mentioned in Theorem 1.1 shows that the estimates that kqkL1 ð0;T;L1 Þ and khkL2 ð0;T;L1 Þ illustrate that the compressible fluid will either accumulate into some points or combust in the finite blowup time. Remark 3. We would like to mention the blowup criterion on compressible Magnethydrodynamics (MHD) flows, see [20–23,31,32] and the references therein. The analysis in this paper can be adapted into MHD flows in our forthcoming work. This paper is organized as follows. In Section 2, we established some useful estimates under the contrary of (12). The key estimate is the uniform bound of kukL2 ð0;T;L1 Þ . In Appendix A, we will mention not only the Lamé system and some related regularity estimates, but also the logarithmic Sobolev inequality. 2. Proof of the main results Let ðq; u; hÞ be a strong solution discussed in Theorem 1.1, we will verify it by using the contradiction argument that suppose that (12) is false, i.e.
lim kqkL1 ð0;T;L1 Þ þ khkL1 ð0;T;L1 Þ 6 M < þ1;
T!T
ð13Þ
where M is independent of T. First, we will establish the fundamental energy estimates. Lemma 2.1. Under the condition (13), it holds that for any 0 6 T < T ,
sup
Z
Z ðqh þ qjuj2 Þdx þ
06t6T
T
jruj2 dx dt 6 C:
0
Proof. Applying standard maximum principle in [5,7] to (1)3 together with h0 P 0 yields to
inf hðx; tÞ P 0:
R2 ½0;T
ð14Þ
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
First, the specific energy E , ct h þ 12 juj2 satisfies the following energy equation
@ t ðqEÞ þ divðqEu þ PuÞ ¼ divðuTÞ þ jDh;
ð15Þ
tr
where T ¼ lðru þ ðruÞ Þ þ ðl þ kÞdivuI is the stress tensor, I is a 3 3 unit matrix. Integrating (15) over X ½0; T, and according to the boundary condition (3), we obtain that
sup
Z
ðqh þ qjuj2 Þdx 6 C:
ð16Þ
06t6T
On the other hand, multiplying (1)2 by u and integrating the resulting equation over X, then by the Young’s inequality and (16), we obtain that
1 d 2 dt
Z
qjuj2 dx þ
Z
ljruj2 þ ðl þ kÞðdivuÞ2 dx 6 C
Z
1
qhjdivujdx 6 Ckhk2L1
6 CkhkL1
Z
qhdx þ ðl þ kÞ
6 CkhkL1 þ ðl þ kÞ
Z
Z Z
1
qh2 jdivujdx jdivuj2 dx
jdivuj2 dx;
which implies
d dt
Z
qjuj2 dx þ 2l
Z
jruj2 dx 6 CkhkL1 ;
ð17Þ
thus, (17) together with (13) and Gronwall’s inequality give that
Z
T
Z
jruj2 dx dt 6 C;
0
which together with (16) gives (14). h Next, the key estimate on ru will be given in the following Lemma 2.2. Lemma 2.2. Under the condition (13), it holds that for 0 6 T < T ,
sup
Z
Z ðqh2 þ jruj2 Þdx þ
06t6T
T
0
krhk2L2 dt 6 C:
ð18Þ
Proof. First, multiplying (1)2 by ut and integrating the resulting equation over X, after integrating by parts, give that
1 d 2 dt
Z
ljruj2 þ ðl þ kÞðdivuÞ2 dx þ
Z
_ 2 dx ¼ qjuj
Z
qu_ ðu rÞudx þ
Z
Pdivut dx;
ð19Þ
The following, we will estimate the two terms of right-hand of (19). On the one hand, with aid of Young’s inequality, we obtain that
Z
qu_ ðu rÞudx 6
1 4
Z
_ 2 dx þ C qjuj
Z
juj2 jruj2 dx;
ð20Þ
on the other hand, a series of directly computation show that
Z
Pdivut dx ¼
d dt
Z
Pdivudx
Z
Pt divudx:
R In order to estimate Pt divudx, we split u into two parts, namely, u ¼ v þ w; null boundary condition, respectively,
ð21Þ
v and w satisfy the following equations with
lDv þ ðl þ kÞrdivv ¼ rP
ð22Þ
_ lDw þ ðl þ kÞrdivw ¼ qu;
ð23Þ
and
