This article was downloaded by: [University of Bath]On: 02 November 2014, At: 07:24Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

Transport Theory andStatistical PhysicsPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/ltty20

Kinetic theory of atoms andphotons: An application to theMilne-Chandrasekhar problemRoberto Monaco a , Jacek Polewczak b & AlbertoRossani aa Dipartimento di Matematica , Politecnico diTorino , Corso Duca degli Abruzzi 24, 10129, Torino,Italy E-mail:b Department of Mathematics , California StateUniversity , Northridge, CA, 91330-8313, USA E-mail:Published online: 20 Aug 2006.

To cite this article: Roberto Monaco , Jacek Polewczak & Alberto Rossani (1998)Kinetic theory of atoms and photons: An application to the Milne-Chandrasekharproblem, Transport Theory and Statistical Physics, 27:5-7, 625-637, DOI:10.1080/00411459808205646

To link to this article: http://dx.doi.org/10.1080/00411459808205646

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for any

losses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

TRANSPORT THEORY AND STATISTICAL PHYSICS, 27(5-7), 625-637 (1998)

Kinetic Theory of Atoms and Photons: An Application to the Milne-Chandrasekhar Problem

Roberto Monaco, Jacek Polewczak, and Alberto Rossani

Roberto Monaco and Alberto Rossani Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24

10129 Torino, Italy (e-mad MONACO@POLITO.IT) (e-mail: ROSSANI@POLITO.IT)

Jacek Polewczak Department of Mathematics

California State University Northridge

CA 91330-8313, USA (e-mail: JACEK.POLEWCZAKQCSUN.EDU)

Mathematics Subject Classification. 76N15, 82C40, 82C70

Keywords. Kinetic theory of reacting gases, Gas radiation, Transport processes

Abstract. We consider a kinetic model describing the interactions between monochromatic photons and gas-particles with two internal energy levels. Using the moment equations we formulate so-called MiIneChandraekhar problem. Next, we provide the existence and uniqueness theorems for the resulting system of partial differential equations.

1. Introduction In a recent paper [RSM], Rossani, Spiga and Monaco have proposed the kinetic model for the study of twdeve l atoms and monochromatic photons. The model incorporates a basic feature that allows a good description of the problem, namely the interplay of inelastic collisions between atoms, on one side, and interaction between gas and radiation, on the other The most remarkable result of such a modelling is that, under thermodynamical equilibrium conditions, Planck’s law of radiation is recovered selfconsistently, without resorting to additional hypotheses. An interesting application for the above kinetic model is the Milne-Chandrasekhar problem [CHA, MIL]. A slab is filled with a gas, illuminated from one side The problem consists in the study of the evolution, within the slab, of radiation field, excited atom density and gas temperature. In this report, after a brief description of the model, we derive the corresponding moment equations for the Milne-Chandrasekhar problem and state existence and uniqueness theorems for the resulting system of partial differential equations. In [MPR], we provide further details on approximate analytical solution to the stationary problem, as well as numerical solutions to the time dependent and stationary prohiems.

2. T h e Kinetic Model

Consider the following physical system:

(a) A gas of atoms A of mass rn endowed with only two internal energy levels El and E z , El < Ez. In what follows we will denote by -4, and A*, respectively, particles A at levels ”1” and ”2” (fundamental and excited levels).

(b) A radiation field of photons p at a fixed frequency

625

Copyright 0 1998 by Marcel Dekker, Inc. www.dekker.com

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

626 MONACO, POLEWCZAK, AND ROSSANI

4 E "= - , A E = E z - E l ,

h being the Planck constant, interacting with the above gas-particles.

The following interactions between particles and between particles and photons take place:

a) Elastic interactions among particles with the same or different energy level.

b) Inelastic mechanism of exchange between the internal energy levels

A i + A i + A z + A , ,

A1 + Az + A2 + Az

c) Interactions between gas-particles and photons

c.1) Absorption

A i + p + A z ,

c.2) Spontaneous emission A z - + A i + p ,

c.3) Stimulated emission A z + p - + 2 p + A i j

where the photons involved in the last process have the same velocity c n , $2 E S , S the unit sphere in R3.

We denote by F ( t , x , n)dxdst the mean number of photons at time t that belong in the volume element dx, with velocity in the direction dst. LRom now on, we shall consider the radiation intensity I ( t , x , st), related to F by

I ( t , x , n ) = c h u F ( t , x , n ) .

