Subsonic solutions of hydrodynamic model for semiconductors

Subsonic Solutions of Hydrodynamic Model

for Semiconductors

Li-Ming Yeh

Department of Applied Mathematics, National Chiao Tung University, 100 Ta Hsueh Road, Hsinchu 30030, Taiwan, R.O.C.

Communicated by B. Brosowski

This paper is concerned with the existence and uniqueness of the steady-state solution of hydrodynamic model for semiconductor devices. Boundary conditions are prescribed by vorticity on inflow boundary as well as by electron density, temperature, and normal component of electron velocity on whole boundary. If the ambient temperature is large, and if both vorticity on inflow boundary and the variation of density on boundary are small, a unique subsonic solution exists.( 1997 by B.G. Teubner Stuttgart—John Wiley & Sons Ltd.

Math. Meth. Appl. Sci., Vol. 20, 1389—1410 (1997) (No. of Figures: 0 No. of Tables: 0 No. of Refs: 15)

1. Introduction

This paper is concerned with the existence and uniqueness of the steady-state solution of hydrodynamic model for semiconductor devices. The model is derived from moments of the Boltzmann’s equation, taken over group velocity space. When coupled with the charge conservation equation, it describes the behaviour of small semiconductor devices and accounts for special features such as hot electrons and velocity overshoots. In steady-state case [6], the model consists of the following equations: + · (o»)"0, (1.1) (» ·+)»# 1 mo+(o¹)! e m+("! » qa, (1.2) !2 3o+ · (b1¹0o+¹)#»+¹# 2 3¹+ · »! 2m»2 3qa # m»2 3qb#¹!¹0qb "0, (1.3) *("e d(o!Z), (1.4)

in a bounded semiconductor domain)LR3. o denotes the electron density, » the average electron velocity, ¹ the temperature in energy units, ( the electrostatic potential. Z (a positive function) is the prescribed ion background density. m, e,b1, ¹0, d are given positive constants. m is the effective electron mass, e the electron charge, ¹0 the constant ambient temperature, d the dielectric constant. qa and qb represent momentum and energy relaxation times, respectively:

qa"b2¹0¹, qb"b2¹02¹

#b3 ¹ ¹0

¹#¹0 (1.5)

for positive constantsb2, b3. Let n be the unit outward normal vector on ­). Define !1:"Mx3­)D»·n(0N to be the inflow boundary, and !2:"Mx3­)D»·n*0N the outflow boundary. The boundary conditions for system (1.1)—(1.4) are

oD­)"oDD_­), ¹D_­)

"¹DD­), (1.6)

»· nD

!1X!2_"»D, curl »D!1"wD. (1.7)

Because of (1.1), the following condition should hold:

P

wD also need to satisfy some compatibility condition on inflow boundary, which will be explained in section 3.

Existence of solutions for a simplified case of the hydrodynamic model, Euler— Poisson equation, in one or two-dimensional cases have been studied by several researchers [1, 5, 8, 13]. Here we consider the existence and uniqueness of a subsonic solution of hydrodynamic model in the three-dimensional case. Boundary conditions are prescribed by vorticity on inflow boundary as well as by electron density, temperature, and normal component of electron velocity on whole boundary. We show that if ambient temperature is large, and if both vorticity on inflow boundary and the variation of density on boundary are small, a unique subsonic solution exists. The strategy to show these results is to write the model in terms of density, vorticity, potential, temperature, and electrostatic potential. That would result in four elliptic systems and one transport equation. One can show that the new differential equations are equivalent to original hydrodynamic model. By fixed point theorem, we prove the new system has a unique solution, so does the hydrodynamic model. To prove the existence of solution for the transport equation, we need to work on a domain with edges because it allows us to reduce the transport equaion to an initial value problem, which is not the case in a smooth domain.

This paper consists of the following sections. In section 2, notations are recalled. In section 3, we discuss the compatibility condition for wD on the inflow boundary. In section 4, we derive auxiliary linear systems which are equivalent to the system (1.1)—(1.7). In section 5, useful lemmas are presented. Existence and uniqueness of a subsonic solution for the system (1.1)—(1.7) is showed in section 6 (see Theorem 6.1). Proofs of lemmas in section 5 are given in section 7.

2. Notation

For convenience, we need the following variables: o"exp(f), curl »"w, ¹

/¹0"E, and ¹D/¹0"ED. ‘ln’ denotes the inverse function of ‘exp’. So o"exp(f) means ln(o)"f. r.h.s. is the abbreviation of right-hand side. Summation convention is used. c is used to denote various constants. Cm,a()) represents the Ho¨lder space. ¼m,p denotes the Sobolev space and if p"2, then ¼m,2())"Hm()). For a function u, u,i:"­u/­xi. diam) is the diameter of ).

In this paper, we consider the model in a non-smooth domain. More precisely, domain) is assumed to be simply connected with one edge ¸"!11W!12, where !1i are the closure of!i (i"1, 2). For any point q3¸, there is a positive dihedral angle h(q) between!1 and !2. (1.7)1 can be written as

»· nD

!1_"»DD!1(0, »_{· n}D

!2_"»DD!2*0. (2.1)

In a neighbourhood of !1 we introduce a curvilinear system of orthonormal co-ordinates. ByMs1(x), s2(x), n(x)N, we denote the orthonormal basis corresponding to the co-ordinate system in such a way that, for x3!1, Ms1(x), s2(x)N are vectors tangent to!1 and n(x) is the unit outward normal vector to !1. For fixed k, l, if we look at the following expressions:

(sk·+)sl"iiklsi#ikln, (sl·+)sk"iilksi#ilkn, then one can show

ikl"ilk on !1. (2.2)

3. Compatibility condition

We now discuss the compatibility condition for wD on the inflow boundary. First let us assume wD·nD!1"

0 and ¹DD!1"

constant. Taking the curl of (1.2) and using (1.5)1, then

(» ·+)w#(+ · »)w!(w · +)»# ¹w b2¹0" !+¹]» b2¹0 # +o]+¹ mo (3.1)

where w :"curl » and ‘]’ is the cross product. Next we write w, » as follows:

w"wnn#wksk, (3.2)

»"»nn#»ksk, (3.3)

where n,sk are the normal and tangential vectors. Summation convention is used. By (3.2) and (3.3), (» ·+)w"(»nn·+)w#(»ksk·+)w " (»nn · +)(wnn#wlsl)#(»ksk · +)(wnn#wlsl) "»n(n ·+wn)n#»nwn(n·+)n#»n(n·+wl)sl#»nwl(n·+)sl #»k(sk·+wn)n#»kwn(sk·+)n#»k(sk·+wl)sl#»kwl(sk·+)sl.

