On the Initial-Boundary Value Problem for the Bipolar Hydrodynamic Model for Semiconductors

2 . 1&max[l, l*]

It follows from duality that &A 2 & W &1, p 2 (0) C(h l&12 +h l*+(1#: )&1 +h l*+(1#; )&1 ) Ä 0, 1

2 . 1+max[l, l*]

h Ä 0,

272

KAI-JUN ZHANG

Then, A 2 is compact in W &1, p2 (0). Thus, we obtain A+R+B+7=A 1 +A 2 +R+B+7 is compact in W &1, p(0), where 11. That is, A+R+B+7 is bounded in W &1, r(0), r>1. So it follows from Lemma 3.4 that A+R+B+7 is compact in H &1 loc (0). This means, '(v) t +q(v) x &S is compact in H &1 loc (0). By the arguments above, we have our desired result. This completes the proof of Theorem 3.1. K Incorporating Theorem 2.1 with Theorem 3.1, we have the following framework theorem of the approximate solutions v h defined in Section 2. Theorem 3.2. Suppose that the initial data (: 0(x), m 0(x), ; 0(x), n 0(x)), # : , # ; > 53 and given functions D(x) satisfy the conditions 0: 0(x), ; 0(x)M 1 , |m 0(x)| M 2 : 0(x), |D(x)| M 3 .

: 0(x), ; 0(x)0, |n 0(x)| M 2 ; 0(x),

THE BIPOLAR HYDRODYNAMIC MODEL

273

Then, the approximate solutions v h satisfy the following estimates and compactness: (1)

There is a constant C(T )>0 such that 0: h(x, t), ; h(x, t)C, |m h(x, t)| C: h(x, t), |n h(x, t)| C; h(x, t),

(x, t) # I T .

(2) For every domain 0/I T and every weak entropy pair (', q), the sequence of enrtopy dissipation measures '(v h ) t +q(v h ) x is compact in H &1 loc (0). Following [4, 10, 11, 30, 31] we have the compactness framework theorem needed in the paper: Theorem 3.3. Assume that approximate solutions v h(x, t)=(: h(x, t), m h(x, t), ; h(x, t), n h(x, t)) satisfy the following framework: (1)

Let 0: h , ; h C, |m h | C: h ,

|n h | C; h , a.e.

for a positive constant C. (2) The sequence of entropy dissipation measures '(v h ) t +q(v h ) x is compact in H &1 loc (0) for every weak entropy pair (', q) and every open bounded set 0/R 2+ . Then, for # : , # ; > 53 , there exists a convergent subsequence, still labeled v h , such that (: h(x, t), m h(x, t), ; h(x, t), n h(x, t)) Ä (:(x, t), m(x, t), ;(x, t), n(x, t)).

4. THE GLOBAL EXISTENCE AND BOUNDARY CONDITION OF WEAK SOLUTIONS In this section, we will prove the convergence of a sequence of the approximate solutions v h derived by the modified Godunov scheme, and

274

KAI-JUN ZHANG

imply the global existence of weak entropy solutions of (1.16)(1.19). At the same time we prove the weak solutions to satisfy the boundary conditions in the sense of traces. Theorem 4.1. Suppose that the conditions of Theorem 3.2 hold. Then, (1) The sequence of the approximate solutions v h =(: h , m h , ; h , n h ) has a convergent subsequence, still labeled v h , such that (: h(x, t), m h(x, t), ; h(x, t), n h(x, t)) Ä (:(x, t), m(x, t), ;(x, t), n(x, t))

a.e.,

(4.1)

and there is a constant C(T )>0 such that 0:(x, t), ;(x, t)C, |m(x, t)| C:(x, t), |n(x, t)| C;(x, t)

(4.2) a.e.

(2) The bounded measurable function v(x, t)=(:(x, t), m(x, t), ;(x, t), n(x, t)) is an entropy weak solution of (1.16)(1.19), i.e., (:(x, t), m(x, t), ;(x, t), n(x, t)) satisfies (1.20)(1.23) and (1.25). Proof. From Theorem 3.3, we obtain a convergent subsequence, still labeled v h , such that (: h(x, t), m h(x, t), ; h(x, t), n h(x, t)) Ä (:(x, t), m(x, t), ;(x, t), n(x, t))

a.e.

