Mathematical Methods in the Applied Sciences, Vol. 19, 1489-1507 (1996) MOS subject classifications: 35 Q 35, 35 L 60, 35 M 20
Well-posedness
of the Hydrodynamic
Model
for
Semiconductors
Li-Ming Yeh
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, R. 0. C
Communicated by B. Brosowski
This paper concerns the well-posedness of the hydrodynamic model for semiconductor devices, a quasi- linear elliptic-parbolic-hyperbolic system. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary conditions for the hyperbolic equations are assumed to be well-posed in L 2 sense. Maximally strictly dissipative boundary conditions for the hyperbolic equations satisfy the assumption of well-posedness in L2 sense. The well-posedness of the model under the boundary conditions is demonstrated.
1. Introduction
This paper addresses the well-posedness of the hydrodynamic model for semiconduc- tors. The model is derived from moments of the Boltzmann’s equation, taken over group velocity space. When coupled with the charge conservation equation, it de- scribes the behaviour of small semiconductor devices and accounts for special features such as hot electrons and velocity overshoots. The model consists of a set of non-linear conservation laws for particle number, momentum, and energy, coupled to Poisson’s equation for the electric potential. It is a perturbation of the drift diffusion model [7]. We consider a ballistic diode problem which models the channel of a MOSFET, so the effect of holes in the model can be neglected. The model is [4,11]
a,n
+
V . ( n V ) = 0, (1.1) 14
I/a,v
+
( V . V ) V+
- V(nT) - - VY = --, m 2 P mn 1 2 2mV2 m V 2 T - T , a,T - - V . ( k n V T )+
VVT+
- T V . I/ - -~+-
+--
- 0, (1.3) n 3 32, 32, 7, ( 1.4)where
k
is a constant, and t E [0, tM], x Ea.
The electron density is labeled n, I/ is theaverage electron velocity, T the temperature in energy units (Boltzmann’s constant
4
AY = - (n - Z ( x ) ) ,
0
CCC 0170-4214/96/181489- 19
0 1996 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.
Received 14 December I994 Revised 2 February I996
has been set to l), m the effective electron mass, q the electron charge, Y the
electrostatic potential, T, the surrounding temperature, Z ( x ) the prescribed ion
background density, o the dielectric constant, and R c R 3 the bounded semiconduc-
tor domain. Electric field E is given by E =
-VY.
The collision terms of equations(1.2-1.3) are approximated in terms of momentum and energy relaxation times, respectively, [2,4]:
TS Ts T Ts
T ' L V = B 2 - + 8 3 - - -
2T T
+
TsTP = B 2 -
for some constants
B2,B3.
The initial conditions of system (1.1-1.4) areV ( 0 , x ) = Vo, n(0,x) = n o > 0, T ( 0 , x ) = T o . (1.6)
In addition, system (1.1-1.4) is supplemented by the following boundary conditions:
where ep = n and the admissible form of M will be explained in section 3. The
hydrodynamic model couples a hyperbolic system of two equations (equations (1.1-1.2)), a parabolic equation (equation (1.3)), and an elliptic equation (equation For steady-state solutions of Euler-Poisson model, a simplified version of the hydrodynamic model, we refer the reader to the works [4,7,11,13,16], and references therein. The well-posedness of the transient model in the two-dimensional space was discussed in [ 131, where maximally strictly dissipative boundary conditions for hyperbolic equations were mainly concerned. For non-linear hyperbolic initial boundary value problems, we refer to [9, lo].
We consider the transient model in three-dimensional space. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary condi-
tions for the hyerbolic equations (1.1-1.2) are assumed to be well-posed in L 2 sense.
Maximally strictly dissipative boundary conditions for the hyperbolic equations are shown to satisfy the assumption of well-posedness in L 2 sense. The well-posedness of
the model under these boundary conditions is then demonstrated. To d o this, we first
establish an existence theorem for a linear hyperbolic system by extending the results of [8,17]. Then the existence and the uniqueness of the solution of the transient hydrodynamic model are proved by employing contractive map and interpolation theorem [6,12]. More precisely, we prove the following:
Theorem 1.1. Under the assumptions A1-8 (see section 3), system (1.1-1.7) has a unique
classical local-in-time solution. (1.4)).
This paper is organized as follows: In section 2, pertinent notation is reviewed. In section 3, we reformulate the differential equations (1.1-1.7) and present the assump- tions A1-8 of Theorem 1.1. In section 4, maximally strictly dissipative boundary conditions for hyperbolic system are shown to satisfy the assumption of well-posed- ness in L 2 sense. In section 5, we establish an existence theorem of a linear hyperbolic system for our application. The existence and the uniqueness of a local-in-time solution of (1.1-1.7) are proved in section 6.
Well-posedness of the Hydrodynamic Model for Semiconductors 1491
2. Notation
Summation convention is used. c denotes various constants, which may not be the
same. That z is a tangential vector field in R means a vector field
z
in Q satisfiesz . 3 = 0 on aR (where 3 = (nl, n 2 , n 3 ) is the unit outward normal vector on aQ).
52' = [O, t M ] x R, aR' = [O, t M ] x
aQ.