The events (c.l), (c.2) and (c.3) are respectively described by means of the so-called Einstein coefficients [RSM] a and P, which will be considered here as identification parameters of the model and therefore are assumed as known.

The evolution equation for I ( t , x , n ) is given by

ar t 0. V I = hu[nz(cr + P I ) - nlPI] = huJ, , (2.1) at

where nl and nz are, respectively, the number densities of particles A1 and A2. For f l = fl(t,x,v) and f2 = fz(t,x,v), the distribution functions of particles A1 and A2, respectively, and v E W3, particle's velocity, the kinetic Boltzmann-like equations for the two particle species Al and A2 have the form

?!! + v . V f k = J; + JL + J; = Jk , at where the J i = J : ( t , x , v ) , e = e, i , r are the collision integrals corresponding to the interaction classes (a), (b) and (c), respectively. In the Appendix we provide the expressions of the collisional integrals J: derived in paper [RSM].

In the 8ame paper [RSM] thermodynamical equilibrium conditions have been derived for the equations (2.1) and (2 .2) . In particular, the elastic collision terms Ji vanish whenever the distribution functions are Maxwellians. that is

(2.3)

In order for the inelastic collision terms JI to vanish, the the number densities must be linked by the following mass action law:

n2Jnl = exp(-4E/keT) . (2.4)

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

627 KINETIC THEORY OF ATOMS AND PHOTONS

Finally, the gas-radiation interaction terms JL and .Ip vanish as well if Planrk’s radiation law is fulfilled

(2.5)

3. Moment Equations and their Approximation A derivation of moment equations proreeds in a standard way; we multiply equations (2.2) by m, nrv, and 4md, respectively, then sun1 over k and finally integrate with respect to dv. This yields,

a &(PU) + V . ( P U @ U + P ) = 0 (3.2)

(3.3) a

,(eth f fp2) i V . { [ ( e th f +pu”)E + IPJ . u + qrh) = S‘AE ,

(I is the unit tensor), where the macroscopic quantities (all functions o f t and x). related to the material gas only, are given by: - maTs density

k

- mean velocity

- stress tensor

- thermal energy density (linked to temperature T)

heat flux

- source of non-excited atoms, due to inelastic interactions only

where the cross sections uii and uii are defined in the Appendix.

Equations (3.1) and (3.2) represent, respectively, mass and momentum conservation for the material gas Equation (3.3) is the kinetic energy balance for atoms.

Next, by integrating the equation of photon transport (2.1) with respect to dn, we obtain

( 3 4 ae, at - f V . j, = hv[n2(4an + cpe,) - 721 cpe,] ,

where 1

e, = - with Z = L I ( n ) d O

is the radiation energy density, and

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

628 MONACO, POLEWCZAK, AND ROSSANI

j, = flI(fl)dfl

is the radiative flux.

An additional independent balance equation can be obtained by considering Eq.(2.1) together with Eqs(2.2). Indeed, after multiplying these equations by +mu2 f Ek, summing them over k, then integrating with respect to dv, and finally by adding this result to Eq.(3.4), we obtain the total energy conservation equation for the whole system:

(3.5) a

z ( e A f er) + v ‘ [(eAH f p) ’ U + 9 t h + 9e.c +jr] = 0

where eA = + eth + eexc

is the total energy for atoms, with

as excitation energy density, and where

represents the diffusion current of excitation energy. Let us observe that all the macroscopic quantities appearing in Eqs(3.4-3.5) are functions o f t and x, as well. The equations (3.1), (3.2), and (3.5) constitute the macroscopic conservation equations of the system. We note that equations (3.1-3.3, 3.5), together with Eq.(2.1), are exact, however, they do not form a closed system of equations. In order to close the above system, we observe that when the characteristic relaxation time of elastic col- lision processes is much smaller compared with the ones relevant to inelastic and gas-radiation interaction processes, one may adopt the standard zero-order Chapman-Enskog approximation.

In this case one can calculate qth , qexc, Uk and S‘ by means of Maxwellians (2.3), namely

9 t h = qexe = 0 , uk = > (3.7)

(3.8) S’ = [nlyl~(T) + nmz(T)][nz - nl exp(-AE/k~T)] ,

where

Using (3.7) and (3.8) in one dimensional case, ie., with

i being the unit vector in the direction z and p = Cl.i E [-l,l]. equations (3.1, 3.2, 3.3, 3.6b) have the form

(3.9)

(3.10)

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

KINETIC THEORY OF ATOMS AND PHOTONS 629

(3.12)

where p = i . P i.