Note wnD!1"

wD·nD!1"0. If we take inner product of (» ·+)w and n on !1, then S(» · +)w, nTD_!1"»n(n ·+wn)#»nwlS(n·+)sl, nT

#»kwlS(sk · +)sl, nTD!1. (3.4)

By a similar argument we can derive, on boundary!1,

S(w · +)», nTD_!1"wk(sk ·+»n)#wk»nS(sk · +)n, nT#wk»lS(sk· +)sl, nTD

!1.

(3.5) Now, we take inner product of (3.1) and n on!1. By (2.2), (3.4) and (3.5), and that ¹DD_!1 is constant, we obtain, in!1, »n(n ·+wn)#»nwkS(n·+)sk, nT!wk(sk·+»n)!wk»nS(sk·+)n, nTD!1"0. (3.6) Since wnD!1"0, 0"+ · wD_!1"(+wn·n#+wk·sk#wk+·sk)D !1. (3.7) By (3.6) and (3.7) and »nD!1( 0 (because »nD!1"»DD!1(0), we obtain »n+wksk#»nwk+ ·sk!»nwkS(n+)sk, nT#wksk+»n#wk»nS(sk+)n, nT"0. In other words, if wD· nD!1" 0 and ¹DD!1" constant, then wk("wD· sk, k"1, 2) have to satisfy the following equation on!1:

+ · (wk»Dsk)!wk»D(S(n+)sk, nT!S(sk+)n, nT)"0. (3.8)

So wk (k"1, 2) have to depend on »D and the geometry of the inflow boundary !1. One trivial solution such that wD satisfies (3.8) is 0. Also note if wD (O0) is a solution of (3.8), then c · wD is a solution of (3.8) for any constant c.

4. Auxiliary system

In this section, we derive auxiliary linear systems for (1.1)—(1.7). One can easily see that a solution of system (1.1)—(1.7) corresponds to a fixed point of the new linear systems. Proof for the other direction will be given in section 6.

Taking the curl of (1.2), we get

(» ·+)w#(+ · »)w!(w · +)»#wE b2" !+E]» b2 #¹0+o]+ E mo (4.1)

where w"curl », E"¹/¹0. Next, taking the divergence of (1.2) and using (1.1), (1.3)—(1.4), we obtain *f!m»+(»*f) ¹0E !

A

B

A

B

whereo"exp(f), »i,j:"­»i/­xj. We then split » in the following way [4]:

»"!+t#p (4.3)

such that

+ · p"0 in ), p · nD_!1X!2"0. (4.4)

So boundary condition (1.7)1 can be written as

+t · nD_{!1X!2"!»DD!1X!2}. (4.5)

Equation (1.1) then becomes

+ · (o+t)"+ · (op), x3). (4.6)

By (4.1)—(4.6), we now define a map F(S, º)"(E, » ) as follows: Given (S, º), solve

*f!mº+(º*f) ¹0S !

A

B

A

B

whereqb"qb(S). Then, using f from (4.7), compute o from

o"exp(f). (4.9)

Then, byo above, we solve

*E#+o+E_o !3º+E 2b1¹0! 3E 2b1¹0qb " !mº2S b1b2¹20 ! º+o b1¹0oS! 3 2b1¹0qb# mº2 2b1¹20qb, (4.10) ED­)"¹D/¹0D­), (4.11)

whereqb"qb(S). Next, by o, E from (4.9)—(4.10), we solve the following, for w and P,

(º ·+)w#(+ · º)w!(w · +)º#wE b2#+P" !+E]º b2 #¹0+o]+ E mo , (4.12) + · w"0, (4.13) wD_!1" wD, +P· nD!1"0, PD !1"0. (4.14)

Variable P and boundary conditions (4.14)2,3 are introduced to let w be divergence free. Next, we solve the following system forp:

curlp"w, x3), (4.15)

+ · p"0, x3), (4.16)

p · n"0, x3!1X!2. (4.17)

Then computet as a solution of the following system:

+ · (o+t)"+ · (op), x3), (4.18)

o+t · n"!oD»D, x3!1X!2, (4.19)

P

(4.20) Finaly, we can compute the velocity » from

»"!+t#p. (4.21)

Existence of a fixed point of the operator F will be shown in section 6. (4.7)—(4.21) forms auxiliary linear systems for the system (1.1)—(1.7). From the derivation, we see that a solution of system (1.1)—(1.7) corresponds to a fixed point of the linear systems (4.7)—(4.21).

5. Auxiliary lemma

In this section, we present four lemmas (proofs are lengthy and will be given in Section 7). They are used to prove the existence of a fixed point for systems (4.7)—(4.21). Part of results can be found in Reference 14.

Domain) considered in this section is a smooth simply-connected domain with edge and­)"!11X!12 (section 2). Lemma 5.1 is a result for a linear elliptic equation and is for solving (4.10)—(4.11). Lemma 5.2 is an existence theorem of a semilinear elliptic equation in ), used to solve (4.7) and (4.8). Lemma 5.3 is to establish an existence theorem for a linear elliptic equation with Neuman boundary condition for computing a solution for (4.18)—(4.20). Lemma 5.4 is an existence theorem for a trans-port equation in) and is for solving (4.12)—(4.14).

Lemma 5.1. Consider the equation

G

gku#(1!gk)+u·n"0, on !k(k"1, 2), (5.1)

wheregk is either 0 or 1 and g1#g2O0. If the following conditions hold: 1. aij, ai, a, f3Cm,a()1), a(x))0, 0(a(1, 0)m,

2. aij"aji, aij(x)mimj*jDmD2 for m3R2 and for some positive constant j,

then there exists a unique solutionu3Cm`2,a()1) and

EuE Cm`2,a()1))c(j, Eaij, ai, a E C_m,a()1))E f E C_m,a()1). (5.2) Proof. Existence of the solution is the Theorem 1 of [2]. (5.2) is obtained by tracing the proof of Theorem 1 of [2]. K

Remark 1. u(q) of condition 3 is obtained as follows: In section 2, we assume domain ) has one edge ¸"!11W!12. For all q3¸, we denote by R1(q) and R2(q) the two one-sided tangential planes which touch!11 and !12 at point q. For a fixed point q3¸ we transform to canonical form the second-order term of the system (5.1)1

aij(q)u,ij"0.

