UNIVERSITÀ' DEGLI STUDI DI SIENA
Dipartimento di Matematica
Via del Capitano, 15
53100 SIENA
Simona MANCINI
Silvia TOTARO
A three dimensionai trans-pori problem with
nonhomogeneous boundary conditions
Rapporto Matematico N.314
DICEMBRE 1996
ATHREEDIMENSIONALTRAMSPORTPROBLEMWITH
NONHOMOGENEOUS BOUNDARY CONDITIONS
Simona Mancini
Dipartimento di Matematica
Università di Firenze
Viale Morgagni 67/A
50134 Firenze
Italy
Silvia Totaro
Dipartimento di Matematica
Università di Siena
Via del Capitano 15
53100 Siena
Italy
ABSTRACT
In this paper an evolution problem arising from a three dimensionai particle transport
problem with nonhomogeneous boundary conditions is studied.
The interaction of particles with thè boundary surface surrounding thè region where thè
physical phenomenon occurs are described by means of thè sum of a linear operator
acting on thè particle flux through thè boundary and a. giyen source terni.
The theory of nonlinear semigroups is used to "gire thè solution of thè problem in an
explìcit forni.
0. INTRODUCTION
In this paper we study thè evolution of a particle problem in a three dimensionai finite
convex region V with nonhomogeneous boundary conditions.
These kind of condirions are of great interest in applications, for instance in astrophysics
and contamination problems (see [2] and its references).
On thè other hand. in [2]. only nonhomogeneous non re-entry boundary conditions were
considered.
Here, we consider thè case in which thè boundary conditions are described in an
abstract way by thè sum of a linear operator A actiug on thè flux of particles through
thè boundary of V. d\~. and a positive source function q defined on dV.
We assume that thè operator A is bounded. positive and :'cut off".
With these assumptions A represents any possible linear boundary conditious
(reilection, diffusion, periodic. Maxwell type. etc..). and so enable us to study a very
generai kind of nonhomogeneous boundary conditions.
As a consequence, thè abstract version ot thè Bolrzmann equation describing thè
physical phenomenon is nonlinear.
Hence. thè nonlinear semigroup theory is appiied to give explicitly thè solution of thè
problem by rneans of thè soiution of thè associated linear homogeneous boundary value
problem and thè source term.
The paper is organized as follows.
In section 1 thè problem is introduced and is split into two simpler problems. one
nonlineax with nonhomogeneous boundary conditions and one linear with homogeneous
boundary conditions.
Section 2 is devoted to thè introduction of thè sets and thè abstract spaces where thè
problems will be studied. Also some results on thè theory of nonlinear semigroups are
recalled. In section 3 thè expression of nonlinear semigroup generated by thè operator
which regulates thè nonlinear problem is found and in section 4 thè solution of thè
problem is expressed.
1. THE PROBLEM
Let V be a convex region of IR3, bounded by thè regular surface dV. Assume that x 6 V,
y € dV and that n(y) is thè outward directed unit vector normal to <9V at y.
Suppose that N(x.u;t) is thè density of particles which at time t > O are in x and have
velocity v=vu. v > O, and that N satisfies thè linear Boltzmann equation:
J-N(x.u:t)= - vu • VN(x,u:t) - vaX(x.u:t) + ^ f N(x,u';t)du'
OÌ
47T J
u 6 S. x e V. t>0: (la)
S
where S={u : u|=l) is thè spherical surface of radius one. Moreover a and crs are
positive constants. which represent interactions between particles and thè host medium
within V: <JS is thè scattering cross section and <J=<T S+<JC is thè total cross section (<T C is
thè capture cross section). Equation (la) is supplemented by thè initial condition:
N(x.u;0)=X 0 (x.ii)
u e S. x e V .
(Ib)
and by thè nonkomogeneous boundary condition:
Nin=ANout+q(y.u)
y 6 <9V. u € S: t > O, q € Xin:
(le)
that wiil be discussed later on . Note that q(y.u) is a time independent nonnegative
function. so that it can be regarded as a sonice of particles on thè boundary <9V.