4 The Milne-Chandrasekhar P rob lem

The problem has its origin in the experiment performed by Hayner in 1925 (see [MIL] and references therein). Suppose that a slab filled with a monoatomic gas (say mercury vapour) is illuminated from one side by light (say from a mercury arc) in resonance with these atoms for a length of time sufficient to establish a steady state. Then we suddenly cut off the slab from the source of illumination. The problem consists in studying two transients cases: before and after the cut off. The latter case wlts studied first by Milne [MIL] in 1926 and then by Chandrasekhar (CHA] in 1949. Both, Milne and Chandrasekhar, considered a material gas at a fixed density. Furthermore, they did not consider any mechanism of exchange between the internal energy levels of atoms. Finally, the equations they actually solved were linearized. Here, the same problem is treated in full generality. In other words,

i) we keep all the sources of nonlinearity, ii) we consider the effects of inelastic interactions,

iii) we do not neglect ( l /c)(aI /a t ) .

In the case of a gas in a slab (-a 5 z 5 a) and a fixed density n, we assume a) For t E (-m,O) the gas at a known number density n is in absolute equilibrium at a temperature TO,

b) The boundaries of the slab are perfectly reflecting mirrors, so that also the radiation field, a t temperature

c) At t = O+ the mirrors are removed and the gas in the slab is subjected to zero radiation a t the left

d) The boundary conditions for the observable u are such that u ( i a ) = 0.

together with the radiation field.

To, is given by the Planck law stated by Eq.(2.5).

boundary z = -a and to a known radiation I ' (p ) at the right one z = a.

Under the above assumptions the system of equations (3.9-3.12) can be written in the form

where nl = n - nz, with the initial data

and with the boundary conditions

(4.5)

Equations (4.2) and (4.3) reduce themselves to the Chandrasekhar model (CHA] when we set y12 = 722 = 0, neglect ( l /c)(aI /&) with respect to p(aI/az), and, under the assumption that n2 is much smaller than n l , approximate nl - n 2 by n l , where 11.1 is treated as a constant.

5 Mathematical analysis of the system: existence and uniqueness results

In a dimensioneless form the system of equations (4.1-4.3) can be written as

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

630 MONACO, POLEWCZAK, AND ROSSANI

aT - at

anz at -

(5.1 - 5.3)

together with the initial conditions

T(O,x) = W x ) , n z ( 0 , x ) = nzo(z), I ( O , Z , P ) = Io(z,p) vz E [-.,a], vp E [-1,1]

and the boundary conditions

I ( t , -a, p ) = 0 v p > 0, ( r ( t , a , p ) = r w vp < o

Here, as before, I ( t , z ) = 2rJ1, I ( t , z , p ) d p , and T ~ z ( T ) , TZZ(T) =: T* when inelastic cross-sections, u i i ( g , x ) , ui:(g,x), are given according to the inverse power potential, with exponent p - 1, and with the standard angular cutoff (see, for example, [GRA]). We recall that so called Maxwell's molecules correspond to p = 5, when TIZ(T), TZZ(T) do not depend on the temperature T . In the present work we consider only this case, i.e., ylz, 722 are given positive constants. Furthermore, in the system above, the densities n2 and nl are normalized to give n ~ ( t , z) + nz(t.2) = 1, for t 2 0 and z E [-a, a], and p, 7 , and X are given positive constants. Finally, while dealing with the mathematical analysis of the problem, we treat more general initial conditions as compared to (4.4).

We look for physically meaningful solutions for (5.1-5.3); this means, among other things, that T ( t , z ) , nz ( t , z ) , I ( t , z , p ) 2 0, for t > 0; z E [-.,a], and p E [-1,1], if the initial conditions To,n2o.I0 and the boundary condition I' (at z = a ) are also nonnegative. In addition, the density, n z ( t , x ) . should satisfy 0 5 nz(t ,z) 5 1 fort 2 0 and z E [-a,.].

