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Global Smooth Solutions to the Spatially Periodic Cauchy Problem for Dissipative Nonlinear Evolution Equations
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
213, 262]274 Ž1997.
AY975535
Global Smooth Solutions to the Spatially Periodic Cauchy Problem for Dissipative Nonlinear Evolution Equations Ling Hsiao* Academic Sinica, Institute of Mathematics, Beijing, 100080, People’s Republic of China
and Huaiyu Jian† Dept. of Applied Mathematics, Tsinghua Uni¨ ersity, Beijing, 100084, People’s Republic of China Submitted by Mark J. Balas Received August 23, 1995
The existence and uniqueness are proved for global classical solutions of the spatially periodic Cauchy problem to the following system of parabolic equations
½
ct s yŽ s y a . c y sux q ac x x u t s yŽ 1 y b . u q nc x q Ž cu . x q bux x ,
which was proposed as a substitute for the Rayleigh]Benard equation and can lead to Lorenz equations. Q 1997 Academic Press
1. INTRODUCTION The aim of this paper is to study the conservation form of Hsieh’s equations which reads as
½
c t s yŽ s y a . c y sux q ac x x u t s yŽ 1 y b . u q nc x q Ž cu . x q bux x ,
Ž 1.1.
* Supported by the National Natural Science Foundation of China. † Supported by the China Postdoctoral Science Foundation. On leaving the Academic Sinica, Institute of Mathematics. 262 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
SPATIALLY PERIODIC CAUCHY PROBLEMS
263
where s , n , a , and b are all positive constants satisfying the relation a - s and b - 1. We refer to Hsieh w1x and Tang w2x for the physical background of Ž1.1.. It is worthy to point out that with a truncation similar to the mode truncation from Rayleigh]Benard equations used by Lorenz in w3x, the system Ž1.1. also leads to the Lorenz equations. In fact, taking
½
c s '2 X Ž t . sin x u s '2 Y Ž t . cos x q 2 Z Ž t . cos 2 x
for Ž1.1. and retaining only the coefficients of sin x, cos x, and cos 2 x, one can obtain Lorenz equations. For the details, see Tang w2x. If we ignore the diffusion terms on the right-hand side of Ž1.1., then the characteristic of the remaining system takes the form
ls
yc "
'c
2
y 4s Ž n q u . 2
.
This means when < c < is small enough, the remaining system is basically elliptic, so one may expect that the inherent instability will cause growth and drive the system into hyperbolic regime. However, the damping and diffusion would diminish the amplitude and draw the system back into elliptic regime. So there might be a ‘‘switching back and forth’’ mechanism, which could possibly lead to some complicated behavior, even chaos. Recently, Tang w2x performed extensive numerical simulation on the system Ž1.1. with spatially periodic Cauchy data. Without loss of generality, one may assume the period L s 1. Then the data read as
Ž c , u . Ž 0, t . s Ž c , u . Ž 1, t . , Ž c x , ux . Ž 0, t . s Ž c x , ux . Ž 1, t . , 0 F t F T Ž 1.2. and
Ž c , u . Ž x, 0 . s Ž c 0 Ž x . , u 0 Ž x . . ,
0 F x F 1.
Ž 1.3.
The numerical results in w2x show that the solutions to Ž1.1. ] Ž1.3. decay to zero for big dissipation. But for smaller dissipation, chaos occurs, i.e., the problem Ž1.1. ] Ž1.3. admits only steady state solutions represented by a set of discrete peaks, and the number of peaks increases as the dissipation decreases. Nevertheless, there are only a couple of rigorous results about Ž1.1. ] Ž1.3., namely, the result obtained by Tang in w2x, which reads that if Ž1.1. ] Ž1.3. does not admit a global smooth solution c Ž x, t ., u Ž x, t .4 , then u must blow-up at first Žsee w2, Theorem 2.2.3x., and the result obtained by Jian w4x which discusses the zero case for both c and ux on the boundaries with artificial assumption H01 u 0 Ž x . dx G yn .
264
HSIAO AND JIAN
As the first step of the program to establish a theoretical analysis on the chaotic nature of the solutions for Ž1.1. with small dissipation, the present paper is devoted to the study of the global existence and uniqueness for Ž1.1. ] Ž1.3.. By combining the energy method and L1-estimate technique and applying the Leray]Schauder fixed point theorem, we will establish the global existence of classical solutions to the spatially periodic Cauchy problem. First, we recall some definitions of Banach spaces for Holder continuous ¨ functions. Let 0 - r F 1, k s 0, 1, 2. By C kq r Žw0, 1x. we denote the Banach space of functions whose derivatives of all orders m F k belong to C r Žw0, 1x. of the Holder continuous functions with exponent r Žor Lipschitz continuous ¨ functions with r s 1.. The norm of this space will be denoted by 5 ? 5 kq r . The set of functions on QT s w0, 1x = w0, T x which are Holder continu¨ ous Žexponent r . with respect to the parabolic distance DŽŽ x, t ., Ž y, t .. s < x y y < 2 q < t y t <41r2 will be denoted by C r Ž QT . with the norm A ? A r . Furthermore, denote C 1, r Ž QT . s u: u, u x g C r Ž QT . 4 , C 2, r Ž QT . s u: u, u x , u t , u x x g C r Ž QT . 4 . The norms of C k, r Ž QT . are imposed by u
kq r
s u
r
q ux
r
q Ž k y 1.
ux x
r
q ut
r
,
k s 1, 2.
For more details and the properties of these spaces, we refer to w5, 6x. Suppose that the initial-boundary data Ž1.3. and Ž1.2. satisfy the following compatibility conditions:
Ž c 0 , u 0 . Ž 0. s Ž c 0 , u 0 . Ž 1. ,
Ž Ž c 0 . x , Ž u 0 . x . Ž 0. s Ž Ž c 0 . x , Ž u 0 . x . Ž 1. . Ž 1.4.
The purpose of this paper is to prove the following: THEOREM 1.1. Let a , b , s , and n all be constants with a and b being positi¨ e. Assume c 0 Ž x . and u 0 Ž x . are in C 2, d Žw0, 1x. Ž0 - d - 1., satisfying Ž1.4.. Then for any T ) 0, there exists a unique solution c Ž x, t ., u Ž x, t .4 g C 2, d Ž QT . = C 2, d Ž QT . to the problem Ž1.1. ] Ž1.3.. Moreo¨ er, the generalized deri¨ ati¨ es c x t and ux t are in L2 Ž QT .. 2. THE PROOF OF THEOREM 1.1 LEMMA 2.1. The problem Ž1.1. ] Ž1.3. has one solution in C 2, d Ž QT . = C QT . at most. 2, d Ž
SPATIALLY PERIODIC CAUCHY PROBLEMS
265
Proof. Let both Ž c 1 , u 1 . and Ž c 2 , u 2 . be C 2, d Ž QT .-solutions to Ž1.1. ] Ž1.3.. Denote
c s c1 y c2 ,
u s u1 y u2 .