For a systemthe partial derivative "a:U(O)" of (2.1) is defined by formally taking i - 1 time
derivatives of the system, solving for 3: U , and evaluating at time t = 0, e.g. "U(0)"
=f,
"at
U(0)" = ( A O ( O ) ) - ' [ F ( O ) - Aj(O)a,f - B ( O ) f ] , etc. That compatibility conditionsfor (2.1) hold up to m - 1 means that, on aR for 0
<
p<
rn - 1,Let Y * be, the formal adjoint of
9,
9* = -AOa,
-
~ j a ,
+
B* - ~ , A O - a , A j ,where B* is the conjugate transpose of the matrix B. The kernel of
A
is the boundarysubspace denoted by N. The adjoint boundary subspace is N
*
= (AaN)', whereAil :=
C
Ajnj. LetA*,
adjoint boundary operator, be a matrix-valued function onR x aR whose kernel is N * .
For p 3 1, ( I f P ) ' is the dual space of H P . For a matrix A, IAl denotes its
operator norm. Suppose A is a positive definite matrix, we define the norm
/I
Uby
3. Statement of the problem
We rewrite the system (1.1-1.4). Letting e P = n, (1.1-1.4) can be written as
T v 9 1 m T P m
a,v
+
( v q v
+
-
v p+
- = -v~
- -v ~ ,
2mV2 mV2 T - T , a,T-
kAT = kVpVT-
VVT -5
TV. V+
-
-
__-
____ 3 (3.3) 3Tp 32, 2, (3.4) 9 AY = - ( e p - Z ( x ) ) . 0The new system (3.1-3.4) is equivalent to the old system (1.1-1.4).
To solve (3.1-3.4), we uncouple the system into a Poisson equation, a parabolic equation, and a linearized symmetric hyperbolic system. Next we show a fixed point of the uncoupled system exists in some weak spaces. Then by interpolation theorem, we show the fixed point is a classical solution of system (3.1-3.4).
System (3.1-3.4) is uncoupled as follows: Given Q, q, S, find Y by solving
9
AY = - (eq - Z(x)), w
Y l l a O = y b .
Then determine T by solving
2mQ’
mQ’
S - T ,a , T - kAT = kVqVS
-
QVS-
:SV.Q+-
-~
- _ _ _3TP(S) 3TW(S) TW(S) ’
Tlant = Tb,
T(0,x) = To.
Finally, we solve the following linearized system for V and p
T T T
m m m
- a , p + - ~ ~ p + - ~ . ~
= o ,
V(0, x) = V,, p(0, x) = In no.
Obviously, a smooth fixed point of system (3.5-3.9) is a classic solution of the system (3.1-3.4) and vice versa. Therefore, we can proceed to work with system (3.5-3.9).
We note that if a smooth solution of the system (1.1-1.4) exists, the following
compatibility conditions hold, for p 2 0,
Well-posedness of the Hydrodynamic Model for Semiconductors 1493
where derivatives are obtained by differentiating (1.1-1.4) with respect to t and setting t = 0.
Define
(q,
p i , Ti)
as follows: When i = 0, it is the initial condition (Vo,lnno, To);When i = 1,2, it is obtained by formally taking i - 1 time derivatives of the system
(1.1-L3), solving for (aiV,afp,afT), and evaluating at time t = 0. Define K6,', by
Let us make the following assumptions
(Al) $2 is open, bounded, and smooth in R3,
(A21 0 <
(Vo,ii)21an
and 0<
l(V0,ii)' - (To/m)I in 82,(A3) 0 < no, To, Tb and Vo,no E H4(R),
(A5) To: I f 5 @ ) ,
Tl
E H4(R), T z E H2(R), and there is aT
satisfyingTIanG
= Tb, (A6) compatibility conditions (3.10-3.11) for system (1.1-1.4) hold up to 2. ( ~ 4 ) T, Ec3(cit),
z ( x ) E H ~ ( R ) , y b E ~ ~ ( a ~B E t )~ 3 ( a q , , M E H ~ ( R ' x129,
a f ~
E L ~ ( H ~ - ' ) , i = 0,1,2,3,a,4T
E L ~ ( H - ' ) ,Remark 1. B y A3-5, we know
( q , p i ,
Ti)
E H3-i(R), i = 0,1,2. B y trace theorems [3],we see K6,t, is not an empty set for some 6.
Given (Q, q, S ) E K d , t M , we are able to solve (3.5-3.6) to obtain T. Then by (Q,q, T ) , we
want to solve (3.7-3.9). For convenience, (3.7-3.9) will be written as Y W := Ao(T)atW
+
A j ( Q , T ) a , W+
B(T) W = F in R',M ( t , X,
Q,
q, T) W = on aR', (3.13)W ( 0 , x ) = (Vo,lnno)' in R,
where
W : = ( V , p ) ' , F : =
Let a,, a , , a2, a3 be constants satisfying
(1) 0
<
a l < Ao(To)<
a. in R',(2) n j A j ( V o , To) is non-singular and 0 < a3 < InjA'(Vo, To)/ < az on aR'.
