On the Cauchy Problem of Some Dissipative Flows

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

218, 479]494 Ž1998.

AY975814

On the Cauchy Problem of Some Dissipative Flows Dehua Wang* Department of Mathematics, Uni¨ ersity of California, Santa Barbara, California 93106 Submitted by Maria Clara Nucci Received June 13, 1997

The Cauchy problem is studied for a system of nonlinear partial differential equations for some dissipative flows in Lagrangian formulation including heat conduction, damping relaxation, and coupling to electric field. The well-posedness of smooth solutions is investigated. It is proved that, for certain large initial data, the solution will develop singularities and shock waves in finite time, which indicates that the Cauchy problem does not have global smooth solutions even if the initial data are smooth, and one has to seek weak solutions. Q 1998 Academic Press

1. INTRODUCTION We are concerned with a system of partial differential equations which arises in semiconductor devices and biophysics. The Lagrangian formulation of the system reads ¨ t y u x s 0,

x g R, t ) 0,

ut y ˆ pŽ ¨ , T . x s Wt y Ž up ˆŽ ¨ , T . . x s

u ¨

Ey

E ¨

y

u

t

,

W y Ž 3r2 . T

t9

Ž 1. Ž 2. q k Tx x ,

Ž 3.

where ¨ , u, T, rˆ, W, and E denote the specific volume, velocity, temperature, pressure, energy, and the electric field, respectively; T ) 0 is the lattice temperature, k ) 0 is the heat conductivity, and t ) 0 and t 9 ) 0 are relaxation times. The electric field E is the gradient of the electric * E-mail address: [email protected]. 479 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

480

DEHUA WANG

potential f satisfying E s fx ,

Ex s ¨ y D Ž x . ,

Ž 4.

where DŽ x . is the doping profile. We consider the Cauchy problem of this system with general large initial data. The issue is in which function space the Cauchy problem is well-posed globally in time. For smooth and small initial data, we can expect the global existence of smooth solutions to this system because of the damping effects in the momentum and energy equations, see w3, 7, 14x. That is, the Cauchy problem is well-posed globally in smooth function space for small initial data. The question is whether or not this problem is well-posed globally in some smooth function space for general large initial data. The answer is not obvious for this system, since this system has complicated coupling effects of damping relaxations, heat diffusion, and electric dissipation. For the hyperbolic system without these terms, it is known that there is no global smooth solutions Žcf. w1, 8, 9, 11x.. For the system with heat and momentum diffusion, there exist global smooth solutions for general large initial data Žcf. w2, 12x.. For our system, the global smooth solutions exist for small initial data. In this paper we investigate if one can expect the smooth solutions globally in time for large initial data. Our results show that the smooth solution can exist only for a finite time, and will eventually develop singularities and shock waves, although this system contains heat diffusion and damping. Therefore one cannot expect the well-posedness of the Cauchy problem in the smooth function space globally but shortly in time, and has to seek global weak solutions in some other discontinuous function spaces. This new result indicates that the heat diffusion together with damping and electric dissipation still can not prevent the formation of singularities for large initial data. Earlier work, such as w4]6, 14x, showed that heat diffusion or damping alone cannot prevent shock waves. It is a challenging problem to prove the global existence of weak solutions to system Ž1. ] Ž4. for general large initial data. This is under investigation for some special case of the system. We will give the notations and state the main result in Section 2. Several essential a priori estimates on the solution will be made in Section 3. This is achieved by certain energy estimates and a similar form of free energy to w4, 6x. The main result will be proved in Section 4. The key idea of this proof is to apply the differential operators along the characteristics to certain Riemann invariants, and then to derive proper differential equations which govern the evolution of the derivatives of the Riemann invariants.

481

SOME DISSIPATIVE FLOWS

2. NOTATIONS AND MAIN RESULT We consider system Ž1. ] Ž4. with the following form of pressure and energy,

ˆp Ž ¨ , T . s p Ž ¨ . q a Ž T y T . ,

WŽ¨, T . s

1 2

u2 q e Ž ¨ , T . ,

with a ) 0 a constant and eŽ ¨ , T . s P Ž ¨ . q 32 T y a T Ž ¨ y ¨ ., ¨ ) 0 a constant, P Ž ¨ . s H¨¨ pŽ j . d j , p9Ž ¨ . ) 0, p0 Ž ¨ . ) 0 for ¨ ) 0, and pŽ ¨ . s 0. We take the following Riemann invariants rsuq

¨

H 'p9 Ž j .

dj ,

¨

ssuy

¨

H 'p9 Ž j . ¨

dj ,

Ž 5.

and the differential operators along the characteristics ) s ­t y

'p9 Ž ¨ .