Then Ž c , u . satisfies
¡ q ac ¢u s yŽ 1 y b . u q nc q Ž cu
~c s yŽ s y a . c y su t
t
x
x
xx 2
q c 1 u . x q bux x
Ž 2.1.
with boundary conditions
Ž c , u . Ž 0, t . s Ž c , u . Ž 1, t . ,
Ž cx , ux . Ž 0, t . s Ž cx , ux . Ž 1, t . , 0 F t F T Ž 2.2.
and initial conditions
c Ž x, 0 . s 0,
u Ž x, 0 . s 0, 0 F x F 1.
Ž 2.3.
Multiply Ž2.1.1 by c and Ž2.1. 2 by u , respectively, and then integrate it over w0, 1x. Using Ž2.2., integration by parts, and Young’s inequality, we have 1 2
1
H0 Ž c
q u 2 . Ž x, t . dx
2
t
1
H0 Ž ac
sy
2 x
q bux2 . Ž x, t . dx
y
1
Ž s y a . c 2 q Ž 1 y b . u 2 Ž x, t . dx
q
1
nc x u y Ž cu 2 q c 1 u . ux y sux c Ž x, t . dx
H0 H0
Fy
1
1
H Ž ac 2 0
2 x
q bux2 . Ž x, t . dx q C
1
H0 Ž c
2
q u 2 . Ž x, t . dx, Ž 2.4.
where C is a positive constant depending only on a , b , s , n , A u 2 A1 , and A c 1 A1. Therefore, it follows from Ž2.3., Ž2.4., and Gronwall’s inequality that 1
H0 Ž c
2
q u 2 . Ž x, t . dx s 0
for all t g w 0, T x ,
which implies c ' 0 and u ' 0 in QT . This proves Lemma 2.1.
266
HSIAO AND JIAN
LEMMA 2.2. Let a, b, and m all be constants with a ) 0 and 0 - m F 1. Let f Ž x, t . be a gi¨ en function such that f Ž x, t . g C m Ž QT .. Suppose u 0 Ž x . g C 2, m Žw0, 1x. satisfying u 0 Ž0. s u 0 Ž1. and uX0 Ž0. s uX0 Ž1.. Then there exists a unique solution uŽ x, t . g C 2, m Ž QT . to the problem
¡u y a u q bu s f Ž x, t . , 0 - x - 1, 0 - t - T ~u Ž 0, t . s u Ž 1, t . , u Ž 0, t . s u Ž 1, t . , 0 - t - T ¢uŽ x, 0. s u Ž x . , 0 F x F 1. 2
t
xx
x
Ž 2.5.
x
0
Moreo¨ er, 1
u Ž x, t . s eyb t
H0
q
u 0 Ž j . G Ž x, t , j , 0 . d j t
1
H0 H0
e bt f Ž j , t . G Ž x, t , j , t . d j dt ,
Ž 2.6.
and u
2q m
F C Ž a, b, T .
f
m
q 5 u 0 5 2q m ,
Ž 2.7.
where G Ž x, t , j , t . s 2
`
Ý
cos 2 kp x ? cos 2 kpj ? eyŽ a2 kp .
2
Ž ty t .
H Ž t y t . Ž 2.8.
ks0
and H Ž z . is the Hea¨ iside’s function, gi¨ en by HŽ z. s
½
1, 0,
zG0 z - 0.
Ž 2.9.
Proof. The uniqueness can be proved by the same arguments as those used in the proof of Lemma 2.1. We omit the details. Due to the theory of Fourier’s series and the expressions in Ž2.6., Ž2.8., and Ž2.9. for uŽ x, t ., which can be constructed by the Fourier method Žsee w7, Sects. 1]2, Chap. 3x., one can easily verify that uŽ x, t . is a classical solution to the problem Ž2.5.. Let A Ž t . s u Ž 0, t . s eyb t
1
H0
q
u 0 Ž j . G Ž 0, t , j , 0 . d j t
1
H0 H0
e bt f Ž j , t . G Ž 0, t , j , t . d j dt ,
and w Ž x, t . s u Ž x, t . y A Ž t . .
Ž 2.10.
267
SPATIALLY PERIODIC CAUCHY PROBLEMS
Then it follows from Ž2.5. that
¡w y a w q bw s f y bAŽ t . y A9Ž t . , 0 - x - 1, 0 - t - T ~w Ž 0, t . s w Ž 1, t . s 0, 0 - t - T ¢w Ž x, 0. s u Ž x . y AŽ 0. s u Ž x . y u Ž 0. , 0 F x F 1. 2
t
xx
0
0
0
Ž 2.11. Applying the standard theory of linear parabolic equations to Ž2.11. Žsee w5x or w6x., we have w g C 2, m Ž QT . and w
2q m
F C Ž a, b, T .
f y bAŽ t . y A9 Ž t .
m
q u0
2q m
.
Thus, Ž2.10. implies that u g C 2, m Ž QT . and u
2q m
F C Ž a, b, T .
f
m
q A
m
q A9
m
q u0
2q m
. Ž 2.12.
Expanding u 0 Ž j . and f Ž j , t . to Fourier’s series, respectively, according to the normal orthogonal base def
E Ž j . s '2 cos 2 kpj , '2 sin 2 kpj : k s 1, 2, . . . 4 j 1 4 , Ž 2.13. we can obtain from the expression of AŽ t . that A9
m
q A
m
F C Ž a, b, T .
f
m
q u0
m
,
which, together with Ž2.12., yields the desired Ž2.7.. Now let m s min d , 1r34 , B s C 1, m Ž QT . = C 1, m Ž QT ., T ) 0. For l g w0, 1x, define the map Pl: B ª B by Pl Ž cˆ , uˆ . ª Ž c , u . , where Ž c , u . is the C 2, m -unique solution to the following linear parabolic problem Ž2.14. ] Ž2.16.:
¡ ¢u s yŽ 1 y b . u q nc q lŽ cuˆˆ.
~c s yŽ s y a . c y lsuˆ q ac t
t
Ž c , u . Ž 0, t . s Ž c , u . Ž 1, t . , c Ž x, 0 . s lc 0 Ž x . ,
x
x
xx x
q bux x ,
Ž 2.14.
Ž c x , ux . Ž 0, t . s Ž c x , ux . Ž 1, t . , 0 F t F T Ž 2.15. u Ž x, 0 . s lu 0 Ž x . , 0 F x F 1.
Ž 2.16.