(A7) There are two constants M 1 , M z such that, for any ( Q i , q i , S i ) E K 6 , t M ,
B y A2-3, above four constants exist. T w o more assumptions for (3.13). i = 1,2,3, the following hold on boundary aRt,
and
*(lVolMl
+
IlnnoIM2) < min{l,c(R)k).(A8) There is a positive constant a. such that, for any a 2 ao, W E H ' ( R x
n),
(a) the followings hold:
(Q,
Y], T ) E &,t,(3.14)
(3.15)
where 0
<
t l < t2<
tM, 9, M are the operators in (3.13), and 9*, M* are the adjointoperators of 9, M (see section 2),
2a1 2
+
a0II
w e IIL2(n)(t2),(b) there are Bk,Mk, and symmetric A;,A: c P ( @ ) satisfying
k - + c o ,
{AkO,Ai,Bk}
-,
{AO(T),Aj(Q, T ) , B ( T ) } inx3(Qt),
Mk + M ( t ,
x,
Q,
?, T ) on ff3(aR'),and inequalities (3.14-3.15) hold for the following
y k w : =
&a,w
+
A : a , W+
BkW = FM k W = g on aRr, (3.16)
W(0,x) = (Vo,lnno)' in R,
in
a',
where 9, M , 2'*, M* in (3.14-3.15) are replaced by y k , M k r 9 t , M:, respectively,
(T;, M: are the adjoint operators of
zk,
Mk).Actually, A8 assume that boundary condition M(t, x, Q, q, T ) for the hyperbolic
system (3.13) is well-posed in L2 sense. In section 6 we will show that if A1-8 hold,
then a fixed point of system (3.5-3.9) exists uniquely in a short-time period, which
implies Theorem 1.1.
4. Example
We give one example for M of (1.7) such that assumptions A1-8 hold. Multiplying
(3.2) by T/m and combining it with equation (3.1) along with initial and boundary
conditions, we obtain
Ao(T)a, W
+
Aj(V,7')ax,W+
B ( T ) W = F in R',W(0,x) = (Vo,lnno)' in R,
Well-posedness of the Hydrodynamic Model for Semiconductors 1495
where W := (V,p)'. Assume that the matrix A-,(=Ajnj) of (4.1) on
ant
has negativeeigenvalues
A,,
p = 1...
t,
and positive eigenvaluesA:,
v =t
+
1...
4, and, corres-ponding to them, orthonormal eigenvectors rp-,r:, p = 1
...
t,
v = t?+
1...
4. Theboundary condition M is taken as
c c 4
M W =
2
( r ; . W)ap,,rp-+
c
1
(r,?. W)bp,,rp-p, Y = I p = l v = c + 1
where up,", bp,,, are smooth functions and (ap,") is an invertible matrix in
an'.
If bp," = 0for all p, v, then M is maximally strictly dissipative boundary condition [ 171.
Remark 2. T h e eigenvalues of A-, for (4.1) are A1 = A2 = V,( := V
.G),
(1
+
T/m)V-,+
J [ V i ( l+
T/m)' - 4T/m(V$ - T / m ) ]2 3 = 9
(1
+
T/m)V-, - J[V,'(1+
T/m)'-
4T/m(Vi - T/m)]2
A4 =
Let z1 := ( T ~ ~ , ~ ~ ~ , T ~ ~ ) , z2 := ( T ~ ~ , T ~ ~ , T ~ ~ ) be two orthonormal tangential vectorsin 0,
and [ be dejned by
T h e corresponding eigenvectors of
A1,A2,A4
are ( T ~ ~ , T ~ ~ , T ~ ~ , O ) , ( ~ ~ ~ , ~ ~ 2 , ~ 2 3 , 0 ) , and (nl , n 2 , n 3 , - m [ / T ) , respectively.Therefore on inflow boundary {(t,x) E aR') Vs(O,x)
<
0}, (1) if V$(O, x) < T,/m, we specify three boundary conditions (because A1, A 2 ,L4
are negative and l 3 positive), ( 2 ) ifT,/m < V$(O, x), M is the identity matrix (since all of the eigenvalues are negative). T h e former corresponds to subsonic case, and the latter to supersonic case. However an outfrow boundary { ( t , x ) E
ant/
V-,(O, x )>
0}, (3 ) if V$(O, x) < T,/m, we need one bound- ary condition (because onlyA,
is negative), (4) i f T o / m < V,"(O,x), M is the zero matrix (because no boundary condition is needed). Similarly, ( 3 ) and (4) correspond to subsonic and supersonic cases, respectively.For example, M W = 39 of (1.7) can be: in case (1)
which means ( W r t , W r 2 , W-, - p m [ / T ) = B f on a0, where Wr, := W . z i , i = 1,2, in case (2) ( W , p ) = 99' on aR, in case ( 3 ) M = (nl ,n2, n 3 , - m [ / T ) , i.e. W, - pmi/T = 99 on
80, and in case (4) M = 0.
If A1-6 and (4.2) hold, and if ( P ( ( O , ~ ) ( ~ ~ , { b p , v ) of (4.2) are sufficiently small, then
they imply assumptions A1-8 hold. A7 holds obviously if JPl(0,x)lan is small. The
Lemma 4.1. Consider (1.1-1.7). Besides assumptions A1-6, ifthe following holds: M of
(1.7) is constructed as (4.2), { a , , } and { b p , v } are in H3(aRt), and they satisfy
max laibl < bC1, max Ibp,J
<
bo, V P , ~ ,4(1/a2
+
l/a3)b;'b;<
l/a2,an1 anc
for some constants b l, b o , then A8 holds for any 6 as long as tM is small enough. Pro05 (3.14) is proved by energy L 2 estimate. By means of a local co-ordinate change
and a partition of unity, it is sufficient to prove (3.14) in a half space. The estimate in
a half-plane can be obtained as follows: Multiplying (3.13) by W integrating the
resulting equation over Rt, and employing assumptions of boundary conditions of this
lemma.
