MATHEMATICAL METHODS IN THE APPLIED SCIENCES Math. Meth. Appl. Sci. 2003; 26:247–271 (DOI: 10.1002/mma.353) MOS subject classi
cation: 35 Q 20; 76 P 05; 82 A 70; 78 A 35; 41 A 60

On spherical harmonics expansion type models for electron–phonon collisions J.-P. Bourgade∗; † MIP; UMR 5640 (CNRS-UPS-INSA); Universit
e Paul Sabatier; 118; route de Narbonne; 31062 Toulouse; Cedex; France

Communicated by H. Neunzert SUMMARY In this paper, we give the rigorous derivation of a diusion model for semiconductor devices, the starting point being a microscopic description of electron transport by means of a kinetic equation of Boltzmann type. The limit of a small mean free path at a large time leads to a diusion equation of ‘SHE’ type (spherical harmonics expansion). We deal with a collision operator that models interactions between electron and phonons. This induces a peculiar form for the diusion tensor: electron–phonon collisions happen to be discontinuous in energy and inelastic, and, as a consequence, the diusion tensor appears as an in
nite dimensional matrix. Copyright ? 2003 John Wiley & Sons, Ltd. KEY WORDS:

semiconductors; Boltzmann equation; spherical harmonics expansion; diusion approximation; electron–phonon collisions

1. INTRODUCTION The evolution of electrons in semiconductor devices can be described at a microscopic level through kinetic equations of Boltzmann type. To that purpose, one introduces the distribution function f(x; v; t) of the electrons, where x ∈ R3 and v ∈ R3 are the electron position and velocity respectively, and t¿0 denotes the time, so that f(x; v; t) d x d v is the number of electrons at time t in an elementary phase-space volume d x d v. Then f is given as the solution of the initial value problem
@t f + v · ∇x f + E · ∇v f = Q(f); x ∈ R3 ; v ∈ R3 ; t¿0 (1) f(x; v; 0) = fI (x; v) ∗

Correspondence to: J.-P. Bourgade, MIP, UMR 5640 (CNRS-UPS-INSA), Universite Paul Sabatier, 118, route de Narbonne, 31062 Toulouse, Cedex, France. † E-mail: [email protected] Contract=grant sponsor: TMR network; contract=grant number: ERB FMBX CT97 0157

Copyright ? 2003 John Wiley & Sons, Ltd.

Received 20 December 2001

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J.-P. BOURGADE

where E = E(x; t) is a force
eld (assumed to be known) and fI an initial condition. The collision operator Q(f) represents the interactions of the particles with the medium, namely, in our case, the collisions of electrons with phonons, which are a quantization of the lattice vibration in the semiconductor. Classically (see Reference [1] for instance), in a low density case, this operator is linear and can be written as  [s(v ; v)f(v ) − s(v; v )f(v)] d v Q(f) = v
∈R3

with s(v; v ) = (v; v )[(N0 + 1)
( −  + ˝!0 ) + N0
( −  − ˝!0 )] where s(v ; v) d v is the transition probability from velocity v to an elementary volume d v about v, N0 is the phonon occupation number given by the Bose–Einstein statistics: N0 = (e˝!0 =kB T − 1)−1 , so that N0 + 1 = e˝!0 =kB T N0 , kB is Boltzmann’s constant, T is the lattice temperature, ˝ is the reduced Planck constant, !0 is a constant approximately equal to the (variable) pulsation of the semiconductor lattice vibrations, so that ˝!0 stands for the phonon energy,  = |v|2 =2 is the kinetic energy of an electron of velocity v. For the sake of simplicity, we suppose that kB T = 1 and we introduce the following transition rate:


(v; v ) = N0 (v; v )e(+ )=2 e(˝!0 =2) The collision operator is
nally written as 
Q(f) = (v; v )[e− f(v ) − e− f(v)][
( −  − ˝!0 ) +
( −  + ˝!0 )] d v v
∈R3

Our aim is to derive from a rescaled version of Equation (1) a macroscopic model of SHE type:  ˜ ˜ · J = N ()Q(F) N ()@t F + ∇  J () = − Dk ()∇˜ F( + k ˝!0 ) k √ where J (x; ; t) is the current of electrons of energy ¿0, N () = 2, F(x; ; t) is the density of electrons of energy , ∇˜ = ∇x +E@=@ is a generalized gradient in the position-energy space, ˜ Q(F) is an operator depending on the collision operator Q, and (Dk ())k is a sequence of diusion tensors. Such diusion asymptotics have been widely investigated in the recent past. The mathematical theory goes back to [2], where the general method relies on an expansion of the solution in powers of a small parameter. This approach has been extensively used in diusion limits of kinetic equations since Hilbert and Chapman-Enskog (see Reference [3]), in particular in semiconductor physics (see References [4–6]), but also in neutron transport (as in Reference [7] or [8]) and in radiative transfer (see Reference [9]). In the case of electron–phonon interactions in semiconductor devices, diusion limits leading to equations depending on the energy have been considered in Reference [10] by means of an explicit diagonalization of the collision operator in terms of spherical harmonics. The mathematical analysis of this scattering operator was developed in Reference [11]. Copyright ? 2003 John Wiley & Sons, Ltd.

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Spherical harmonics expansion (SHE) models appeared to be a good compromise between a very accurate description of physical phenomena by kinetic models and less numerically expensive macroscopic models such as drift-diusion or energy-transport models. Whereas kinetic equations deal with functions of seven variables (six variables in the phase space, and time) and classical macroscopic models deal with functions of four variables (three in the position space, and time), SHE models introduce an intermediate one dimensional variable replacing the velocity and which appears as the kinetic energy associated to the velocity of a particle. This makes of SHE models quite reliable models at a rather low numerical cost. (For a formal derivation of SHE models and their situation in a hierarchy of models from the Boltzmann equation to drift-diusion equations, see References [6,12], for their application to semiconductor physics, see e.g. References [13,14].) A second step in the mathematical approach to these models was to attempt to take the inelastic nature of common collisions into account. This was made on both formal and rigorous points of view in [15] and led to ‘coupled SHE models’, i.e. models of SHE type with a diusion matrix coupling dierent levels of energy—and thus rendering the non-locality in energy of inelastic collisions. Our purpose is to realize the derivation of a coupled SHE model in the case of electron– phonon collisions. More precisely, we suppose that the typical energy of electrons is not too large compared with the phonon energy ˝!0 . This is the case when the potential dierence applied to the semiconductor is low enough. Then the inuence of interactions with phonons on the energy of the electrons cannot be neglected. It follows from the non-classical nature of these collisions that the energy transfers are discontinuous: the interaction with a phonon of energy ˝!0 swaps the electron energy  for the energy  = + ˝!0 or  =  −˝!0 whether the electron absorbs or emits a phonon, respectively. This induces the main dierence between the collision operator we will consider and that studied in Reference [15]. The shape of the diusion matrix in the current equation is somewhat modi
ed. In Reference [15] the diusion equation was of the form  ∞ (;  )∇˜ F( ) d  J () = −D()∇˜ F() − 0

so that the current J () depended on the value of the generalized gradient of F at any energy level  ¿0, while, in our case, it will take the following form  Dk ()∇˜ F( + k ˝!0 ) (2) J () = − k

The series introduces the coupling of the energy level  at which we evaluate J with all the energies (and only these) that could be reached by absorbing or emitting phonons of energy ˝!0 . We recover the non-locality of the electron–phonon interaction, the current J at the energy level  being inuenced by the gradient of the density F at each level of energy  + k ˝!0 (k ∈ Z s.t.  + k ˝!0 ¿0). Section 2 is devoted to general properties of the collision operator. We introduce two splittings of this operator into ‘elastic’ and ‘inelastic’ parts, which will not lead to the same physical and mathematical characteristics. In Section 3, we give a formal derivation of the SHE model. We will see that, combining the two splittings given in Section 2 the right way gives rise to SHE type models with energy coupling. Namely, the current equation will be proven to take the form (2). The computation of the diusion coecient (which is in this case an in
nite dimensional matrix) will make necessary to solve an auxiliary problem. This task will Copyright ? 2003 John Wiley & Sons, Ltd.