268
HSIAO AND JIAN
Noticing that Ž2.14.1 is a single linear parabolic equation, we know from Lemma 2.2 that for any given Ž cˆ, uˆ. g B, there exists a unique solution Ž c , u . to Ž2.14. ] Ž2.16. such that Ž c , u . g C 2, m Ž QT . = C 2, m Ž QT . and AcA2q m F C Ž a , s , T . l Ž AuˆxAm q 5 c 0 5 2q m . , AuA2q m F C Ž b , n , T . Ac xAm q l AcˆA1q m q AuˆA1q m q 5 u 0 5 2q m
ž
/
.
Thus, we have AcA2q m q AuA2q m F C1 l AcˆA1q m q AuˆA1q m q 5 c 0 5 2q m q 5 u 0 5 2q m ,
ž
/
Ž 2.17. where C1 s C Ž a , b , s , n , T . is a positive constant. Due to the fact that Ž2.17. holds for any c 0 , u 0 g C 2, m Žw0, 1x. and any cˆ, uˆg C 1, m Ž QT ., and the fact that the injection of C 2, m Ž QT . into C 1, m Ž QT . is compact, we obtain LEMMA 2.3. The map Pl: w0, 1x = B ª B is well-defined and possesses the following properties: Ž1. For fixed l g w0, 1x, Pl: B ª B is completely continuous; Ž2. For e¨ ery bounded subset A ; B, the family of maps TlŽ cˆ, uˆ.: Ž cˆ, uˆ. g A 4 is uniformly equicontinuous in l g w0, 1x; Ž3. The map T0 has precisely one fixed point in B. LEMMA 2.4. There exists a constant C depending only on the constants a , b , n , s , T, and the known functions c 0 and u 0 such that for any possible fixed point Ž cl, ul . of Pl ,
cl
1q m
q ul
1q m
F C.
Moreo¨ er, the generalized deri¨ ati¨ es Ž cl . x t and Ž ul . x t are in L2 Ž QT .. Pro¨ ing this will require a long sequence of estimates; we will carry it out in the next section. Now, we complete the proof of Theorem 1.1. By virtue of Lemma 2.1, it suffices to prove the existence. Due to Lemmas 2.3 and 2.4, the map Pl satisfies the assumptions of the Leray]Schauder fixed point theorem Žsee w8, Theorem 3.1x.. Applying this theorem, we obtain a fixed point Ž c , u . of P1 , which is in C 2, m Ž QT . = C 2, m Ž QT . and is a global classical solution to Ž1.1. ] Ž1.3.. Furthermore, the fixed point Ž c , u . g C 2, d Ž QT . = C 2, d Ž QT ..
269
SPATIALLY PERIODIC CAUCHY PROBLEMS
In fact, it follows from Lemma 3.1 in w5, IIx that the fact Ž c , u . g C 2, m Ž QT . = C 2, m Ž QT . implies Ž cu . x , c x , and ux are all in C 0, 1 Ž QT .. Thus, applying Lemma 2.2 to Ž1.1. ] Ž1.3. Žsee Ž2.7.., one can obtain that c and u are in C 2, d Ž QT .. This proves Theorem 1.1. 3. A PRIORI ESTIMATES Suppose that Ž c , u . is a fixed point of Pl defined in Lemma 2.3. We will prove Lemma 2.4 in this section. In the sequel, C will denote a generic constant, depending only on a , b , s , n , T, and the functions c 0 and u 0 . Without loss of generality, we assume l s 1. This means that Ž c , u . is a solution to Ž1.1. ] Ž1.3., moreover, both c and u are in C 2, m Ž QT .. We will deduce, by a sequence of lemmas, the following estimate which may be stronger than that in Lemma 2.4:
c
1q 1r3
q u
1q1r3
F C.
Ž 3.1.
The fact that the generalized derivatives c x t and ux t are in L2 Ž QT . will be shown in Lemma 3.4. LEMMA 3.1. There exists a constant C such that for all t g w0, T x,
u Ž ?, t .
L1 w0, 1 x
F C.
Ž 3.2.
Proof. Denote uˇŽ x, t . s u Ž x, t . q n . Then Ž1.1. 2 is turned to
uˇt s n Ž 1 y b . y Ž 1 y b . uˇq Ž cuˇ. x q buˇx x .
Ž 3.3.
'
Choose « g Ž0, 1. and let w« s uˇ2 q « . Multiply Ž3.3. by uˇrw« and integrate it with respect to x over w0, 1x. Using Ž1.2. ] Ž1.3., integration by parts, and the equality
Ž uˇrw« . x s «uˇxrw«3
ˇˇxrw«3 s Ž y1rw« . x , uu
and
we obtain 1
H0
1
Ž w« . t Ž x, t . dx q Ž 1 y b . H
0
sy
1
Fy
1
H0 H0
ˇˇx «cuu w«3
«c x w«
uˇ2 w«
1
Ž x, t . dx y n Ž 1 y b . H
0
1
Ž x, t . dx y «b H
0
Ž x, t . dx,
uˇx2 w«3
Ž x, t . dx
uˇ w«
Ž x, t . dx
270
HSIAO AND JIAN
which, integrated over w0, t x, implies that for any t g w0, T x, 1
H0
w« Ž x, t . dx q Ž 1 y b . F
1
H0
w« Ž x, 0 . dx y
t
1
H0 H0 t
1
H0 H0
ž
uˇ2 w«
«c x w«
y
nuˇ w«
/
Ž x, t . dx dt
Ž x, t . dx dt .
Letting « ª 0q and noticing that
uˇ w«
F 1,
uˇ2
ª < uˇ< ,
w«
«
and
w«
ª 0,
we have 1
H0
uˇŽ x, t . dx F
1
H0
uˇ0 Ž x . dx q C Ž b , n , T . 1 q
t
1
H0 H0
uˇŽ x, t . dx dt ,
Ž 3.4. which, applied to Gronwall’s inequality, yields 1
H0
for all t g w 0, T x .
uˇŽ x, t . dx F C1 Ž b , n , T .
This immediately implies Ž3.2.. LEMMA 3.2. 1
H0 Ž c
2
For all t g w0, T x, q u 2 . Ž x, t . dx q
t
1
H0 H0 Ž c
2 x
q ux2 . Ž x, t . dx dt F C.
Proof. For any t g w0, T x, using Theorem 2.2 and Remark 2.1 in w5, IIx for the function F s u y H01 u Ž x, t . dx, one immediately obtains
u Ž ?, t .
L`w0, 1 x
FC
u Ž ?, t .
L1 w0, 1 x
q ux Ž ?, t .
2r3 L2 w0, 1 x
? u Ž ?, t .
1r3 L1 w0, 1 x
,
which, combined with Lemma 3.1, implies 1
H0
u 4 Ž x, t . dx F C u Ž ?, t . FC 1q
1
H0
3 L`w0, 1 x
ux2 Ž x, t . dx
for all t g w 0, T x . Ž 3.5.