(3.15) can be proved in a similar way as (3.14). A8(b) holds by using Friedrichs' mollifiers and arguing as the proof for (3.14).
5. Existence results for linear problems
In this section, we present existence results for a parabolic equation (Lemma 5.1)
and for a linear symmetric hyperbolic system (Lemma 5.4). These are used to show the existence of a fixed point of system (3.5-3.9) in section 6. Lemma 5.1 is a standard
result. Lemma 5.4 is established by Lemmas 5.2, 5.3. Lemma 5.2 gives an a priori
estimate for a linear symmetric hyperbolic system (5.3) by energy method. Lemma 5.3
is to present approximate systems of (5.3) so that the solutions of these approximate
systems exist. Lemma 5.4 is to show that the solutions of the approximate systems converge (by a priori estimate obtained in Lemma 5.2). Furthermore, we show the
limit is a solution of (5.3). Domain R will be assumed in Rd.
Lemma 5.1. Consider the following system
a,U - aAU = F in R',
u = g
on aR', U ( 0 , X )=f
in R,where a is a constant. I f F E H2(Rt), a:F E L 2 ( H - ' ) and if there exists a function
@ satisfying
@laat
= 9,a;@
E L ~ ( H ~ - ~ ) , i = 0,1,2,3,a;'@
E L ~ ( H - l),af(U -@ ) ( O , X ) E H ; ( R ) , i = 0 , 1 , 2 ,
a:(u
- @ ) ( O , X ) E L ~ ( R ) , then there is a U 6 X3(R') such thataf
U E L2(H4-'), i = 0,1,2,3, andi = O L i = O
Well-posedness of the Hydrodynamic Model for Semiconductors 1497
Lemma 5.2. Consider the system
(5.3)
I f the following conditions hold:
1. R is open, bounded, and smooth in Rd,
2. A', A', B E X,(R') are j x j matrices, A%' E H"(aR'), and s 2 [d/2]
+
2,3. A', A' are symmetric and 0 < a, < A'
<
a. in R',4. Aa :=
1
A j n j is nonsingular on aR' and 0 < a3<
I
AhI
<
a2 onaR',
5. F E Hm(R'), f E Hm(R), g E Hm(aR'), 1 G m d S,
6. for any U E H'(R x
n),
0 d t l < t 2<
t ~and , alcl 2 ~ ( l l L ( ( ( ~ , ~ ~ ~ ) , (3.14) holds for(5.3), i.e.
(5.4)
Proof: By means of a local co-ordinate change and a partition of unity, it is sufficient
to prove (5.5) in a half-space. The estimate in a half plane will be obtained by
differentiating the equation (5.3), to estimate tangential derivatives and then, using det Aa # 0 to solve for the normal variables in terms of the tangential ones.
For convenience, we assume (5.4) holds in a half plane. (5.5) is proved by induction.
Proof of this lemma is close to that of Lemma 3.2 in [17], so similar arguments will
not be repeated here. The main difference between the two proofs is the estimate for
/I
Aa,Tx,u
- a Y x , ( A U ) IIO:anr,n.where
a/,x,
is the y derivative with respect to t , x' = (x2,.. .
, xd) and I y ( = 0,1,.. . ,
s. By(5.41, (5.6)
+
a0II
a : x J J ll;n(O). (5.7)By Holder inequality, embedding theorem, using det Aa # 0 to solve for the normal
variables in terms of the tangential ones, and an argument similar to (3.20-3.28) of
[17], one can show that, for 0 6 t 6 t M ,
Next we assume the following inequality holds:
+
a0I1
u
Il;n(O). (5.10)By Holder inequality, (5.9), and arguing as (3.31-3.38) of [17], one can show that
11
~ a T x ~ u
-a y x , z u
ll;nhWell-posedness of the Hydrodynamic Model for Semiconductors 1499
m - 1
<
cC
I I a ~ ~ p ~ I I t ~ ~ ( a n ~ ) I / a [ x , U e - * ' IItyant), (5.11)where for (i) m = s, p = 0 case we pick p = 1, q = co in (5.11), and for (ii) m < s, or for (iii) m = s, p > 0 case we pick p , q 2 1 in (5.1 1) in such a way that
p = o
1 1 1 d - 2 ( s - m + p ) 1 d - 2 ( m - p - l / 2 )
d - 1
I = - + - , - 2 >
P 4
P
d - 1For case (i), inequality (5.1 1 ) implies that
llAa?x,U - aG,.,d~llo2,an~,, G max lU12
lla?x~~llo2,iJnh
n'
(5.12)
For cases (ii), (iii), inequality (5.1 1 ) implies that
By using det A, # 0 to estimate
lla~x,,x,
U Il&e-2"' as (3.39-3.42) in [17] and combin-ing (5.10-5.13), we complete the proof.
Remark 3. Consider system (5.3). Besides the assumptions of Lemma 5.2, we assume: 1. compatibility conditions for (5.3) hold up to m - 1, 2
<
m<
s,2. there are B k , A k , and symmetric { A t , A : } c C m ( n f ) such that, as k -+ co,
{ A ~ , A : , B ~ } -+ ( A O , A ~ , B } in
x,(cY),
A k+ 4 on Hs(aQt).