then, by virtue of Proposition 3, we obtain that
and
krv kL3 6 CkqhkL3
ð24Þ
pffiffiffiffi _ L2 : kr2 wkL2 6 Ck quk
ð25Þ
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
Now
Z
Pt divudx ¼
Z
Pt divv dx þ
Z
Pt divwdx:
Herein, using (15) and (22), after integrating by parts, we obtain that
Z
Pt divv dx ¼
Z
rPt v dx ¼
Z
ðlDv t þ ðl þ kÞrdivv t Þv dx ¼
1 d 2 dt
Z
ljrv j2 þ ðl þ kÞðdivv Þ2 dx
ð26Þ
and
Z
Z
Z R 1 RðqhÞt divwdx ¼ ðqEÞt divwdx ðqjuj2 Þt divwdx ct 2 Z R ðqEu þ Pu jrh lru u lu ru kudivuÞ rdivwdx ¼ ct Z R 1 divðquÞjuj2 qut u divwdx þ ct 2 Z R 1 qjuj2 u þ ðct þ RÞqhu jrh lru u lu ru kudivu rdivwdx ¼ ct 2 Z R 1 ðqu_ udivw þ qujuj2 rdivwÞdx ct 2 Z R ððct þ RÞqhu jrh lru u lu ru kudivuÞ rdivwdx ¼ ct Z Z Z R _ ¼ I1 þ I2 : qu_ udivwdx 6 C ðqhjuj þ jrhj þ jujjrujÞjrdivwjdx þ C qjujjujjdivwjdx ct
Pt divwdx ¼
Z
For the first term I1 , we get the following estimate by Young’s inequality, (13) and (14),
I1 ¼ C
Z
ðqhjuj þ jrhj þ jujjrujÞjrdivwjdx 6 ekr2 wk2L2 þ C
Z
qh2 juj2 dx þ Ckrhk2L2 þ C
Z
juj2 jruj2 dx:
ð27Þ
ð28Þ
For the second one, a series of directly computations and Hölder inequality yield to
I2 ¼ C
Z
pffiffiffiffi
_ L2 kukL1 kdivwkL2 6 e _ qjujjujjdivwjdx 6 Ck quk
Z
_ 2 dx þ Ckuk2L1 kruk2L2 þ krv k2L2 ; qjuj
ð29Þ
thus, substituting (20) and (24)–(29) into (19), we obtain that
d dt
Z
ljruj2 þ ðl þ kÞðdivuÞ2 2Pdivu þ ljrv j2 þ ðl þ kÞðdivv Þ2 dx þ
Z
_ 2 dx qjuj
Z 6 C 1 krhk2L2 þ Ckuk2L1 kruk2L2 þ krv k2L2 þ qh2 dx :
ð30Þ
Next, multiplying (1)3 by h and integrating the resulting equation over X lead to
d dt
Z
qh2 dx þ 2jkrhk2L2 ¼ 2
Z
qhdivudx þ 4l
6 CkhkL1
Z
Z
jDðuÞj2 hdx þ 2k
Z
2
ðdivuÞ hdx
jdivujdx þ CkhkL1 kruk2L2 6 CkhkL1 kruk2L2 þ 1 ;
ð31Þ
due to interpolation inequality. Choose a constant C 2 suitable large that satisfies the following conditions,
(
C 2 P C12þ1 j ; l 2
kruk2L2 2Pdivu þ C 2 qh2 P qh2 :
Thus, adding (31) multiplied by the constant C 2 to (30), we obtain that
d dt
Z
ljruj2 þ ðl þ kÞðdivuÞ2 2Pdivu þ ljrv j2 þ ðl þ kÞðdivv Þ2 þ C 2 qh2 dx þ
Z 6 CkhkL1 kruk2L2 þ 1 þ Ckuk2L1 kruk2L2 þ krv k2L2 þ qh2 dx :
Z
_ 2 dx þ qjuj
Z
jrhj2 dx ð32Þ
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Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
Let
Z t pffiffiffiffi pffiffiffiffi _ 2L2 þ krhk2L2 ds: wðtÞ ¼ e þ sup kruðsÞk2L2 þ k qhðsÞk2L2 þ krv ðsÞk2L2 þ k quk s2½0;t
ð33Þ
0
It follows from (32) and (33) and Gronwall’s inequality that
Z
wðTÞ 6 CwðsÞ exp
T
CkuðsÞk2L1 ds
s
ð34Þ
for any 0 6 s 6 T < T . Now, we have to establish the estimate of kukL2 ðs;T;L1 Þ . In fact, using of Lemma 2.3 shows that
kuk2L2 ðs;T;L1 Þ 6 C 1 þ kuk2L2 ðs;T;H1 Þ lnðe þ kukL2 ðs;T;W 1;3 Þ Þ 6 C 1 þ kruk2L2 ðs;T;L2 Þ lnðe þ kukL2 ðs;T;W 1;3 Þ Þ ;
ð35Þ
due to the Poincaré inequality. By virtue of (22)–(25), we obtain that
kukW 1;3 6 CkukL3 þ CkrukL3 6 CkukH1 þ Ckrv kL3 þ CkrwkL3 6 CkrukL2 þ Ckwk 2;6 þ CkPkL3 W 5 pffiffiffiffi pffiffiffiffi _ 6 þ CkrukL2 þ CkhkL3 6 Ck quk _ L2 þ CkrukL2 þ CkrhkL2 þ C; 6 Ck quk L5
ð36Þ
where we have used the Poincaré inequality, interpolation inequality and Sobolev embedding theorem. The estimate of khkL3 is stated as follows,
khkL3 6 CkrhkL2 þ C: due to
kh hkL2 ðXÞ 6 CkrhkL2 ðXÞ and