271
SPATIALLY PERIODIC CAUCHY PROBLEMS def
Multiplying Ž1.1. 2 by u and integrating over Q t s w0, 1x = w0, t x, by virtue of Ž1.2., Ž1.3., and integration by parts, we have 1
H0
u 2 Ž x, t . dx q b FC 1q q F«
1 2 t
t
t
1
H0 H0 t
1
H0 H0 1
H0 H0
1
H0 H0
ux2 Ž x, t . dx dt
u 2 Ž x, t . dx dt q n
t
1
H0 H0
c x u dx dt
c x u 2 dx dt
u 4 Ž x, t . dx dt q C Ž « .
= 1q
t
1
H0 H0 Ž u
2
q c x2 . Ž x, t . dx dt
for any «)0. Ž 3.6.
Similarly, multiplying Ž1.1.1 by c and integrating it over Q t , we obtain 1
H0
c 2 Ž x, t . dx q
a
t
1
HH 2 0 0
c x2 Ž x, t . dx dt FC 1q
t
1
H0 H0 Ž c
2
q u 2 . Ž x, t . dx dt .
Ž 3.7.
Combining Ž3.5. ] Ž3.7. together and noticing the smallness of « , one can obtain that for all t g w0, T x, 1
H0 Ž c
2
q u 2 . Ž x, t . dx q
t
1
H0 H0 Ž c
2 x
q ux2 . dx dt FC 1q
t
1
H0 H0 Ž c
2
q u 2 . dx dt .
Now, application of Gronwall’s inequality to the above inequality immediately implies Lemma 3.2. LEMMA 3.3. 1
H0 Ž c
2 x
For all t g w0, T x, q ux2 . Ž x, t . dx q
t
1
H0 H0 Ž c
2 xx
q ux2x . Ž x, t . dx dt F C.
272
HSIAO AND JIAN
Proof. Using integration by parts and Ž1.1. ] Ž1.3., we have 1
H0
1
c x c x t Ž x, t . dx s y
H0
s ya
c x x c t Ž x, t . dx 1
H0 1
q
c x2x Ž x, t . dx
Ž s y a . cc x x q sux c x x Ž x, t . dx.
H0
Integrating this equality over w0, t x and using Young’s inequality and Lemma 3.2, we obtain 1
1
H 2 0
c x2 Ž x, t . dx q
FC 1q
t
1
a
t
1
HH 2 0 0
H0 H0 Ž c
2
c x2x Ž x, t . dx dt
q ux2 . Ž x, t . dx dt
for all t g w 0, T x .
FC
Ž 3.8.
It is well known that if v Ž?, t . g W 1, 2 Žw0, 1x., then for any « ) 0, max v 2 Ž ?, t . F « w0, 1 x
1
H0
v x2 Ž x, t . dx q C Ž « .
1
H0
v 2 Ž x, t . dx.
Ž 3.9.
This, together with Ž3.8. and Lemma 3.2, implies that max c Ž x, t . F C. QT
Ž 3.10.
By the same argument as that used to get Ž3.8. we arrive at 1
1
H 2 0
ux2 Ž x, t . dx q
FC 1q
t
1
b
t
1
HH 2 0 0
H0 H0 Ž u
2
ux2x Ž x, t . dx dt
q c x2 q ux2c 2 q u 2c x2 . Ž x, t . dx dt . Ž 3.11.
It can be shown, by using Ž3.8. ] Ž3.10. and Lemma 3.2, that T
1
H0 H0 Ž u c 2
2 x
q c 2ux2 . Ž x, t . dx dt
273
SPATIALLY PERIODIC CAUCHY PROBLEMS T
FC FC
H0
max u 2 Ž ?, t . dt q w0, 1 x
T
1
H0 H0 Ž u
2
T
1
H0 H0
ux2 Ž x, t . dx dt
q ux2 . Ž x, t . dx dt
F C, which, combined with Ž3.11. and Ž3.8. together, implies the conclusion of Lemma 3.3. Due to Lemmas 3.2 and 3.3, it follows, by applying Ž3.9. to u , that max
Ž x , t .gQ T
u Ž x, t . F C.
Ž 3.12.
LEMMA 3.4. The distributional deri¨ ati¨ es c x t and ux t are in L2 Ž QT .. Moreo¨ er, for all t g w0, T x, 1
H0
c t2 Ž x, t . q u t2 Ž x, t . dx F C,
Ž 3.13.
1
c x2x Ž x, t . q ux2x Ž x, t . dx F C,
Ž 3.14.
H0
T
1
H0 H0 Ž c
2 xt
q ux2t . Ž x, t . dx dt F C.
Ž 3.15.
Proof. Differentiate Ž1.1.1 and Ž1.1. 2 with respect to t formally Žthe rigorous proof can be established by finite difference., then multiply by c t and u t and integrate over Q t at last, respectively. Using Ž1.2. ] Ž1.3., integration by parts, Young’s inequality, Lemmas 3.2 and 3.3, Ž3.10., Ž3.12., and Gronwall’s inequality, we can obtain Ž3.13. and Ž3.15.. The details are just the same as the proof of Lemma 3.6 in w4x. Inequality Ž3.14. is a direct consequence of Ž1.1., Ž3.13., Ž3.12., Ž3.10., and Lemma 3.3. Due to the results obtained above, the proof of Ž3.1. can be completed by a routine argument Žsee, for example, the proof of w8, Lemma 3.7x.. The details are the same as those given at the end of w4x.
REFERENCES 1. D. Y. Hsieh, On partial differential equations related to Lorenz system, J. Math. Phys. 28 Ž1987., 1589]1597. 2. S. Q. Tang, ‘‘Dissipative Nonlinear Evolution Equations and Chaos,’’ Doctoral Thesis, Hong Kong Univ. Sci. Techn., 1995. 3. E. N. Lorenz, Deterministic non-periodic flows, J. Atom. Sci. 20 Ž1963., 130]141.
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4. H. Y. Jian, Global smooth solutions to a system of dissipative nonlinear evolution equations, J. of Partial Differential Equations 10, No. 2 Ž1997., to appear. 5. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva, ‘‘Linear and Quasilinear Equations of Parabolic Type,’’ Amer. Math. Soc., Providence, 1968. 6. A. Friedman, ‘‘Partial Differential Equations of Parabolic Type,’’ Prentice Hall, Englewood Cliffs, NJ, 1964. 7. L. S. Jiang and Y. Z. Chen, ‘‘Lectures on Equations of Mathematical Physics,’’ China High Education Press, Beijing, 1986. wIn Chinesex. 8. C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in onedimensional nonlinear thermoviscoelasticity, Nonlinear Anal. 6, No. 5 Ž1982., 435]454. 9. D. Henry, Geometric theory of semilinear parabolic equations, in ‘‘Lecture Notes in Math.,’’ Vol. 840, Springer-Verlag, New York, 1980.