Clearly, 0:(x, t), ;(x, t)C, |m(x, t)| C:(x, t), and |n(x, t)| C;(x, t) a.e. For every function  # C (I T ) satisfying (x, T )=0, and (0, t)= (1, t)=0, we consider the integral identity T

| | 0

1

(: h  t +m h  x ) dx dt+

|

: h  dx=A :()+R :(),

(4.3)

t=0

0

where A :()=: i, j

R :()=: i, j

| (: &: )  dx, i h

i j

i

ti+1

| | (m &m )  h

ti

h

x

(4.4) dx dt.

(4.5)

275

THE BIPOLAR HYDRODYNAMIC MODEL

For 53 <# : , # ; 2, using Holder's inequality, (3.1), and the fact that : ij is the average value of : h on cell (x j&1 , x j ), we have

} | (: &: )( & ) dx }  : | | & | dx \ + \: | |: &: | dx+ h && \ : | |x&x | dx+ \ : | |: &: | dx+

|A :()| = :

i h

i j

i

i j

i, j

12

i

12

i 2 j

i h

i, j

i 2 j

i, j

12

12

C1

i h

j

i, j

Ch

12

12

i 2 j

i, j

&& C 1 Ä 0,

h Ä 0.

as

(4.6)

For # : , # ; >2, by (3.13), we have

}

| (: &: )( & ) dx }

|A :()| = :

i h

i, j

: i, j

i j

i

i j

| | & | |: &: | dx i

i j

h && C 1 : i, j

i h

i j

| |: &: | dx i h

i j

Ch 1#: && C 1 Ä 0,

as h Ä 0.

(4.7)

It follows from the uniform bound of v :h that

}

R :()= : i, j

: i, j

ti+1

| | (m &m )  h

h

x

ti+1

| |_ ti

i, j

: i, j

ti+1

| | \ |m ti+1

| | \C$ ti

1 k {

\ +

}|

t

e &(t&s){

ti

|

1

0

G x(: h(!, s) 

}& | | dx dt x

| (1&e &t&ti { )+: h 

h

ti

C$ 1+

}

|m h | (1&e &(t&ti ){ )+: h  

&; h(!, s)&D(!)) d! ds  :

dx dt

ti

|

t

+

e &(t&s){ ds C$ | x | dx dt

ti

|t&t i | +C${(1&e &(t&ti ){ ) | x | dx dt {

+

T

| | 0

Ch && C 1 Ä 0,

1

| x | dx dt 0

as

h Ä 0.

(4.8)

276

KAI-JUN ZHANG

Then, (4.6)(4.8) imply that lim

hÄ0

_|

T

|

0

1

(: h  t +m h  x ) dx dt+

|

&

: h  dx =0.

t=0

0

(4.9)

Using the dominated convergence theorem in (4.9), we have T

| | 0

1

(: t +m x ) dx dt+

|

: 0(x)  dx=0.

(4.10)

t=0

0

For every function  # C 1(I T ) satisfying (0, t)=(1, t)=0 for t0 and (x, T)=0 for 0x1, we consider the integral identity T

| | 0

1 0

(m h  t + f 2(v :h )  x +V :2(v :h ) ) dx dt+

|

m h  dx

t=0

=A :()+R :(),

(4.11)

where f :2(v : )=m 2:+: #: # : and V :2(v : )=:  10 G x (:&;&D) d!& m{ , A :()=: i, j

ti+1

xj

ti

xj&1

| (m &m )  dx+: | | i h

i j

i

i, j

V :2(v h )  dx dt, 

(4.12)

and R :()=: i, j

ti+1

| | ti

xj xj&1

[(m h &m h )  t +( f :2(v :h )& f :2(v :h ))  x  

+(V :2(v :h )&V :2(v :h )) ] dx dt. 

(4.13)

In terms of the uniform bound of v :h , and |m h &m h | ( k{ |m h | +C$k |: h |),    we have ti+1

xj

||

_ | | + | | |{f (v +%(v &v ))| + { & |m | _  +C$: k dx dt \{ +

|R :()| :

i, j

| | ti

t

x

xj&1

: 2

: h

: h

: h

h

h

Ch && C 1 Ä 0, where % # [0, 1].

as

h Ä 0,

(4.14)

277

THE BIPOLAR HYDRODYNAMIC MODEL

We decompose A :() into three parts,

{

A :()= : i, j

{

= {

|

(m ih &m ij )( i & ij ) dx + :  i, j ti+1

| | (V (v )&V (v

+ :

: 2

ti

i, j

: h

: 2

:i h

ti+1

| | V (v )(& ) dx dt= : 2

ti

))  ij dx dt

: h

i j

=

#A :1()+A :2()+A :3().