Since point q is fixed, this is an equation with constant coefficients. After the transformation, the planes R1(q) and R2(q) will be transformed to other planes that intersect at an angleu(q).

Lemma 5.2. Consider the equation

G

uD­)"uDD_­). (5.3)

If the following conditions hold:

1. aij, ai, a, f3Cm,a()1), 0(a(x), 0(a(1, 0)m,

2. aij"aji, aij(x)mimj*jDmD2 for m3R2 and for some positive constant j, 3. g3Cm`1(R), g@'0, there are constants u, uN satisfying g(u))f/a(x))g(uN), 4. uD3Cm`2,a()1), u1)uD(x))u2 for all x3­),

5. u(q)(2n/(m#2#a) for all q3!11W!12 (see Remark 1), then there exists a unique solutionu3Cm`2,a()M) satisfying

min(u1, u))u(x))max(u2, u6) ∀x3),

EuE Cm`2,a)c(jEaij,ai, aEC<sub>m,a</sub>)P(EuDEC<sub>m`2,a</sub>,E f E C_m,a,EgE C

m`1,(-1)#Eg~1E C

0(-2)),

where P is a polynomial with P(0, 0, 0)"0 and-1, -2 denote ranges of u, f/a over ). Lemma 5.3. Consider the equation

+ · (a(x)+u)"f in ),

G

P

(5.4) If the following conditions are satisfied:

1. a3Cm`1,a()M), 0)m, 0(j(a(x) for some positive constant j, 2. f3¼m,p()), g3¼m`1~1@p,p(!i), :₎f dx#+2

i/1:!g ds"0, p*2,

then there is a unique solutionu3¼m`2,p()) satisfying EuE Wm`2,p()))c(EaE C_m`1,a()1), 1/j)

A

i/1EgE Wm`1~1@p,p(!1i)

B

. (5.5)

In next lemma,Mb, º, f, gN are vector functions and " is a matrix function. Lemma 5.4. Consider the system

(b ·+)º#"º"f in ), º"g on!1.

If the following conditions are satisfied: 1. b3¼l_{`1,p()), b · nD}

!1(!j(0, b · nD

!2*0, 3(p(4, 2)l,

2. ", f3¼l_{,p()), g3¼}l

,p(!1), " is a positive-definite matrix, 3. "(x)'"mI in ), "m!i(p, EbE Wl_`1,p()), 1/j)""d'0,

wherei is a continuous positive function of its orguments, I is identity matrix, and j, "m, "d are some positive constants, then there exists a unique solution º3¼l_{,p()) and, for}

0)s)l,

EºEpWs,p()))c(E" E Wl_,p()),EbE Wl_`1,p()), 1/j, 1/"d)(E f E pW_s,p())#EgE pW_s,p(!1)). (5.6)

6. Existence and uniqueness of a subsonic solution

In this section, we prove existence and uniqueness of a subsonic solution of (1.1)—(1.7). To do this, we first define a set A]D (see Remark 2), and show that operator F defined by (4.7)—(4.21) is a map from A]D to itself (see Lemmas 6.1—6.5). Next, we show that the map F is continuous in some weaker space (see Lemma 6.6). Then by fixed point theorem, we conclude that a fixed point of (4.7)—(4.21) exists. Moreover, we see that if the ambient temperature is large and if the variation of density on boundary is small, a unique fixed point exists. Then we show a fixed point of system (4.7)—(4.21) corresponds to a solution of (1.1)—(1.7) (see Theorem 6.1). Now let us make the following assumptions:

(A1) 3(p(4 and 0(a(1 such that ¼1,p())˝C0,a()M) (continuous

imbed-ding),

(A2) )LR3 is a smooth simply-connected domain with edge ¸ :"!11W!12;

­)"!11X!12; 0(h(q)(n/(5!2/p) and h(q)On/j for all q3¸ and j3N (positive integer),

(A3) Z3C1,a()1), 0(2o1)Z(x))o2/2 for all x3),

(A4) oD3C3,a()1), 0(o1)oD(x))o2 for all x3),

(A5) »D3¼3~1@p,p(!i),+2

i/1:!ioD»Dds"0, »DD!1(!

vd(0)»DD!2,

(A6) wD3¼2,p(!1), wD ·nD!1"

0 and wk :"wD · skD!1(k"1, 2) satisfy (3.8),

(A.7) ¹D3C3,a()1), 0(E1)¹D(x)/¹0 for all x3), ¹DD!1 and ¹0 are positive

Remark 2. For constants cm('1) and ºm, we define A and D as follows: A"MS3C2,a()M)D SD­)"¹D/¹0D­),E+SE C1,a()1))

Em:"E1/cmN, D"Mº3¼3,p())Dº · nD_{!1X!2"»D, EºE W}3,p())_)ºmN.

To show that operator F is a map from A]D to itsel, constant Em, ºm cannot be chosen arbitrary. As we will see below, Em, ºm,+2

i/1E»DE W3~1@p,p(!i), and E1 are

related to each other. We state our main result below:

Theorem 6.1. ºnder (A1)—(A7), there exist two continuous functionsi1, i2 such that if 1. ¹he following hold for Em, ºm,+

iE»DE W3~1@p,p(!i), and E1:

(a) ¹here is a constanto3 satisfying 0(o3)S(x) ∀x3), S3A,

(b) 0(o1)!¹0Sd/e2F3)4(S, º))o2, ∀(S, º)3A]D (see (4.7) for

F3)4(S, º)),

(c) i1(1/o1, ElnoDEC3,a,EoDEC3,a,EZEC1,a)+2_i/1E»DE W_{3~1@p,p(!i)(ºm},

(d) ¹here is a constant ks satisfying 0(ks(E1(1!diam )/cm)/b2!i2(ºm, 1/vd), 2. 1/¹0#E+ ln(oD)EC2,a()1)#EwDEW

2,p(!1)is small, then a unique subsonic solution of

system (1.1)—(1.7) exists. In other words, there exists (¹, »,o, () uniquely such that (1)o, ¹3C3,a()M), », (3¼3,p()); (2) (¹, », o, () satisfies (1.1)—(1.7), and (3) D»D2(x))¹(x)/m for all x3).