B y {A:, A:, B k , A k } above we consider the following:
(5.14)
I t is clear that if k is large, assumptions 2-4 of Lemma 5.2 hold in (5.14), where
(A', A', B , 4 , A,} are replaced by { A t , A { , Bk,J%Lk, Ak,,( : = A [ n j ) } . Let S?:,
A:
be the3. for U E H'(R x
Q,
0<
t l < t2<
t M , and alcl 3 C ( ( ~ A O , A ~ , B ~ ~ ~ ~ , ( ~ I , , (5.4) and the following hold for (5.14):Zat 2
+
a0II
u
e llL2(n,(tz).Assumption 3 above is similar to A8(b) but for (5.14).
(5.15)
Lemma 5.3. Under assumptions of Lemma 5.2 and Remark 3, there exist a sequence of functions {Fk,fk,gk} such that Fk E H"+'(R'),fk E
ff"+2(n),
g k E Hrn''(afif),Fk -+ F in H"(R')
fk+f in H"(R) as k - i c o ,
g k -+g on Hm(aRt)
and,
if
we consider the following (replacing F 7 J g of (5.14) by F k , f k , g k )z k uk= A j a t u k f Akjax, u k
+
Bkuk
= Fk ina',
A k uk= g k on aili (5.16)
u k (07 = f k ( x ) in R,
then (5.4), (5.15), and compatibility conditions up to m also hold in (5.16).
Remark 4. First let us give definition (for (5.3)) for Gi,f,:
Gi := -(ai[(Ao)-1AJ])8j - a:[(Ao))- ' B ] , i 3 0,
fo := U ( 0 , x ) =f;
f p : =
pi'
( p7
l ) G i ( O , x ) f p - l - i+
(ap-'[(Ao)-'F])(O,x), p 2 1. (5.17)i = O
By definition, f , corresponds to "apU(0,x)". W e define B,, E, as Bo f := f o = f and B, f
+
E,(A0)-'F := f,, p 2 1, whereB, f := G t ( 0 , x ) f
+
term of the form GG G I ,. . .
GZq(O, x ) f (5.18) w i t h y + i , +...
+ i , d p - l a n dE,(Ao)-'F := sum ofthe terms of the form GLG,,
...
Glq(0)C$'[(Ao)-lF](O)(5.19) with y
+
i,+
...
+ i,
+
p b p - 1Proofi ( 1 ) That (5.4), (5.15) hold in (5.16) is because (5.4), (5.15) hold in (5.14). By assumption 2 of Remark 3, there are smooth functions A;,Akj,Bk,Ak which converge to A o , A J , B , A . Consider system (5.14) for each k . Let us use f k $ Bk,p,Ek,p to
Well-posedness of the Hydrodynamic Model for Semiconductors 1501
denote the f p , B,, E , of (5.14) (see (5.17-5.19)). By the assumption of compatibility
conditions for (5.3), on boundary asZ, for 0
<
p<
rn - 1,where f p - p is the ( p - p)th derivative, "af-"U(O,x)", of (5.3). Since
we may choose k such that
Let q 2 rn
+
2 be an integer. For fixed k, we choose sequences F: E H"+q(sZ'),4;
E H"+q(R), and g: E Hm+q(aC2f) such that(5.21)
Then we want to find i ( k ) and such that, for 0
<
p<
rn,(5.22) 1 2k' hi'k' E H"+2(R),
jl
h pIIHyn)
d -i
( ~ ) a : & k ( B k , p - p ( - h i ( k '+
4;'")
+
E k , p - p ( A t ) - l F L ( k ) ) -a
p
g k i'k) = 0 9 p = O (5.23)for ( t , x) E {O] x
an.
Suppose is available, we definef k i ( k ) = - h l ( k ) +
4 i ( k )
Then we see, when i(k) is large,
I/
F f ( k ) - FI I H ~ R ' )
< c / kII fki(k)
-f
Ilwyn) <c / k II
g:'k' - IIHyaR') <c / k
and system (5.16) satisfies compatibility conditions up to m. So Lemma 5.3 holds true.
Next step is to search for
(2) Equation (5.23) can be written as, for 0
<
p<
m, ( t , x ) E (0) xan,
Let Rk be the inverse of d kwhen it is restricted to the orthogonal complement of the
kernel of d k . Note a:gf(o,x) is in the image of &k. For
o <
p<
m on (t,x) E ( 0 ) xan,
to solve (5.24) is sufficient to solve(5.25) The left-hand side of equation (5.25) can be written as
P - 1
p = 1 y = o
Bk,ph:
+
R k (;)ardkBk,p-uhL =Dkqaa:hf+
ck,p,p-ya$hi,where Dk,ii:= -(A;)-'A{nj, a;h: denotes the normal derivative of h: on
an,
andC k , p , p - an operator of order ( p - y) which only contains derivatives tangential to
an.
So equation (5.25) becomes(5.26) where aqhh := bp,i,k. Since a p . I , . k E Hmtq-p-1/2 (an), it follows that
(an). b . E ~ m + q - ~ - 1 / 2
p . i , k
Also by (5.20-5.21) and (5.25-5.26), there is a large i ( k ) such that IIbp,i(k),k I l H m - p - 1 / 2 ( a q < c/k, 0 d
P
d m - 1.Furthermore, by a trace theorem [lo], there is a u k E Hm+q(Q) such that
= bp,i(k),k, 0 d p d m - 1,
//
u k / / H y qd
c/k.Finally, we write hLfk' = u k
+
wk, where wk is required to satisfy wk E H~(R), and,/I
wk/IH"(R)
<
c / k ; arwklan = bm,i(k),k-
arVk; a $ W k l a f i = 0, 0<
p<
m - 1.To find such a wk, we refer the reader to the proof of Lemma 3.3 of [8]. Therefore the
Lemma 5.4. Under the assumptions of Lemma 5.3, system (5.3) has a unique solution U E X,(n') n Hm(i3Qt) a n d there is a polynomial P such that
desired hi(k' can be obtained.