The proof of the existence result for ( 5 . 1 5 3 ) is divided into several steps: 1. Derivation of a priori estimation for the problem. 2. Solution of an approximated problem. 3. Passage to the limit with the sequence of approximated solutions and the proof that the limit is a

solution to the original problem.

5.1 A priori estimations In this section we state several a priori estimations for the system (3.1-5.3) that will be used later in the section. The proofs of these estimations as well as some other theorems are quite technical and are postoponed to the forthcoming paper of the same authors.

Proposition 5.1. Suppose that the system (5.2-5.3) has smooth and nonnegative solutions T ( t , 5); n2(t. z),

I ( t , z , p ) , and 0 5 n 2 ( 0 , z ) 5 1, for x E [-a.a]. Then

0 5 nz( t .2) 5 1; t > 0, z E [ - a a ] . (5.4)

Proof: We observe that under the assumptions of the proposition the equation

First, we prove the inequality lln&.= = sup,.. In?(t,z)l 5 3/2 and then that actually Iln211~- 5 1.

is equivalent to the equation ( 5 . 2 ) , if l (n2 j l~- 5 1. On the other hand. for t 2 0, equation (*) can be written as

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

KINETIC THEORY OF ATOMS AND PHOTONS 63 1

where h ( s ) = p^(1211 - nz(s)l + mzznz(s) + 47rXq + ZqZ(s). Therefore, since 0 5 nz(0,z) 5 1,

(47rXq + ZqZ(s)) ds

and we obtain the inequalities (since nz(t,z) 2 0) ;

We note that the proof of the inequality (5.4) does not depend on a choice of nonnegative constants

P,T, A, Y12>722.

In our numerical simulations we will often deal with the cases when both the initial value, nzo(z), and the solution, na(t ,z) , should satisfy 0 5 nzo(z),nz(t,s) I $, for t > 0 and z E [-a,.]. Under certain conditions on the parameters p, q , A,yl~,yzz the following result holds:

Proposition 5.2. Suppose that the system (5.1-5.3) has smooth and nonnegative solutions T ( t , x) , nl(t , I), I ( t , z . p ) , and 0 5 nz(0,z) 5 d, for z E [-.,a], where 4 5 d 5 1. Then

0 5 nz(t,z) 5 d, t > 0, x E [-.,.I, (5.5)

if (i - 1) pmax(y lz ,m) I 4 n ~ 7 . (5.6)

Proof: By Proposition 5.1, Iln~11~- I 1, as well as 111 - nz11~- 5 1. Now, identity (**) yields

nz(t ,z) 5 dexp(- 1' ~ ( s ) ) d s

+ d ~ ' $ { r n l ~ l l - nz(s)l + p y ~ m ( s ) + q z ( s ) ] exp(- [ A ( r ) d r ) d s ,

where as before, h ( s ) = wlzll -nz(s ) l +pyz2nz(s) +4aAq+2qZ(s). that

The proof will he complete if we show

;{ P y l Z l l - nz(s)l + rnzznz(s) + qZ(s) 5 WlZll - nz(s)l + p7%nz(s) + 4ax7 + ZqZ(s). I (: 1 ( 3

However, by (5.6),

( 5 - 1) { m 2 / 1 - m ( s ) ~ + mzznz(s)~ 5 - - 1 Pmax(y lz ,m) 5 47r~7 + 2 - - 71,

since by the assumption, (2 - 1/14 2 0. This completes the proof.

The next two estimations together with Propositions 5.1 (or 5 .2 ) give bounds on the temperature field T ( t , z)

and the photon intensity I ( t , z , p ) .

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

632 MONACO, POLEWCZAK, AND ROSSANI

Proposition 5.3. Suppose that the system (5.1-5.3) has smooth and nonnegative solutmns T(t. z), n2(t . T),

I ( t , z ,p ) , and nz(O,z),T(O,s) 2 0. Then

Proof: By adding equation (5.1) to the equation (5.2) multiplied by 3/2, and then by adding and sub- stracting the expression (3/2)T (47rX1) + 2qT), we obtain the identity

1 [ I ’ 3 +I’ { [qI + ? ~ ( s , z ) ( 4 n ~ q + 2 q ~ ( s , z ) ) exp - ( 4 x ~ q + 2 q 2 ( r 3 s ) ) d r

and using nonengativity of nz,

Next, we need the following generalization of Gronwall’s inequality [bIPF, Theorem 1, p. 3581:

Lemma [H. Movljankulov and A. Filatov]. Let u(t) be real, continuous, and nonnegatir-e such that for t 2 to

1 .

u(t) 5 c + lo k ( t , s)u(s) ds, c > 0,

where k ( t , s ) is a continuously differentiable function in t and continuous in s with k ( t , s) 2 0 fort _> s 2 to. Then

Now, for each fixed 2, we apply the above Lemma with to = 0 and

1 d k ( t , s) = exp [ - l f (47rXq + ZqZ(r,z))dr

to obtain

exp{lof [ E ( s , ~ ) + ~ ~ ’ ~ ( s , r ) d r ] d s ] = e x p [ l - e x p ( - ~ f ( 4 a h ) + 2 $ ( r , z ) ) dr >I . This completes the proof

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

KINETIC THEORY OF ATOMS AND PHOTONS 633

Proposition 5.4. Assume that 0 5 nz(0,z) 5 1 and 0 5 T(O,z), I ( O , z , p ) 5 C, for some C > 0, and for z E [-a,.], p E [-1,1]. In addition, assume that 0 I‘(p) 5 C, for -1 5 p 5 0. Finally, suppose that the system (5.1-5.3) has smooth and nonnegative solutions T(t ,z) ,nZ(t ,z) , I ( t ,z ,p) on the time interval [O ,T ] . Then there is C, > 0 such that

0 5 I ( t , z , p ) 5 C 7 , t E [ O , T l , zE[-a,al, c l E [ - l , l l . (5.8)

- Proof: By adding a suitable source term, I*(t, z, p), (see, for example, [GMP, p. 413-4161) to the right hand side of (5.3), we can transform the nonhomogeneous boundary condition into the homogeneous boundary condition, and write I ( t , z, p ) in the form

where S( t ) IS the evolution operator associated with the operator -& and the corresponding homogeneous boundary condition. The boundedness of I’ implies the boundedness of ? ( t , z , p ) for t E [O,T], z E [-a,.], and p 6 [-1,1]. Finally, since 1 5 nz(t ,z) 5 1, Gronwall’s lemmma completes the proof.

We end this section with the following energy estimation:

Proposition 5.5. Suppose that the system (5.1-5.3) has smooth and nonnegative solutions T ( t , z), nz(t, z), I ( t , z , p ) , and I*(p) 20, for p E [-1,0]. Then, for t 2 0,

Proof: The following energy identity together with nonnegativity of I ( t , z , p ) completes the proof

5.2 The approximated problem The main idea is to construct a suitable approximated (or modified) problem to the original system (5.1-5.3) that retains the a priori estimations of section 5.1 and can be solved relatively easy.

For k 2 1, we consider the following sequence of approximated problems for t E [ O , T ] , T > 0:

(5.10 - 5.12)

together with the initial conditions

and the boundary conditions

In addition, we set ni(t , z) = nzo(z)

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

634 MONACO, POLEWCZAK, AND ROSSANI

It is well known (see, for example, [GMP, pp. 413-4161 that the above nonhomogeneous boundary conditions, with 0 5 I' E L"(-l,O), can he transformed into the homogeneous boundary condition by adding a nonhomogeneous (source) term, F ( t , z , p ) , to the right hand side of equation (5.12). In addition, ? ( t , z , p )

can be constructed in such a way that 0 5 r^. E L"((0,T) x (-a,.) x ( -1 , l ) ) . From now on this system will be denoted as (5.10 - 5.12)o.

For each fixed k , we consider the system (5.10 - 5.12)o as a semilinear problem in the Banach space X = L'(-a,a) xL ' ( -a ,a ) x L ' ( ( - a , a ) x ( - 1 , l ) ) . In order toavoidcumbersome notation, the index k, appearing in T , nz, and I , will be dropped from now on. Next, let X+ be a subset of X consisting of elements u = (T,n2,1) such that T E L' ( -a ,a) , '112 E L'(-a,a), and I E L'((-a,a) x ( -1 , l ) ) are nonnegative (almost everywhere) functions on (-.,a), ( - a , a ) , and ((-a,.) x (-1, l)), respectively. Let D c X be defined as

