On Stationary and Time-Dependent Solutions
to the Linear Boltzmann Equation
Rolf Pettersson
Department of Mathematics
Chalmers University of Technology
S-412 96 Göteborg, SWEDEN
Abstract. This paper considers the stationary linear, space-dependent Boltzmann equation in the case of an interior source
term together with an absorption term and general boundary reflections. First, mild L1 -solutions are constructed as limits of
iterate functions. Then an H-theorem on a relative entropy function of two different solutions is studied. Finally a generalized
inequality (of Dorrozes-Guiraud type) for the boundary terms in the H-theorem is proved.
INTRODUCTION
The linear Boltzmann equation is frequently used for mathematical modelling in physics, (e.g. for discribing the
neutron distribution in reactor physics, cf. [1]-[4]).
One fundamental question concerns the large time behavior of the function f x v t , representing the distribution
of particles; in particular, the problem of convergence to a stationary equilibrium solution, when time goes to
infinity. In our earlier papers [5]-[8] we have studied such convergence to equilibrium for the space-dependent linear
Boltzmann equation with general boundary conditions and general initial data, under the assumption of existence of
a corresponding stationary solution. For the proofs we use iterate functions, defined by an exponential form of the
equation together with the boundary conditions, and we also use a general relative entropy functional for the quotient
of the time dependent and the stationary solutions.
Then a fundamental question in kinetics concerns the existence and uniqueness of stationary solutions to the spacedependent transport equation, with general collision mechanism (including the case of inverse power forces), together
with general boundary conditions, (including the periodic, specular and diffuse cases). We will study this problem
in the angular cut-off case, using our earlier methods with iterate functions F n x v, (representing the distribution of
particles having undergone at most n collisions, inside the body or at the boundary).
This will be done in the case of an interior source term α 0 G x v, where α0 0 is a constant and G is a given
function, together with an absorption term α F x v α 0, and general boundary reflections. We will also prove an
H-theorem for a relative entropy function of two solutions with different (absorption) coefficients; in order to study
the problem of convergence, when the coefficients go to zero. Finally we give a generalization of an inequality of
Dorrozes-Guiraud type for the boundary terms in the H-theorem.
For bounded gain operators the problem of existence and uniqueness for solutions to the linear Boltzmann equation
has been studied earlier by a different technique, cf. ref. [3]. But in our approach unbounded operators also are
included, e.g. the case of hard inverse power collision forces.
PRELIMINARIES
We consider the stationary transport equation for a distribution function F x v, depending on a space variable
x x1 x2 x3 in a bounded convex body D with (piecewise) C 1 -boundary Γ ∂ D, and depending on a velocity
variable v v1 v2 v3 V Ê3 . The stationary linear Boltzmann equation in the case of given interior source
α0 G x v, where α0 0 is a constant and G 0 is a given (measurable) function, together with an absorption term
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
51
α F x v, α
0, is in the strong form
α F x v v∇xF x v α0 G x v QF x v
(1)
The collision term can be written
QF x v VΩ
Y
x v F x v Y x v F x v B θ v v d θ d ζ dv
(2)
where Y 0 is a known distribution function, and B 0 is given by the collision process. Here v v are the velocities
before and v v the velocities after a binary collision, and Ω θ ζ : 0 θ θ̂ 0 ζ 2π is the impact plane.
In the angular cut-off case with θ̂ π2 the gain and the loss term can be separated
QF x v K xv
V
vF
x v dv
L x vF
x v
(3)
where L is the collision frequency
L x v VΩ
w v v
B θ wY x v d θ d ζ dv
(4)
In the case of nonabsorbing body we have
L x v V
K x v v dv
(5)
One physically interesting case is that with inverse k-th power collision forces
k5
k1
with hard forces for k 5, Maxwellian for k 5, and soft forces for 3 k 5.