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be achieved in Section 4. It will appear as the inversion of an in
nite tridiagonal matrix on an appropriate space of sequences: this is the consequence of the discretization in energy of the operator Q, resulting from the discontinuous nature of electron–phonon collisions. Section 5 will be dedicated to the rigorous proof (in a force-free case) of the convergence of solutions of a rescaled Boltzmann equation to the solutions of the SHE set of equations. Existence and uniqueness for these asymptotic equations will be proven. The central point in this section will thus be to establish Theorem 5.10. The dual structure of the current equation will be underlined. 2. TWO ELASTIC=INELASTIC SPLITTINGS In this section we propose to split the collision operator into two parts: one part will be referred to as the ‘elastic’ part—it does not change the number of particles at a given energy level—, while the other one will be called ‘inelastic’. According to Reference [15], it is useful to consider the convex combination of two splittings in order to obtain more physical accuracy. Indeed, we will see in Sections 3 and 4 that by using such a combined operator we get a coupled (in energy) current equation in a diusion limit, while considering only the operator arising in the
rst splitting (see below the splitting Q0 − Qin0 ) would lead to a ‘classical’ uncoupled SHE system (as in Reference [6] for instance). Since we want to base these splittings on energy considerations, it is relevant to interpret the velocity of a particle in terms of its energy. This is why we introduce the following change of variables in R3 :  v  !=   | v |    2 |v| =   2    √  v = |v|! = 2! = N ()! in order to have, for any integrable function ’(v):    ’(v) d v = ’(; !) d !N () d  v∈R3

¿0

!∈S2

In this change of variables, ! represents the angular part of the velocity, and  the energy of an electron in the frame of a parabolic band diagram approximation. In the remainder of our work, we will assume the cross section  to be symmetric. More accurately, we suppose that (; !;  ; ! ) = ( ; ! ; ; !) = ( ; !; ; ! ) We
rst give a series of notations:  ∀f = f(!), f will denote the angular average of f on the sphere S2 (f = 1=4 !∈S2 √  f(!) d !); N = N () = 2; M = e− ; + =  + ˝!0 , − =  − ˝!0 ;
+ =
( − + ),
− =
( − − ); I = I(˝!0 ; +∞) (), where IA denotes the indicatrix function of the set A; + = + (; !; ! ) = (; !; + ; ! )N (+ ); − = − (; !; ! ) = (; !; − ; ! )N (− )I; ∀f = f(), f = f( ), f+ = f(+ ), f− = f(− ), except, of course, for f =  and for f =
, where the notations are those given previously. Copyright ? 2003 John Wiley & Sons, Ltd.

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With these notations, the collision operator may formally be split into an elastic contribution Qel and an inelastic contribution Qin : 

Q0 (f)(v) = Qel0 (f)(v) = 

Qin0 (f)(v) =

or




¿0

!
∈S2

!
∈S2

(+ e−+ + − e−− )[f(; ! ) − f(; !)] d !


(; !;  ; ! )N  [e− f( ; ! ) − e− f(; ! )]

 [
+ +
− ] d   d !  Q1 (f)(v) = Qel1 (f)(v) = (+ e− [f(+ ; ! ) − f(+ ; !)] !
∈S2

+ − e− [f(− ; ! ) − f(− ; !)]) d !  
1 Qin (f)(v) = (; !;  ; ! )N  [e− f( ; !) − e− f(; !)] 
¿0

!
∈S2

 [
+ +
− ] d   d !

We will see that Q0 has good mathematical properties (positivity for instance) that we lose when considering the operator Q1 . On the other hand, the operators Q0 and Q1 do not keep the same information on the levels of energy (while Q0 operates on f(v) only through the angular part ! of the velocity v and keeps the energy constant, Q1 also operates on the energy level ), so that we can hope to get a coupled-in-energy model by studying a convex combination of these operators: (1 − )Q0 + Q1 , and we can expect to keep the good mathematical properties of Q0 if we take  small enough. We introduce two additional hypotheses on the cross-section : Hypothesis (H1) There exists two positive constants C0 and C1 and a function of  only ’() such that





C0 ’’ e(+ +˝!0 )=2 6(; !;  ; ! )6C1 ’’ e(+ +˝!0 )=2 Hypothesis (H2) There exists a positive constant ’ such that ’¿’¿0 and

’N ¿’¿0

In addition, we underline that if the cross-section  is bounded from above, the operators Q0 and Q1 are bounded in L2 (R3 ; M −1 dv). From now on, we make this hypothesis of boundedness. We note: L2M −1 = L2 (R3 ; M −1 d v) and L2 = L2 (R3 ; d v). Now, we give a few classical properties of the elastic operator Q0 : Copyright ? 2003 John Wiley & Sons, Ltd.

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Lemma 2.1 (i) For any functions f(; !) and g(; !), we have   1 Q0 (f)(v)g(v)M −1 d v = Q0 (g)(v)f(v)M −1 d v = − (+ e−+ + − e−− ) 2 3 3
2 v∈R v∈R ¿0;!;! ∈S



× [f(; ! ) − f(; !)][g(; ! ) − g(; !)]M −1 d ! d ! d 

(3)

As a consequence, −Q0 is a self-adjoint positive operator on the Hilbert space L2M −1 :   dv 1 0 = − Q (f)f (+ e−+ + − e−− )[f(; ! ) M 2 ¿0;!;!
∈S2 v∈R3 −f(; !)]2 d ! d !N

d

M

¿0

(4)

the null-space of Q0 and its orthogonal in L2M −1 are given by N (Q0 ) = {f ∈ L2 (R3 ; M −1 d v): f(; !) = f()}

(5)

N (Q0 )⊥ = {f ∈ L2 (R3 ; M −1 d v): f() = 0 ∀¿0}

(6)

(ii) If we suppose (H1) and (H2), we have the following estimates on Q0 :   ∀f ∈ N (Q0 )⊥ ; − Q0 (f)fM −1 d v¿8C0 ’2 f2 (v)M −1 d v v∈R3

(7)

v∈R3

Therefore, the range R(Q0 ) of Q0 is closed and, by the self-adjointness of Q0 , R(Q0 ) = N (Q0 )⊥ . (iii) For all functions f(; !) and g(), we have Q0 (gf) = gQ0 (f). Remark In fact, all the properties we gave in the functional space L2M −1 would still be valid in L2 , but, as we shall see, it is not the case for the analogous statements for the elastic operator Q1 for which we shall have to work in the weighted space L2 (R3 ; M −1 d v). Proof Equations (3) and (4) follow from a simple computation, (5) and (6) are obvious consequences of (4). Now, we prove (7). For any f ∈ N (Q0 )⊥ , according to (H1), we have (omitting the dependences on  for f):  −

Q0 (f)fM −1 d v = ¿

1 2



(+ e−+ + − e−− )[f(! ) − f(!)]2 M −1 N d  d ! d !

1 C0 2 − C0

 

(’’+ N+ + ’’− N− e˝!0 I)[f(! )2 + f(!)2 ]M −1 N d  d ! d ! (’’+ N+ + ’’− N− e˝!0 I)f(! )f(!)M −1 N d  d ! d !

Copyright ? 2003 John Wiley & Sons, Ltd.