I//
uIII&(Qfi
+
II
u
II&aolle 2 a h
<
~ ~ ( l l l ~ l I l X S ( * ~ ) 9 l l ~ l l ~ ~ ~ ~ ~ ~ ) ~+
~lIglI~-(aa) 2 ~ ~ l l+
~llflli~i*)l~
l l i ~ i * ~ )a1
(5.27)
Well-posedness of the Hydrodynamic Model for Semiconductors 1503 Proof: (1) By Lemma 5.3, there exist
&
and symmetric matrices A f , A : E Cm(Cl'),d kE
cm(aQ'),
Fk E H""(a'),f, E H"+'(R), and g k E f f m + 2 ( 8 Q * ) such that( A : , A:,
&}
-
{A', A',B} in X,(CY)d k--t &? on Hs(aR')
Fk --* F in H"(Rt) as k +co.
.ti
-f
in Hm(R)gk
-
on Hm(aRt)Furthermore (5.4), (5.15), and the compatibility conditions up to m hold for system
(5.16) for each k . Consider the system (5.16). By Proposition 5.1 of [ 8 ] , one can show
that system (5.16) has a solution
uk
EX,, By Lemma 5.2, { U k ) is bounded, i.e.,]/I
U k l l l X ,+
11
Uk lIH"(af2') 6 c, v k . (5.28)Let us consider the following:
p(vk -
u/)
= Fk - FI+
(9 - 9 k ) U k - (9 - 9 l ) U l in R',d ( U k - U , ) = ( A - A k ) uk
+
(Ai
-d )
u,
+
gk - gl on aiz',( U k - U/)(o,x) = f k
-fi
iniz.
(5.29)
Inequality (5.4) of Lemma 5.2 applied to { Uk - U , } shows that {
uk}
converges inC ( L 2 ) . Let U be the limit of { U , } , then U E C(L'). By (5.28),
I1
U IIHm(an~) 6 c.By (5.28) and the Sobolev's interpolation theorem [l, 61, we have for arbitrary v
>o,
1 - v i m
11
Uk - U / llHm-*'(Q)(t) 6/I
Uk - U /11
Uk - uI \/H"(R) ( t ) ?so
Uk + U in C ( H m - " ) as k -+a.
By (5.3) and (5.16), we see that
m - 1
Uk -+ U E C i ( H m - i - v ) as k +a,
i = O
a#
E L " ( H m - i ) , i = 0...
m.By
( 5 . 3 ,
we see U satisfies the estimate (5.27). Equation (5.27) implies solution of (5.3) is unique. So the proof is complete as long as we knowaf
U E C ( H m P i ) , i = 0...
m.( 2 ) That a f U E C ( H m - i ) , i = 0
...
m, is proved by three steps:Step 1: As t -+to, afU(t;) -+ a l U ( t 0 ; ) weakly in H m - i ( Q ) for all i 6 m.
Step 2: a r x U ( t ; ) is continuous at t = 0 in L2(iz). Step 3: U E Xm(Cl').
Proofof Step 1. If q < m and € ( H q ( Q ) ) ' , by (5.30), $ l ( u k ( t , . ) ) - + $ t ( U ( t , . ) ) uni- formly in t as k -+ co. Now if $ E (H")', there is a $ I E ( I f q ) ' , for some q < m, such that
1)
$ - $1Il(Hm),
d 1/2k for any k . By (5.28), for all t,/ $ ( U k ( t ? ' ) ) -
$('(t,'))I
6
l$(uk)
-
$l(uk)l+
l $ l ( u k )-
$ l ( U ) l+
1$1(U) - $ ( U ) /6 C / 2 k - '
+
I $ l ( U k )-
$ l ( U ) l .This implies that $(&) -+ $ ( U ) uniformly in t. Because $(&) is continuous in t, $( V )
is also continuous in t for all $ E (H"(R))'; that is, as t --+ t o , V ( t ; ) -+ LJ(t0;) weakly in
H"(R). Similarly, one can show that if t + to and 1 d i < m, then
af
U ( t ) -+af
U ( t o )weakly in H"-'(R) by (5.30). Then, by (5.3), we see that a?U(t;) - a ~ U ( t 0 ; ) weakly
in L2(Q) as t -+ to.
Proof of Step 2. To prove this step, we will show the convergence result in an
equivalent norm
// .
I/O,AO(0).
By differentiating (5.16) with respect to tangential vectorsor time a:,,, we have
- % k a ~ , t U k = a ~ , t - % k U k + ( - % k a ~ , t u k - a y , t - % k U k ) in R',
&ka;.tuk =a:,t(&kUk)
+
( A h a y , t u k - a y , t ( & k u k ) ) on (5.31)a;,, U k ( 0 , x) =
a:,,
uk(o,x)
in R.Multiplying above by a;,, V k , and integrating the resulting equation over R, we derive,
for
I y I
d m,(5.34) Now, since U k ( t , ' ) converges weakly to U ( t ; ) in H"(R) for each t, U k ( t ; ) converges
weakly to U(r;) in H"(Q) in an equivalent norm for each t. So for all I y I d m,
(5.35)
Well-posedness of the Hydrodynamic Model for Semiconductors 1505
Let RE be the intersection of R and a E-neighborhood of a n , and let $& be a half bell
function satisfying cl/"(x) = 1 for x E aR and with support in
a'.