213, 262]274 Ž1997.
AY975535
Global Smooth Solutions to the Spatially Periodic Cauchy Problem for Dissipative Nonlinear Evolution Equations Ling Hsiao* Academic Sinica, Institute of Mathematics, Beijing, 100080, People’s Republic of China
and Huaiyu Jian† Dept. of Applied Mathematics, Tsinghua Uni¨ ersity, Beijing, 100084, People’s Republic of China Submitted by Mark J. Balas Received August 23, 1995
The existence and uniqueness are proved for global classical solutions of the spatially periodic Cauchy problem to the following system of parabolic equations
½
ct s yŽ s y a . c y sux q ac x x u t s yŽ 1 y b . u q nc x q Ž cu . x q bux x ,
which was proposed as a substitute for the Rayleigh]Benard equation and can lead to Lorenz equations. Q 1997 Academic Press
1. INTRODUCTION The aim of this paper is to study the conservation form of Hsieh’s equations which reads as
½
c t s yŽ s y a . c y sux q ac x x u t s yŽ 1 y b . u q nc x q Ž cu . x q bux x ,
Ž 1.1.
* Supported by the National Natural Science Foundation of China. † Supported by the China Postdoctoral Science Foundation. On leaving the Academic Sinica, Institute of Mathematics. 262 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
SPATIALLY PERIODIC CAUCHY PROBLEMS
263
where s , n , a , and b are all positive constants satisfying the relation a - s and b - 1. We refer to Hsieh w1x and Tang w2x for the physical background of Ž1.1.. It is worthy to point out that with a truncation similar to the mode truncation from Rayleigh]Benard equations used by Lorenz in w3x, the system Ž1.1. also leads to the Lorenz equations. In fact, taking
½
c s '2 X Ž t . sin x u s '2 Y Ž t . cos x q 2 Z Ž t . cos 2 x
for Ž1.1. and retaining only the coefficients of sin x, cos x, and cos 2 x, one can obtain Lorenz equations. For the details, see Tang w2x. If we ignore the diffusion terms on the right-hand side of Ž1.1., then the characteristic of the remaining system takes the form
ls
yc "
'c
2
y 4s Ž n q u . 2
.
This means when < c < is small enough, the remaining system is basically elliptic, so one may expect that the inherent instability will cause growth and drive the system into hyperbolic regime. However, the damping and diffusion would diminish the amplitude and draw the system back into elliptic regime. So there might be a ‘‘switching back and forth’’ mechanism, which could possibly lead to some complicated behavior, even chaos. Recently, Tang w2x performed extensive numerical simulation on the system Ž1.1. with spatially periodic Cauchy data. Without loss of generality, one may assume the period L s 1. Then the data read as
Ž c , u . Ž 0, t . s Ž c , u . Ž 1, t . , Ž c x , ux . Ž 0, t . s Ž c x , ux . Ž 1, t . , 0 F t F T Ž 1.2. and
Ž c , u . Ž x, 0 . s Ž c 0 Ž x . , u 0 Ž x . . ,
0 F x F 1.
Ž 1.3.
The numerical results in w2x show that the solutions to Ž1.1. ] Ž1.3. decay to zero for big dissipation. But for smaller dissipation, chaos occurs, i.e., the problem Ž1.1. ] Ž1.3. admits only steady state solutions represented by a set of discrete peaks, and the number of peaks increases as the dissipation decreases. Nevertheless, there are only a couple of rigorous results about Ž1.1. ] Ž1.3., namely, the result obtained by Tang in w2x, which reads that if Ž1.1. ] Ž1.3. does not admit a global smooth solution c Ž x, t ., u Ž x, t .4 , then u must blow-up at first Žsee w2, Theorem 2.2.3x., and the result obtained by Jian w4x which discusses the zero case for both c and ux on the boundaries with artificial assumption H01 u 0 Ž x . dx G yn .
264
HSIAO AND JIAN
As the first step of the program to establish a theoretical analysis on the chaotic nature of the solutions for Ž1.1. with small dissipation, the present paper is devoted to the study of the global existence and uniqueness for Ž1.1. ] Ž1.3.. By combining the energy method and L1-estimate technique and applying the Leray]Schauder fixed point theorem, we will establish the global existence of classical solutions to the spatially periodic Cauchy problem. First, we recall some definitions of Banach spaces for Holder continuous ¨ functions. Let 0 - r F 1, k s 0, 1, 2. By C kq r Žw0, 1x. we denote the Banach space of functions whose derivatives of all orders m F k belong to C r Žw0, 1x. of the Holder continuous functions with exponent r Žor Lipschitz continuous ¨ functions with r s 1.. The norm of this space will be denoted by 5 ? 5 kq r . The set of functions on QT s w0, 1x = w0, T x which are Holder continu¨ ous Žexponent r . with respect to the parabolic distance DŽŽ x, t ., Ž y, t .. s < x y y < 2 q < t y t <41r2 will be denoted by C r Ž QT . with the norm A ? A r . Furthermore, denote C 1, r Ž QT . s u: u, u x g C r Ž QT . 4 , C 2, r Ž QT . s u: u, u x , u t , u x x g C r Ž QT . 4 . The norms of C k, r Ž QT . are imposed by u
kq r
s u
r
q ux
r
q Ž k y 1.
ux x
r
q ut
r
,
k s 1, 2.
For more details and the properties of these spaces, we refer to w5, 6x. Suppose that the initial-boundary data Ž1.3. and Ž1.2. satisfy the following compatibility conditions:
Ž c 0 , u 0 . Ž 0. s Ž c 0 , u 0 . Ž 1. ,
Ž Ž c 0 . x , Ž u 0 . x . Ž 0. s Ž Ž c 0 . x , Ž u 0 . x . Ž 1. . Ž 1.4.
The purpose of this paper is to prove the following: THEOREM 1.1. Let a , b , s , and n all be constants with a and b being positi¨ e. Assume c 0 Ž x . and u 0 Ž x . are in C 2, d Žw0, 1x. Ž0 - d - 1., satisfying Ž1.4.. Then for any T ) 0, there exists a unique solution c Ž x, t ., u Ž x, t .4 g C 2, d Ž QT . = C 2, d Ž QT . to the problem Ž1.1. ] Ž1.3.. Moreo¨ er, the generalized deri¨ ati¨ es c x t and ux t are in L2 Ž QT .. 2. THE PROOF OF THEOREM 1.1 LEMMA 2.1. The problem Ž1.1. ] Ž1.3. has one solution in C 2, d Ž QT . = C QT . at most. 2, d Ž
SPATIALLY PERIODIC CAUCHY PROBLEMS
265
Proof. Let both Ž c 1 , u 1 . and Ž c 2 , u 2 . be C 2, d Ž QT .-solutions to Ž1.1. ] Ž1.3.. Denote
c s c1 y c2 ,
u s u1 y u2 .