Considei now thè following particle densities: w(x.u;t). i.e., thè density of those particles
that at time t have not yet interacted with thè host medium (" first flight particles")
and m(x,u;t), i.e., thè density of those particles that. at time t. have already undergone
at least one intera.ction with thè host medium
Then. it is possible to split thè given problem into thè following two probiems:
-^-w(x.u:t)= - vu - Vw(x.u:t) - vcrw(x.u:t)
u £ S. x e V. t>0,
(2a)
with initial condition and nonhomogeneous boundary condition:
w(x.u:0)=w 0 (x.u)
. u e S. x 6 V .
^ n =Aw out +q(y.u) . y e dV. u 6 S. t > 0. q € Xin;
(2b)
(2c)
and.
€S,x€V,t>0,(3a)
4
with initial condition and homogeneous boundary condition:
m(x,u;0)=m 0 (x,u) . u 6 S. x e V ,
(3b)
m in =Am out , y e dV, u e S, t > 0.
(3c)
Remark 1 Note that summing (2) and (3) we obtain problem (1). provided that:
w 0 +m 0 =N 0 . Moreover. thè nonhomogeneous boundary condition affects directly only
the density of first flight particles, see (2c). whereas thè scattering phenomenon
transforms w-particles into m-particles. Finally note that problem (3) has a linear
boundary condition and its solution is known , ([I], [4]). D
2. MATHEMATICA! PRELIMINAREES
In order to write down thè abstract version of problem (2) and (3). we introduce thè
following sets:
S0ut=Sout(y)={*eS: u - n ( y ) > 0}.
Sìn=Sin(y)={-a€S:u.n(y)<0},
(4)
(5)
where y is a givtn element of <5'V ;
<9V out (u)={y £ 3Y: u - n ( y ) > 0}.
9V in (u)={y€aV:u-n(y)<0},
where u is a given element of S;
(6)
(7)
K o u t ={(y,u):yedV,ueS o u J,
K in ={(y,u):y€dV,ueS in }.
(8)
(9)
In (4) through (9) d\~ is thè (regular) surface that bounds thè region V C R3. S is thè
unit sphere. and n(y) is thè outward directed norma! at y 6 3V. see section 1.
We shall also use thè following Banach spaces:
X=L!(VXS ; dx du) . ||f Ilx= "dx f du |f(x.u) ;
V S
Xout=L1(7out; (u-n(y)) da, du) , If° u t IL=J ^y{ du (u-n(y)) |f(y : u)| ;
. dV Sout
Xin-L^F- u-n(y)| day du) , IHin= [ d<ry [ du |u-n(y)| |f(y.u)| ;
av sin
(10)
(11)
(12)
5
where for instance || • ||x is thè norm in X and dcrv denotes thè surface element centred
at y 6 <9V.
The apices "in" and "out" will be dropped when thè variables and their domaiu are
explicitly indicated.
Finally. we denote thè positive cones of X. X out , X in by X , X^ t , Xj^
respectively.
Remaxk 2 We consider Lj-spaces because thè norm of thè function which represents thè
particle density gives thè number of particles present at t in thè region V.
Moreover Fubini's theorem enables to identify spaces (11) and (12) with thè following
ones:
X^^Lidly.u): y 6 <9Vmn. u 6 S}: f u - n ( y ) ) d<r y du) :
(13)
X^LjdCy.u): y £ 5Vin, u e S}: j u - n(y)| d<ry du) . D
(14)
Definition 1 A linear operator A : X out —> X in is said to be "cut off. if thè followiug
holds. Given 0<e«l, let any f o u l =f(y,u). y € <9V. u - n ( y ) > 0 . be decomposed
accordine; to (15) and (16)
f(y.u) a.e. Vy e <9Y: O < u- n(y)<e
I
O
O
a.e. Vy 6 d V : u- n(y) > e
a.e. \/y£ d\~: O < u - n(y)<e
f(y.u) a.e. Vy 6 d\~ : u - n(y) > e .
Then. f out —p ut _[-f^ ut
Àf out =Àf^ ut ,
an d
A operates as follows
(17)
i.e.
A f ut =0. D
It is now possible to define thè operator A- that describes thè interaction of thè particles
with thè boundary of dV.
We assume that D(A)=X o u t . R(A)=X i n , A is linear, bounded with ||A||< 1. positive,
i.e. A maps X^t in XjJ" and cut off. (i.e., A satisfies (17)).