Multiplying (5.31) byI,P, then applying (5.4) to the resulting equation, one can show that, for any E,
lim
1's
(e-2af$Ea;,,Uk(2dodtk - m 0 an
<
c(il$%
u
IILa(n)(0)+
1' 1
le-2afa;,,g12 d a d t+
R(z)dz , (5.38) for some constant c. To obtain the result (5.38), we need to control the boundary term11
uk
- a;,r(Akuk)
/ 1 , & 1 , ~ . It can be estimated by the same technique as(5.1 2-5.13). (5.38) implies that
o an
l
)
_ -
lim lim
1' 1
la:,
ukAk,aay,t
ukl
d o d t<
cjlpay,,
u
i ~ ~ ~ ( ~ , ( o ) .
5-0' 5-0' n!
(5.39)
5 + 0 + k + ~ o an
The right-hand side goes to 0 as E approaches 0. Since the left-hand side is independent
of E, it equals 0. By the continuity of A', we get, for JyI Q rn,
lim
In
a;,tuAOa;,,u(~,X)dx =lim
a~,tu(r,x)~O(o,x)ay,,u(~,X)dx.
(5.40) By (5.37-5.40), we have, for J y J Q rn,
lim
S,
a:,
t ~( (7x ) ~ o ( ~ x ) ~ , t~ ( 5 ,
dx Q11
a:,,
~ ( 0 ) 1102,Ao(o).5-0'
(5.41)
Combining the result of Step 1, Theorem 3.8 in [l], and (5.41), we see
a:,,
U ( t ; ) iscontinuous at t = 0 in L2(R) for all / y J = m.
Next we show that a?,U(t;) is also continuous at t = 0 in L2(R). By partition of
unity, we can reduce the problem to the boundary aR. By using det Aa # 0 on aR', we
can solve for the normal variables in terms of the tangential ones and time. Because
a:,,
U(t;) is continuous at t = 0 for ( y I = rn, a;,U(t;) is also continuous at t = 0 in L 2around the boundary aR. Therefore, we conclude that U(t,. ) is continuous at t = 0
in L2(R).
ProofofStep 3. By the same argument as that in Step 2, one can prove strong right
continuity at any point t E [O,t,). Note that equation (5.3) and the argument of
Step 2 are reversible in time, so the proof of strong right continuity in [0, t M ) implies
strong left continuity on (0, t M ] . Therefore, we conclude that U E X,(R').
6. Existence of a unique local-in-time solution
In this section we prove the existence of a unique local-in-time solution of the system (1.1-1.7) under assumptions Al-8 of section 3. That is equivalent to showing the existence of a fixed point of the system (3.5-3.9). By A3-5, we know K6,1, (see (3.12)) is not an empty set for some 6. Given ( Q , q , S ) E K 6 , t y , we are able to solve
n(Q, q, S) = ( V , p , T ) . The existence of a unique local-in-time solution of the system (1.1-1.7) is proved as follows: First we show that, as tM is small enough, there exists
a 6 such that
II
is a map from K 6 , [ , to itself (Lemma 6.1). Next we prove thatn
isa contractive map in some weak spaces. So we obtain a fixed point in these weak spaces. Then we show the fixed point is a smooth solution of the system (3.5-3.9) (Lemma 6.2). The existence and uniqueness of classical solution of system (1.1-1.7) follows the result of Lemma 6.2.
Lemma 6.1. Under assumptions A1-8 of section 3, as tM is small enough, there exists
6 such that (3.5-3.9) is uniquely solvable and the solution of (3.5-3.9) is in K6,[, for all
ProoJ: Given (Q, q, S) E we solve the system (3.5-3.9) to obtain ( V , p, T ) by A1-8
of section 3 and Lemmas 5.1, 5.4. By Lemma 5.1, the solution of (3.6) satisfies
(Q, V , S) E Kd,r,.
T E X ~ ( C F ) , afTEL2(H4-i), i=O,1,2,3,
afT(o,x) =
Ti,
i = 0,1,2.By (5.2),
111
T -T o ,
T ,
,T 2 ,
T,, no, Vo but is independent of 6.+
11
T - To ( I H 3 ( a n t ) d dl. If tM is small enough, d1 depends onBy Lemma 5.4 for d = m = s = 3, we see that the solution (3.7-3.9) satisfies
( ~ , p ) ~ x ~ ( n I ) n ~ ~ ( a n I ) , (a:v,a:p)(o,x)
=(c,pi),
i = 0,1,2.BY (5.271,
IIt(v,p)
-
VO,1nno)III&+
II(V,p) - (~O,lnno)li&an~, G 6 2 . If tM is smallenough, we see that d2 is dependent on no, Vo, To,"* but independent of 6. If we take
do := max{6,61,62}, repeat the above procedure, and let tM smaller, we see
( V , p, T ) E KGo,tm for all (Q, q, S) E KdOlfy. That is,
I
I
is a map from K d , , t M to itself.W
Lemma 6.2. Under assumptions A1-8 of section 3, system (3.5-3.9) has a uniquefixedpoint
if
tM is small enough. Moreover, the solution satisfies(afv,ajp,afT)E L " ( H ~ - ~ ) , a f T E L ~ ( H ~ - ~ ) ,
afw
E L " ( H ~ - ' ) , (6.1) for i = 0,1,2,3.Proof. By Lemma 6.1, II(Q,y, S) = ( V , p , T ) is a map from Kd,t, to for some 6.