D = { u E A'+ : 0 5 n2(z) 5 1 almost everywhere in z E ( - n , a ) } (5.13)

and C([O,T],X) be the Banach space of continuous functions from [O,T] --t S. For k 2 1, the system (5.10 - 5.12)o is equivalent to the following abstract semilinear problem on the closed set D c X:

u' ( t ) = Au(t ) + Bk(t ,u( t ) ) for t E ( O , T ] , u(0) = uo E D, (5.14)

where for u = (T,nz,I) ,

(5.15)

with the domain D ( A ) defined by

ar ax D ( A ) = {u E X : p- E L'((-a,a) x (-1,l)) plus the homogeneous boundary condition},

and, for ( t ,u ) E [ O , T ] x D,

$ p { y d l - 7 ~ 2 ) ~ + YZZ(~Z)~ - [m(l- nz)' + Y Z Z ( ~ - n2)n2] exp (-+)}

hz + 2nz& + r*(t) p { [m(1 -n# + m(1 - n2)m] exp (-+) -ylp(l - n~)n2 - Y Z ~ ( ~ Z ) ~ } + r

with

Z(x) = 21iJ: I ( z , p ) d p , and exp (-+) = 0, for T = 0 We remark that the homogeneous boundary condition appearing in the domain of the linear operator A is understood in the sense of trace class (see, for example, [CES] or [GMP]). The existence and uniqueness result is based on the following properties of the initial value problem (5.14): (1) The linear operator A generates a strongly continuous semigroup U ( t ) in X such that U ( t ) : D + D ,

(2) The nonlinear operator B(t ,u) is a continuous mapping from [O,T] x D into X that is also Lipschitz for t 2 0.

continuous on D , i.e.,

II&(t,ul) - &(t,uz)llx I L(k)llu~ - uzllx for .dl ( t , u ~ ) , ( t , u ~ ) E [O,T] x D,

for some Z ( k ) > 0. Here, / / . 1/x denotes any equivalent norm on X. (3) The set D c X is invariant for the evolution problem (5.14).

Property (1) is the standard result that can be found in [GMP, Theorem 2.2, p. 4101. Property (2) follows after somewhat lenghtly but easy computations; here, the crucial is the fact that nz E Lm(-a,a). Finally, property (3) follows from the following observation (see, [MAR, Theorem 2.1, p. 3351):

Proposit ion 5.6. For a fixed k 2 1 , and all (t ,u) E [0, T] x D

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

KINETIC THEORY OF ATOMS AND PHOTONS 635

dist(u+ hBk( t .u) ;D) h

=0 lim h i l l +

(5 .16)

where dist(r.. D) = infttEn /It' - u / / y is thr dlstance of a n rfernrrrt r t S from the SCT D

The limit in (3.16) is a different formulation of the first a priori estimation (Proposition 3.1) for the approxi- mated problem (.5.10-5.12). I t also implies that the evolution problem (5.15) grnelatcs nonnegative solutions if we start with nonnegative initial values.

We have (see. Theorrm 2.1, p. 333. and Remark 2.1. p. 337 of [ X I R ] )

Theorem 5.7. For each 110 E D thrre is a unique continuous function 11 : [O.r] --t D that sat1sfic.s

~ ( t ) = C(t )uo + c(t - s ) B b ( s . u ( s ) ) ds for all t E [O. 71. J,' where the integral in (5.1 7) IS the RiPmalln integral in S

A funrtioii I L that satisfies (5.17) is said to be a mild solution to (5.14) on [U.T].

lye note that C ( t ) has the folloiving representation for u = ( T . n ? . I ) :

T c-(t)u = ( 112 )

c-o( t )J

(3.17)

(5 .18)

where Vo(f) is a strongly continuous semigroup in L'((-a.a) x (-1.1)). generated by the oprrator -,A% - I with the homogeneous boundary condition (see. [GMP. Theorem 2.2. p. 4101).

Next, it follows tha t . for earh k 2 1. and 110 E D n D ( A ) . Propositions 5.3-5.4 still hold for u ( t ) given by (5.17). On the other hand. for u g E D . we choosr a sequence { u o ) ) C X surh that uol; -+ uo in S. as k + x, with tL0li = ( T o k , & . I o ~ ) E DnD(.4) , and TO^ E L"( -a .a) ; Ioe t L x ( ( - a , a ) x (-1.1)) (unifi,rrlil?. in k 2 1). Then again. hk- Propositions 3.3-3.4. the solution (now indexed h\- k). u k ( t ) = ( T A ( t ) , n ; ( t ) . I k ( f ) ) . satisfies Tk( . ) .n$( . ) E LX([O.7] x ( - u . R ) ) and Ik(.) E L1([O.r] x ( - a . a ) x (-1.1)). uniformly i n k 2 1.