­x ,

9 s ­t q

­x .

Ž 6.

Ž T q u . u x q H Ž ¨ , u, u . ,

Ž 7.

'p9 Ž ¨ .

Set u s T y T. Then Eq. Ž3. can be rewritten into

ut y

2 3

kux x s

2a 3

where H Ž ¨ , u, u . s ŽŽ2t 9 y t .r3tt 9. u 2 y Ž2r3t 9.Ž P Ž ¨ . y a T Ž ¨ y ¨ .. y urt 9. We investigate the Cauchy problem of Ž1. ] Ž4. with the initial data

Ž ¨ , u, T , f . Ž x, 0 . s Ž ¨ 0 Ž x . , u 0 Ž x . , T0 Ž x . , f 0 Ž x . .

Ž 8.

satisfying the compatibility condition: f Y0 Ž x . s ¨ 0 Ž x . y DŽ x .. Take u Ž x, 0. s u 0 Ž x . s T0 Ž x . y T. Then we have the following theorem: THEOREM 1. Assume ¨ 0 , u 0 , u 0 g C 2 ŽR., and u 0 g L2 ŽR., ¨ 0 y ¨ , u 0 g L R. l L2 ŽR., ­x¨ 0 , ­x u 0 , ­x u 0 , ­x f 0 ª 0, as x ª "`, 1Ž

<¨0 y ¨ < - d , `

Hy` Ž < ¨

0

< u0 < - d ,


Ž 9.

2 2 Ž x . y ¨ < 2 q u 0 Ž x . q u 0 Ž x . q < ¨ 0 Ž x . y ¨ < q < u 0 Ž x . < . dx - d 2 ,

Ž 10 . QŽ x . s

x

Hy` Ž ¨

0

Ž j . y D Ž j . . d j g L` l L2 Ž R . ,

Ž 11 .

482

DEHUA WANG

and for some positi¨ e constant K 1 , sup y­x u 0 Ž x . y

ž

x

'p9 Ž ¨

0

q sup y­x u 0 Ž x . q

ž

x

Ž x . . ­ x¨ 0 Ž x . /

'p9 Ž ¨

0

Ž x . . ­ x¨ 0 Ž x . / - K 1 .

Ž 12.

Then, for some d ) 0, gi¨ en any L ) 0, there exists K 2 ) 0 depending on L, ¨ , T, d , K 1 , such that, if sup ­x u 0 Ž x . q x

ž

'p9 Ž ¨

0

q sup ­x u 0 Ž x . y x

ž

Ž x . . ­ x¨ 0 Ž x . /

'p9 Ž ¨

0

Ž x . . ­ x¨ 0 Ž x . / ) K 2 ,

Ž 13 .

the maximal length of the time inter¨ al of existence of any smooth solution is less than L. To prove this theorem, we will first make a priori estimates on the solutions in Section 3.

3. ESTIMATES OF SOLUTIONS Assume that we have a C 2 solution Ž ¨ , u, u , f . on R = w0, L x for some positive L - 1, and uŽ x, ? . g L2 ŽR., ¨ Ž x, ? . y ¨ , u Ž x, ? . g L1 l L2 ŽR., and ¨ x , u x , ux , E ª 0 as x ª "`. For sufficiently small L - 1 and d minŽ ¨ , T, 2.r2, we have < ¨ Ž x, t . y ¨ < - ¨ r2, < uŽ x, t .< - 1, < u Ž x, t .< - Tr2 for any Ž x, t . g R = w0, L x. We will prove that for small d , L can be selected a priori such that the above holds for any solution with the initial conditions satisfying Ž9. ] Ž11., which is a consequence of Lemma 2 and Lemma 3 below. LEMMA 1. There exists a constant L ) 0, such that, for any x g R and t g w0, L x, `

Hy` Ž u

2

2

q Ž ¨ y ¨ . q u 2 . dx q < E Ž x, t . < F L ,

x

Hy`

`

t

H0 Hy` u u ¨

2 x

Ž x, s . dxds F L d 2 ,

E Ž j , t . dj F L .