(4.15)

For A :1() and A :2(), by Lemma 3.1, we have, for 53 <# : 2,

}

|A :1()| = : i, j

| (q +m &m )( & ) dx } i h

i

i j

i

i j

12

12

+ _\ + + : | (m &m ) dx \ + & C$h && \N : | |x&x | dx+ (C$ &q & C$ :

i, j

| ( & ) i

i 2 j

:

i, j

|q

2 i

dx

12

1

i, j

dx

i h

0

i 2 j

12

12

C1

j

0

i



+C$)

j

Ch 12 && C 1 Ä 0,

as h Ä 0,

(4.16)

where q i = m ih (1&e &k{ ) & : ih  tti e &ti &s {  10 G x ( : h(!, s) & ; h(!, s) & i&1     D(!)) d! ds. For # : >2, from Lemma 3.6, we have

}

|A :1()| = : i, j

: i, j

| (q +m &m )( & ) dx } i

i h

i j

i

i j

| \ |2 | + |m &m | ) | & | dx i h

i

i j

i

i j

h && C 1 (C$+C$h (1#: )&1 ) (hC$+h 1#: ) && C 1 Ä 0,

as

h Ä 0,

(4.17)

where q i =m ih (1&e &k{ )&: ih  

|

ti ti&1

e &ti &s{

|

1

0

G x (: h(!, s)&; h(!, s)&D(!)) d! ds,  

278

KAI-JUN ZHANG

and |A :2()| : i, j

||

| i & ij |

\ |x&x |

|V 2(v :h )| 

h+

j

Ch && C 1 Ä 0,

|(x, t)& i | k dx dt |t&t i |

+

as h Ä 0.

(4.18)

By Lemma 2.4, Lemma 2.5, and Lemma 2.1, we have

}

|A :3()| = :  ij i, j

|| (V (v )&V (v : 2

C &&  : i, j

C &&  : i, j

C &&  : i, j

: h

: 2

:i h

)) dx dt

|| ( |: &:

:i h

|| |: &:

| dx dt

h

h

|

ti+1

i h

}

| + |m h &m ih | ) dx dt  

=h dt

(by Lemma 2.4)

(by Lemma 2.5)

ti

C= &&  ,

(4.19)

where =>0 is an arbitrarily small constant. It follows from (4.14)(4.19) that lim

hÄ0

_|

T 0

|

1 0

(m h  t + f :2(v :h )  x +V :2(v :h ) ) dx dt+

|

&

m h  dx =0.

t=0

(4.20) Using the dominated convergence theorem, we have T

| | 0

1 0

\

T

| |

1

+

0

0

m 2 : #:  x dx dt + : #:

\ \: |

m t +

+ +

1

0

G x (:&;&D) d!&

m  dx dt+ {

+

|

m 0(x)  dx=0. t=0

(4.21)

We can similarly get (1.22)(1.23). For every weak and convex entropy pair (', q) and every nonnegative smooth function  that has compact support in region I T , we consider the integral identity T

| | 0

1

('(v h )  t +q(v h )  x ) dx dt=A( )+R( )+7( )+S( ), 0

where A( ), R( ), 7( ), and S( ) are similar to those of (3.15).

(4.22)

279

THE BIPOLAR HYDRODYNAMIC MODEL

Since (', q) is a convex entropy pair and  0, as [51], we have 7( )0,

(4.23)

| (v &v ) { '(! )(v &v ) dx +: | ('(v )&'(v ))( & ) dx

A( )=:  ij

i h

i T j

2

i j

i j

i

i h

i j

i, j

i h

i j

i, j

: i, j



{

| ('(v )&'(v ))( & ) dx i h

i j

i

i j

1
&Ch l&12 && C l0 ,

5 <# : , # ; 2 , 3

&C(h l&12 +h l+(1#: )&1 +h l+(1#; )&1 ) && C l0 ,

{

max 1&

1 1 , 1&
=

# : , # ; >2. (4.24)

As in (3.28), we have S()&Ch && H 10 .