(8)

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253

But, since C0 (’’+ N+ + ’’− N− e˝!0 I) =: K() is independent of !, we have  d − C0 (’’+ N+ + ’’− N− e˝!0 I)f(! )f(!)N d ! d ! M  d 2 =0 K()ffN = −(4) M ¿0 for f ∈ N (Q0 )⊥ . Therefore, by (8):   − Q0 (f)fM −1 d v¿4C0 (’’+ N+ + ’’− N− e˝!0 I)f2 (v)M −1 d v

(9)

R3

which proves (7) under hypothesis (H2). The closedness of R(Q0 ) is a straightforward consequence of (7). We can establish some analogous results for Q1 . The following expression shows that Q1 is self-adjoint in L2 (R3 ; M −1 d v) since NN  [
+ +
− ] is symmetric in ,  :   dv 1 = − Q1 (f)g NN  [
+ +
− ][f( ; ! ) − f( ; !)] M 2 × [g(; ! ) − g(; !)] d ! d  d ! d 

(10)

We also have N (Q0 ) ⊂ N (Q1 ). At last, we underline that in general we do not have Q1 (gf) = gQ1 (f) for f(; !) and g(). This is again the same remark on Q1 : it operates on the energy level while Q0 did not. There is no obvious positivity property for −Q1 . Meanwhile, we can establish a coercivity inequality for a convex combination Q = (1 − )Q0 +Q1 if  is not too large. More precisely, the following lemma stands for Q : Lemma 2.2 (i) Q is self-adjoint in the weighted space L2 (R3 ; M −1 d v). (ii) If hypotheses (H1), (H2) and Hypothesis (H3) 0 ) stand, we have  ∈ [0; C0C+C 1  dv − Q (f)f ¿2C ’2 f2L2 M M

(11)

with C = 4C0 (1−(C0 +C1 )=C0 ). Therefore the null spaces of Q and Q0 coincide, the range of Q is closed and, by the self-adjointness of Q , R(Q ) = N (Q )⊥ = N (Q0 )⊥ . (iii) The restriction of −Q to N (Q )⊥ (also denoted by −Q ) is an invertible linear operator from N (Q )⊥ onto N (Q )⊥ . The inverse operator (that we note (−Q )−1 ) is bounded from N (Q )⊥ onto N (Q )⊥ , of operator norm 6(2C ’2 )−1 . Proof Point (i) follows from the self-adjointness of Q0 and Q1 in L2 (R3 ; M −1 d v). Copyright ? 2003 John Wiley & Sons, Ltd.

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We can imagine that estimate (11) on Q is based on the analogous statement on Q0 established in Lemma 2.1. Meanwhile, we have to control Q1 if we want its lack of coercivity to be compensated by the positivity of Q0 . Hypothesis (H3) naturally arises when we try to obtain this compensation.  First, we notice that we also can express the product Q1 (f)f in terms of  and + only: 

(; !;  ; ! )N 
+ [f( ; ! ) − f( ; !)][f(; ! ) − f(; !)]N d ! d  d ! d  

=

( ; !; ; ! )N
− [f(; ! ) − f(; !)][f( ; ! ) − f( ; !)]N  d ! d  d ! d 

so that, by (10)   dv = + [f(+ ; ! ) − f(+ ; !)][f(; ! ) − f(; !)]N d ! d ! d  − Q1 (f)f M for instance. We now give a bound on this product: ∀f ∈ N (Q0 )⊥ ; f = 0 and consequently:   − Q1 (f)f d v 6 |+ |(|f(+ ; ! )f(; ! )| + |f(+ ; !)f(; !)| + |f(+ ; ! )f(; !)| M

+ |f(+ ; !)f(; ! )|)N d ! d ! d    ’’+ N+ ˝!0 =2 6 8C1 e |f(+ ; ! )f(; ! )| d ! N d  1=2 ¿0 (MM+ ) S2 by (H1) using f = 0  

= 8C1

¿0

f(; ! ) (’N’+ N+ e˝!0 )1=2 (’N’+ N+ )1=2 1=2 M 2 S

f( ; ! ) + d ! d  M+1=2

 6 4C1 

= 4C1

R3

f(+ ; !)2 ˝!0 ’N’+ N+ e d! d + M+



f(; !)2 ’N’+ N+ d! d M



(’− N− e˝!0 I + ’+ N+ )f(v)2 ’M −1 d v

Then, under hypothesis (H1) and using estimate (9) on Q0 , we have for Q : ∀f ∈ N (Q0 )⊥ ; −

 C0 + C1 Q (f)fM −1 d v ¿ 4C0 1 −  C0 R3  (’+ N+ + ’− N− e˝!0 I)f2 ’M −1 d v



R3

Copyright ? 2003 John Wiley & Sons, Ltd.

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which, under hypotheses (H2) and (H3), leads to (11). Consequently, the range of Q is closed and the boundedness of (−Q )−1 follows straightforwardly. This lemma ends this brief study of two elastic–inelastic splittings of the collision operator.

3. FORMAL APPROACH TO THE DIFFUSION APPROXIMATION We now consider the rescaled Boltzmann equation (with a classical diusion scaling): 1 1 @t f + [v · ∇x + E · ∇v ]f = 2 (Q (f ) + 2 Qin (f ))  

(12)

with Qin = (1 − )Qin0 + Qin1 . Thus, we assume that elastic collisions dominate inelastic ones at order 1=2 . We want to study the asymptotic behaviour of the solutions f of this equation as  → 0. This approach will formally give rise to a set of diusion equations to be viewed as a SHE system. In fact, this formal derivation will not be completed in this section since an ‘auxiliary problem’—the resolution of which will be postponed to Section 4—will arise during this study. We assume that hypotheses (H1)–(H3) hold. Let us suppose that f → f0 in a smooth way. Then, if we multiply (12) by 2 and take the limit  → 0, we obtain Q (f0 ) = 0. According to Lemma 2.2, this means that f0 is a function of energy only (in the velocity space): f0 (x; v; t) = f0 (x; ; t). Therefore, it is quite natural to expect that f be mainly given by a function of  only at order 0 in , for instance the macroscopic quantity f . We set: F  = f . So we expect that there exists a function g (x; v; t) such that: f (x; v; t) = F  (x; ; t) + g (x; v; t) After introducing the current of energy J  = 1=N ()vf , (so that J  = N ()vg ), we integrate (12) on a sphere of constant energy. It leads to the following continuity equation (we recall that Q (f ) is of angular average equal to zero): N ()@t F  + ∇˜ · J  = N ()Qin (f ) with ∇˜ = ∇x + E@=@. Indeed, we have     (v · ∇x + E · ∇v )(F  + g ) d ! = v d ! · ∇˜ F  + ∇x · vg d ! + E · ∇v g d ! S2

S2

=

4 ∇x · J  + E · N

S2

 S2

∇v g d !



S2

v d! = 0

for S2

Moreover, for any test function (), a straightforward computation yields:    @J   ∇v g d !N () d  = 4  d @ ¿0 S2 ¿0  which
nally proves: 1=4 (v · ∇x + E · ∇v )f d !N = ∇˜ · J  . Copyright ? 2003 John Wiley & Sons, Ltd.

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Next, we have to
nd a current equation giving a relation between J  and the density F  :  g (M
((v) − 0 )v)M −1 d v J  (0 ) = N (0 )(g v)(0 ; :) = (4)−1 = (4)−1

v∈R3

 v∈R3

= (4)−1



v∈R3

g (−Q )[(−Q )−1 (M
( − 0 )v)]M −1 d v −Q (g )(−Q )−1 (M
( − 0 )v)M −1 d v

by the self-adjointness of Q in L2M −1 . We underline that applying (−Q )−1 to M
( − 0 )v is not justi
ed up to now. We temporarily admit that there exists 0 a (measure) solution of the equation −Q (0 ) = MN
( − 0 )!