The equation (1) is supplemented with (general) boundary conditions
B θ w b θ wγ
nṽ
R x ṽ vF
V nv
F x v 1 β where β is a constant, 0 β
γ
(6)
x ṽd ṽ
(7)
1. The function R 0 satisfies
R x ṽ vdv 1
V
(8)
and n n x is the unit outward normal at x Γ. The functions F and F represent the ingoing and outgoing trace
functions corresponding to F. Furthermore, in the specular reflection case, the function R is represented by Dirac
measure R x ṽ v δ v ṽ 2n nv, and in the diffuse reflection case R x ṽ v nvW x v with some
given function W 0 (e.g. Maxwellian function).
Now using differentiation along the characteristics, the equation (1) can formally be written
d
F x tv v α0 G x tv v
dt
K x tv v
V
Let
vF
x tv v dv
α L x tv vF
tb tb x v inf τ : x τ v
τ Ê (9)
x tv v
D
and xb xb x v x tbv. Here tb represents the time for a particle going with velocity v from the boundary point
xb x tbv to the point x.
Then we have the following mild form of the stationary linear Boltzmann equation
F x v F xb v tb
0
QF x τ v v α0G x τ v vd τ
(10)
and the exponential form
F x v F xb ve
tb
0
Êτ
e 0 α
Êt
b
0
α Lx
sv vds
Lxsv vds
α0 G
x τ v v 52
V
K x τv v
vF x τ v v dv d τ (11)
CONSTRUCTION OF STATIONARY SOLUTIONS
We construct mild L1 -solutions to our problem as limits of iterate functions F n , when n ∞. Let first F 1 x v 0
for all x v Ê3 . Then define, for given function F n1 the next iterate F n , first at the ingoing boundary (using the
appropriate boundary condition), and then inside D and at the outgoing boundary (using the exponential form of the
equation);
Fn x v 1 β V
nṽ
R x ṽ vF n
nv
F n x v Fn x tbv ve
tb
Êτ
e 0 α
Lxsv vds
0
Êt
b
0
α0 G
D Γ v v
x
α Lx
1
sv vds
x ṽd ṽ
V
Ê3 ;
(12)
x tauv v V
Γ ∂D v
nv 0 x
Ê3 V
K x τv v
vF n
1
x τ v v dv d τ
(13)
Let also F n x v 0 for x Ê 3 D.
Now we get a monotonicity lemma, which is essential in the following, and which can be proved by induction.
F n x v F n1 x v
Lemma 1
x
D
v
V
Æ.
n
Using differentiation along the characteristics, we get by (13) that
d n
F x tv v α0 G x tv v α F n x tv v
dt
K x tv v
V
vF n
1
x tv v dv
L x tv vF n
(14)
x tv v
Then integrating (14), it follows by Green’s formula that
α
F n x vdxdv DV
α0
ΓV
G x vdxdv DV
F n x vnvdvdΓ ΓV
Fn x vnvdvdΓ DV
L x vF n1 x v F n x vdxdv
(15)
where by (8) and (12)
ΓV
Fn x vnvdvdΓ 1 β ΓV
F n1 x vnvdvdΓ
(16)
Now by Lemma 1 it follows that
α
So, if G
DV
F n x vdxdv β
L1 D V , then we have for all α
DV
ΓV
F n x vnvdvdΓ α0
G x vdxdv
(17)
DV
0 that
F n x vdxdv α0
α
DV
G x vdxdv ∞
(18)
Then Levi’s theorem (on monotone convergence) gives existence of a mild (defined by (10)) L 1 -solution F x v limn∞ F n x v to the stationary linear Boltzmann equation (1) with (3), (7), and F F α β α satisfies for all
0
α α0 0 0 β 1, the inequality
α
DV
F x vdxdv β
ΓV
F x vnvdvdΓ α0
G x vdxdv
(19)
DV
Furthermore, if L x vF x v L 1 D V , then we get equality in (19) together with uniqueness in the relevant
function space, cf. [6] and also [3].
53
So, for instance, if β
ρ
α
DV
α0 α 0 ρ
0, then
F x vdxdv ρ
ΓV
F x vnvdvdΓ G x vdxdv
(20)
DV
In summary, we have the following existence theorem for solutions to our stationary linear Boltzmann equation with
general boundary reflections.