(4.25)

Using the fact that : h(x, t)=: h(x, t),  ; h(x, t)=; h(x, t),  and m h(x, t)=m h(x, t) e &t&ti { +: h   _

|

e &t&s{

ti

G x(: h(!, s)&; h(!, s)&D(!)) d! ds,  

n h(x, t)=n h(x, t) e &t&ti { &; h  

|

t

1

0

_

|

|

t

t i&1 t
e &t&s{

ti

1

0

G x(: h(!, s)&; h(!, s)&D(!)) d! ds,  

t i&1 t
280

KAI-JUN ZHANG

we obtain 1 |V 2(v ih )&V 2(v ih )|  m ih (e &ti &ti&1 { &1)+: ih {   

}

|

|

ti

e &ti &s{

ti&1

1

G x (: h(!, s)&; h(!, s)&D(!)) d! ds  

_

0

1 + n ih (e &(ti &ti&1 {) &1)&; ih {  

}

|

|

ti

e &ti &s{

ti&1

1

G x(: h(!, s)&; h(!, s)&D(!)) d! ds  

_

0

k (|m ih | + |n ih |) 2 +(: ih +; ih ) {    

|

}

ti

}

e &ti &s{ ds

ti&1

C$ Ch. {

Then,

||

R( )=:

1

0

i, j

{'(v ih +%(v ih &v ih ))(v ih &v ih ) d%  i dx  

| \|

=:

1

0

i, j

|

1

_

|

0 1 0 1

_

ti

e &ti &s{

ti&1

&

0

_

' n(v ih +%(v ih &v ih )) d% } n ih (1&e &k{ )+; ih   

i

_

|

ti

| \| 0

1 0

e &ti &s{

ti&1

' m(v ih +%(v ih &v ih )) d% 

e &ti &s{

ti&1

1

|

0

G x (: h(!, s)&; h(!, s)&D(!)) d! ds  

& +

&m ih (1&e &k{ )  i dx+: } ; ih 

ti

& +

1

} : ih 

|

G x(: h(!, s)&; h(!, s)&D(!)) d! ds  i dx  

&Ch&:

_

|

G x(: h(!, s)&; h(!, s)&D(!)) d! ds  i  

+

|

_

' m(v ih +%(v ih &v ih )) d% } m ih (1&e &k{ )&: ih   

|

ti ti&1

e &ti &s{

|

i

1

| \| 0

1

0

' n(v ih +%(v ih &v ih )) d% 

1

0

& +

G x(: h(!, s)&; h(!, s)&D(!)) d! ds  

+n ih (1&e &k{ )  i dx.

(4.26)

281

THE BIPOLAR HYDRODYNAMIC MODEL

It follows from (4.23)(4.26) that T

| | 0

1

1

| \|

('(v h )  t +q(v h )  x ) dx dt+: 0

0

i

_

} : ih 

|

ti

e &ti &s{

ti&1

|

_

|

e &ti &s{

ti&1

' m(v ih +%(v ih &v ih )) d% 

G x (: h(!, s)&; h(!, s)&D(!)) d! ds  

0

&

ti

0

1

&m ih (1&e &k{ )  i & } ; ih 

1

|

|

1 0

' n(v ih +%(v ih &v ih )) d% 

1

G x (: h(!, s)&; h(!, s)&D(!)) d! ds  

0

& +

+n ih (1&e &k{ )  i dx



{

&Ch l&12(&& C l0 +h 32&l(1+&& H 10 )), 1 5 2. (4.27) max 1& , 1& #: #;

{

=

As [51], letting h Ä 0 in (4.27) and using the fact that v h Ä v a.e., we obtain the entropy condition T

| | 0

T

1

('(v)  t +q(v)  x ) dx dt+ 0

| | 0

1

{'(v) V 2(v)  dx dt0,

(4.28)

0

where

\

V 2(v)= 0, :

|

1

G x (:&;&D) d!&

0

m , 0, &; {

|

1

G x (:&;&D) d!&

0

n . {

+

This completes the proof of Theorem 4.1. K Now we turn to the boundary conditions of weak solutions. Let v(x, t)=(:(x, t), m(x, t), ;(x, t), n(x, t)) be a weak solution of (1.16)(1.19) obtained in Theorem 4.1. We introduce the generalized function B: C 10(R 2 ) Ä R 2 as follows: for  # C 10(R 2 ), T

| |

1

B()=&

0

0

(v t + f (v)  x +F) dx dt,

282

KAI-JUN ZHANG

where m 2 : #: n 2 ; #; + , n, + : #: ; #;

\ F= 0, : | \

f (v)= m,

1

0

T

+,

m G x (:&;&D) d!& , 0, &; {

|

1

G x (:&;&D) d!& 0

n {

T

+.