6
Assumption (17) means that particles impinging on dV with velocity v such that thè
angle betweeu v and n(y) is dose to a right angle are not reflected by <9V. Thus, this
condition 1-cuts off thè possibility of having two successive reflections at y and y7 with
y — y*) small. In any case, from a physical point of view thè mimber of this kind of
interactions per unit time would be small compared with thè total number of
interactions at y G 3V.
We shall also assume that, for thè given e. <9V is such that :
6= inf I
inf
s0(y,u)}>0
(18)
y €d V [ u - n ( y ) > e
J
where s0(x,u) is such that y=x — s0(x,u)u e dV. i. e.. s0(x.u) is thè distance of x from thè
boundary dV along thè direction of u.
Assumption (18) is certainly satisfied if d\~ is smooth enough.
Define now thè thè streaming operarors T ^ „ and T\ Q and thè scattering operator K
as follows:
TA>qf = -vu-Vf-v<rf ,
(19a)
D(T Y )={f G X. T v f €X, fn=Afout+q! f n=f | yv £ Xm, fut=f| y
m
6 Xout } .
* out
J
R(T A > q )CX;
(19b)
( T A _ 0 f ) = - vu - Vf - vai ,
( 20a)
n
\ _ Jf1 €
a AV T
' - \ o'-j
- -1 \ o1feeA~Y- t -P=At \fout
Ti/T1
u
rin
R(T A > 0 )CX,
x.u') du' . D(K)=R(K)=X .
ri
H i/"
ec
K
m
"-•\-
r°ut _ ti
'v
e
K
out
r- V
ut
° i •1
(20b)
(21)
S
Remark 3 Note that T\ is non linear because its domain is not a linear subspace of X,
whereas T\ Q is a linear operator. (see [3] for details). Note also that TQ Q is thè
streaming operator with non re-entry boundary conditions. In contrast, TQ is thè
streaming operator with nonhomogeneous boundary condition studied in [2j.
Finalty, thè scattering operator K is a linear, bounded operator ( ||K||=1) .D
If we now consider w(t)=w(. . . :t) and m(t) — m(. . . ;t) 35 functions from [O, +00) into
thè space X, thè abstract versions of problems (2) and ( 3 ) . read as follows
.t>0,
(22)
[w(0)=w 0 ,
f-J-m(t)=T, 0 m ( t ) + V ( T s K ( m ( t ) + w ( t ) ) , t>0.
Jdt
-'
m(0)=m 0 .
( 23 )
wdiere ^— and ^^ are derivatives in thè strong sense.
dt
dt
Problems (22) and (23) will be studied by using thè theory of linear and nonlinear
semigroups: note that thè scattering operator K in (23a) can be considered as a bounded
perturbation of thè linear operator T ^ n.
We recali here some basic definitions and properties of nonlinear semigroups that we
shall use in thè sequel of thè paper.
Let X1 be a Banach space over thè field of real or complex numbers with nomi i • |j.
The operator A with domain D(A) C X t and range R( A) C X T is called dissipative if thè
operator J z =(zl — A)" 1 , exists for each z > 0. D( J Z )=X} and
| J 2 f, - J,f21! < I fi - f, || .V z > O , f^ f 2 (E Xj.
(24)
Let X0 be a subset of thè Banach space X-,. a family of operators
|Y(t). t > 0}
defined on X0 is calied a nonlinear semigroup of contractions on X0 if thè following hold:
51) Y(t) g l - Y(t)g 2 < ||gi -g2|| ,V t > O . Y(0) gl = gl . V gl . g, 6 X0;
52) Y(t+s)=Y(t)Y(s)=Y(s)Y*(t). V t,s > 0;
53) lim + Y(t)g=g - V g e X0.
It is easy to see that thè map Y(t)g: [0,+oc)—>X 0 is continuous for each g 6 X0 (if X0 is
a linear set and each Y(t) is a linear operator, then thè family
{ Y(t). t > 0} is a
semigroup of linear contractions on X0, [(
Theorem 1 ([6]) Let A be a dissipative operator with domain D(A) dense in Xì
<D(A)=Xl): then there is a nonlinear semigroup of contractions {Y(t). t> 0} defined on
n
(
X-±. such that Y(t)g= lim [I—
j ij A ìV g, Vg£X1. t> O and thè convergente is uniform on
bounded intervals of fO,+oo).