Then Ž c , u . satisfies
¡ q ac ¢u s yŽ 1 y b . u q nc q Ž cu
~c s yŽ s y a . c y su t
t
x
x
xx 2
q c 1 u . x q bux x
Ž 2.1.
with boundary conditions
Ž c , u . Ž 0, t . s Ž c , u . Ž 1, t . ,
Ž cx , ux . Ž 0, t . s Ž cx , ux . Ž 1, t . , 0 F t F T Ž 2.2.
and initial conditions
c Ž x, 0 . s 0,
u Ž x, 0 . s 0, 0 F x F 1.
Ž 2.3.
Multiply Ž2.1.1 by c and Ž2.1. 2 by u , respectively, and then integrate it over w0, 1x. Using Ž2.2., integration by parts, and Young’s inequality, we have 1 2
1
H0 Ž c
q u 2 . Ž x, t . dx
2
t
1
H0 Ž ac
sy
2 x
q bux2 . Ž x, t . dx
y
1
Ž s y a . c 2 q Ž 1 y b . u 2 Ž x, t . dx
q
1
nc x u y Ž cu 2 q c 1 u . ux y sux c Ž x, t . dx
H0 H0
Fy
1
1
H Ž ac 2 0
2 x
q bux2 . Ž x, t . dx q C
1
H0 Ž c
2
q u 2 . Ž x, t . dx, Ž 2.4.
where C is a positive constant depending only on a , b , s , n , A u 2 A1 , and A c 1 A1. Therefore, it follows from Ž2.3., Ž2.4., and Gronwall’s inequality that 1
H0 Ž c
2
q u 2 . Ž x, t . dx s 0
for all t g w 0, T x ,
which implies c ' 0 and u ' 0 in QT . This proves Lemma 2.1.
266
HSIAO AND JIAN
LEMMA 2.2. Let a, b, and m all be constants with a ) 0 and 0 - m F 1. Let f Ž x, t . be a gi¨ en function such that f Ž x, t . g C m Ž QT .. Suppose u 0 Ž x . g C 2, m Žw0, 1x. satisfying u 0 Ž0. s u 0 Ž1. and uX0 Ž0. s uX0 Ž1.. Then there exists a unique solution uŽ x, t . g C 2, m Ž QT . to the problem
¡u y a u q bu s f Ž x, t . , 0 - x - 1, 0 - t - T ~u Ž 0, t . s u Ž 1, t . , u Ž 0, t . s u Ž 1, t . , 0 - t - T ¢uŽ x, 0. s u Ž x . , 0 F x F 1. 2
t
xx
x
Ž 2.5.
x
0
Moreo¨ er, 1
u Ž x, t . s eyb t
H0
q
u 0 Ž j . G Ž x, t , j , 0 . d j t
1
H0 H0
e bt f Ž j , t . G Ž x, t , j , t . d j dt ,
Ž 2.6.
and u
2q m
F C Ž a, b, T .
f
m
q 5 u 0 5 2q m ,
Ž 2.7.
where G Ž x, t , j , t . s 2
`
Ý
cos 2 kp x ? cos 2 kpj ? eyŽ a2 kp .
2
Ž ty t .
H Ž t y t . Ž 2.8.
ks0
and H Ž z . is the Hea¨ iside’s function, gi¨ en by HŽ z. s
½
1, 0,
zG0 z - 0.
Ž 2.9.
Proof. The uniqueness can be proved by the same arguments as those used in the proof of Lemma 2.1. We omit the details. Due to the theory of Fourier’s series and the expressions in Ž2.6., Ž2.8., and Ž2.9. for uŽ x, t ., which can be constructed by the Fourier method Žsee w7, Sects. 1]2, Chap. 3x., one can easily verify that uŽ x, t . is a classical solution to the problem Ž2.5.. Let A Ž t . s u Ž 0, t . s eyb t
1
H0
q
u 0 Ž j . G Ž 0, t , j , 0 . d j t
1
H0 H0
e bt f Ž j , t . G Ž 0, t , j , t . d j dt ,
and w Ž x, t . s u Ž x, t . y A Ž t . .
Ž 2.10.
267
SPATIALLY PERIODIC CAUCHY PROBLEMS
Then it follows from Ž2.5. that
¡w y a w q bw s f y bAŽ t . y A9Ž t . , 0 - x - 1, 0 - t - T ~w Ž 0, t . s w Ž 1, t . s 0, 0 - t - T ¢w Ž x, 0. s u Ž x . y AŽ 0. s u Ž x . y u Ž 0. , 0 F x F 1. 2
t
xx
0
0
0
Ž 2.11. Applying the standard theory of linear parabolic equations to Ž2.11. Žsee w5x or w6x., we have w g C 2, m Ž QT . and w
2q m
F C Ž a, b, T .
f y bAŽ t . y A9 Ž t .
m
q u0
2q m
.
Thus, Ž2.10. implies that u g C 2, m Ž QT . and u
2q m
F C Ž a, b, T .
f
m
q A
m
q A9
m
q u0
2q m
. Ž 2.12.
Expanding u 0 Ž j . and f Ž j , t . to Fourier’s series, respectively, according to the normal orthogonal base def
E Ž j . s '2 cos 2 kpj , '2 sin 2 kpj : k s 1, 2, . . . 4 j 1 4 , Ž 2.13. we can obtain from the expression of AŽ t . that A9
m
q A
m
F C Ž a, b, T .
f
m
q u0
m
,
which, together with Ž2.12., yields the desired Ž2.7.. Now let m s min d , 1r34 , B s C 1, m Ž QT . = C 1, m Ž QT ., T ) 0. For l g w0, 1x, define the map Pl: B ª B by Pl Ž cˆ , uˆ . ª Ž c , u . , where Ž c , u . is the C 2, m -unique solution to the following linear parabolic problem Ž2.14. ] Ž2.16.:
¡ ¢u s yŽ 1 y b . u q nc q lŽ cuˆˆ.
~c s yŽ s y a . c y lsuˆ q ac t
t
Ž c , u . Ž 0, t . s Ž c , u . Ž 1, t . , c Ž x, 0 . s lc 0 Ž x . ,
x
x
xx x
q bux x ,
Ž 2.14.
Ž c x , ux . Ž 0, t . s Ž c x , ux . Ž 1, t . , 0 F t F T Ž 2.15. u Ž x, 0 . s lu 0 Ž x . , 0 F x F 1.
Ž 2.16.