We take ` 0 , ` T , ' 0 , ' 1 # C 10(R) with ` 0(0)=1,

` 0(T)=0;

` T (0)=0,

` T (T)=1;

' 0(0)=1,

' 0(1)=0;

' 1(0)=0,

' 1(1)=1.

For any / # C 10(R), we define the generalized functions, v C( } , 0)(/)=B(/ } ` 0 )&/(0) B(' 0 } ` 0 )&/(1) B(' 1 } ` 0 ), v C( } , T )(/)=&B(/ } ` T )+/(0) B(' 0 } ` T )+/(1) B(' 1 } ` T ), f C(v)(0, } )(/)=B(' 0 } /), f C(v)(1, } )(/)=&B(' 1 } /), where (/ } ` 0 )(x, t)=/(x) ` 0(t) and so on mean the tensor product. Then we can define the trace of v along the segments (0, 1)_[0] and (0, 1)_[T], and the trace of f (v) along the segments [0]_(0, T ) and [1]_(0, T ), respectively, as v C( } , 0), v C( } , T ), f C(v)(0, } ), f C(v)(1, } ). Similarly, for any t # (0, T ), we also can define v C( } , t) as the trace of v along the segment (0, 1)_[t]. For any x # (0, 1), define f C(v)(x, } ) as the trace of f (v) along the segment [x]_(0, T ). It is similar to [19], we have Lemma 4.1. Let v be a weak solution of (1.16)(1.19). Then v C( } , 0)| (0, 1) , v C( } , T)| (0, 1) # L  loc(0, 1); f C(v)(0, } )| (0, T ) , f C(v)(1, } )| (0, T ) # L  loc(0, T), and for any  # C 10(R 2 ), T

| | 0

1

(v t + f (v)  x +F) dx dt 0

=

|

1

v C(x, T ) (x, T) dx&

0

|

T

+

0

|

f C(v)(1, t) (1, t)&

1

v C(x, 0) (x, 0) dx

0

|

T

0

f C(v)(0, t) (0, t) dt.

(4.29)

283

THE BIPOLAR HYDRODYNAMIC MODEL

Theorem 4.2. Let v h(x, t)=(: h(x, t), m h(x, t), ; h(x, t), n h(x, t)) be the approximate solutions of (1.16)(1.19) constructed in Section 2 and v(x, t)= (:(x, t), m(x, t), ;(x, t), n(x, t)) is the limit function of v h as h Ä 0. Then v(x, t) satisfies the initial-boundary conditions, m C(0, t)=m C(1, t)=0,

t # (0, T )

n C(0, t)=n C(1, t)=0,

t # (0, T )

v C(x, 0)=v o(x),

x # (0, 1)

(4.30) (4.31)

Proof. From the proof of Theorem 4.1, is easy to get, for any  # C 10(R 2 ), that T

lim

hÄ0

_| | 0

1

(: h  t +m h  x ) dx dt+

|

: h  dx&

t=0

0

|

&

: h  dx =0,

t=T

which implies T

| | 0

1

(: t +m x ) dx dt+

|

: 0(x)  dx& lim

hÄ0

t=0

0

|

: h  dx=0.

t=T

(4.32)

Inserting (4.29) into (4.32), we have

|

1

: C(x, T ) (x, T ) dx&

0

|

1

: C(x, 0) (x, 0) dx

0

|

T

+

|

m C(1, t) (1, t)&

0

|

T

m C(0, t) (0, t) dt& lim

hÄ0

0

|

: h  dx t=T

1

+

\(x, 0) (x, 0) dx=0.

0

Take (x, t)=`(x) /(t) # C 10(R 2 ) with `, / # C 10(R), and /(0)=1, `(1)= `(0)=0, /(T)=0 in (4.33). We get

|

1

0

: C(x, 0) `(x) dx=

|

1

: 0(x) `(x) dx,

0

which implies : C(x, 0)=: 0(x) on (0, 1). Similarly, holds that m C(x, 0)=m 0(x), ; C(x, 0)=; 0(x), and n C(x, 0)= n 0(x) on (0, 1).