Remark 4 Theorem 1 is a simplified version of a more generai theorem (Th 4.2 page 77)
of [6]. If A is a closed linear operator, Theorem 1 reduces to thè well known Hille-Yosida
s
theorem: thè family { Y(t), t > 0} is thè semigroupj exp(tA), t > 0} generated by A and
A belongs to thè class (j(1.0:X). [l].[5].[7].D
We say that u=u(t): [O.Fj—>-X is a strict solution (or simply a solution [6]) of thè Cauchy
problem (CP)
-f u(t) = Au(t) . t > O
(CP)
u(0)=g
if thè following holds
i) u(0)=g;
ti] u(t) is Lipschitz continuous on [0,t]:
aòu(t)
is
strongly
differentiable
almost
everyAvhere
in
[O.tl
and
satisfies
A function u=u(t) [O.^-oo)—>-X is called a solution of (CP)^ if for ali t > 0. thè
restriction of Ti(t') to [O .Fi , n-(t). O < t < t. is a solution of ( C P ) in [O .Fi.
Theorem 2 ( [6], Th 4.10 page 88') Lei A be a dosed dissipatine operator. D(A)—X and
lei Y/t) be thè nonlinear semigroup of contractions defined in Theorem, 1. If g^D(A)
and thè map Y(t)g: [Q,+oo)—*X is strongly differentiable alinosi everywhere in [O,+OG),
then thè function Y(t)g :[0,+eo)-+X is thè unique solution of
3. THE NONLINEAR SEMIGROUP GENERATED BY TA
In order to write thè solution of problem (22) by means of thè theory of nonlineax
semigroups (see Theorem I and Theorem 2) it is necessary to investigate thè structure
of (I — zT ^ )"n . Hence. we study thè equation:
(I-zTA,q)f=,g.
(25)
where z>0 . g is a given element of X aiid thè uiiknown f must be sought in Di 1 ' \a ).
Owing to definition (19a).(19b) equation (25) can be written in thè following way
(1+vcrz) f(x — su.uj+vzu• Vf(x — su.u)=g(x — su.u).
(26)
9
Since ti • Vf(x — su,u)= — j-f(x — su.u) , we have:
vz
( ~ su'u) -
f x
( ~ su - u ) = 4 §(x ~ su'u) •
f x
By integrating (27) we obtain
(2
/ \
ol x > u j
ex
f(x,u) = f(x - s0(x.u)u.u) exp[ - bs o (x,u)] + ^
P[ ~ bs] g( x -
where b=
s
. Since f must belong to thè domain of T A
vz
su u
; ) ds
(28)
, (28) must also holds
for y E <9V and so we have:
s0(y,u)
f(y.ti) = f(y-s 0 (y,u)u.u) exp[-bs 0 (y.u)] + i I exp[-bs] g(y-su.u) ds .
O
Relation (28) and (29) suggests to deiine thè following operators:
(29)
(A z c^)(x.u) = ?(* ~ s 0 (x.u)u.u) exp[ - bs 0 (x,u)| . (f 6 D( A z )=X ì n , R ( A 7 ) C X:
(30)
s 0 (x,u)
(B z g)(x ; u)=i I" exp(-bs) g(x-su.u) ds . D(B Z )=X . R ( B Z ) C X :
O
(31)
- s 0 (y.u)u,u) exp[-bs 0 (y.u)] . D(^4 z )=X i n ,
)= 4 f
O
R(^z)cXout:
exp( - bs) g(y - su.u) ds . D(B Z )=X . R(BZ) C Xout.
(32)
(33)
Remark 5 The operators A z . Bz. Az. ^àz are linear, bounded and positive, moreover
Az. *$>z are formally identica! to A z . B z , but their range is restricted to Xout.
Roughly speaking. Az transfers an incoming par.ticle density (p E Xin to thè "opposite
boundary" as an element of Xout and. in an analogous way °&z transfers an element of X
:o thè "outgoing boundary" as an element of X out (for details on this kind of operators
see [3]).