(13)

This existence result is not immediate and will be proven in Section 4. Then, by (12), we have, for any test function (v):     1 @t f  d v + (v · ∇x + E · ∇v )f  d v = Q (f ) d v +  Qin (f ) d v   R3 R3 R3 R3 and thus



Q (g ) d v =

R3

Therefore 

R3



J (0 ) = −



¿0

v · ∇˜ F   d v + O()

˜ F  ()N () d  + O() M −1 0 (; !) ⊗ v∇

Taking the limit  → 0
nally gives a system of coupled equations of SHE type: N@t F + ∇˜ · J = N Qin (f0 )  J () = − D(;  )∇˜ F( ) d 

(14) (15)


¿0

where J = lim J  ; F = lim F  , and: D(;  ) =

N 2    ( ; !) ⊗ ! M

(16)

It only remains to prove the existence of  , which will be achieved in Section 4. We underline that the diusion matrix D( ; ) has general properties (symmetry, positivity) that will be given in Section 5 (see Lemma 5.5). Copyright ? 2003 John Wiley & Sons, Ltd.

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4. THE DIFFUSION TENSOR: AN INFINITE DIMENSIONAL MATRIX This section is devoted to the calculation of the diusion tensor. We have seen in Section 3 that in order to give a rigorous meaning to Equation (16), we have to solve (13), or, more generally: −Q (f) = g with g = (!)
( − 0 )

and

 =0

(17)

with 0 ¿0
xed. The diusion coecient in the current equation will then appear as an in
nite dimensional matrix: this matrix will couple any two levels of energy separated by a gap of energy k ˝!0 with k ∈ Z; this is a consequence of both the inelastic nature of Q1 and the discontinuity in energy that appears in electron–phonon collisions. According to the right hand side of (17) and to the computation of Q (
( − 0 )), it is quite natural to search f as an in
nite sum of Dirac measures such as  fk (!)
( − k ) with fk  = 0; fk ∈ L2 (S2 ) (18) f(; !) = k¿−k0 (0 )

where k = 0 + k ˝!0 , and k0 (0 ) = −E(1 − 0 = ˝!0 ) is the only integer such that 0¡0 − k0 (0 )˝!0 6˝!0 (E denotes the integer part function). The restriction to integers superior to −k0 is necessary to deal only with positive values of k , which are the only physically meaningful ones. To the same purpose we introduce the symbol 5: it will denote I[k¿−k 0 (0 )+1] . With these notations, we have f(+ ; !) = k¿−k0 fk+1
( − k ) and f(− ; !) = k¿−k0 fk−1
( − k )5, and the actions of operators Q0 and Q1 on such a function f are, respectively, written as  [(k; k+1 Nk+1 Mk+1 + k; k−1 Nk−1 Mk−1 5)fk  (4)−1 Q0 (f) = k¿−k0

− k; k+1 Nk+1 Mk+1 + k; k−1 Nk−1 Mk−1 5fk ]
( − k )

(4)−1 Q1 (f) =



(19)

[(k; k+1 Nk+1 Mk fk+1  + k; k−1 Nk−1 Mk 5fk−1 )

k¿−k0

− (k; k+1 Nk+1 Mk fk+1 + k; k−1 Nk−1 Mk 5fk−1 )]
( − k )

(20)

where we have used the following notations: ∀g() except ; gk = g(k ), and j; k = (j ; !; k ; ! ). We now see that our problem will be solved by identifying terms in
( − k ) for the same k in equation −Q (f) = (!)
( − 0 ), which can come into view as the inversion of an in
nite tridiagonal matrix, the diagonal terms of which come from (19) and the infraand supradiagonal terms of which come from (20). Then, the following spaces of sequences appear as a suitable frame of work:


  2 2 2 N 2 Nk gk d !¡∞ LSN=M = (gk )k∈N ∈ L (S ) : Mk S2 k¿−k 0

0 L2;SN=M = {(gk )k∈N ∈ L2SN=M : gk  = 0 ∀k ¿ − k0 }

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We denote by (f|g)SN=M the inner product in L2SN=M and by gSN=M the corresponding norm. 0 We now precise the operator we have to invert in the space L2;SN=M . We can express it as 2; 0 2; 0 2; 0 0 0 1 that satisfy: a convex combination of two operators A : LSN=M → LSN=M and A : LSN=M → L2;SN=M 2; 0 ∀f = (fk )k ∈ LSN=M (A0 f)k = ((k; k+1 Nk+1 Mk+1 + k; k−1 Nk−1 Mk−1 5)fk  − k; k+1 Nk+1 Mk+1 + k; k−1 Nk−1 Mk−1 5fk )k

according to the kth term of sum (19), and (A1 f)k = (k; k+1 Nk+1 Mk fk+1  + k; k−1 Nk−1 Mk 5fk−1 ) − (k; k+1 Nk+1 Mk fk+1 + k; k−1 Nk−1 Mk 5fk−1 )k

according to the kth term of sum (20). We naturally set A = (1 − )A0 + A1 in order to have −1  (4) Q (f) = k¿−k0 (A f)k
( − k ). The following lemma proves that we can invert A 0 . in L2;SN=M Lemma 4.1 (i) A is a self-adjoint operator in L2SN=M (ii) If hypotheses (H1), (H2) and Hypothesis (H4) 0 ,  ∈ [0; C0 +eC˝!0 0 C1 ) stand, we have ∀f ∈ L2;SN=M (−A f|f)SN=M ¿2C˜  ’2 f2SN=M

(21)

with C˜  = C0 (1 − (C0 + e˝!0 C1 )=C0 ) and therefore, under these assumptions, the null 0 0 is reduced to zero, the range R(A ) is closed in L2;SN=M and thus we space N (A ) in L2;SN=M 2; 0   ⊥ have (by self-adjointness) R(A ) = N (A ) = LSN=M (where the orthogonal was taken 0 in L2;SN=M ). (iii) Consequently, if hypotheses (H1), (H2) and (H4) are satis
ed, −A is an invertible 0 0 0 onto L2;SN=M , the inverse operator (−A )−1 is bounded on L2;SN=M of operator from L2;SN=M operator norm 6(2C˜  ’2 )−1 . Proof As in the case of operator Q , the proof consists in giving a coercivity inequality on A0 and controlling |(A1 f|f)| in order to obtain a coercivity estimate for the convex combination A . We begin with the estimate on A0 . We note: c0 = c0 (k; !; ! ) = k; k+1 (!; ! )Nk+1 Mk+1 + ∀h(!), h = k; k−1 (!; ! )Nk−1 Mk−1 5. Throughout this proof, we shall use the convention:  h(! ). Then, the self-adjointness of A0 is given by: 2 × 4(−A0 f|g)SN=M = k¿−k0 c0 (fk − fk )(gk − gk ) d ! d !(Nk =Mk ), which in turn gives, for g = f:   Nk 0 c0 (fk − fk )2 d ! d ! 2 × 4(−A f|f)SN=M = M
2 k k¿−k !; ! ∈S 0

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SPHERICAL HARMONIC EXPANSION TYPE MODELS

and, since fk  = 0 ∀k, we have, by hypothesis (H1) and remarking that e˝!0 ¿1:   Nk (’k ’k+1 Nk+1 + ’k ’k−1 Nk−1 5) fk2 d ! (−A0 f|f)SN=M ¿C0 M 2 k S k¿−k

(22)

0

Now, we turn our attention to operator A1 :    k; k+1 Nk+1 Mk (fk+1 − fk+1 )(gk − gk ) 4(−A1 f|g)SN=M = k¿−k0

!; !
∈S2

 − fk−1 )(gk − gk ) d ! d ! + k; k−1 Nk−1 Mk 5(fk−1

=

  k¿−k0

Nk Mk

 k; k+1 Nk+1 Mk (gk+1 − gk+1 )(fk − fk )

!; !
∈S2

 − gk−1 )(fk − fk ) d ! d ! + k; k−1 Nk−1 Mk 5(gk−1

Nk Mk

0 so that this operator is self-adjoint in L2;SN=M . We give the estimate on A1 that will enable us  to prove the coercivity of A :

   k; k+1 Nk+1 Mk (fk+1 − fk+1 )(fk − fk ) 8|(−A f|f)| =
2 !; ! ∈S 1

k¿−k0

 + k; k−1 Nk−1 Mk 5(fk−1

−

fk−1 )(fk

Nk − fk ) d ! d ! Mk 

 1   {k; k+1 Nk+1 Mk (fk+1 6 − fk+1 )2 + k; k+1 Nk+1 Mk (fk − fk )2 2 k¿−k 0

 + k; k−1 Nk−1 Mk 5(fk−1 − fk−1 )2 + k; k−1 Nk−1 Mk 5(fk − fk )2 } d ! d !