Note that if Em, ºm,+

iE»DE W3~1@p,p(!i) are small, and if E1 is large, then the four

constants satisfy 1. (a), (b), (c), (d). To show that operator F is a map from A]D to itself, let us first consider the solvability of (4.7)—(4.9).

Lemma 6.1. ºnder (A1)—(A7), there exist constants Em, ºm such that as 1/¹0 is small enough, (4.7)—(4.9) has a unique solution o"exp(f)3C3,a()1) for all (S, º)3A]D. Moreover,

0(o1)o(x), ∀x3), (6.1)

EoEC3,a()1))

k3

A

¹0E1 ,¹0ºmE1,¹0EmE1

B

KK

KK

)k4

A

ºm, Em,_E11 ,E ln(oD)EC3,a,EZ, ln ZEC1,a)

]

A

#E+ ln(oD)EC_2,a

B

where k3, k4 are continuous functions of its arguments.

Proof. Let us take constants Em, ºm such that the following hold:

1. There exists a constanto3 satisfying 0(o3)S(x) ∀x3), S3A, (6.4)

2. 0(o1)(!¹0Sd/e2)F3)4(S, º))o2, ∀(S, º)3A]D

Next, we look at the second-order terms of (4.7). Setting Aij"dij!mºiºj/¹0S, then the eigenvalues of (Aij) are 1, 1, 1!mDºD2/¹0S. If 1/¹0 is small enough, then

1!mDºD2(x)/(¹0S)*os'0 for (, º)3A]D and some constant os. (6.6)

Also if 1/¹0 is small enough, by (A1) and (A2) and Remark 1, u(q)(2n/(3#a) for q3!M1X!M2. By Lemma 5.2, (4.7) and (4.8) has a unique solution f3C3,a()1) and f satisfies

ln(o1))f(x))ln(o2), (6.7)

EfEC3,a()1))

c1

A

¹0E1 ,¹0ºmE1,¹0EmE1

B

(6.8) and Lemma 5.2. K

Next, we consider the solvability of (4.10) and (4.11).

Lemma 6.2. ºnder the same assumptions as ¸emma 6.1, (4.10) and (4.11) has a unique solution E3C3,a()1) for all (S, º)3A]D and o from (4.9). Moreover,

E+EEC2,a()1))

k5(E+o/oEC1,a, ºm, EEDEC1,a) (E+EDEC2,a#

1/¹0), (6.9)

where k5 is continuous function of its arguments.

Proof. By (A1)—(A2) and Lemma 5.1, (4.10) and (4.11) has a unique solution E3C3,a()M) and

EE!EDEC3,a()M))

c3(E+o/oEC2,a, ºm, EEDEC1,a) (E+EDEC2,a# 1/¹0), which implies (6.9). K

By º3D ando, E from above two lemmas, we are ready to solve (4.12)—(4.14). Let us also define an operator M(wJ )"w as follows: Given wJ, solve the following for P;

*P"!+E · wJ/b2!+·(+E]º/b2), x3), (6.10)

+P · nD_!1"0. PD

!2"0 (6.11)

By P from (6.10) and (6.11), compute from

(º ·+)w#(+ · º)w!(w · +)º#Ew b2"!+P! +E]º b2 #¹0+o]+ E mo , (6.12) wD_!1" wD. (6.13)

Lemma 6.3. Besides the assumptions of Lemma 6.1, if+ · º/2#Ew/b2'0 in ), then a solution of (4.12)—(4.14) is a fixed point of (6.10)—(6.13) and vice versa.

Proof. This lemma only shows the equivalence between a solution of (4.12)—(4.14) and a fixed point of (6.10)—(6.13). Existence of a fixed point of (6.10)—(6.13) is in next lemma.

Taking the divergence of (4.12), we get (6.10). So one side is done. The other side is equivalent to show that+ · w"0 in ). Taking inner product of (6.12) and n on !1, we obtain (by (A6)—(A7), (6.11), and using a similar argument in section 3)

ºn(n ·+wn)#ºnwkS(n·+)sk, nT!wk(sk·+ºn)!wkºnS(sk·+)n, nTD!1"0, where ºn :"º ·n. By (A6) (i.e. (3.8)), we see +· wD!1"0. Taking the divergence of (6.12), we obtain, by (6.10),

º+(+ · w)#(+ · º#E/b2)+·w"0 in ), (6.14)

+ · w D_!1"0. (6.15)

Multiplying (6.14) by+ · w and doing integration by parts, we see that (6.14)—(6.15) implies+ · w"0 in ) because + · º/2#Ew/b2'0. So a fixed point of (6.10)—(6.13) is a solution of system (4.12)—(4.14). K

We now prove the solvability of the system (4.12)—(4.14).

Lemma 6.4. ºnder (A1)—(A7), there is a continuous functioni2 such that if

0(ks((1!diam)/cm)E1/b2!i2(ºm 1/vd) for some constant ks (6.16)

(Em, ºm are chosen as Lemma 6.1 and cm is defined in Remark 2) and if 1/¹0 is small, then (4.12)—(4.14) has a unique solution w3¼2,p()) and

EwEW2,p()))

k6

A

1

E1,E+¹DEC2,a()M)

B

2,p(!1)), (6.17)

where k6 is a continuous function of its arguments.

Proof. By Lemma 6.3, it is equivalent to proving the existence of a fixed point of system (6.10)—(6.13). Given wJ 3¼2,p()), by (A1)—(A2) and Lemma 5.1 (6.10) and (6.11) is uniquely solvable, and

EPEC3,a)

c4E+EEC2,a(EwJEW2,p#EºEW_3,p). (6.18)

By Lemma 5.4, (A5) and (6.16), we see (6.12) and (6.13) has a unique solution

w3¼2,p()) and

EwEW2,p()))

c5(EEEW2,p, ºm, 1/vd, 1/ks) (Er.h.s. of (6.12)EW2,p()1)#EwDEW

2,p(!1)).

(6.19) So we can define a map M : ¼2,p())P¼2,p()) by M(wJ)"w.

Next, we claim M is a contractive map. If w*

1, w*2 are given and if M(w*_{1)"w1, M(w}*_{2)"w2, then by (6.10) and (6.11).}

EP1!P2EC3,a)

c6E+EEC2,aEw*1!w*2EW2,p. (6.20)

By (6.12), (6.13) and (6.20),

Ew1!w2EW2,p)c7(EEEW_2,p, ºm, 1/vd, 1/ks)E+EEC_2,aEw*1!w*2EW_2,p. Because 1/¹0 is small, M is a contractive map.