We remark also that with thè notation L'(h(x))(x') we mean that thè operator L acts on
a function h depending on x and thè range element is a function depending on x7. D
10
By using defmitions (32) and (33) . relation (29) becomes
(34)
If we impose thè boundary condition specified in thè domain of T^ . we bave:
(35)
and formally
(36)
That (36) niakes sense is proved in thè following lemma.
Lenima 1 Lei A be a cui off operator with \ A||< 1. then (I—A.^iz)~l
and is bounded.
: X-m— >Xin exists,
Proof From thè defmition of thè operator nomi and (17) we bave
TÌ-# J,ut < I A || |
, W- (E Xout
(37)
Then. given any ^=^(y,u) 6 Xhl. by using assumption (18) definition (32) and (37) Ave
bave
>#
,n<ì|A|i
UL
.11
, |^
~-7I r.'JJ_
aju1i r
(38)
Hence by using (38) and assumption (17) we obtain
(39)
We conclude that j AJ.Z||<1 if exp( — W) ||A||<1 (this is always true if ||A||< 1); hence
(I — A-Az)" 1 exists and thè lemma is proved. D
Consider now thè abstract formulation of (28):
f=A z f n +B z g.
(40)
where we used definition (30) and (31).
Since f=(I — zT^ pl^g substitution of (36) into (40) gives
(I - zT A _ q )- 1 g={A z (I - A^ z )- 1 A^ z +B z }g+A z (I - A^ z )- J q .
(41)
Remaxk5 The following relations can be obtained witbout difficulty from (30), (31),
(32), (33):
11
(I-zT 0 , 0 r 1 g=B z g,
(42)
(43)
,
(I - ZTA)0)-1g={Az(I - A^y-^+B^g.
(44)
Relation (42) is well known; as far as (43) and (44) are concerned see [2] and [3] respectively.D
By using thè a.bove relations (41) can be rewritten in thè following way:
(I - zTA)q)-1g=(I - zT Ai0 r 1 g+ A Z (I - A^r'q.
(45)
By iterating (45) we finally have
(I - zT ^ q r tt g=(I - zTA O r n g+ £ A Z (I - A-Az^q,
j=o
Lemma 2 Lei A a cui off operator with || A|| < 1, then :
(46)
n.
A.OJ
V gl .g 2 6 X,
] c x+ v z >o. q e Xi+ ,
iv) D(T^ ) =X . i. e.. D ( T ^ ) is dense in X.
Proof For thè proof of i) see [3], «i. u t ) , i-v) follows from (41). in an anaiogous way
(see also [2]).D
Lemma 2 shows that T ^ satisfies thè assumptions of theorem 1 for thè generation of a
nonlinear semigroup of contractions. To find thè explicit expression of exp(tT.. ) it is
necessary to put z = t/n in (46) and then take thè limit a.s n—>-+oo.
n-l
To do this. we first transform thè term ^ A Z (I — A_4.z)"1q appearing on thè right-hand
side of (46) as follows.
Let us define thè operators
(Cp)(x.u)=p(x-s 0 (x.u)u,u)exp[-<7s 0 (x.u)] . D(C)=X i n . R ( C ) C X .
(47)
(Cp)(y.u)=p(y - s 0 (y,u)u.u) exp[ - as 0 (y.u)] . D(C)=X ÌI1 ,
(48)
R(C) C Xout.
Note that (47). (43) are similar to (30) and (32) with <r inst^.a.d of b=b(z)= 1 "^° rz .
Then thè following lemma holds.
12
Lemma 3 // thè operator
A z , B z , -4Z, Sz, C, C are defmed by (30), (31), (32), (33),
(47), (48) , then we have
LÌL] (I — AC)" 1 exists and it is bounded and positive.
Proof i) By using (31) and (47) we have
s 0 (x.u)
B z (Cp) = ^2
e x p ( — bs) exp[ — £7s0(x — su.ti)] p(x — ? 0 (x — su.u)u — su.u) ds.
(49)
O
Since s 0 (x — su,u)=s 0 (x.u) — s. we obtain from (49):
-b+a 0 (y,ii) = Cp _ ^
(Cp)=ì exp[ - crs 0 (y,u)j p(y.u) * -«*( -b+a)s
(5Q)
n) can be obtained in an anaìogous way.
tu) can be proved as lemma 1, because (48) implies || AC|| < l.D
By lemma 3 and relation (44) with g— Cp we have
A z p=(l - (I - zT A _ 0 )- 1 ) (C P )+A Z (I - A^zJ-^ACp - Ajtzp) VP e X in .