 

=

(k; k+1 Nk+1 Mk + k; k−1 Nk−1 Mk 5)(fk − fk )2 d ! d !

k¿−k0



6 C1

’k (’k+1 Nk+1 e

˝!0



+ ’k−1 Nk−1 5)

(fk − fk )2 d ! d !

k¿−k0

= 8C1



’k (’k+1 Nk+1 e

˝!0



+ ’k−1 Nk−1 5)

k¿−k0

6 8e

˝!0

C1



fk2 d !

Nk Mk



’k (’k+1 Nk+1 + ’k−1 Nk−1 5)

k¿−k0

Copyright ? 2003 John Wiley & Sons, Ltd.

S2

fk2 d !

Nk Mk

Nk Mk

Nk Mk Nk by (H1) Mk

for fk  = 0 since 1¡e˝!0

(23)

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Consequently, we have the following coercivity inequality for A (by (22) and (23), under hypothesis (H1)):

  C0 + e˝!0 C1  Nk ’k (’k+1 Nk+1 + ’k−1 Nk−1 5) fk2 d ! (−A f|f)SN=M ¿C0 1 −  C0 M 2 k S k¿−k 0

which leads to (21) under assumptions (H2) and (H4). Point (iii) follows as in Lemma 2.2. Remark We point out that hypothesis (H4) implies hypothesis (H3) since e˝!0 ¿1. This lemma enables us to prove the following proposition: Proposition 4.2 0 ∀¿0, ∀ ∈ L2 (S2 ) such that   = 0, there exists a unique sequence (fk )k ∈ L2; SN=M such that the function f de
ned by (18) solves Equation (17). Moreover, the diusion tensor (16) takes the form D(;  ) =

N 2    ( ; !) ⊗ ! M

(24)

where  is the unique solution (of type (18)) of Equation (17) with Equation (2) holds if we set

= N ()M ()!.

Dk () = k (!) ⊗ !N ( + k ˝!0 )2 M −1 ( + k ˝!0 ) where (k )k is such that  ( ; !) = k¿−k0 () k (!)
( − k ). Proof The existence and uniqueness follow from Lemma 4.1(iii). One has just to consider (fk )k = (−A )−1 ())k where the sequence  is given by k0 (0 ) = =4 and k = 0 if k = k0 (0 ). Then the solution is written as  fk (!)
( − (0 + k ˝!0 )) f(; !) = k¿−k0 (0 )

Now, if we take for solution, with

(!) the function: N (0 )M (0 )!, we call 0 (; !) the corresponding 

0 (; !) =

k0 (; !)
( − (0 + k ˝!0 ))

k¿−k0 (0 )

Then the diusion matrix given in (16) takes the following form: N 2   (!) ⊗ !
( − ( + k ˝!0 )) D(;  ) =  M k¿−k () k 0

Then we obtain Equation (2) by setting Dk () = k (!) ⊗ !N ( + k ˝!0 )2 M −1 ( + k ˝!0 ). Therefore, the form of the diusion tensor is the expected one. Nevertheless, we underline that, if the uniqueness was proven for the solutions of the auxiliary problem (17), it was only in a restricted class of solutions: those that can be written as a sum of Dirac measures. This Copyright ? 2003 John Wiley & Sons, Ltd.

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SPHERICAL HARMONIC EXPANSION TYPE MODELS

261

is not a real problem, since we have seen in Section 3 that all that was required was the existence of such a solution. Proposition 4.2 makes the formal approach to the derivation complete. It remains to see in what sense this derivation can be made rigorous. 5. RIGOROUS APPROACH TO THE DIFFUSION APPROXIMATION We are now concerned with the rigorous asymptotic study of the solution f of the rescaled Boltzmann equation:   @t f + 1 v · ∇x f = 1 Q (f )  2 (25)  f (t = 0) = f I where we have supposed Qin = 0 and E = 0. The aim of this section is to prove the convergence Theorem 5.10. The following proof if inspired from Reference [15]: the approach developed in Reference [15] relies on functional theoretic properties of the diusion operator, which can be adapted to the present case. We also improve the results of Reference [15], following the work [16], and obtain regularity of the solution without additional hypothesis. 5.1. Convergence and continuity equation Classical semigroup theory (see e.g. References [17,18]) gives the following existence result for the initial value problem (25). Lemma 5.1 Problem (25) admits a unique solution f (t; x; v) ∈ C 1 ((0; ∞); L2 (R6 ; M −1 d x d v)) ∩ C 0 ((0; ∞); D(A)) where D(A) denotes the domain of the dierential operator A = (1=)v ·∇x − (1=2 )Q in L2 (R6 ; M −1 d x d v), i.e. D(A) = {f ∈ L2 (R6 ; M −1 d x d v): v · ∇x f ∈ L2 (R6 ; M −1 d x d v)}. Furthermore, the solutions f satisfy the following weak form of Equation (25) (∀T ¿0):    T 1 T − f @t ’M −1 d x d v d t − f v · ∇x ’M −1 d x d v d t  0 0    1 T − 2 Q ’f M −1 d x d v d t = fI ’(0)M −1 d x d v (26)  0 ∀’ ∈ S = { ∈ L1loc (R+ ; L2 (R6 ; M −1 d x d v)): @t ∈ L1loc (R+ ; L2 (R6 ; M −1 d x d v))

v · ∇x ∈ L1loc (R+ ; L2 (R6 ; M −1 d x d v))} s:t: ’(T ) = 0 As we have seen in Section 3, one can expect the sequence (f ) to tend to the density F(x; ; t), while (N=vf ) should approach the current J (x; ; t). To give a rigorous meaning to this, we have to consider peculiar weighted functional spaces. We note: L 2 = L2 (R6 ; d v d x), and L 2 = L2 (R3 × (0; ∞); d  d x), where is the weight. Copyright ? 2003 John Wiley & Sons, Ltd.