Therefore, the fixed point of (6.10)—(6.13) exists uniquely. If w is the fixed point of (6.10)—(6.13), by (6.19), we see w satisfies (6.17). K

Next, we prove operator F is a map from A]D to itself.

Lemma 6.5. ºnder (A1)—(A7), there exist two continuous functionsi1, i2 such that if 1. Em, ºm (chosen as Lemma 6.1)+2

i/1E»DEW3~1@p,p(!i)and E1 satisfy

(a) i1(1/o1,ElnoDEC_3,a,EoDEC_3,a,EZ EC_1,a)+2

i/1E»DEW3~1@p,p(!i)(ºm, (b) (6.16) hold,

2. 1/¹0#E+ ln(oD)EC2,a()1)#EwDEW

2,p(!1)is small,

3. E+¹DEC_2,a()1)is bounded,

the operator F of (4.7)—(4.21) is a map from A]D to itself.

Proof. For any (S, º)3A]D, if 1/¹0 is small by Lemma 6.1, (4.7)—(4.9) has a unique solution o3C3,a()1) and (6.1)—(6.3) hold. If 1/¹0 is small and if E+¹DEC2,a()1) is

bounded, by Lemma 6.2, (4.10)—(4.11) has a unique solution and E+EEC2,a()1))

Em, E'0, (6.21)

i.e. E3A. By Lemma 6.4, system (4.12)—(4.14) has a unique solution w3¼2,p()) and w satisfies (6.17). With w from (4.12)—(4.14), by Theorem 10.3 of [15] and (A2), the system (4.15)—(4.17) is uniquely solvable,p3¼3,p()), and

EpEW3,p()))c8EwEW_3,p()). (6.22)

Next, we consider (4.18)—(4.20). By (A5), Lemmas 5.3, 6.1, and let 1/¹0 small, we obtain

EtEW4,p()))

c9(1/o1, EoEC3,a) (E+opEW2,p())#+

i EoD»DEW3~1@p,p(!i)

). (6.23)

Let »"!+t#p. By (6.22)—(6.23), if 1/¹0 is small enough, E»EW3,p)k1(1/o1, Eln oDEC_3,a,EoDEC_3,a,EZEC_1,a) (EwEW_2,p

#+

i E»DEW3~1@p,p(!i) ).

By (6.17) and above assumptions, as long as 1/¹0#E+ ln(oD)EC2,a()1)#EwDEW

2,p(!1)is

small enough,

E»EW3,p_)ºm. (6.24)

So »3D. By (6.21) and (6.24), we conclude that F maps A]D to itself. K Next we show F is a continuous map in C1,a]¼2,p. If it is true, by Schauder fixed point theorem we know the operator F has a fixed point in A]D.

Lemma 6.6. By assumptions of Lemma 6.5, F is a continuous map in C1,a]¼2,p. F is a contractive map if 1/¹0#E+ ln(oD)EC2,a()1)is even smaller.

Proof. This is shown by a straightforward way, so we only sketch the proof. First let us given a notation:

c11"1/¹0#E+ ln(oD)EC2,a()1),

c12 is a constant which depending on EoD, EDEC3,a()1), ºm, 1/o1, 1/E1,EZEC1,a()1),

1/vd, 1/ks.

Given (Sa, ºa), (Sb, ºb), by solving (4.7)—(4.21), we obtain (fa, oa, Ea, wa, pa, ta, »a) and (fb, ob, Eb, wb, pb, tb, »b). F(Sa, ºa)"(Ea, »a), F(Sb, ºb)"(Eb, »b). By (4.7)—(4.9), we can derive Efa!fbEC2,a)c12 ¹0[ESa!SbEC1,a #Eºa!ºbEC_1,a]. (6.25) By (4.10)—(4.11) and (6.25), we obtain EEa!EbEC2,a)c12 ¹0[ESa!SbEC1,a #Eºa!ºbEC_1,a]. (6.26)

By Lemmas 5.4, 6.1, 6.2, 6.4, and (6.25)—(6.26), we have

Ewa!wbEW1,p)

c11c12[ESa!SbEC1,a#Eºa!ºbEW_2,p]. (6.27)

By Reference 15 and (4.15)—(4.17),

Epa!pbEW2,p)

c13Ewa!wbEW1,p. (6.28)

By Eq. (4.18) and Lemma 5.3, we get the estimate

E+(ta!tb)EW2,p)c14(1/o1, oD) [Epa!pbEC_1,a#Eoa!obEC_2,a]. (6.29) By (6.27)—(6.29),

E»a!»bEW2,p)

c11c12[ESa!SbEC1,a#Eºa!ºbEW_2,p]. (6.30)

By (6.29) and (6.30), we see that F is a continuous map in C1,a]¼2,p. If c11 is even smaller, F is a contractive map. K

Proof of ¹heorem 6.1. By Lemmas 6.5, 6.6, we know that F maps A]D to itself and that F is continuous in C1,a]¼2,p. Since A]D is a compact, convex subset of

C1,a]¼2,p, by Schauder fixed point theorem, a fixed point exists. If 1/¹0#

E+ ln(oDEC2,a()1)is small enough, fixed point of F exists uniquelybecause F is

contrac-tive map.

Next, we show a solution of (1.1)—(1.7) corresponds to a fixed point of (4.7)—(4.21) and vice versa. Suppose (o, », ¹, () is a solution of the system (1.1)—(1.7), by tracing the derivation of (4.7)—(4.21), it is easy to see that (¹/¹0, ») is a fixed point of the system (4.7)—(4.21). On the other hand, let us assume that (E, ») is a fixed point of (4.7)—(4.21). Define ¹"E¹0'0. By (4.9) and (4.21), we obtain (o, V, T). (o, », ¹) satisfy equations (1.1), (1.3), (1.6)2, (1.7)1, (1.7)2 by (4.18), (4.10), (4.11), (4.19), (4.14)1 respectively. By (4.15) and (4.21), (4.12) can be written as

curl

C

mo+(o¹)#

¹»

b2¹0

D

By (4.14)2,3, we see (6.31) implies P"0 in (4.12). Since domain is simply connected, (6.31) implies there is a function( such that

(» ·+)»# 1 mo+(o¹)! e m+(# ¹» b2¹0"0, (6.32)

i.e. (1.2). Next taking divergence of equaton (6.32), and comparing with (4.7), we obtain equations (1.4) and (1.6)1. Therefore, we conclude a fixed point (¹/¹0, ») of (4.7)—(4.21) also corresponds to a solution (o, », ¹, () of (1.1)—(1.7). Uniqueness of the fixed point of (4.7)—(4.21) is equivalent to uniqueness of the solution of (1.1)—(1.7).