284

KAI-JUN ZHANG

Take (x, t)=`(x) /(t) # C 10(R 2 ), `, / # C 10(R), /(T )=/(0)=0, `(0)=1, `(1)=0 in (4.33), and one can get

|

T

m C(0, t) /(t) dt=0.

0

Thus m C(0, t)=0 on (0, T). It is similar to show that m C(1, t)=0, n C(1, t)=0, n C(0, t)=0 on (0, T ). This completes the proof of Theorem 4.2. K

5. RELAXATION LIMITS In this section we deal with the relaxation limit of entropy weak solutions of (1.16)(1.19) as { Ä 0. By making the change of the scale t= {s , we can transform (1.16)(1.19) into the scaled form : {s +m {x =0,

\

({ 2m { ) s + { 2 =: {

|

1

(m { ) 2 + p(: { ) :{

+

(5.1)

x

G x(: {(!, s)&; {(!, s)&D(!)) d!&m {,

(5.2)

0

; {s +m {x =0,

\

({ 2n { ) s + { 2 =&; {

(n { ) 2 + p(; { ) ;{

|

1

+

(5.3)

x

G x(: {(!, s)&; {(!, s)&D(!)) d!&n {,

(5.4)

0

where s

\ {+ , 1 s m (x, s)= m x, , { \ {+ : {(x, s)=: x, {

s

\ {+ , 1 s n (x, s)= n x, . { \ {+

; {(x, s)=; x, {

From [50], we have the following div-curl lemma.

(5.5)

(5.6)

285

THE BIPOLAR HYDRODYNAMIC MODEL

Lemma 5.1 (div-curl Lemma). Given two sequences [U { ] and [V { ] uniformly bounded in L 2loc and assume that [div U { ] and [curl V { ] belong to a bounded set of L 2loc independent of {. Then U { } V { ( U } V in D$, where U=w&lim U { and V=w&lim V {. Lemma 5.2. Let (: {, m {, ; {, n { ) be the entropy solutions of (5.1)(5.4). Then for any T>0, there exists a constant C=C(T) independent of { such that (n { ) 2(x, t) dx dtC(T ), ; {(x, t)

(5.7)

&{: {& L ([0, 1]_[0, T]) +&{; {& L ([0, 1]_[0, T]) C(T ),

(5.8)

&{ 3m {& L ([0, 1]_[0, T ]) +&{ 3n { & L ([0, 1]_[0, T]) C(T ),

(5.9)

T

| | 0

1

0

(m { ) 2 (x, t) dx dt+ : {(x, t)

T

| | 0

1 0

where { satisfies (1.8) Proof. It is easy to get (5.8) and (5.9) from Theorem 4.1. We prove (5.7) now. Consider the classical mechanical entropy-entropy flux pair '*=

: #: 1 n2 ; #; 1 m2 + + + , 2 : # :(# : &1) 2 ; # ;(# ; &1)

q*=

1 m3 1 1 n3 1 #: &1 + m+ + : ; #; &1n. 2 2 2: # : &1 2; # ; &1

From the entropy inequality, we have, for almost every t0, t

|

1

'*(x, t) dx 0

|

1

0

\

(m&n)

|

1

G x (:&;&D) d!& 0

m2 n2 (x, t) dx. & :{ ;{ (5.10)

+

Let (t)=

|

1

0

f (t)=

|

\

m2 n2 + (x, t) dx, : ;

+

1

'*(x, t) dx.

0

Then we have df  C$ - & , dt {

(5.11)

286

KAI-JUN ZHANG

which comes from the deduction

|

1

(m&n) 0



|

1

\|

1

G x (:&;&D) d! dx 0

m2 dx :

12

1

+ \ | \| n + | \ ; dx+ \| ; \| 0

1

0

2

2

1

:

0

12

1

0

0

1

0

12

+ + G (:&;&D) d! dx + +

G x (:&;&D) d!

dx 2

12

x

C$ - (t). Here we have used Lemma 2.6 and Theorem 2.1, and C$ is independent of {. Now let  ,(s)=

s

\{+ =|

{2

F(s)= f

1

0

\

(m { ) 2 (n { ) 2 + { (x, s) dx, :{ ;

+

s

\ {+ .