Proof. We note that the internal energy e and the pressure ˆ p satisfy e¨ s ˆ p y Tp ˆT , then there exist a free energy

s s s Ž¨, T . s PŽ¨ . q a Ž¨ y¨ .u q

3 2

T Ž ln T y ln T . q

3 2

T,

483

SOME DISSIPATIVE FLOWS

and a corresponding entropy

h s h Ž ¨ , T . s ya Ž ¨ y ¨ . y

3 2

Ž ln T y ln T . ,

satisfying s¨ s ˆ p, s T s yh , and s s e y Th. Equation Ž3. implies

ž

s q uh q s

u

u2 2

2

u2 T

Ey

¨

3

y

tT

T

kT

q

/

2

T

t

ux2 y up ˆq k

ž

u Ž W y Ž 3r2 . T .

y

t 9T

uux T

/

x

.

Integrate it over R = w0, t x to get `

Hy`

ž

u2

s q uh q

s

`

Hy`

ž

q

t

2

3

y

`

ž

u ¨

/

2

u2

s q uh q

H0 Hy`

T Ž x, t . dx q y

2

3 2

u2 T

Ey

kT T2

ux2 dxdt

/

T Ž x, 0 . dx

y

tT

`

t

H0 Hy`

u Ž W y Ž 3r2 . T .

/

t 9T

dxdt.

There exist two positive constants C1 , C2 , such that, for < ¨ y ¨ < - ¨ r2 and < u < F Tr2, 2

C1 Ž Ž ¨ y ¨ . q u q u 2

2

. F s q uh q

u2 2

y

3 2

T

2

F C2 Ž Ž ¨ y ¨ . q u 2 q u 2 . . Integrating Eq. Ž4., and using Ž1. and Ž11., we have Es

t

H0 u Ž x, s . ds q Q Ž x . ,

and then by Jensen’s inequality, `

Hy`

u ¨

`

E dx F C

Hy` u

FC

`

Hy`

FC

`

2

dxds q C

`

u 2 dx q C

Hy` u

`

Hy` E t

ž

Hy` H0

2

dx q C

t

`

2

dx 2

u Ž x, s . ds

H0 Hy` u

2

/

dxds q C.

dx q C

`

Hy` Q Ž x .

2

dx

484

DEHUA WANG

The following estimate is immediate from the form of the energy, y

t

`

H0 Hy`

u Ž W y Ž 3r2 . T .

dxds F C

`

t

H0 Hy` Ž u

t 9T

q u 2 q Ž¨ y¨.

2

2

. dxds.

Therefore, from the above estimates and the initial conditions, we obtain `

Hy` Ž Ž ¨ y ¨ .

2

q u 2 q u 2 . dx q

F CL q C

t

`

H0 Hy` Ž Ž ¨ y ¨ .

`

t

H0 Hy` u

2

2 x

Ž x, s . dxds

q u 2 q u 2 . dxds,

When L - d 2 , Gronwall’s inequality yields `

Hy` Ž Ž ¨ y ¨ .

2

q u 2 q u 2 . dx q

t

`

H0 Hy` u

2 x

Ž x, s . dxds F L d 2 .

Lemma 1 follows. LEMMA 2.

For small L - 1,

< ¨ Ž x, t . y ¨ < F L d ,

Ž x, t . g R = w 0, L x ,

< u Ž x, t . < F L d ,

for some constant L ) 0. Proof. Applying the differential operators Ž6. to Ž5., we have r) s

E ¨

y

u

t

q aux ,

s9 s

E ¨

u

y

q aux .

t

Ž 14 .

Set

x s x Ž x, t . s

x

Hy`

ž

1 2

3

u2 q P Ž ¨ . q

2

u dj ,

/

then integrating Eq. Ž3. over Žy`, x . with respect to x, one has

xt y u Ž p Ž ¨ . q a Ž u q T . . s

x

x

u

Hy` ¨ Edj y t 9 q ku

x

q

aT

x

H Ž ¨ y ¨ . dj , t 9 y`

and then

kux s x ) q s x9 y

'p9 Ž ¨ . 'p9

ž Ž . ž ¨

1 2 1 2

u2 q P Ž ¨ . q u2 q P Ž ¨ . q

3 2 3 2

u qM

/ /

u q M,

Ž 15 .