Because of the fixed point (¹/¹0, »)3A]D, solution (o, », ¹, () of (1.1)—(1.7) satisfyo, ¹3C3,a()M), », (3¼3,p()). Because of (6.6), we see that solutin (o, », ¹, () satisfy D» D2(x))¹(x)/m for all x3), i.e. the solution (o, », ¹, () is a subsonic solution. So we complete the proof.

7. Proof of auxillary lemmas

In this section, we prove Lemmas 5.2—5.4. Lemma 5.2 is proved by employing the Leray—Schauder fixed point theorem [7] and Lemma 5.1, Lemma 5.3 is proved by results in [11] and method of continuity [7]. Lemma 5.4 is proved based on Lemmas 7.1—7.3. Lemma 7.1 is an extension theorem, and its proof is similar to that of Theorem 7.25 in [7]. By energy method, Lemma 7.2 gives a priori estimate for a transport equation in a bounded smooth domain. Then an existence result of a transport equation in a bounded smooth domain is shown in Lemma 7.3. Finally, by Lemma 7.3, we prove Lemma 5.4.

Proof of ¸emma 5.2. Set 0(c)a, Km"max(DuD, DuND), KM"sup­)DuDD#Km,

yKM(x) :"

G

if D y(x)D'KM, if D y(x)D)KM.

Next, define a map M : C0,c()M)][0, 1]PC0,c()M) by M(y, t)"z, where z is the solution of

L(z) :"aijz,ij#aiz,i"t(a(x)g(yKM)!f (x)) in),

zD_­)"tuDD_­).

Observe that if yn convergences to y in C0,c, then yn,KMalso converges to yKMin C0,c.

By Lemma 5.1, we set that

EzEC2,c)tc(j, Eaij, aiEC_0,a) (EuDEC_2,c#Ea(x)g(y

KM)EC0,c#E f (x)EC0,c).

So M is a continuous and compact operator. If zt is a fixed point of the following system:

L(zt)"t(a(x)g(yKM)!f (x)),

ztD­)"tuDD_­),

then zt3C2,c()1) [2]. By De Giorgi—Nash theorem [7], there is a positive number l such that EzlEC0,l is bounded, and the bound is independent of y, t. Note l is independent ofc, so we may assume c"l. Therefore by Leray—Schauder fixed point theorem [7], a fixed point,u, of M( · , 1) exists.

Suppose the fixed pointu satisfies Du(x)D)KM in ), then u is a solution of (5.3).

Next we proveDu(x)D)KM in ).

(a) Claim. Ifu is a fixed point of M( · , 1) then Du(x) D)KM in ).

Proof. The set )`-) of points at which u(x)'Km holds is open in ), and the boundary of )` consists of points x at which either u(x)"Km or the point is

contained on ­). We assume that )` is non-empty. Let x*3)`. We denote the

maximal connected component of)` containing x* by )*`. Define

uKm,KM(x) :"      KM if u(x)'KM, u(x) if Km)u(x))KM, Km if u(x)(Km. Then,u D_)* `satisfies L(u)"a(x)g(u<sub>Km, KM</sub>)!f (x) in )*`, uD­)*_`"uDD_­) or Km.

Note a(x)g(Km)*f (x), so L(u)*0. We obtain sup)*_{`</sub>u(x))sup­)*<sub>`}u(x) by maximal

principle [7]. Therefore sup_)*

`u(x))KM, which implies u(x))KM in ). A similar

argument can be used to prove the other side, i.e.u(x)*!KM in ). Therefore, we conclude thatEuEL=₍₎₎)

KM.

(b) By Claim (a), the fixed point u is a solution of (5.3) and is in C2,l₍)1). So

g(uK

Uniqueness of solution is obtained by the maximal principle [7]. The upper and lower bounds ofu(x) are obtained as follows: Define I :"min(u1, u), so

aij(x)(u!I),ij#ai(u!I),i!a(x)(g(u)!g(I))"!f (x)#a(x)g(I), u!ID­)*0.

Since !f (x)#a(x)g(I))0 for all x3), min(u1, u))u(x) by the maximal principle. By a similar argument, we can proveu(x))max(u2, u6) for all x3).

By (5.2), we see that solutionu satisfies

EuECm`2,a)c(j, Eaij, ai, aEC<sub>m,a</sub>) (EuDEC<sub>m`2,a</sub>#E f EC_m,a#Eg(u)EC_m`1). (7.1) By interpolation inequality [7, p. 176] and (7.1), one can derive

EuECm`2,a)c(j, Eaij, ai, aEC<sub>m,a</sub>) P(EuDEC<sub>m`2,a</sub>E f EC_m,a,EgEC

m`1(-1)

#Eg~1EC

0(-2)). K

Proof of ¸emma 5.3. This lemma will be proved by method of continuity [7]. We first consider the case g"0. Define

B"

G

K

P

H

G

K

P

H

Then B and V are Banach spaces. Let L0u:"*u and L1u :"+ · (a(x)+u). Then L0 and L1 are bounded linear operators from B to V. Define Lt, t3[0, 1], as follows:

Ltu :"(1!t)*u#t+ · (a(x)+u).

By [11], L0 is a one-to-one and onto map. Suppose EuEWm`2,p)cELtuEW_m,p, by method of continuity, L1 is also one-to-one and onto. Then the theorem holds true.

To show thatEu EWm`2,p)cELtuEW_m,pfor allu3B, we note

*u" Ltu

(1!t)#ta(x)!

t+a(x)

(1!t)#ta(x)+u.