Given ( Q l , q l , S l ) and ( Q 2 , q 2 , S 2 ) , by solving (3.5-3.9) we get two solutions
( V l , p l , T l , Y l ) and ( V 2 , p z , T 2 , Y 2 ) . Subtracting one solution from the other, multi-
plying the difference of (3.6) by ( T I - T2)e-2"', integrating the resulting equations
over Q', using (5.4) for the difference of system (3.7-3.9), we see, if c( is large and t M is
small enough,
II
is a contractive map in V , p E C(L2)nL2(aR'), T E L2(Q'). That is,/I
Ti -GIl;n',.
+
Il(v1
- V2,PI - P2)e-2a'11Z~cn,+
IlVI
- V29Pl-
Pz)ll;ant,. 2at 2< c
/I
Si - S 2 Il;n',,+
II
(Qi - Q 2 , ~1 - V Z ) ~ - 1 / ~ 2 ( n )2
+
ll(Qi
- Q 2 , ~ 1 - ~ 2 ) 1 / 0 . a n ~ , ~ ~for some constant c < 1.
By the above, if we define n(Vk- 1 , pk- 1, Tk- 1 ) = ( v k , pk,
Tk),
then sequence { Vk, P k , Tk} converges to the unique fixed point { V , p , T } wherev,pEc(L2)nL2(af21), TeL2(fit). Since {Vk,pk,Tk} c ( V , ~ , T ) E
Well-posedness of the Hydrodynamic Model for Semiconductors 1507 a f T E L 2 ( H 4 - i ) , i = 0,1,2,3. By Sobolev's interpolation theorem [1,6], we have for arbitrary v
>
0 (set w k := ( V k , P k ) )11
w k-
IIH"-'(12)(t)I/
wk
- wiIIti:l2)(s
11
wk-
w i i i , $ i A { 3 ( c t ) 7which implies that I/, p E C ( H 3 - " ) . By equation (3.7), we see
a:
V , alp EC(H3-v-i).
Therefore { V , p , T ) is the unique classical solution of system(3.5-3.9). Moreover, by (3.1-3.4), we see the fixed point satisfies, for i = 0,1,2,3,
(afv,afP,a:T)EL")(H3-'),
a f ~
€ L ~ ( H ~ - ~ ) ,a f y
P(H~-').Theorem 1.1 is a direct result of Lemma 6.2.
Acknowledgement
The author would like to thank Professors Jim Douglas Jr., Peter Markowich, and Anton Arnold for their valuable suggestions while preparing this paper.
References
1. Adams, R. A,, Soboleu Spaces, Aademic Press, New York, NY, 1975.
2. Baccarani, G. and Wordeman, M. R., An investigation of steady-state velocity overshoot effects in Si 3. Brezzi, F., Finite Element Handbook, McGraw-Hill, New York, 1987.
4. Gardner, C. L., Jerome, J. M. and Rose, D. J., Numerical methods for the hydrodynamic device model: subsonic flow, I E E E Trans. Computer-Aided Des., 5, 501-507 (1989).
5. Gilbarg, D. and Trudinger, N. S., Elliptic Partial DlfSvrential Equations of Second Order, Springer,
Berlin, 2nd edn., 1983.
6. Majda, A,, Compressible Fluid Flow and Systems of Conservations Laws in Several Space Dimensions,
Springer, Berlin, Heidelberg, New York, 1984.
7. Markowich, P. A,, On steady-state Euler-Poisson models for semiconductors, 2. Angew. Math. Phys., 8. Massey, F. and Rauch, J., Differentiability of solutions to hyperbolic initial boundary value problems, 9. Metivier, G., Problemes mixtes non lineaires et stabilite des chocs multidimensionnels, Seminaire 10. Mokrane, A., Problemes Mixtes Hyperboliques Non Lineaires, PhD thesis, Rennes University, France, 11. Odeh, F. and Rudan, M., Multi-dimensional discretization scheme for the hydrodynamic model of 12. Schochet, S., The compressible Euler equations in a bounded domain: existence of solutions and the 13. Thomann, E., Odeh, F., On the well-posedness of the two-dimensional hydrodynamic model for 14. Treves, F., Basic Linear Partial Diflerential Equations, Academic Press, New York, 1975.
15. Valli, A. and Zajaczkowski, W. M., Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case, Commun. Math. Phys., 103, 259-296 (1986).
and GaAs devices, Solid State Electron., 28, 404-416 (1985).
42, 389-407 (1991).
Trans. Amer. Math., 189, 303-318 (1974).
Bourbaki, 671, 37-53 (1986-87).
671, 37-53 (1987).
semiconductor devices, C O M P E L , 3, 149-183 (1986). incompressible limit, Comm. Math. Phys., 104, 49-75 (1986). semiconductor devices, C O M P E L , 9, 45-57 (1990).
16. Yeh, L. M., O n a steady state Euler-Poisson model for semiconductors, accepted b y Comm. P.D.E. 17. Zajaczkowski, W. M., Non-characteristic mixed problems for non-linear symmetric hyperbolic systems,