485

SOME DISSIPATIVE FLOWS

where M s yu Ž p Ž ¨ . q a Ž u qT . . y

x

u

x

Hy` ¨ Edj q t 9 y

aT

x

H Ž ¨ y¨ . d j . t 9 y`

Set x 0 s x Ž x, 0. and

wsry

a k

Ž x y x0 . ,

cssy

a k

Ž x y x0 . ,

Ž 16 .

then, using Ž15. to eliminate ux in Ž14., we get

w) s

E ¨

=

c9s

E ¨

=

y

ž

1 2

y

ž

1 2

u

q

t

a k

'p9 Ž ¨ .

u2 q P Ž ¨ . q u

y

t

a k

3 2

uy

1

u 20 y P Ž ¨ 0 . y

2

3

u0 q

/

2

a k

M

Ž 17 .

'p9 Ž ¨ .

u2 q P Ž ¨ . q

3 2

uy

1

u 20 y P Ž ¨ 0 . y

2

3

u0 q

/

2

a k

M.

Integrating Ž ¨ y ¨ . t y u x s 0 over Žy`, x . = w0, t x and using the initial condition, we have, for L - d , x

Hy` Ž ¨ y ¨ . dx

F

`

Hy` < ¨

t

0

Ž x . y ¨ < dx q H < u < ds F Cd , 0

and then integrate Ž7. over R = w0, t x and use integration by parts and Lemma 1, `

Hy` u dx

F

`

t

H0 Hy`

FC

t

ž

2a 3 `

uux dxds q

H0 Hy` Ž u

2

F Cd q Cd 2 q

1

t

q ux2 . dx ds q

/

1

t

`

H H u dx t 9 0 y`

`

H H u dx t 9 0 y`

1

t

ds q Cd `

H H u dx t 9 0 y`

ds q Cd

ds.

By Gronwall’s inequality, for t g w0, L x, L - 1, the following holds `

Hy` u dx

F Cd .

486

DEHUA WANG

Equation Ž7. indicates that u has a representation in terms of the heat kernel, and we integrate it with respect to x and use the above estimates to get x

Hy` u Ž y, t . dx

; x g R, t g w 0, L x .

F Cd ,

Therefore, Lemma 1 implies < x < F Cd . From Ž16. and Ž5., us

wqc

y

2

a k

¨ s By1 Ž w y c . ,

Ž x y x0 . ,

where By1 is the inverse of the operator B Ž ¨ . s 2 H¨¨ For each t g w0, L x, define the Lipschitz functions F Ž t . s max < w Ž x, t . < ,

'p9 Ž j .

dj .

C Ž t . s max < c Ž x, t . < .

x

x

Fix t g Ž0, L x. There are two points ˆ x, ˇ x g Žy`, `., such that F Ž t . s < wŽ ˆ x, t .<, C Ž t . s < c Ž ˇ x, t .<. ŽThis is guaranteed by the properties of the solution at infinity.. For any h g Ž0, t x, F Ž t y h. G w ˆ x q h p9 Ž ¨ . , t y h ,

' ž / C Ž t y h. G c ž ˇ x y h'p9 Ž ¨ . , t y h /

,

then d dt

d

F F < w) Ž ˆ x, t . < ,

dt

C F < c 9Ž ˇ x, t . < .

Ž 18 .

By Ž17. and using Lemma 1, the properties of the solution, and the above estimates, we have d dt

Ž F q C . F L1 q F L2 q F

uŽ ˆ x, t .

t

q

uŽ ˇ x, t .

t

wŽ ˆ x, t . q c Ž ˆ x, t . 2t

wŽ ˆ x, t . q c Ž ˇ x, t . t

q

wŽ ˇ x, t . q c Ž ˇ x, t . 2t

q L2

F LŽ F q C. q L . The initial conditions imply that F Ž0. F L d and C Ž0. F L d . If L - d , then F q C F L d , that is, < ¨ y ¨ < F L d , < u < F L d . For d small enough, < ¨ y ¨ <) F L d - ¨ r2, < u < F L d - 1. This completes the proof of Lemma 2.

487

SOME DISSIPATIVE FLOWS

LEMMA 3. There exists a constant L ) 0, such that, for Ž x, t . g R = w0, L x, < u Ž x, t . < F L d . Proof. Multiply Ž7. by 2 nu 2 using the integration by parts, d

`

H u dt y`

2n

Fy q q

y1

n

, and integrate it over Žy`, `. to obtain,

Ž x, t . dx 2k 3

`

ny 1

Ž ­x u 2 . y`

Ž 4 y 2 2y n . H

2k 3

`

Hy` Ž ­ u x

2aT 3

2 ny 1

`

2

n 2 n y1

dx

dx q 2 L d 2n

.