By [11] and interpolation theorem [7],

EuEWm`2,p)c(EaEC<sub>m`1,a</sub>, 1/j) ( ELtuEW_m,p#EuEL_p). (7.2) Next, we want to showEuE¸p)cELtuEWm,p for allu3B. If not, then there exists

a sequenceMun,t

nNLB such that Eun,tnE¸p"1 and ELt

nun,tnEWm,pP0. By (7.2),

Eun,tnEWm`2,p)c(EaECm`1,a, 1/j) ( ELtnun,tnEWm,p#Eun,tnE Lp),

i.e.Eun,t

nEWm`2,p is bounded. Since tn3[0, 1], we assume tnPt*. Because ¼m`2,p is

a reflexive Banach space [3], there exists a subsequenceMu@n,t

u@n,tnPu*3¼m`2,p()) weakly. One can see Eu*ELp"1. However,

Lt*u*"0 in ),

+u* · n"0 on !1X!2,

P

(7.3) Equation (7.3) impliesu*"0 [9], which is a contradiction. So EuEL_p)cELtuEW_m,p. Therefore, by (7.2), we have

EuEWm`2,p)c(EaEC<sub>m`1,a</sub>, 1/j)ELtuEW_m,p, ∀u3B. (7.4) So L1: BPV is an one-to-one and onto map, that is, solution of (5.4) exists uniquely for g"0. By (7.4), (5.5) holds for the case g"0.

We now consider the case gO0. By [11], one can find G3¼m`2,p()) such that +G · nD<sub>!i</sub>"g/a, i"1, 2, EGEW<sub>m`2,p())</sub>)c(+ Eg/aEW

m`1~1@p,p(!i)). (7.5)

Consider the following

+ · (a(x)+uL)"f!+ · (a(x)+G), in ), a(x)+uL · n"0, on !1X!2,

P

(7.6) By the result of previous case g"0, (7.6) has a unique solutionuˆ3¼m`2,p()) and by (7.4) and (7.5),

EuˆEWm`2,p)c(EaEC<sub>m`1,a</sub>, 1/j) ( E f EW_m,p#+ EgEW

m`1~1@p,p(!i)E). (7.7)

Let us define

u"uˆ#G! 1 D )D

P

G dx,

whereD)D is the volume of ). Then u3¼m`2,p()) is the unique solution of (5.4), and it is easy to check, by (7.7), (5.5) holds. Thus the conclusion of this lemma follows. K Remark 3. Next, we give an extension theorem for a domain with edge. Let)LR3 be a Ck~1,1 domain with an edge ¸ (see Remark 1). Then we can find a bounded smooth domain)@ such that (1) )L)@; (2) !1L­)@; (3) for all q3¸"!11W!12, there exists a neighborhood N(q) of q such that N(q)W) is smoothly (Ck~1,1) homomorphic to the intersection between a unit ball and a quadrantR3/4 :"M(x1, x2, x3)Dx1'0, x3'0N; (4) N(q)W)@ is smoothly (Ck~1,1) homomorphic to the intersection of a unit ball and the half-spaceR3`:"M(x1, x2, x3)Dx1'0N.

Lemma 7.1. For any ) and )@ in Remark 3, there exists a bounded linear operator E : ¼k,p())P¼k,p()@) such that E(º)(x)"º(x) for x3) and

EE(º)EWk,p()@))cEºEW_k,p()), ∀º3¼k,p()), c"c(k,), )@).

Proof (see Theorem 7.25 [7]). As in (7.56) of [7], we define an extension in half-space as follows: E0º(x)"      º(x), k + i/1ciº(x@,!x3/i), x3'0, x3(0,

where x"(x@, x3) and c1, x2,2,ck are constants determined by the system k

+

i/1ci(!1/i)m"1, m"0, 1,2, (k!1). If º3C=(R3/4)W¼k,p(R3/4), then

E0º3Ck~1,1(R3`)W¼k,p(R3`) and EE0ºEWk,p(R3`₎)cEºEW_k,p(R3/4). By approximation, one see that the domain of E0 can be extended to ¼k,p(R3/4), i.e.

E0 : ¼k,p(R3/4)P¼k,p(R3`) and EE0ºEWk,p(R3`₎)cEºEW_k,p(R3/4).

Then, by partition of unity and following the argument of Theorem 7.25 [7], we can

show this lemma true. K

Next, we derive a priori estimate for a transport equation. Domain) considered in the next two lemmas are smooth domains,!~ is a closed subset of ­), Mb, º, f, gN are vector functions, and" is a matrix function.

Lemma 7.2. Consider the system (b ·+)º#"º"f, ),

º"g, !~, (7.8)

where!~:"Mx3­)Db·n(0N. If the following conditions are satisfied

1. ) is a bounded smooth domain, H1())˝¸p()), ¼1,p())˝¸=()),

2. b3¼l

`1,p()), 2)l, b · n(!j(0 on !~, 3. f,"3¼l,p()), g3¼l

,p(!~), º3Hl`1()),

4. "(x)'"mI in ), "m!i1(p,EbEWl_`1,p()), 1/j)""d'0,

wherej, "m, "d are constants, I is identity matrix, and i1 is a continuous function of its arguments, then the following estimate holds, 0)s(l:

EºEpWs,p()))c( p,E"EWl_,p,EbEWl_`1,p, 1/j, 1/"d)(E f EpW_s,p())#EfEpW

Proof. By means of a partition of unity, a local co-ordinate change, and Lemma 1.1.1 in [10], it is sufficient to prove (7.9) in a half-space. Estimate in a half-plane is obtained by differentiating the equation (7.8)1 to estimate tangential derivatives and then, using b · nO0 on!~ to solve for the normal variables in terms of the tangential ones. For convenience, we assume assumptions of this lemma hold in a half place and­)"!~. We use the following notations: b"(b1, b@)"(b1, b2, b3), )"MxDx"(x1, x@)" (x1, x2, x3), x1*0N, ­sx@º" + Dc D)s ­c2 ­xc2 2 ­c3 ­xc3 3 º, ­s x1, x@º" + Dc D)s ­c2 ­xc2 1 ­c3 ­x@c3º, DcD"c2#c3, ci*0.

Consider the following in a half-space; Lº :"(b ·+)º#"º"f, ºDx

1/0"g. (7.10)

We prove the inequality (7.9) by method of induction.

Case s"0: Multiply (7.10)1 by DºDp~2º and integrate over ):

P

P

P

fDºDp~2º dx. (7.11)

Note

P

P

P

(+ · b)Dº Dp dx. (7.12) By (7.11) and (7.12) and Ho¨lder’s inequality,

P

("!(+ · b)/p!( p!1)/p)DºDp dx)1/p E f EpL_p#1/p

P

!~D b · n EgDp dx@.