Theorem 2 Assume that K L and R are nonnegative, measurable functions, such that (5)
and (8) hold, and L x v
L1loc D V . Let α α0 0 and 0 β 1 be constants, and G x v L 1 D V with Gdxdv 0.
a) Then there exists a mild L1 -solution F x v to the problem (1)-(4) with (7). This solution, depending on α α 0 and
β , satisfies the inequality (19).
b) Moreover, if L x vF x v L 1 D V , then the trace of the solution F satisfies the boundary condition (7) for
a.e. x v Γ V . Furthermore, mass conservation, giving equality in (19), holds, together with uniqueness in
the relevant L1 -space.
Remarks:.
1) For the case α α0 0 β 0, we have in an earlier paper obtained uniqueness of mild L 1 -solutions to the
stationary linear Boltzmann equation, using a general entropy functional cf. [8]; cf. also Section 4 in this paper.
2) The statement in Theorem 2 (b) on existence of traces follows e.g. from Proposition 3.3, Chapter XI, in [3].
3) The assumption LF L 1 D V is for instance, satisfied for the solution F in the case of inverse power collision
forces, cf. (6), together with specular or diffuse boundary reflections. This follows from a statement on global
boundedness of higher velocity moments, cf. Theorem 4.1 in [9],
DV
1 v2σ 2 F x vdxdv Cσ
if 1 v2σ 2 G x v
σ
∞
max 0 γ 1 γ k 5 k 1 1
(21)
L1 D V .
For a proof of (21) we can multiply equation (14) by 1 v 2σ 2 , and integrate, (using Green’s formula) getting an
equation analogous to (15). Then the gain and loss terms can be estimated, using an inequality for the velocities in a
binary collision, cf. [5],
1 v 2 σ 2 1 v2σ 2 C1 w cos θ 1 vmax1 σ 1 1 v2
with constants C1 C2 0 and σ
σ 2
2
C2w cos2 θ
1 v2 σ 1
2
0, together with some elementary estimate,
wγ 1 1 vγ
1
2
1
1 v2 γ 1
2 The function Y in (2) is here assumed to satisfy the following conditions
V
1 vγ
max2 σ sup Y x v dv ∞
xD
inf Y x v dv 0
V xD
For further details on boundedness of higher velocity moments, see [9], and also our earlier papers [5]-[8].
AN H-THEOREM (FOR A RELATIVE ENTROPY FUNCTION)
In this section we will prove an entropy theorem for the quotient of two solutions from the previous section, F̄ and F,
with coefficients ᾱ , ᾱ0 β̄ and α α0 β respectively. So we start from the equations
d
F̄ x tv v ᾱ0 G x tv v ᾱ F̄ x tv v
dt
vF̄ x tv v dv L x tv vF̄
nṽ
1 β̄ R x ṽ vF̄ x ṽd ṽ
nv
K x tv v
F̄ x v 54
x tv v
(22)
and
d
F x tv v α0 G x tv v α F v tv v
dt
vF x tv v dv L x tv vF
nṽ
1β
R x ṽ vF x ṽd ṽ
nv
K x tv v
F x v x tv v
(23)
To prove an H-theorem for the convex function ϕ z z 1 2 z F̄ F, we begin with the following calculations
2
d F̄ x tv v
1 F x tv v dt F x tv v
d F̄ 2
F̄ d F̄ F̄ 2 dF
d F̄ dF F 2F̄ F x tv v 2
2 x tv v dt F
F dt
F
dt
dt
dt
F̄
F̄ 2
2
α0 G α F KF v dv
1 ᾱ0 G ᾱ F̄ K F̄ v dv LF̄ 1 F
F
LF where we have shortened the notations to the essential variables.
Now some (elementary) calculations give
2 d F̄
1 F x tv v dt F
ᾱ
F̄ ᾱ 2
F̄
2
ᾱ α 2
0
1 α0 G 0 α0 G F 2ᾱ α α0
F α0
2ᾱ α
F
K v
v
F̄ v F̄ v 2
F v dv
F v F v K v
v
F̄ v F v
(24)
1
ᾱ 2
F
2ᾱ α
2
F v dv
L v
F̄ v
F v
1
2
F v
where L v K v v dv .