Hy` 2 u

2

`

2

2a 2 3k

n

x

n 2 n y1

Hy` 2 u

u x dx q

max u 2

H Ž ¨ , u, u . dx.

From integration by parts and Lemma 1, 2aT 3

`

n 2 n y1

Hy` 2 u

F F

2aT 3 2aT 3 q

F

Ž 2 n y 1. 2 n

3

3

uux u 2

žH

< u
Ž 2 n y 1 . 2 ny1d 2

2aT

2aT

u x dx

n

y2

ž

`

Hy` u

y2

y`

ž

k aT2

H< u <)d `

Hy` u

ny 1

uu 2 ­x u 2

`

2n

dx q

dx q

2 x

`

Ž 2 n y 1 . 2 ndy1 2 1yn H

Ž 2 n y 1. 2 n

n

Hy` Ž ­ u x

2 ny 1

2

.

1 n q d 2 y2 2

2

dx

ny 1

dx q `

Hy` u

uux u 2

2 x

n

y2

dx

/

/

dx

aT k

max u 2

dx q

n

x

L 2

d2

n

/

,

488

DEHUA WANG

By Lemma 2, we have the estimate `

n 2 n y1

Hy` 2 u s

2 3

H Ž ¨ , u, u . dx

2n

`

Hy` u

y 2n F

2 3

2n

2 n y1

`

2n

Hy` u

H< u
2 q 2n 3

ž

2t 9 y t 2tt 9

1

u2 y

Ž PŽ¨ . y aT Ž¨ y ¨ ..

t9

/

dx

dx

2 n y1

ž

2t 9 y t 2tt 9

2 n y1

H< u <)d u `

F C2 nd 2

ny 1

Hy` u

F C2 nd 2

ny 1

Hy` u

`

ž

u2 y

2t 9 y t

1

t9

u2 y

2tt 9

`

2n

Ž PŽ¨ . y aT Ž¨ y ¨ .. 1

t9

dx

Ž PŽ¨ . y aT Ž¨ y ¨ ..

2 ny 1

q C2 n

2 ny 1

q C2 n max u 2 q C2 n

Hy` u

/

/

dx

dx

n

x

ž

2

`

Hy` u

2 ny 1

/

.

For any « ) 0, n

max u 2 s max x

x

F«

x

Hy` 2 u

`

Hy` Ž ­ u

F«

2 ny 1

2 ny 1

x

`

Hy`

Ž ­x u 2

ny 1

2

dx q «y1

2

dx q

. .

Ž j , t . ­x u 2

1 2

ny 1

Ž j , t . dj

`

Hy` u

2n

dx

max u 2 q 2y1«y2 n

x

2

`

žH

y`

u2

ny 1

/

dx .

Then max u x

2n

F 2«

`

Hy` Ž ­ u

2 ny 1

x

2

.

y2

dx q «

`

žH

y`

2

u

2 ny 1

/

dx .

Choosing « small enough, we obtain d

`

H dt y`

n

u 2 Ž x, t . dx F C2 6 n

ž

`

Hy`

q C2 2 nd 2

2

u2 n

ny 1

y2

dx `

q C2 nd 2

/

Hy` u

2 x

ny 1

`

Hy` u n

dx q C2 2 nd 2 .

2 ny 1

dx

489

SOME DISSIPATIVE FLOWS

Integrate it over w0, t x and use the initial condition and Lemma 1 to get `

Hy`

n

n

u 2 Ž x, t . dx F Cd 2 q C2 6 n q C2 nd 2

ny 1

t

`

ž

H0 Hy` t

`

H0 Hy` u

2

u2

ny 1

2 ny 1

Ž x, s . dx

/

ds n

Ž x, s . dxds q C2 2 nd 2 .

` Define A n s max 0 F t F L Hy` u 2 Ž x, t . dx, n s 1, 2, 3, . . . . Then n

ny 1

n

A n F C2 6 nA2ny1 q C2 nd 2 A ny1 q C2 2 nd 2 . By induction on n, there is a constant L ) 0, such that A n F n 2y6 nLy1r2 Ž L d . 2 . Then 5 u 5 2 n F L d . Letting n ª `, < u Ž x, t .< F L d for any Ž x, t . g R = w0, L x. For d small enough, < u < F L d - Tr2. This completes the proof of Lemma 3.