(7.13) Case s"1: Differentiate (7.10)1 with respect to ­x@, we have

(b ·+)­x@º#"­x_@º"­x_@f!(­x_@b ·+)º!(­x_@")º. (7.14) Applying (7.13) to (7.14), we obtain

P

P

#c( p)E(­x_@b ·+)ºEpL_p#c( p)E(­x_@") ºEpL_p. (7.15) By (7.10)1,

By (7.15) and (7.16), we obtain

("m!i1(p, 1/j, EbEW_1,=())))E­x_@ºEpL_p)

P

!~Db · nD D ­x@

gDp dx@

#c( p, 1/j, E+bEL₌)E f EpW_1,p#c( p,E"EW_1,=, 1/j, E+bEL₌)EºEpL_p. (7.17) By (7.16) and (7.17),

"dE­xºEpLp)c( p,E"EW_1,=, 1/j, EbEW_1,=)

A

P

!~D­x@

gDp ds

B

EºEpW1,p)c( p,E"EW_1,=, 1/j, EbEW_1,=, 1/"d)(E f Ep¼1,p#EgEp¼1,p(!~)). (7.19)

So we prove (7.9) for cases s"0, 1.

Suppose (7.9) holds for k"0, 2 , s!1, we plan to show (7.9) holds for k"s, s)l. Differentiate (7.10)1 with respect to ­sx@, then

(b ·+)­sx@º#"­sx{º"­sx_@f#[L,­sx_@]º, where [L,­sx@] º :"(b ·+)­sx@º#"­sx{º!­sx_@((b ·+)º#"º). By (7.13),

P

P

gDp dx@#c(p)E­sx@fEpLp#c( p)E[L, ­sx_@] ºEpL_p, which implies

("m!c(p, 1/j, EbEWl_`1,p))E­sx_@ºEpL_p )c( p)

P

!~D b · n D D ­sx@

gDp dx@#c(p)E­sx@fEpLp

#c( p,EbEWl_{`1,p</sub>,E" EWl<sub>,p</sub>)EºEpW<sub>s~1,p</sub>#c( p,EbEWl<sub>`1,p})E­sx

1x@ºEpLp. (7.20)

By (7.10)1, we have ­s~1_{x ­x1}º"­s~1

x@ ( f!b@­x@º!"º)/b1. (7.21)

So

E­s~1_x@ ­x₁ºEL_p)c(1/j, EbEWl_{`1,p</sub>,E"EWl<sub>,p</sub>) (EºEW<sub>s~1,p</sub>#E f EW<sub>s~1,p</sub>) #c(1/j, EbEWl<sub>`1,p})E­sx_@ºEL_p.

Moreover, one can show

E­sx1,x@ºELp)c(1/j, EbEWl`1,p,E"EWl,p) (EºEWs~1,p#E f EWs~1,p)

#c(1/j, EbEWl_`1,p)E­sx_@ºEL_p. (7.22)

By (7.20), (7.22), and assumption of method of induction,

"dE­sx@ºEpL_p)c( p,EbEWl_`1,p,E"EWl_,p, 1/j, 1/"d)(E f EpW_s,p#EgEpW

s,p(!~)).

(7.23) By (7.22) and (7.23),

EºEpWs,p)c( p,E" EWl_,p,EbEWl_`1,p, 1/j, 1/"d)(E f EpW_s,p#EgEpW

s,p(!~)).

So we prove (7.9). K

Lemma 7.3. Besides assumptions 1, 2, 4 of Lemma 7.2, if 1. f,"3¼l_{,p()), g3¼}l

,p(!~),

then the system (7.8) has a unique solution º3¼l_{,p()) and}

EºEpWl_,p()))c( p,E"EWl_,p,EbEWl_`1,p, 1/j, 1/"d)(E f EpW_s,p())#EgEpW

s,p(!~)).

(7.24) Proof. By density theorem [7], we can choose the sequences Mbk, "k, fkNLC=()1), MgkNLC=(!~) such that

bkPb in ¼l

`1,p()), "k, fkP", f in ¼l

,p()), gkPg in ¼l

,p(!~) as kPR. Let k be large enough such thatMbk, "k, fkN satisfy the assumptions of Lemma 7.3. Then for each k, the system

(bk · +)ºk#"kºk"fk in ), ºk"gk on !~,

has a unique weak solution ºk3Hl`1()) [12]. By Lemma 7.2, the following holds:

EºkEpWl_,p)c( p,E"kEWl_,p,EbkEWl_`1,p, 1/j, 1/"d)(E fk EpWl_,p#EgkEpWl

,p(!~)).

Since MEbkEWl_`1,p,E"kEWl_,p,E fk EWl_,p,EgkEWl_,pN are bounded, EºkEWl_,p()) is bounded. Because ¼l

,p is a reflexive Banach space [3], there is a subsequence Mº@kN

of MºkN such that º@kPº* weakly in ¼l

,p, which implies Eº*EWl_,p) lim infk?= EºkEWl_,p. Passing k to R, we see that º* is a classical solution of (7.8) and satisfies (7.24). K

Based on the results of Lemmas 7.1—7.3, we now prove Lemma 5.4.

Proof of Lemma 5.4. First, we extend the domain) to a smooth simply connected domain)@ (see Remark 3) such that !1L!@1L­)@. For function b, we extend it to b* in)@, by Lemma 7.1, such that !@1"Mx3­)@Db*(x)·n(0N,

Eb*EWl_{`1,p()@)</sub>)cEbEWl<sub>`1,p())}, b* · nD

We also extend f," to f *, "* in )@ and g to g* on ­)@ such that E f *EWl_,p()@))cE f EWl_,p()), Eg*EWl

,p(!@1))cEgEWl

,p(!1),

E"*EWl_,p()@))cE"EWl_,p()), "*(x)'"m'0, ∀x3).

Furthermore, b*, f *, "*, g* satisfy the assumptions of Lemma 7.3. Consider the following

(b* ·+)º*#"*º*"f * in )@, º*"g* on!@1.

Then, by Lemma 7.3, there exists a solution º*3¼l_{,p()@) in the above equation and}

Eº*EpWl_,p()@))c( p,E"*EWl_,p,Eb*EWl_`1,p, 1/j, 1/"d)(E f *EpWl_,p()@)#Eg*EpWl_,p(!@1)). Then º"º*D) is the required solution. Note that Eº*EpWl_,p()@))Eº*EpWl_,p()@). Next by adjusting the coefficients of the above inequality, we obtain estimate (5.6) for s"l case. Other cases of estimate (5.6) can then be obtained by tracing the proof of Lemma 7.2. Uniqueness can be proved simply by energy method. K

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