Then, in view of (5.11), we get dF C$ - ,&,, ds

(5.12)

which leads to F(t)+

|

T

,(s) dsF(0)+C$ 0

|

T

- ,(s) ds.

0

Thus we obtain, by F(t)0,

|

T

0

,(s) dsF(0)+C$ - T

\|

T

,(s) ds 0

+

12

.

That is,

|

T

0

,(s) ds 14 (C$ - T+- (C$) 2 T+4F(0)) 2.

This completes the proof of Lemma 5.2. K

THE BIPOLAR HYDRODYNAMIC MODEL

287

From Lemma 5.2, can be shown there exist :~, ;, p~, and q~ being in L ([0, 1]_[0, T]) such that p~ =w C &lim p({: { ),

q~ =w C &lim q({; { )

as { Ä 0,

(5.13)

:~ =w C &lim {: {,

; =w C &lim {; {

as { Ä 0,

(5.14)

Now we give a key result in this section. Lemma 5.3. Suppose # : , # ; 2. Then q~ =q( ; ),

p~ = p(:~ ), {: {p({: { ) ( :~p(:~ ), Proof.

{; {q({; { ) ( ;q(; ),

as

(5.15) { Ä 0 in D$,

(5.16)

For # : , # ; =2, we rewrite (5.1)(5.4) into the new form ({: { ) s +({m { ) x =0,

\

{ 3({m { ) s + { 4 ={: {

|

1

(m { ) 2 + p({: { ) :{

+

(5.17)

x

G x({: {(!, s)&{; {(!, s)&{D(!)) d!&{ 2m {,

(5.18)

0

({; { ) s +({n { ) x =0,

\

{ 3({n { ) s + { 4 =&{; {

(n { ) 2 +q({; { ) ;{

|

1

+

(5.19)

x

G x({: {(!, s)&{; {(!, s)&{D(!)) d!&{ 2n {.

(5.20)

0

We define U {: =[{: {, {m { ],

U {; =[{; {, {n { ],

(5.21)

and

{

V {: = &{ 4

(m { ) 2 & p({: { ), { 4m { , :{

=

{

V {; = &{ 4

(n { ) 2 &q({; { ), { 4n { . ;{ (5.22)

=

288

KAI-JUN ZHANG

From (5.1) and (5.3) we get div U {: =0 and div U {; =0 which lead to the boundedness of [div U {: ] and [div U {; ] in L 2loc . According to Lemma 5.2, one can get &{ 12m {& L2 ([0, 1]_[0, T ]) &{: {& 12 L ([0, 1]_[0, T])

"

(m { ) 2 :{

"

12 L1 ([0, 1]_[0, T])

independent of {,

C(T )

&{ 12n {& L2 ([0, 1]_[0, T]) &{; {& 12 L ([0, 1]_[0, T])

(n { ) 2 ;{

" "

(5.23) 12 L1 ([0, 1]_[0, T])

independent of {,

C(T )

(5.24)

which imply &U {: & L2 ([0, 1]_[0, T]) , &U {; & L2 ([0, 1]_[0, T]) C(T ),

(5.25)

and that there exist functions m~, n~ and :~, ; being in L 2([0, 1]_[0, T]) such that { 12m { ( m~,

{ 12n { ( n~

in

L 2([0, 1]_[0, T]),

(5.26)

{: { ( :~,

{; { ( ;

in

L 2([0, 1]_[0, T]).

(5.27)

As far as V {: and V {; are concerned, we have

"

&curl V {: & L2 ([0, 1]_[0, T]) = {: {

1

|

"

G x ({: { &{; { &{D) d!&{ 2m {

0

L2 ([0, 1]_[0, T])

C(T),

"

&curl V {; & L2 ([0, 1]_[0, T]) = {; {

|

1

"

G x ({: { &{; { &{D) d!+{ 2n {

0

(5.28)

L2 ([0, 1]_[0, T])

C(T). On the other hand, by means of Lemma 5.2 and Theorem 5.1, we get

"

{4

(m { ) 2 :{

"

"

{ { 2 L2 ([0, 1]_[0, T])

m{ :{

C(T ) {

"

L ([0, 1]_[0, T])

&{m {& L2 ([0, 1]_[0, T]) , (5.29)

THE BIPOLAR HYDRODYNAMIC MODEL

289

and

"

{4

(n { ) 2 ;{

"

L2 ([0, 1]_[0, T])

n{ ;{

" "

{ { 2

L ([0, 1]_[0, T])

&{n {& L2 ([0, 1]_[0, T]) ,

C(T ) {

(5.30)

which mean that &V {: & L2 ([0, 1]_[0, T]) , &V {; & L2 ([0, 1]_[0, T]) C(T ).