4. PROOF OF THEOREM 1 Recall that, by applying the differential operators Ž6. to Ž5., we obtain Eq. Ž14.. We now set w s r x and v s s x . Then u x s Ž w q v .r2, ¨ x s p9Ž ¨ .y1 r2 Ž w y v .r2, and ¨ ) s v , ¨ 9 s w. Taking the partial derivative with respect to x in Ž14., one has 1 1 y1 y1 w) s y p9 Ž ¨ . p0 Ž ¨ . ¨ )w q p9 Ž ¨ . p0 Ž ¨ . w 2 4 4 y

v9 s y

E 2¨ 1 4

2

p9 Ž ¨ .

p9 Ž ¨ . E

y1

y1 r2

Žw y v. y

p0 Ž ¨ . ¨ 9v q y1 r2

1 4

wqv 2t

p9 Ž ¨ .

y1

q 1 y ¨ y1 D Ž x . q aux x , p0 Ž ¨ . v 2

wqv

q 1 y ¨ y1 D Ž x . q aux x . 2t 2¨ Multiplying the above by the integrating factor p9Ž ¨ .1r4 , we get 1 2 Ž p9 Ž ¨ . 1r4 w . ) s 4 p9 Ž ¨ . y5r4 p0 Ž ¨ . Ž p9 Ž ¨ . 1r4 w . y

2

p9 Ž ¨ .

y y

Žw y v. y

E 2¨ 2

p9 Ž ¨ .

p9 Ž ¨ .

1r4

q a p9 Ž ¨ .

y1 r4

Žw y v.

Žw q v.

2t 1r4

ux x ,

q p9 Ž ¨ .

1r4

Ž 1 y ¨ y1 D Ž x . .

490

DEHUA WANG

1

Ž p9 Ž ¨ . 1r4 v . 9 s 4 p9 Ž ¨ . y5r4 p0 Ž ¨ . Ž p9 Ž ¨ . 1r4 v . y y

E 2¨ 2

p9 Ž ¨ .

p9 Ž ¨ .

1r4

y1 r4

Žw y v.

Žw q v.

q p9 Ž ¨ .

2t

q a p9 Ž ¨ .

1r4

2

1r4

Ž 1 y ¨ y1 D Ž x . .

ux x .

From Ž7.,

ux x s

3

a

3

H Ž ¨ , u, u . 2k a 3 1r2 s u) q p9 Ž ¨ . ux y Ž T q u . u x y H Ž ¨ , u, u . 2k 2k k 2k 3 3 a 3 1r2 s u9 y p9 Ž ¨ . ux y Ž T q u . u x y H Ž ¨ , u, u . , 2k 2k k 2k then by Ž15., one has 3 1r4 p9 Ž ¨ . ux x s Ž k p9 Ž ¨ . 1r4u q p9 Ž ¨ . 3r4 x . ) 2k 2 3 y3 r4 1r2 y p9 Ž ¨ . p0 Ž ¨ . Ž ku q 3 p9 Ž ¨ . x . v 8k 2 a 1 1r4 1r4 y Ž T q u . p9 Ž ¨ . u x y p9 Ž ¨ . k k 2k 3

ut y

ŽT q u . ux y

k 3

=

ž

y

u2

y

t 3 2k 2

1

t9

p9 Ž ¨ .

ž

u2 2

3r4

q PŽ¨ . q

3 2

u y aT Ž¨ y ¨ .

u pŽ ¨ . q a Ž u q T . .

žŽ

x

x

u

Hy` ¨ Edj y t 9

q q 3

3 2k

2

p9 Ž ¨ .

5r4

1r4

ž

1 2

u2 q P Ž ¨ . q 3r4

3 2

u

/

/

Ž k p9 Ž ¨ . u y p9 Ž ¨ . x . 9 2k 2 3 y3 r4 1r2 y p9 Ž ¨ . p0 Ž ¨ . Ž ku y 3 p9 Ž ¨ . x . w 8k 2

s

//

491

SOME DISSIPATIVE FLOWS

y = q

a k

ž

1r4

Ž T q u . p9 Ž ¨ .

u2

1

y

t 3

2k 2

u2

ž

t9

p9 Ž ¨ .

q

3 2k

2

p9 Ž ¨ .

p9 Ž ¨ . 3 2

1r4

u y aT Ž¨ y ¨ .