These estimations alone imply that the sequence of solutions Q(.) = (Tk(.).n$(.).Ik(.)). k = 1 , 2 . . . . is weakly compact in L1([0.7]:X)). Here, the weak compactness in L ' ( [ 0 . r]:S)) means that the seqnmces {&(.)}, { T I ; ( . ) } are weakly compact in L'([O.r] x ( - a , a ) ) , and the sequence { I k ( . ) } is weakly compact in

This type of weak convergence (after passing to a subsequence) is not enough to pass to the limit (when k -+ cc) in nonlinear terms of (5.10-5.12). An additional argument will provided in the next section.

5.3 Passage to the limit We start with the following compactness result that is crucial in the passage to the limit in (5.10.5.12)

Proposition 5.8. [BGPS. Proposition I , see also V L ~ ] or o CR.'". an open, ronnected set. let f E

and f l ( a ~ x s , + ) - = I r t L ' r ( ( O , ~ ) x ( 8 0 x S.')-). 0 < m < inf(l /p, 1 - l/p), and

L1( [O,T] X ( - U , Q ) X (-1.1))

L p ( ( 0 , ~ ) x 0 x S"), p > 1 . 8 f / 8 t + ff. Tzf E LP((0.r) x O x S"). fit=" = f o E L"(O x S'v). Then J'. f ( t . ~ - , C 2 ) d $ l t II

We apply the above result with 0 = (-a,.) and for .V = 0, when S,\ becomes the interval [-1,1]. Since th? imbeddingWn',P c LPisrompact, .and { I ( . ) & } t L ~ ( [ 0 , r ] x ( - n , a ) x ( - l ~ l ) ) n L ' ( [ 0 , r ] x ( - a , o ) x ( - l , 1 ) ) , uniformly in k, Proposition 5.8 implies

Theorem 5.9. Assume that ug = (To.ll.ro,Io) E D with To E L=(-n,a) and l o E L x ( ( - n , a ) x (-1, I ) ) . Furthermore, assume that 0 5 I' E L"(-1,l). Then. for the sequence of mild solutions u k ( t ) , k >_ 1, the sequence { Z k ( . ) } E L"-([O,r] x (-a,.)) nL'([O,r] x (-.,a)) isielatir.eJ?cr)niparr in L'([U.T] x ( -a .uJ ) .

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

636 MONACO, POLEWCZAK, AND ROSSANI

In particular, when k -+ DO, and after passing to a subsequence if necessary,

&(t , z ) - - t I ( t , z ) , almost everywhere in (42) t [ O , T ] x ( -%a) . (5.20)

Next, we combine (5.20) and nonnegativity of solutions to show L'- convergence of the sequences {Tk(.)}} and {n$( )]. Indeed, after adding equation (5.11) to the equation (5.10) multiplied by 3/2, and integrating with respect to t we obtain

We note that all terms on both sides of identity (5.21) are nonnegative. Furthermore, since the sequence {&(.)} E LcO([0,i]x (-a,a))nL'([O,r]x ( -a ,a ) ) is relativelycompact in L 1 ( I 0 , 7 ] x ( - a , a ) ) , t,heDominated Convergence Theorem applied to the sequence

i + ~ ~ ~ ~ . ~ implies that (after passing to a subsequence if

necessary) the last term on the right hand side of (5.21) converges almost everywhere in ( t , z) E [0, r] x (-a, a),

Therefore. using nonnegativity of the terms on the left hand side of (5.21), we obtain pointwise convergence almost cverywhere of the sequences { T k ( - ) ] } and {n,"(.)] in ( t , z ) t [O,T] x (-a,a).

Thus, we have

Theorem 5.10. Let U O ~ 4 uo in X , as k 4 DO, with Ung = ( T ~ b , & , I o k ) E D n D ( A ) , and TO^ E L"(-a ,a) , 10k ELm((-a,a)x(-l,l))(uniformlyink> 1). Furthermore,assumethatO< 1 8 ~ L m ( - l , 1 ) . Then, for the sequence of solutions uk( t ) = (Tk(t),n$(t),Ik(t)) of (5.14), the sequences { T k ( , ) } , In$(.)} E Lm([O,s] x ( - a , a ) )nL ' ( [O ,~ ] x (-a,aj) arerelativelycompactinL'([O,r] x ( -a , . ) ) .