(5.31)

Thus, we can show V {: ( [/ : , 0],

V {; ( [/ ; , 0].

(5.32)

By (5.13), (5.29), and (5.30), we get / : =&p~, / ; =&q~. Making use of Lemma 5.1, we finally obtain U {: } V {: =&{: {p({: { ) ( U : } V : =&p~:~

in D$,

U {; } V {; =&{; {q({; { ) ( U ; } V ; =&q~;

in D$.

By Minty's monotonicity arguments [29] we obtain (5.15). For # : , # ; >2, (5.1)(5.4) can be transformed into the new form ({: { ) s +{ 12({ 12m { ) x =0,

\

{ 32+#: ({ 12m { ) s + { 2+#: ={ #: &2({: { )

|

1

(m { ) 2 + p({: { ) :{

+

x

G x({: {(!, s)&{; {(!, s)&{D(!)) d!&{ #: &12({ 12m { ),

0

(5.34) ({; { ) s +{ 12({ 12n { ) x =0,

\

{ 32+#; ({ 12n { ) s + { 2+#; =&{ #; &2 ; {

|

1 0

(5.33)

(n { ) 2 +q({; { ) ;{

+

(5.35)

x

G x({: {(!, s)&{; {(!, s)&{D(!)) d!&{ #; &12({ 12n { ). (5.36)

We take U {: =[{: {, {m { ],

U {; =[{; {, {n { ]

(5.37)

290

KAI-JUN ZHANG

and

{ V = &{ {

V {: = &{ 2+#: { ;

(m { ) 2 & p({: { ), { 2+#: m { , :{

=

{ 2

2+#;

(n ) &q({; { ), { 2+#; n { . ;{

(5.38)

=

As done in the case of # : , # ; =2, we can get what we want. This completes the proof of Lemma 5.3. K Lemma 5.4. Let p, q # C 2(R) be such that p", q">0. Let [: {, ; { ] be any sequence of L , such that p(: { ) ( p(:)

in L  weak C,

(5.39)

q(; { ) ( q(;)

in L  weak C,

(5.40)

where :=weak C&lim : {, ;=weak C&lim ; { as { Ä 0. Then : { Ä :, p ; { Ä ; in L loc strongly for all p # (1, ). The proof of Lemma 5.4 is the same as the Proposition 4.3 of [33]. Now we are in a position to prove the main result of this section. Theorem 5.1 Assume that the conditions of Theorem 4.1 hold. Let (: {, m {, ; {, n { ) be the sequence of solutions of (5.1)(5.4). Then there exist :~, ; # L  and m~, n~ # L 2 such that {: { Ä :~,

{; { Ä ;

{ 12m { ( m~,

{ 12n { ( n~

a.e. as

{ Ä 0,

as { Ä 0

in D$.

(5.41) (5.42)

The limit function (:~, m~, ;, n~ ) satisfies the simplified drift-diffusion equation, for # : , # ; =2, ; s =0,

:~ s =0, p(:~ ) x =:~

|

(5.43)

1

G x (x, !)(:~(!, s)&;(!, s)) d!,

(5.44)

0

q(; ) x =&;

|

1

0

G x (x, !)(:~(!, s)&;(!, s)) d!,

(5.45)

THE BIPOLAR HYDRODYNAMIC MODEL

291

and for # : , # ; >2, :~ s =0,

; s =0,

(5.46)

p(:~ ) x =0,

q(; ) x =0,

(5.47)

in the sense of distributions. From the dominated convergence theorem, Theorem 4.1, and Lemma 5.4, it is easy to get the proof of Theorem 5.1.

ACKNOWLEDGMENTS The author is thankful to his supervisors Professors Ling Hsiao and Pierre Degond for their warm help and guidance. Professor Peter A. Markowich also deserves thanks for his kind support and guidance. The author's work was carried out while staying at Mathematiques pour l'Industrie et la Physique, Universite Paul Sabatier, Toulouse, France, and was also supported by the Erwin Schrodinger International Institute for Mathematical Physics in Vienna while visiting. He expresses his sincere thanks for their warm hospitality and support.

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