//

u pŽ ¨ . q a Ž u q T . .

x

u

x

k

žŽ

Hy` ¨ Edj y t 9

q

1

q PŽ¨ . q

2

3r4

ux y

5r4

1

ž

2

/

u2 q P Ž ¨ . q

3 2

u .

/

Introduce two functions f s p9 Ž ¨ .

1r4

g s p9 Ž ¨ .

1r4

wy

vy

3a 2k 2 3a 2k 2

p9 Ž ¨ .

1r4

Ž ku q p9 Ž ¨ . 1r2 x . ,

p9 Ž ¨ .

1r4

Ž ku y p9 Ž ¨ . 1r2 x . ,

then w s p9 Ž ¨ .

y1 r4

v s p9 Ž ¨ .

y1 r4

ž ž

fq gq

3a 2k 2 3a 2k 2

p9 Ž ¨ .

1r4

Ž ku q p9 Ž ¨ . 1r2 x .

,

p9 Ž ¨ .

1r4

Ž ku q p9 Ž ¨ . 1r2 x

,

/ ./

and u x s p9 Ž ¨ .

y1 r4

fqg 2

q

3a 2k

u.

Combine the above calculations to get f) s

1 4 y y y

p9 Ž ¨ . E 2¨ 2 3a 8k 2

ž

y5 r4

p9 Ž ¨ .

p0 Ž ¨ . f 2 q y1 r2

p9 Ž ¨ .

y1

a 2ŽT q u . 2k

3a 4k 2

p9 Ž ¨ .

y1

p0 Ž ¨ . Ž ku q p9 Ž ¨ .

Ž f y g.

p0 Ž ¨ . Ž ku q 3 p9 Ž ¨ . q

1 2t

/

Ž f q g.

1r2

x.g

1r2

x.f

492

DEHUA WANG

q y y = y q g9 s

1

9a 2 4k 3a

4

2 kt 3a 3 2k

2

u2

ž

3a 2k 2 3a 2

p9 Ž ¨ .

4 y y

E

p9 Ž ¨ .

1r4

p9 Ž ¨ .

1r4

1

8k 2

p0 Ž ¨ . x 2 y

u q p9 Ž ¨ .

3r4

p9 Ž ¨ .

5r4

y5 r4

ž ž

y1 r2

y1

1 2

a

p9 Ž ¨ .

k 3 2

1r4

u y aT Ž¨ y ¨ .

u2 q P Ž ¨ . q 3a 4k 2

x

Ž 1 y ¨ y1 D Ž x . .

3 2

p9 Ž ¨ .

x

// u

x

Hy` ¨ Edj y t 9

/

u ,

/

y1

p0 Ž ¨ . Ž ku q 3 p9 Ž ¨ . q

2k

9a 2

1r4

p0 Ž ¨ . Ž ku q p9 Ž ¨ .

1r2

x.g

Ž f y g.

a 2ŽT q u .

ž

p9 Ž ¨ .

uŽ pŽ ¨ . q a Ž u q T . . q

p0 Ž ¨ . g 2 q

p9 Ž ¨ .

2k 2¨ 2

q PŽ¨ . q

2

p9 Ž ¨ .

1r4

3a E

ŽT q u .u y

u2

ž

t9

p9 Ž ¨ .

2¨ 2 3a

y

1r4

y

t

2k

p9 Ž ¨ .

1 2t

/

1r2

x.f

Ž f q g.

1r4

p9 Ž ¨ . p0 Ž ¨ . x 2 4k 4 3a E 3a 1r4 1r4 1r4 y p9 Ž ¨ . x y p9 Ž ¨ . u q p9 Ž ¨ . Ž 1 y ¨ y1 D Ž x . . 2 2 2 kt 2k ¨ q

y = q q

3a 3 2k

ž

2

u2

t 3a

2k

2

3a 2k

2

p9 Ž ¨ . y

1

t9

1r4

ž

ŽT q u .u y

u2 2

p9 Ž ¨ .

3r4

p9 Ž ¨ .

5r4

q PŽ¨ . q

a k 3 2

p9 Ž ¨ .

1r4

u y aT Ž¨ y ¨ .

u pŽ ¨ . q a Ž u q T . . q

žŽ ž 1 2

u2 q P Ž ¨ . q

3 2

u .