Now, t,he passage to the limit can be easily achieved. The following two theorems hold:

Theorem 5.11. Under the assumptions of Theorem 5.10, and after passing to a subsequence in { u k } , if necessary,

(-1'

Bt( . ,uk) + B(. ,u) , weaklyin L'([O,r] x (-a,a) x (-1,l))) as k + DO, (5.22)

where u k ( . ) is the sequence of solutions to (5.14) converging wedly in Li([0,7] x (-a,.) x (-1,l))) to

Furthermore, u(.) : [0, T] + D C X is continuous and satisfies (5.1 7). u(.) E D.

Theorem 5.12. Assume that uo = (To,nzo,lo) E D with TO E Lm(-a,a) and10 E Lm((-a,a) x (-1,l)) . Furthermore, let 0 5 I' E Lm(-l, 1). Then (5.17) has a unique solution in X .

APPENDIX

Inelastic rollision terms

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14

KINETIC THEORY OF ATOMS AND PHOTONS 631

1 1 2 2

where v; = - (v + w +g*O'), w; = -(v + w - g*O'),

6' = 4AE/m, g = Iv - wI, 0 = (v - w)/g, g* = m, c o s x = 0. O',

U being the Heaviside function. Note t h a t v and w are the velocities of the colliding partic1c.s before interaction, whereas the prime indicates quantities after collision. Moreover the following microreversibility relations link the inelastir cross-sections u;;, utf and m i : . rf: (subscripts: incoming particles; superscripts: outcoming particles)

92d:k7>x) = U ( g - E)g1u::(g-,x)

s 2 U : : ( % x ) = i X ( g + , x ) 3

Radiation interaction terms

Elastic collision terms

(.4.3)

u;! being the cross-section of the elastic k - e interaction ( u ; ~ = .II). Note tha t in (A.l -A.4) , for t he sake of simplicity, t and x have been dropped from the argnments of fi an 1.

REFERENCES

[GRA] H. Grad, Asymptotic theory of the Boltzmann equation. 11, in J. A. Laurmann (ed.) Third Symposium on Rarefied Gas Dynamics, I , New York, Academic Press, 1963, pp. 26-59.

[MPFJ D. S. MitrinoviC, J . E. PecariC, and A. M Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.

[GMP] W Greenberg, C. van der Mee, V. Protopopescu, Boundary Value Problems in Abstract Kinetic Theory.

[MAR] R. H. Martin, Nonlinear Operators and Differential Equations in Banach spaces, John Wiley & Sons. NPIV York. Birkhauser Verlag, Basel, 1987.

1976. [BGPS] C . Bardos, F. Golse, B Perthame, and R. Sentis, The nonaccretive Radiative Bansfrr Equations EXZSIPIIW of

Solutions and Rosseland Approximation, J. Functional Analysis, 77 (1988), pp 434-460

[CES] M. Cessenat, Theoreme de trace pour des espaces de fonctions de la neutronique, C R Arad. Sri Paris. 300(1)

[CHA] S. Chandrasekhar, Radiative Transfer. New York: Dover Publications, 1960 [MIL] E A. Milne (1926) The diffusion of imprisoned radiation through a gas J. London Math. Sor.. 1. 40 (192G)

[RSM] A. Rossani, G. Spiga and R. Monaco (1997) Kinetic approach for two-level atoms interacting with monodiromatlr

(1985), 89-92

photons, Mech. Research Comm., 24: 237 [MPR] R. Monaco, J. Polewczak, A Rossani (1997) Kinetic theory of atoms and photons Moiiieirt equations and theii

[VLA] V S. Vladimirov, Mathematical Problems of the uniform-speed theory of transport, (Russiari) Trudy Mat. Inst Stehlov

application to the Milne-Chandrasekhar problem, submitted to ZAMP

61: 1.58 (1961)

Received: 01 October 1997 Revised: 03 March 1998 Accepted: 10 March 1998

Dow

nloa

ded

by [

Uni

vers

ity o

f B

ath]

at 0

7:24

02

Nov

embe

r 20

14