268
HSIAO AND JIAN
Noticing that Ž2.14.1 is a single linear parabolic equation, we know from Lemma 2.2 that for any given Ž cˆ, uˆ. g B, there exists a unique solution Ž c , u . to Ž2.14. ] Ž2.16. such that Ž c , u . g C 2, m Ž QT . = C 2, m Ž QT . and AcA2q m F C Ž a , s , T . l Ž AuˆxAm q 5 c 0 5 2q m . , AuA2q m F C Ž b , n , T . Ac xAm q l AcˆA1q m q AuˆA1q m q 5 u 0 5 2q m
ž
/
.
Thus, we have AcA2q m q AuA2q m F C1 l AcˆA1q m q AuˆA1q m q 5 c 0 5 2q m q 5 u 0 5 2q m ,
ž
/
Ž 2.17. where C1 s C Ž a , b , s , n , T . is a positive constant. Due to the fact that Ž2.17. holds for any c 0 , u 0 g C 2, m Žw0, 1x. and any cˆ, uˆg C 1, m Ž QT ., and the fact that the injection of C 2, m Ž QT . into C 1, m Ž QT . is compact, we obtain LEMMA 2.3. The map Pl: w0, 1x = B ª B is well-defined and possesses the following properties: Ž1. For fixed l g w0, 1x, Pl: B ª B is completely continuous; Ž2. For e¨ ery bounded subset A ; B, the family of maps TlŽ cˆ, uˆ.: Ž cˆ, uˆ. g A 4 is uniformly equicontinuous in l g w0, 1x; Ž3. The map T0 has precisely one fixed point in B. LEMMA 2.4. There exists a constant C depending only on the constants a , b , n , s , T, and the known functions c 0 and u 0 such that for any possible fixed point Ž cl, ul . of Pl ,
cl
1q m
q ul
1q m
F C.
Moreo¨ er, the generalized deri¨ ati¨ es Ž cl . x t and Ž ul . x t are in L2 Ž QT .. Pro¨ ing this will require a long sequence of estimates; we will carry it out in the next section. Now, we complete the proof of Theorem 1.1. By virtue of Lemma 2.1, it suffices to prove the existence. Due to Lemmas 2.3 and 2.4, the map Pl satisfies the assumptions of the Leray]Schauder fixed point theorem Žsee w8, Theorem 3.1x.. Applying this theorem, we obtain a fixed point Ž c , u . of P1 , which is in C 2, m Ž QT . = C 2, m Ž QT . and is a global classical solution to Ž1.1. ] Ž1.3.. Furthermore, the fixed point Ž c , u . g C 2, d Ž QT . = C 2, d Ž QT ..
269
SPATIALLY PERIODIC CAUCHY PROBLEMS
In fact, it follows from Lemma 3.1 in w5, IIx that the fact Ž c , u . g C 2, m Ž QT . = C 2, m Ž QT . implies Ž cu . x , c x , and ux are all in C 0, 1 Ž QT .. Thus, applying Lemma 2.2 to Ž1.1. ] Ž1.3. Žsee Ž2.7.., one can obtain that c and u are in C 2, d Ž QT .. This proves Theorem 1.1. 3. A PRIORI ESTIMATES Suppose that Ž c , u . is a fixed point of Pl defined in Lemma 2.3. We will prove Lemma 2.4 in this section. In the sequel, C will denote a generic constant, depending only on a , b , s , n , T, and the functions c 0 and u 0 . Without loss of generality, we assume l s 1. This means that Ž c , u . is a solution to Ž1.1. ] Ž1.3., moreover, both c and u are in C 2, m Ž QT .. We will deduce, by a sequence of lemmas, the following estimate which may be stronger than that in Lemma 2.4:
c
1q 1r3
q u
1q1r3
F C.
Ž 3.1.
The fact that the generalized derivatives c x t and ux t are in L2 Ž QT . will be shown in Lemma 3.4. LEMMA 3.1. There exists a constant C such that for all t g w0, T x,
u Ž ?, t .
L1 w0, 1 x
F C.
Ž 3.2.
Proof. Denote uˇŽ x, t . s u Ž x, t . q n . Then Ž1.1. 2 is turned to
uˇt s n Ž 1 y b . y Ž 1 y b . uˇq Ž cuˇ. x q buˇx x .
Ž 3.3.
'
Choose « g Ž0, 1. and let w« s uˇ2 q « . Multiply Ž3.3. by uˇrw« and integrate it with respect to x over w0, 1x. Using Ž1.2. ] Ž1.3., integration by parts, and the equality
Ž uˇrw« . x s «uˇxrw«3
ˇˇxrw«3 s Ž y1rw« . x , uu
and
we obtain 1
H0
1
Ž w« . t Ž x, t . dx q Ž 1 y b . H
0
sy
1
Fy
1
H0 H0
ˇˇx «cuu w«3
«c x w«
uˇ2 w«
1
Ž x, t . dx y n Ž 1 y b . H
0
1
Ž x, t . dx y «b H
0
Ž x, t . dx,
uˇx2 w«3
Ž x, t . dx
uˇ w«
Ž x, t . dx
270
HSIAO AND JIAN
which, integrated over w0, t x, implies that for any t g w0, T x, 1
H0
w« Ž x, t . dx q Ž 1 y b . F
1
H0
w« Ž x, 0 . dx y
t
1
H0 H0 t
1
H0 H0
ž
uˇ2 w«
«c x w«
y
nuˇ w«
/
Ž x, t . dx dt
Ž x, t . dx dt .
Letting « ª 0q and noticing that
uˇ w«
F 1,
uˇ2
ª < uˇ< ,
w«
«
and
w«
ª 0,
we have 1
H0
uˇŽ x, t . dx F
1
H0
uˇ0 Ž x . dx q C Ž b , n , T . 1 q
t
1
H0 H0
uˇŽ x, t . dx dt ,
Ž 3.4. which, applied to Gronwall’s inequality, yields 1
H0
for all t g w 0, T x .
uˇŽ x, t . dx F C1 Ž b , n , T .
This immediately implies Ž3.2.. LEMMA 3.2. 1
H0 Ž c
2
For all t g w0, T x, q u 2 . Ž x, t . dx q
t
1
H0 H0 Ž c
2 x
q ux2 . Ž x, t . dx dt F C.
Proof. For any t g w0, T x, using Theorem 2.2 and Remark 2.1 in w5, IIx for the function F s u y H01 u Ž x, t . dx, one immediately obtains
u Ž ?, t .
L`w0, 1 x
FC
u Ž ?, t .
L1 w0, 1 x
q ux Ž ?, t .
2r3 L2 w0, 1 x
? u Ž ?, t .
1r3 L1 w0, 1 x
,
which, combined with Lemma 3.1, implies 1
H0
u 4 Ž x, t . dx F C u Ž ?, t . FC 1q
1
H0
3 L`w0, 1 x
ux2 Ž x, t . dx
for all t g w 0, T x . Ž 3.5.