(51)
On thè other hand. thè identity I+(I - AJ.Z)"1A-4Z=(I - A^)"1. gives:
A Z (I - \AzrP = (!-(!- zT^o)'1) (Cp)+A z (I - A^ Z )- J ACp VP e Xin.
(52)
By ut) of lemma 3 with p=(I — AC) ~ q. we get from (52)
AZ(I - ^-^(l - (I - zT^^o)- 1 ) C(I - AC)-'q .
(53)
and so by substituting (53) into (46) we have:
(! - zTAiq)-g=(I - zTA)0r(g - C(I - AC)- 1 q)+C(I - AC)-^.
(54)
Finally, if Ave put z=t/n and take thè limit as n—*-fcc in (54). then thè nonlinear
semigruop generated by T ^
A50 )
is found to have thè form:
(g - C(I - AC)- 1 q)+C(I - AC)'^.
(55)
13
Thus, we have proved thè following theorem.
Theorem 3 The nonlinear operator T.
given by (55). D
a
generates a nonlinearsem.igroup and exp(tT>
) zs
PO
Remark 6 Since (I — AC)" 1 = Y"" (AG)-" . this term can be interpretaci as thè operator
j=0
of thè "successive interactions" of thè particles with thè surface <9V.
4. CONCLUDING REMARKS
Theorem 2 shows that thè solution of probìem (2) can be written as follows
w(t)=exp(tTA?q)g=exp(tTAi0) (w0 - C(I - AC)-1q)+C(I - AC^q.
(56)
because e x p ( t T ^ Q J is thè linear semigroup generated by T ^ Q that is strongly
differentiable for eacli t > 0.
Substituting w ( t ) in probìem (3). thè solution of this probìem is found to have thè forni:
t
m(t)=exp[t(T A 0 +vo- s K)]mo+ exp[(t - t')(TA>0+v<rsK)]v<TsK(C(I - ACj^q) dt'+
0
+(exp[t(TAi0+v(rsK)] - exp[tT A _ 0 j) (w0 - C(I - ACÌ^q).
(57)
Finally thè solution X=w-j-m of probìem (1) is obtained frorn (56) and (57):
X(t)=exp[tT A-0 +v<7 s K]
t
-f Jexp[(t - t')(TA50+v<rsK)]vasK(C(I - AC^q) dt/.
(58)
O
Remaxk 7 The first term on thè right-hand side of (58) is thè effect of thè initial density
X0 of thè particles and it approaches zero, as t goes to oc, because:
exp[t(T Y Q+v<7 s K)] < exp( — vcr c t). Vt > 0. where crc=cr — crs.
The second term is due to first flight particles and thè last term is due to thè scattering
with thè host medium. D
14
REFERENCES
[1] Belleni Morante A., "A Concise Guide to Semigroups and Evolution Equations",
Series on Advances in Math. and Appi. Sci. Voi. 19, World Scientific (1994).
[2] Belleni Morante A.,''A nonlinear evolution problem arising from particle transport
with nonhomogeneous boundary conditions ". Nonlinear Anal. (to appear).
[3] BorgioliG.TotaroS., "3-Dstreamingoperator with generai multiplyingboundary
conditions: semigroup generation properties ", Semigroup Forum, (to appear).
[4] Greenberg W., van der Mee C.V.M., Protopopescu V., "Boundary value problems in
abstract kinetic theory", Birkhauser, Basel 1987.
[5] Kato T., "Perturbation Theory for Linear Operators77, Springer-Verlag. New York (1976).
[6] Miyadera I., "Nonlinear Semigroups". American Math. Soc., Providence, R..I. (1992).
[7] Pazy A., "Semigroup of Linear Operators and Applications to Partial Differential
Equations". Springer-Verlag. New York (1983).
ACKNOWLEDGEMENTS
Tnis *,vcrk was partially supported by thè Itaiian "Ministero denTuiversila e della.
Ricerca Scientifica e Tecnologica" 40% and 60% research funds. as well as by GNFM of
Itaiian CNR.