Then by integration of (24) using Green’s formula, and also by the relation
VV
K v v dv
F̄ v
F v
1
2
F vdv K v
VV
v
F̄ v F v
1
2
F v dvdv
(25)
it follows that
F̄ x v
ΓV
α0
F x v
F̄ x v
DV
DVV
1
F x v
2
F̄ x v
F x vnvdvdΓ F x v
ΓV
ᾱ0 2
G x vdxdv 2 ᾱ α α0
1
2
F x vnvdvdΓ F̄ x v
F x v
DV
F̄ x v F̄ x v 2
K x v v F x v dxdvdv ᾱ0 α0
α0
F x v
2
DV
because of the massrelation (19).
For the boundary terms we use, if β
F̄
V
ᾱ α 2
2ᾱ α
ᾱ0 α0 2
α0
RHS F
β̄
F x vdxdv
ᾱ α 2 α0 α 2ᾱ α G x vdxdv
DV
a Darrozes-Guiraud inequality,
2
F̄
F nvdv 55
(26)
DV
0,
1
ᾱ 2
F x vdxdv 2ᾱ α
F x v
G x vdxdv where (right-hand side)
V
F
1
2
F nvdv
(27)
More generally, for β
αρ
F̄ v
F v
V
Now, in the case β
β̄
0,
0 β̄
1
ᾱρ
2
0, it follows from Theorem 4 below, with ϕ z z 1 2 and K 1, that
F vnvdv 1 β V
2
1 β̄ F̄ v
1 F vnvdv
1 β F v
the following H-theorem (on relative entropy) holds, cf. (26), (27)
F̄ x v
F̄ x v
ᾱ0 2
ᾱ 2
G x vdxdv 2 ᾱ α F x vdxdv α0
2ᾱ α
DV F x v
DV F x v
F̄
x v F̄ x v 2
K x v v
F x v dxdvdv F x v F x v
DVV
ᾱ0 α0 2 α0 ᾱ α 2 G x vdxdv
α0
α 2ᾱ α DV
α0
Furthermore, let now α 0 α ᾱ0 ᾱ , and define δ by ᾱ α 1 δ , i.e. δ
Then, after division by α α 0 in the H-theorem (29), it follows that
F̄ x v
1
α
1δ
2
G x vdxdv 1 2δ F̄ x v
1
(29)
ᾱα , and suppose that 0 δ 12.
1 δ 2
F x vdxdv F x v
1 2δ
DV F x v
1 F̄ x v F̄ x v 2
K x v v G x vdxdv (30)
F x v dxdvdv δ 2 1 F x v F x v
1 2δ
DVV
DV
DV
(28)
Further calculations (among others using an intermediate step through α ᾱ0 between α α0 and ᾱ ᾱ0 ) give the
following version of the H-theorem (when ρ 0), with a constant C 0,
F̄ x v
2
F̄ x v
1
α
G x vdxdv 1
2
F x vdxdv F x v
x v
F̄ x v 2
F̄
K x v v F x v dxdvdv C δ 2
F x v F x v
DVV
F x v
DV
1
DV
G x vdxdv
(31)
DV
In the rest of this section we will study a sequence of stationary solutions FN x v∞
N 2 by choosing the coefficients
α and ᾱ as follows. Let
α
αN 1
N
Then the H-theorem (31) with F̄
ᾱ
αN 1 FN 1
F
1
N 1
FN
δ
δN
1
ᾱ
α
1
N 1
can be written (for N 2 3
N 2 3 4
(32)
) as
F
2
F
2
x v
N 1 x v
1 G x vdxdv 1 FN x vdxdv FN x v
FN x v
DV
F
x v FN 1 x v 2
C
K x v v N 1
G x vdxdv
FN x v dxdvdv 2
F
x
v
F
x
v
N
DVV
DV
N
N
N 1
DV
N
From (33) it follows (among others), if 0 G x v
lim
N ∞
(33)
L 1 D V that
FN 1 x v
FN x v
1
(34)
where αN N1 0 when N ∞.