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2 We now assume the family of initial values (fI ) to be bounded in LM −1 . Then the next proposition holds:

Proposition 5.2 Up to the extraction of subsequences, we have f * F;

2 L∞ (0; T ; LM weak ∗ −1 )

(27)

F  * F;

2 L∞ (0; T ; LNM weak ∗ −1 )

(28)

J  * J;

L∞ (0; T ; LN2 −3 M −1 )3

weak ∗

(29)

and the asymptotic density F and current J satisfy the following weak form of the continuity Equation (14):  T d NF@t  + J · ∇x  d x dt M 0 R3x ; ¿0  d NFI |t=0 d x = 0 ∀ ∈ S s:t : (T ) = 0 (30) + M 3 Rx ; ¿0 where FI is the limit as  → 0 of the (bounded) sequence (FI  ) = (fI ) . Proof We formally multiply (25) by f =M and integrate:  t   1 t   −1 @t f f M d v d x d t + v · ∇x f M −1 d v d x d t  0 R6x; v 0 R6x; v   1 t − 2 Q f f M −1 d v d x d t = 0  0 R6x; v i.e. f (t)2L2

M −1

−

2 2

 t 0

R6x; v

Q f f M −1 d v d x d t = fI 2L2

M −1

(31)

 for v · ∇x f f M −1 d x d v d t = 0 by oddity. Therefore, using the coercivity inequality (11), we have the following estimates: f 2L∞ (0; ∞; L2

M −1

 2 ) 6fI L2

M −1

; P ⊥ f 2L2 (0; t; L2

M −1

)6

2 f  2 2 4C ’2 I LM −1

where P ⊥ denotes the orthogonal projection on N (Q )⊥ . By the uniform boundedness of (fI ) , these estimates respectively lead to: f (x; t; v) * f(t; x; v);

2 L∞ (0; T ; LM weak ∗ −1 )

and 2 ˜ x; ) ∈ L2 (0; T ; LNM f(t; x; v) = F(t; −1 )

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SPHERICAL HARMONIC EXPANSION TYPE MODELS

263

2  since P ⊥ f → 0 strongly in L2 (0; T ; LM −1 ). Moreover, remarking that F L∞ (0; T ; L2 )6 NM −1 f L∞ (0; T ; L2 ) yields (28). One can easily check that F = F˜ and thus obtain (27). M −1

On the other hand, we have to study the convergence of the current J  = N=f v. We have 2

   1  2 −1 −3  N 2 M −1 N −3 d  d x |J | M N d  d x 6 f v d ! (4)2 R3x ; ¿0 R3x ; ¿0 S2  1 6 (f )2M −1 d v d x6fI 2L2 6C 42 R6x; v M −1 Thus, J  is bounded in L∞ (0; ∞; LN2 −3 M −1 )3 and there exists J s.t. (29) holds.   2 In addition, the boundedness of fI implies that FI  → FI LNM −1 —weak, where FI = fI . Using the weak form (26) of Equation (25) with  = (x; ; t) we have  T   1 T d d f d !@t (t; x; )N d x dt + vf d ! · ∇x N d x dt M  M 3 2 3 2 0 0 Rx ; ¿0; !∈S Rx ; ¿0; !∈S    1 T Q(f ) d !NM −1 d  d x d t + fI (0)M −1 d v d x = 0 + 2  0 R3x ; ¿0; !∈S2 6 Rx; v i.e.



T





R3x ; ¿0

0

NF @t M



+ R3x ; ¿0

−1

 d d x dt + 0

NFI  (0)M −1 d  d x = 0

T

 R3x ; ¿0

J  · ∇x M −1 d  d x d t (32)

where  was taken in the admissible set S, independent of ! and such that (T ) = 0. Finally, taking the limit  → 0 in (32) gives the continuity equation for F and J in the weak sense stated in (30). 5.2. Current equation We now wish to prove that the density F and the current J given by Proposition 5.2 satisfy a current equation. According to Section 3, this equation is expected to be (15). Following Reference [15], we could establish a weak formulation of this equation, namely (see Reference [15] for more details): Denition 5.3 2 J ∈ L∞ (0; T ; LN2 −3 M −1 )3 and F ∈ L∞ (0; T ; LNM −1 ) are said to satisfy the current equation (15) 2 in the weak sense if ∀ ∈ H = { ∈ L (0; T ; LN2 3 M −1 )3 : ∇x · v(−Q )−1 ( · v) ∈ L2 (0; T ; 2 LNM −1 )},  T  T J · M −1 d  d x d t = F(x; ; t)∇x · v(−Q )−1 ( · v)NM −1 d  d x d t 0

R3x ; ¿0

0

Copyright ? 2003 John Wiley & Sons, Ltd.

R3x ; ¿0

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J.-P. BOURGADE

One can prove (see Reference [15]) that the current and density arising in Proposition 5.2 actually satisfy Equation (15) in the weak sense. But we will see that, in order to prove a uniqueness result, ∇x F will be asked to belong to L2 (0; T ; LN2 3 M −1 )3 . This regularity was obtained in Reference [15] under an additional hypothesis (namely, the density of H in L2 (0; T ; LN2 3 M −1 )3 was assumed) which is satis
ed at least in the case of an isotropic cross section (; !;  ; ! ) = ’()’( ). Meanwhile, in a dierent context (with the hypothesis of an a priori discrete distribution of energy levels, which naturally appears in multigroup neutron transport theory) T. Goudon and A. Mellet developed a strategy of proof in two steps (see Reference [16]). First, they established an approximate form of the current equation by a truncation-in-energy method, and then proved the convergence of the solutions of this approximate equation to the solution of a dual form of the current equation (concerning the formulation of the current equation by means of duality considerations, see also Reference [15]). Following Reference [16], we
rst give an approximate weak formulation of the current equation. From now on, we note ∀ ∈ LN2 3 M −1 ;

K  = (−Q )−1 (v · )

2 We point out that ∀ ∈ LN2 3 M −1 , v ·  ∈ LM −1 and satis
es the solvability condition v ·  = v·  ⊥  = 0. Therefore, v ·  ∈ N (Q ) and K is a well de
ned bounded linear operator from (LN2 3 M −1 )3 onto N (Q )⊥ . Since we want to establish an approximate current equation by means of truncation in energy, we also de
ne a truncation function R by R () = (=R) with ∈ C ∞ (R) with values in [0; 1] satisfying () = 0 if ||¿2, and () = 1 if ||61. With these notations, we have:

Lemma 5.4 The pair (J; F) given by Proposition 5.2 satis
es an approximate weak form of the current Equation (15). Namely, we have ∀R¿0 and ∀ ∈ H:  T  T 4

R F ∇x · K  vNM −1 d  d x d t = − Q (K  R )gM −1 d x d v d t (33) 0

R3x ; ¿0

0

R6x; v

where g = lim g = lim 1=(f − F  ) satis
es N vg = J

(34)

Remark Equation (33) is indeed an approximate weak form of the current equation since −Q (K  R ) can be viewed as an approximation of −Q (K  ) = v ·  and one has N gv = J by (34). Proof 2 2 Let  ∈ H. Then a straightforward computation yields K  ∈ { ∈ L2 (0; T ; LM −1 ): @t ∈ L (0; T ; 2 2 2  LM −1 ); v · ∇x ∈ L (0; T ; LM −1 )} ⊂ S. Then, R K is an admissible test function. Introducing it in (26) gives  T  T   −1

R f @t K M d x d v d t +

R f v · ∇x K  M −1 d x d v d t  0

+

0

1 



T



Q (K  R )f M −1 d x d v d t = −

0

Copyright ? 2003 John Wiley & Sons, Ltd.