271
SPATIALLY PERIODIC CAUCHY PROBLEMS def
Multiplying Ž1.1. 2 by u and integrating over Q t s w0, 1x = w0, t x, by virtue of Ž1.2., Ž1.3., and integration by parts, we have 1
H0
u 2 Ž x, t . dx q b FC 1q q F«
1 2 t
t
t
1
H0 H0 t
1
H0 H0 1
H0 H0
1
H0 H0
ux2 Ž x, t . dx dt
u 2 Ž x, t . dx dt q n
t
1
H0 H0
c x u dx dt
c x u 2 dx dt
u 4 Ž x, t . dx dt q C Ž « .
= 1q
t
1
H0 H0 Ž u
2
q c x2 . Ž x, t . dx dt
for any «)0. Ž 3.6.
Similarly, multiplying Ž1.1.1 by c and integrating it over Q t , we obtain 1
H0
c 2 Ž x, t . dx q
a
t
1
HH 2 0 0
c x2 Ž x, t . dx dt FC 1q
t
1
H0 H0 Ž c
2
q u 2 . Ž x, t . dx dt .
Ž 3.7.
Combining Ž3.5. ] Ž3.7. together and noticing the smallness of « , one can obtain that for all t g w0, T x, 1
H0 Ž c
2
q u 2 . Ž x, t . dx q
t
1
H0 H0 Ž c
2 x
q ux2 . dx dt FC 1q
t
1
H0 H0 Ž c
2
q u 2 . dx dt .
Now, application of Gronwall’s inequality to the above inequality immediately implies Lemma 3.2. LEMMA 3.3. 1
H0 Ž c
2 x
For all t g w0, T x, q ux2 . Ž x, t . dx q
t
1
H0 H0 Ž c
2 xx
q ux2x . Ž x, t . dx dt F C.
272
HSIAO AND JIAN
Proof. Using integration by parts and Ž1.1. ] Ž1.3., we have 1
H0
1
c x c x t Ž x, t . dx s y
H0
s ya
c x x c t Ž x, t . dx 1
H0 1
q
c x2x Ž x, t . dx
Ž s y a . cc x x q sux c x x Ž x, t . dx.
H0
Integrating this equality over w0, t x and using Young’s inequality and Lemma 3.2, we obtain 1
1
H 2 0
c x2 Ž x, t . dx q
FC 1q
t
1
a
t
1
HH 2 0 0
H0 H0 Ž c
2
c x2x Ž x, t . dx dt
q ux2 . Ž x, t . dx dt
for all t g w 0, T x .
FC
Ž 3.8.
It is well known that if v Ž?, t . g W 1, 2 Žw0, 1x., then for any « ) 0, max v 2 Ž ?, t . F « w0, 1 x
1
H0
v x2 Ž x, t . dx q C Ž « .
1
H0
v 2 Ž x, t . dx.
Ž 3.9.
This, together with Ž3.8. and Lemma 3.2, implies that max c Ž x, t . F C. QT
Ž 3.10.
By the same argument as that used to get Ž3.8. we arrive at 1
1
H 2 0
ux2 Ž x, t . dx q
FC 1q
t
1
b
t
1
HH 2 0 0
H0 H0 Ž u
2
ux2x Ž x, t . dx dt
q c x2 q ux2c 2 q u 2c x2 . Ž x, t . dx dt . Ž 3.11.
It can be shown, by using Ž3.8. ] Ž3.10. and Lemma 3.2, that T
1
H0 H0 Ž u c 2
2 x
q c 2ux2 . Ž x, t . dx dt
273
SPATIALLY PERIODIC CAUCHY PROBLEMS T
FC FC
H0
max u 2 Ž ?, t . dt q w0, 1 x
T
1
H0 H0 Ž u
2
T
1
H0 H0
ux2 Ž x, t . dx dt
q ux2 . Ž x, t . dx dt
F C, which, combined with Ž3.11. and Ž3.8. together, implies the conclusion of Lemma 3.3. Due to Lemmas 3.2 and 3.3, it follows, by applying Ž3.9. to u , that max
Ž x , t .gQ T
u Ž x, t . F C.
Ž 3.12.
LEMMA 3.4. The distributional deri¨ ati¨ es c x t and ux t are in L2 Ž QT .. Moreo¨ er, for all t g w0, T x, 1
H0
c t2 Ž x, t . q u t2 Ž x, t . dx F C,
Ž 3.13.
1
c x2x Ž x, t . q ux2x Ž x, t . dx F C,
Ž 3.14.
H0
T
1
H0 H0 Ž c
2 xt
q ux2t . Ž x, t . dx dt F C.
Ž 3.15.
Proof. Differentiate Ž1.1.1 and Ž1.1. 2 with respect to t formally Žthe rigorous proof can be established by finite difference., then multiply by c t and u t and integrate over Q t at last, respectively. Using Ž1.2. ] Ž1.3., integration by parts, Young’s inequality, Lemmas 3.2 and 3.3, Ž3.10., Ž3.12., and Gronwall’s inequality, we can obtain Ž3.13. and Ž3.15.. The details are just the same as the proof of Lemma 3.6 in w4x. Inequality Ž3.14. is a direct consequence of Ž1.1., Ž3.13., Ž3.12., Ž3.10., and Lemma 3.3. Due to the results obtained above, the proof of Ž3.1. can be completed by a routine argument Žsee, for example, the proof of w8, Lemma 3.7x.. The details are the same as those given at the end of w4x.
REFERENCES 1. D. Y. Hsieh, On partial differential equations related to Lorenz system, J. Math. Phys. 28 Ž1987., 1589]1597. 2. S. Q. Tang, ‘‘Dissipative Nonlinear Evolution Equations and Chaos,’’ Doctoral Thesis, Hong Kong Univ. Sci. Techn., 1995. 3. E. N. Lorenz, Deterministic non-periodic flows, J. Atom. Sci. 20 Ž1963., 130]141.
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4. H. Y. Jian, Global smooth solutions to a system of dissipative nonlinear evolution equations, J. of Partial Differential Equations 10, No. 2 Ž1997., to appear. 5. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva, ‘‘Linear and Quasilinear Equations of Parabolic Type,’’ Amer. Math. Soc., Providence, 1968. 6. A. Friedman, ‘‘Partial Differential Equations of Parabolic Type,’’ Prentice Hall, Englewood Cliffs, NJ, 1964. 7. L. S. Jiang and Y. Z. Chen, ‘‘Lectures on Equations of Mathematical Physics,’’ China High Education Press, Beijing, 1986. wIn Chinesex. 8. C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in onedimensional nonlinear thermoviscoelasticity, Nonlinear Anal. 6, No. 5 Ž1982., 435]454. 9. D. Henry, Geometric theory of semilinear parabolic equations, in ‘‘Lecture Notes in Math.,’’ Vol. 840, Springer-Verlag, New York, 1980.