Furthermore, we get (from (33) when ρ 0) also the following theorem for the sequence of our solutions
FN x v∞N2 . The case ρ 0 can be proved in the same manner, cf. (28).
Theorem 3 Let FN x v be the solutions from Theorem 3.2 corresponding to (absorption) coefficients α α k 1k k 2 3 , and let α 0 α β αρ ρ 0, and assume that G x v L x vFk x v L1 D V . Then the
following estimate holds (with a positive constant C)
56
F
∞
∑
2
x v
1 G x vdxdv Fk x v
F
k 1
k N
DV
DVV
F
K x v v k
2
x v
1 Fk x vdxdv
Fk x v
k 1
DV
C
x v Fk 1 x v 2
Fk x v dxdvdv Fk x v Fk x v
N
(35)
1
Remark:. We observe that α N αN 1 1 δN α2 ∏Nk3 1 δk α2 exp ∑Nk3 ln 1 δk N
N
α2 exp ∑k3 δk O 1 ∑k3 δk2 , using MacLaurin’s formula, where O 1 is a bounded function, when N
So we find that αN 0, when N ∞, if and only if ∑∞
3 δk ∞.
∞.
AN INEQUALITY (OF DORREZES-GUIRAUD TYPE) FOR THE BOUNDARY TERMS
(IN AN H-THEOREM)
We will here prove a generalization of an inequality (cf. [2], p.115, “A remarkable inequality”), concerning the relation
between ingoing and outgoing boundary terms in an H-theorem for the relative entropy functional with general convex
function.
Suppose that F̄ and F are two functions satisfying the following boundary conditions, cf. (7)
F̄ v c̄
F v c
nṽ
R ṽ vF̄
nv
nṽ
R ṽ vF
V nv
ṽd ṽ
(36)
ṽd ṽ
(37)
V
with (for simplicity) c̄ 1 β̄ c 1 β 0 β̄ β 1.
Let
(38)
W ṽ cR ṽ vnṽF ṽ nvF v
Then V W ṽd ṽ 1, so W ṽ is a weight-function, because of the boundary conditions (37) for F.
Suppose ϕ z is a (general) strictly convex continuous function. Then we have, because of Jensens inequality, that
ϕ
g ṽW ṽdv V
ϕ g ṽW ṽd ṽ
(39)
F̄ v
F v
(40)
V
Choose now the function g k FF̄ , with a constant k 0, so
g ṽ k
F̄ ṽ
F ṽ
g v k
nv 0 nṽ
Then we can prove the following generalization of the Dorrezes-Guiraud inequality for the boundary terms.
Theorem 4 Suppose F̄ and F satisfy the boundary conditions (36), (37), and let ϕ z be a strictly convex continuous
function. Then (for any constant K 0)
F̄ v V
ϕ K
F v
nvF
vdv c
c̄F̄ v V
ϕ K
cF v
nvF
vdv
Proof.. By the boundary conditions (36) for F̄ we get, cf. (38) and (40), that
g v k
c̄
c
F̄ v
F v
kc̄
V
nṽR ṽ vF̄
cR ṽ vF ṽk
ṽd ṽ nvF v F̄ ṽ
c̄
d ṽ nvF v F ṽ
c
57
W ṽg ṽd ṽ
(41)
Multiplying by cc̄ , and taking ϕ
ϕ
of both sides, we find, by Jensens inequality (39), that
c
c̄
g v
ϕ g ṽW ṽd ṽ ϕ g ṽcR ṽ vnṽF ṽd ṽ nvF v
Then multiplying by nvF v, and integrating
V
W ṽg ṽd ṽ ϕ
ϕ
c
c̄
dv, we get
g v nvF vdv c
ϕ g ṽnṽF ṽd ṽ
which gives the inequality (41), if we use (40) and set k cc̄ K.
Remark.. If k c c̄ 1, then we get the inequality in [2].
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