R fI K|t=0 M −1 d x d v

(35)

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SPHERICAL HARMONIC EXPANSION TYPE MODELS

265

∀R¿0, ∀¿0. Since (f ) and (fI ) are bounded, taking the limit  → 0 results in:  T

R f @t K  M −1 d x d v d t → 0  



0 

R fI K|t=0 M −1 d x d v → 0; ∀R¿0

On the other hand, f = F  + g (where we set g = 1=(f − F  ) as in Section 3) and, since F  ∈ N (Q ) = R(Q )⊥ , we have    T 1 T    −1 Q (K R )f M d x d v d t = Q (K  R )g M −1 d x d v d t  0 0 Since we have Q (f )f = 2 Q (g )g , and since g ∈ N (Q )⊥ , Lemma 2.2 gives  T 2 − Q (g )g M −1 d x d v d t ¿C2 g 2L2 (0; T ; L2 ) M −1

0

which proves (using estimate (31) and the boundedness of (fI ) ) that there exists g ∈ L2 (0; T ; 2  2 2 LM −1 ) such that (up to a subsequence) g * gL (0; T ; LM −1 )—weak. So we have  T  T Q (K  R )g M −1 d x d v d t → Q (K  R )gM −1 d x d v d t 0

0

2  for Q (K  R ) ∈ L2 (0; T ; LM −1 ). The boundedness of (g ) , (28) and the identity





T



R f v · ∇x K M 

−1

 d x dv dt =

0



T

R F  v · ∇x K  M −1 d x d v d t

0



+

T



R g v · ∇x K  M −1 d x d v d t

0

lead to



T



R f v · ∇x K  M −1 d x d v d t →

0

0

T



R Fv · ∇x K  M −1 d x d v d t

0

for v · ∇x K ∈ L ∀ ∈ H:  T 4 



2

2 (0; T ; LM −1 )

R3x ; ¿0

since K ∈ S. Finally, we asymptotically have (R being
xed) 

R F ∇x · K  vNM −1 d  d x d t = −

 0

T

 R6x; v

Q (K  R )gM −1 d x d v d t

It remains only to prove equality (34), where J is the current given by Proposition 5.2. We have 

 f  = N v(g − g ) for vF   = 0 N vg − J = N v g −  and thus N vg − J  * 0 in L2 (0; T ; LN2 −3 M −1 )3 —weak. Copyright ? 2003 John Wiley & Sons, Ltd.

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J.-P. BOURGADE

We remark that, while it was required to have  in H for the sake of the convergence  → 0, Equation (33) is meaningful for less regular functions . For instance, if  ∈ E = { ∈ L2 (0; T ; LN2 3 M −1 )3 : ∇x · vK  N ∈ L2 (0; T ; LN2 −3 M −1 )}, this equation still holds since, by truncation in 2 energy, R F ∈ LN2 3 M −1 ∀R¿0 as soon as F ∈ LNM −1 . Therefore, we can de
ne on E a linear mapping FR by  T ∀ ∈ E; FR () = −4

R F ∇x · vK  NM −1 d  d x d t (36) R3x ; ¿0

0

By regularization in space, E is dense in N vK   ∈ LN2 −3 M −1 . Moreover, by Equation (33), 

T



FR () =

R6x; v

0

L2 (0; T ; LN2 3 M −1 )

gQ (K  R )M −1 d x d v d t

since

∀ ∈ LN2 3 M −1 ,

∀ ∈ L2 (0; T ; E)

2 2 By the continuity of Q on LM −1 , we have, ∀ ∈ L (0; T ; E):

|FR ()|6gL2 (0; T ; L2 −1 ) Q (K  R )L2 (0; T ; L2 −1 ) 6C gL2 (0; T ; L2 −1 ) K  L2 (0; T ; L2 −1 ) M

M

M

M

for R 61. But: K  = (−Q )−1 (v · ) and, by the continuity of (−Q )−1 (by Lemma 2.2): |FR ()|6C gv · L2 (0; T ; L2 −1 ) = C gL2 (0; T ; L2 3

N M −1

M

)

(37)

Therefore, FR is a continuous linear form on L2 (0; T ; LN2 3 M −1 ) (since E is dense in LN2 3 M −1 ); it stands for a weak form of ∇x F R . To obtain a dual form for the current equation, it is wishful to introduce a duality correspondence between L2 (0; T ; LN2 3 M −1 ) and linear forms on this space. Then it will remain to prove the convergence of a family ( R )R ∈ LN2 3 M −1 — corresponding to (∇x F R )R —, at least in a weak way. This will give rise to a dual form of the asymptotic current equation (15). Moreover, we will get some regularity on F if we can prove the convergence of R in L2 (0; T ; LN2 3 M −1 ). First, we give a precise meaning to the duality between L2 (0; T ; LN2 3 M −1 ) and its dual space: Lemma 5.5 Under hypotheses (H1), (H2) and (H4), the expression:    D(; ) = T ()D(;  )( )M −1 () d  d  d x R3x

¿0


¿0

where D(;  ) is given by (24), de
nes a continuous bilinear symmetric form on (LN2 3 M −1 )3 . Furthermore, D is coercive, namely, there exists C¿0 such that ∀ ∈ (LN2 3 )3 ;

D(; )¿C 2L2

N 3 M −1

(38)

The proof of Lemma 5.5 can be found in Reference [15]. From now on, we identify LM2 −1 = L2 ((0; ∞) × R3x ; M −1 d  d x) with its dual space. Then (LN2 3 M −1 ) = LN2 −3 M −1 and the symmetric continuous positive bilinear form D de
nes (by Riesz’ representation theorem) a linear bicontinuous one-to-one map F from LN2 3 M −1 onto Copyright ? 2003 John Wiley & Sons, Ltd.

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SPHERICAL HARMONIC EXPANSION TYPE MODELS

LN2 −3 M −1 . Precisely, we require:



∀;  ∈ (LN2 3 M −1 )3 ;

D(; ) = −

R3x

 ¿0

F() · M −1 d  d x

(39)

Now, the duality frame is set up, and we can complete the derivation of the current equation. We give: Denition 5.6 2 J ∈ L∞ (0; T ; LN2 −3 M −1 )3 and F ∈ L∞ (0; T ; LNM −1 ) are said to satisfy the current Equation (15) 2 2 3 in the dual sense if ∇x F ∈ L (0; T ; LN 3 M −1 ) and J = F(∇x F). Then we have: Proposition 5.7 The density F and the current J given by Proposition 5.2 satisfy the current equation in the dual sense. To prove Proposition 5.7, we need the following density result: Lemma 5.8 The set K = {N vK  :  ∈ L2 (0; T ; LN2 3 M −1 )} is dense in LN2 −3 M −1 . Proof We prove that K⊥ = {0}, where the orthogonal space is taken in L2 (0; T ; LN2 3 M −1 ). Let ∈ L2 (0; T ; LN2 3 M −1 ) such that 

T

0

 i:e: 0

 i:e:

Thus,

T 0

T

T

 R6x; v





R3x

¿0

· vK  NM −1 d  d x d t = 0 ∀ ∈ LN2 3 M −1

· v(−Q )−1 (v · )M −1 d v d x d t = 0 ∀ ∈ LN2 3 M −1

D(; ) d t = 0

∀ ∈ LN2 3 M −1 according to the de
nition of D(;  )

0

D( ; ) d t = 0 and, since by (38)

T 0

D( ; ) d t ¿C  2L2 (0; T ; L2

N 3 M −1

)

, = 0.

Proof of Proposition 5.7 Estimate (37) allows us to see FR as a point in L2 (0; T ; LN2 3 M −1 ) thanks to the dual correspondence F. More accurately, there exists ( R )R ∈ L2 (0; T ; LN2 3 M −1 ), uniformly bounded with respect to R, such that  T  T ∀R¿0; FR () = −4 F( R ) · M −1 d  d x d t = 4 D( R ; ) d t 0

R3x ; ¿0

0

The uniform boundedness of ( R )R is a consequence of (37) and the continuity of F−1 . Therefore, maybe up to a subsequence, there exists ∈ L2 (0; T ; LN2 3 M −1 ) such that R * L2 (0; T ; Copyright ? 2003 John Wiley & Sons, Ltd.