khkL3 6 CkhkH1 6 CkrhkL2 þ CkhkL2 ; R h ¼ jX1 j hdx. where Substituting (35) and (36) into (34), we show that
wðTÞ 6 CwðsÞðCwðTÞÞ
C 3 kruk22
L ðs;T;L2 Þ
ð37Þ
:
With aid of (14), (37) and choosing some s which is close enough to T , we obtain
lim C 3 kruk2L2 ðs;T;L2 Þ 6
T!T
1 ; 2
then
wðTÞ 6 Cw2 ðsÞ < 1: Thus, we finish the proof of Lemma 2.2. h Next, we would like to improve the higher regularity estimates on q; u and h, which play the important role to extend the local strong solution to a global one. The proof is standard, and we would like to point it out to the readers in [29]. Proposition 2. Under the condition (13), it holds that for 0 6 T < T , then
sup
Z Z _ 2 dx þ jrhj2 þ qjuj
06t6T
sup 06t6T
0
Z
qh_ 2 dx þ
Z
T
0
T
pffiffiffiffi _ 22 þ kruk _ 2L2 dt 6 C; k qhk L
_ 22 dt 6 C; krhk L
sup kqkH1 \W 1;q þ krukH1 þ krhkH1 6 C:
ð38Þ
ð39Þ ð40Þ
06t6T
Proof of Theorem 1.1. Suppose that (12) was false, that is, (13) holds. In fact, due to Lemmas 2.1–2.3 and Proposition 3, the functions ðq; u; hÞðx; T Þ , limt!T ðq; u; hÞðx; tÞ satisfy the conditions imposed on the initial data (7) at the time t ¼ T . Fur_ qh_ 2 Cð½0; T; L2 Þ, which imply thermore, we have that qu;
_ _ _ qhÞðx; _ qhÞðx; T Þ ¼ lim ðqu; tÞ 2 L2 : ðqu; t!T
Y. Wang, Q. Zhang / Applied Mathematics and Computation 232 (2014) 719–726
725
Hence, we have
ðlDu ðl þ kÞrdivu þ RrðqhÞÞjt¼T ¼
pffiffiffiffi
qðx; T Þg 1 ðx; T Þ;
pffiffiffiffi
jDh þ 2ljDðuÞj2 þ kðdivuÞ2 jt¼T ¼ qðx; T Þg 2 ðx; T Þ;
where
( g 1 ðxÞ ,
1
_ q2 ðx; T ÞðquÞðx; T Þ; for x 2 fxjqðx; T Þ > 0g; 0; for x 2 fxjqðx; T Þ ¼ 0g
and
( g 2 ðxÞ ,
1
q2 ðx; T Þ½qh_ þ Rqhdivuðx; T Þ; for x 2 fxjqðx; T Þ > 0g; 0; for x 2 fxjqðx; T Þ ¼ 0g:
It is clear that g 1 ; g 2 2 L2 are due to the estimates (38)–(40). Thus, ðq; u; hÞðx; T Þ satisfy compatibility conditions (8). Therefore, the local strong solution beyond T can be extended by taking ðq; u; hÞðx; T Þ as the initial data and using Proposition 1, which contradicts the maximality of T . Then the assumption (13) does not hold, and we complete the proof of Theorem 1.1. h Acknowledgments The authors would like to thank the referees to giving many helpful suggestions to improve our paper. Appendix A In the appendix, we introduce the following Lamé system, which has been used in the a priori estimates,
lDU þ ðl þ kÞrdivU ¼ F; in X; UðxÞ ¼ 0;
ð41Þ
on @ X:
Suppose U 2 H10 is a weak solution to the Lamé system, we have some regularity estimates for system (41), the proof has been shown in [28]. Proposition 3. Let q 2 ð1; 1Þ, then there exists some constant C depending only on k;
l; p and X such that
(1) if F 2 Lp , then
kUkW 2;p 6 CkFkLp ; (2) if F 2 W
1;p
ð42Þ p
(i.e, F ¼ div f with f ¼ ðfij Þ; ðfij 2 L ðXÞÞ, then
kUkW 1;p 6 Ckf kLp :
ð43Þ
Finally, in order to acquire the bound of kukL2 ð0;T;L1 Þ in Section 2, we introduce the logarithmic Sobolev inequality, and we would like to refer its proof in [14]. Lemma 2.3. Let X be a bounded domain in R2 , and f 2 L2 ð0; T; H10 \ W 1;p Þ with p > 2. Then there exists a constant C depending only on p such that
kf k2L2 ð0;T;L1 Þ 6 C 1 þ kf k2L2 ð0;T;H1 Þ lnðe þ kf k2L2 ð0;T;W 1;p Þ Þ : References [1] [2] [3] [4] [5] [6] [7] [8]
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