/

x

// u

x

Hy` ¨ Edj y t 9

/

493

SOME DISSIPATIVE FLOWS

Define Lipschitz functions on w0, L x F Ž t . s max < f Ž x, t . < ,

G Ž t . s max < g Ž x, t . < ,

x

x

"

"

F Ž t . s max Ž "f Ž x, t . . ,

G Ž t . s max Ž "g Ž x, t . . .

x

x

Clearly, F Ž t . F FqŽ t . q FyŽ t ., GŽ t . F GqŽ t . q GyŽ t .. Fix t g w0, L x, choose ˆ y, ˇ y g R, such that FyŽ t . s yf Ž ˆ y, t ., GyŽ t . s yg Ž ˇ y, t ., then similar to Ž18., we have d dt

d

Fy Ž t . F yf ) Ž ˆ y, t . ,

dt

Gy Ž t . F yg9 Ž ˇ y, t . ,

and by Ž19., Ž20., and Lemmas 1]3, d dt

Ž Fy Ž t . q Gy Ž t . . F C Ž Fy Ž t . q Gy Ž t . . q C Ž Fq Ž t . q Gq Ž t . . q C,

then, integrate it to get Fy Ž t . q Gy Ž t . F C

t

q

Ž s . q Gq Ž s . . ds q C Ž Fy Ž 0 . q Gy Ž 0 . . q C

t

q

Ž s . q Gq Ž s . . ds q L K 1 q L .

H0 Ž F

FC

H0 Ž F

Fix t g w0, L x, choose ˆ x, ˇ x g R, such that FqŽ t . s f Ž ˆ x, t ., GqŽ t . s gŽ ˇ x, t ., then d dt

d

Fq Ž t . G f ) Ž ˆ x, t . ,

dt

Gq Ž t . G g 9 Ž ˇ x, t . ,

and similarly, by Ž19., Ž20., and Lemmas 1]3, we have d dt

2 Ž Fq Ž t . q Gq Ž t . . G m Ž Fq Ž t . q Gq Ž t . . y L Ž Fq Ž t . q Gq Ž t . .

y L Ž Fy Ž t . q Gy Ž t . . y L 2

G m Ž Fq Ž t . q Gq Ž t . . y L Ž Fq Ž t . q Gq Ž t . . yL

t

H0 Ž F

q

Ž s . q Gq Ž s . . ds y L K 1 y L , Ž 21 .

with m ) 0 a constant. By Ž13., FqŽ 0. q GqŽ 0. G L K 2 y L. Then for sufficiently large K 2 , from Ž21., one can check that FqŽ t . q GqŽ t . will

494

DEHUA WANG

blow up at a finite time which is less than L. The proof of Theorem 1 is now completed.

ACKNOWLEDGMENT I thank the referee for careful reading of the manuscript and valuable comments.

REFERENCES 1. S. Alinhac, ‘‘Blowup for Nonlinear Hyperbolic Equations,’’ Birkhauser, Boston, 1995. ¨ 2. J. Bebernes and D. Eberly, ‘‘Mathematical Problems from Combustion Theory,’’ Appl. Math. Sci., Vol. 83, Springer-Verlag, New York, 1989. 3. G.-Q. Chen, J. Jerome, and B. Zhang, Existence and the singular relaxation limit for the inviscid hydrodynamic energy model, in ‘‘Proceedings of Workshop on Modeling and Computation for Applications in Science and Engineering,’’ Oxford Univ. Press, London, in press. 4. C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44 Ž1986., 463]474. 5. H. Hattori, Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations, Quart. Appl. Math. 40 Ž1982., 113]127. 6. W. J. Hrusa and S. A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111 Ž1990., 135]151. 7. L. Hsiao, Nonlinear system of conservation laws with dissipation, in ‘‘Nonlinear Variational Problems and Partial Differential Equations,’’ Longman, HarlowrNew York, 1995. 8. F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 Ž1974., 377]405. 9. W. Kosinski, Gradient catastrophe in the solutions of nonconservative hyperbolic systems, J. Math. Anal. Appl. 61 Ž1977., 672]688. 10. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva, ‘‘Linear and quasilinear ˇ equations of parabolic type,’’ in Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, 1968. 11. P. D. Lax, Development of singularities in solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5 Ž1964., 611]613. 12. A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 Ž1983., 445]464. 13. M. Rudan and F. Odeh, Multi-dimensional discretization scheme for the hydrodynamic model of semiconductor devices, COMPEL 5 Ž1986., 149]183. 14. M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional, nonlinear thermoelasticity, Arch. Rational Mech. Anal. 76 Ž1981., 97]133.