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LN2 3 M −1 )—weak. On the other hand, we have: ∀ ∈ L2 (0; T ; LN2 3 M −1 ), 

T

FR () = 4

 D( R ; ) d t = 4

0



T

R3x ; ¿0

0



T



= −4

R3x ; ¿0

0

R · vK  NM −1 d  d x d t

( R F)∇x · vK  NM −1 d x d  d t

by (36)

Thus, R and R ∇x F actually coincide on the set K = {N vK  :  ∈ L2 (0; T ; LN2 3 M −1 )}. We know by Lemma 5.8 that K is dense in L2 (0; T ; LN2 −3 M −1 ). Therefore, we can conclude that R coincides with R ∇x F as a linear form on L2 (0; T ; LN2 −3 M −1 ). Furthermore, the boundedness of ( R )R in L2 (0; T ; LN2 3 M −1 ) and the convergence R ∇x F → ∇x F in a distributional sense yield the expected outcome R ∇x F * ∇x FL2 (0; T ; LN2 3 M −1 )—weak. Consequently, ∀ ∈ L2 (0; T ; LN2 3 M −1 ),  T  T FR () → 4 D(∇x F; ) d t = −4 F(∇x F) · M −1 d  d x d t 0

R3x; v ; ¿0

0

Taking the limit R → ∞ in (33) gives rise to a weak formulation of the current equation:  − 4 0

T

 R3x ; ¿0

F(∇x F) · M

−1

 d d x dt =



T

gQ (K  )M −1 d x d v d t

R6x; v

0





T

=−

R6x; v

0



T

= −4 0

vg · M −1 d x d v d t

 R3x ; ¿0

J · M −1 d x d  d t

since J = N vg. We deduce from this the equality in L2 (0; T ; LN2 −3 M −1 )3 : J = F(∇x F). 5.3. Uniqueness of the asymptotic current and density As a by product we have proved in Proposition 5.7 that ∇x F ∈ L2 (0; T ; LN2 3 M −1 )3 . This enables us to prove the uniqueness of J and F in the very spaces where their existence was established. Proposition 5.9 2 The density F given by Proposition 5.2 is in C([0; T ]; LNM −1 ) and satis
es F 2L∞ (0; T ; L2

M −1

)

+ ∇x F 2L2 (0; T ; L2

N 3 M −1

2 ) 6C F|t=0 L2

M −1

(40)

Proof The proof relies on the following general embedding: W (0; T ; Y; Y ) ⊂ C([0; T ]; L) 

(41)



where W (0; T ; Y; Y ) = {f ∈ L (0; T ; Y): @t f ∈ L (0; T ; Y )}, L and Y are Hilbert spaces, L is identi
ed to its dual space and Y is dense in L. Inclusion (41) is continuous. 2

Copyright ? 2003 John Wiley & Sons, Ltd.

2

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SPHERICAL HARMONIC EXPANSION TYPE MODELS

In our case, since we want to recover the continuity of F with values in LN2 (or, if we 2 2 see F as a function of v, in LM −1 ), we shall identify LM −1 with its dual space. The previous results give
2 F ∈ L2 (0; T ; LM −1 ) ∇x F ∈ L2 (0; T ; (LN2 3 M −1 )3 ) = L2 (0; T ; (LN2 2 M −1 )3 )

We set Y = {f ∈ LM2 −1 : ∇x f ∈ (LN2 2 M −1 )3 }. A straightforward computation leads to Y = 3 { =1 ||=1 @x J  : J ∈ (LN2 −2 M −1 )3 }. In addition, (30) yields N@t F + ∇x · J = 0 in a distributional sense, and thus, @t F ∈ L2 (0; T ; Y ). Consequently, we have F ∈ W (0; T ; Y; Y ). Since Y 2 2 is obviously dense in LM −1 , inclusion (41) holds with L = LM −1 , and we have F ∈ C([0; T ]; 2 LM −1 ). By classical properties of W (0; T ) spaces, we have   d 1 F 2L2 = 2@t F; F Y
×Y = −2 ∇x · J; F dt N M −1 Y
×Y     1 1 J; ∇x F =2 = F(∇x F); ∇x F N N L2 ×L2 L2 ×L2 

= R3x ; ¿0

N −2 M −1

N 2 M −1

N −2 M −1

N 2 M −1

F(∇x F) · ∇x FM −1 d  d x

since we actually deal with functions of  only. By (39), this in turn gives d = d t F 2L2 −D(∇x F; ∇x F). An integration in time yields:  T F(t)2L2 + D(∇x F; ∇x F) d t = F|t=0 2L2 M −1

0

M −1

=

M −1

By the coercivity inequality (38), this gives the expected estimate (40). The uniqueness follows from (40) by considering F|t=0 = 0. This uniqueness result guarantees that the whole sequences converge, without extracting subsequences. The following theorem gives a synthesis of this section: Theorem 5.10 The initial value problem (25) admits a sequence of solutions (f ) such that, under hypotheses (H1), (H2), (H4) and an assumption of boundedness on (fI ) , the sequences of approximate density (F  = f ) and current (J  = N f ) converge to a pair (F; J ) ∈ {G ∈ L2 (0; T ; 2 2 2 3 2 2 LNM −1 ): ∇x G ∈ L (0; T ; LN 3 ) } × L (0; T ; LN −3 ). These asymptotic current and density satisfy a set of equations of SHE-type, coupled in energy:  N@t F + ∇x · J = 0      D(;  )∇x F( ) d  J =−  
¿0    F|t=0 = FI in the weak following sense: Copyright ? 2003 John Wiley & Sons, Ltd.

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continuity equation: 2 2 2 ∀ ∈ { ∈ L2 (0; T ; LM ∈ L2 (0; T ; LM ∈ L2 (0; T ; LM −1 ): @t −1 ); v · ∇x −1 )}

such that (T ) = 0,  T 0

R3x ; ¿0

NF@t  + J · ∇x M

−1

 d d x dt + R3x ; ¿0

NFI |t=0 M −1 d  d x = 0

current equation: J = F(∇x F) Moreover, the pair (F; J ) is uniquely determined in the class of solutions {G ∈ L2 (0; T ; 2 ∇x G ∈ L2 (0; T ; LN2 3 )3 } × L2 (0; T ; LN2 −3 ), and F ∈ C([0; T ]; LNM −1 ).

2 LNM −1 ):

6. CONCLUSION In this paper, we studied the rigorous derivation of a SHE type system of equations, coupled in energy, from a kinetic equation arising in electron transport in semiconductors. The derivation is based on the analysis of the convex combination of two elastic operators (coming from two dierent splittings of the collision operator). This combination appears as a useful means to take into account the inelastic nature of collisions through a non-local dependence on the energy. We considered a collision operator rendering interactions between electrons and phonons (which we assumed to be dominant). These interactions are discontinuous in energy: it gave rise to the peculiar form of the (coupling) diusion tensor in the asymptotic current equation. Not only did it appear to couple dierent levels of energy (as in Reference [15] for instance); it is overall characterized in that it only couples energies separated by gaps—namely, only those that can be joined by phonon emission or absorption. Thus the diusion tensor takes the form of an in
nite dimensional matrix in the energy space. The convergence proof was given in a force-free case. We also obtained an existence and uniqueness result for the SHE system. The general formal duality between currents and densities (that appears in any equation of the form of a Fourier law) was given a rigorous meaning and proved to be fundamental in the convergence proof. In this work, we dealt with a linear collision operator: in the case of a low density of electrons, the non-linear contribution of Pauli’s exclusion terms can be neglected. We will dedicate future work to the study of the general case of a non-linear collision operator for electron–phonon interactions. ACKNOWLEDGEMENTS

This work has been supported by the TMR network No. ERB FMBX CT97 0157 on ‘Asymptotic methods in kinetic theory’ of the European Community. REFERENCES 1. Markowich PA, Ringhofer C, Schmeiser C. Semiconductor Equations. Springer: Berlin, Wien, 1990. 2. Bensoussan A, Lions JL, Papanicolaou G. Boundary layers and homogenization of transport processes. Publication of RIMS Kyoto University 1979; 15:53 –157. Copyright ? 2003 John Wiley & Sons, Ltd.

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