Eternal Solutions of the Boltzmann Equation
A. V. Bobylev∗ and C. Cercignani†
∗
Division of Engineering Sciences, Physics and Mathematics, Karlstad University, Karlstad, Sweden
†
Dipartimento di Matematica, Politecnico di Milano, Milano, Italy
Abstract. The Boltzmann equation for molecules of the Maxwell class (cross section inversely proportional to the relative
speed) is shown to possess solutions with infinite energy. The same result is conjectured for power law potentials, but only for
Maxwell molecules these solutions are eternal.
INTRODUCTION
We consider a class of solutions of the Boltzmann equation which can be constructed starting from rather peculiar,
self-similar solutions with infinite energy [1]. We were led to considering these solution by the interesting question
of extending the solution for the structure of an infinitely strong shock wave from the case of hard spheres (or cutoff
potentials) to that of molecules interacting at distance . We show that the problem of the upstream asymptotics can be
formulated in terms of the space homogeneous BE. Moreover it is connected with the old question concerning large
time asymptotics of solutions with infinite energy. We first consider Maxwell molecules and find that these self-similar
solutions exist and are eternal, i.e. they exist from −∞ < t < ∞. For other power law potentials the property of being
eternal is lost.
CONNECTION WITH THE SHOCK-WAVE PROBLEM.
Recently Grad’s conjecture [2] that there is a solution of the Boltzmann equation describing the structure of an infinite
Mach number shock wave, was successfully tested by numerical methods of both Monte Carlo [3] and deterministic
nature [4]. For molecules without angular cutoff, the assumption by Grad (a delta plus a regular function) does not
hold. Monte Carlo simulations [5] seem to indicate, however, that a solution exits in this case as well.
This revived the interest in a paper by Pomeau [6], who proposed an approach where the delta term is replaced by an
approximate self-similar solution. Pomeau’s idea is crystal-clear but his exposition is not. Here we make his conjecture
precise.
The idea is that there is an asymptotic solution for x → −∞ of the Boltzmann equation having the following form:
f (x, ξ ) = |x|3λ F(v|x|λ sgn(x))
where v = ξ − u0 i and F is normalized to a constant:
Z
F(w)dw = c
Then we also have
Z
f (ξ )d ξ = c
If λ > 0, for x → −∞, f tends to a delta (multiplied by c). Let us insert this in the Boltzmann equation:
(u0 + v1 )(3λ |x|3λ −1 F + λ |x|4λ −1 v · Fw ) = Q(F.F)|x|3λ (1− s−1 )
s−5
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
19
where the differentiation in Fw and integration in the collision term are made with respect to the variable
w = v|x|λ sgn(x)
Let us rewrite the equation in terms of w:
(u0 + w1 |x|−λ )(3λ |x|3λ −1 F + λ |x|3λ −1 w · Fw ) = Q(F.F)|x|3λ (1− s−1 )
s−5
The fact that F is only an asymptotic and not an exact solution arises from the fact that we cannot make x disappear.
In order to do that we must neglect the second term in the factor (u0 + w1 |x|−λ ). This term tends to zero for a fixed
value of w when x → −∞. Then x disappears if we choose
λ=
In fact, we have:
s−1
s−5
u0 (3λ F + λ w · Fw ) = Q(F.F)
The method does not seem to work for Maxwell molecules (s = 5), but in this case one can make the alternative ansatz:
f (x, ξ ) = e−3λ x F(ve−λ x )
to obtain
(u0 + w1 eλ x )(−3λ e−3λ x F − λ e−3λ x w · Fw ) = Q(F.F)e−3λ x
If we neglect again the second term in the first factor, we get rid of x.
−u0 (3λ F + λ w · Fw ) = Q(F.F)
The approximation is not uniformly valid. This has the result that the second order moment cannot exist. In fact if we
multiply by |w|2 and integrate formally, we have a contradiction.
We remark that, if we go back to the x and ξ variables, f now satisfies
u0 fx = Q( f , f )
which looks as the space homogeneous Boltzmann equation with time t = x/u0 , as already indicated by Pomeau [6],
who from this inferred that the solution should not have finite energy.
BASIC EQUATIONS
Let f (v,t) (where v ∈ ℜ3 and t ∈ ℜ+ are the velocity and time variables) be a distribution function normalized by
Z
ℜ3
dv f (v) = 1
and satisfying the homogeneous Boltzmann equation for Maxwell’s molecules:
µ
¶
Z
V·n
dwdng
ft =
[ f (v0 ) f (w0 ) − f (v) f (w)],
|V|
ℜ3 ×S2
(1)
(2)
where, for simplicity, we do not indicate the time dependence of f in the collision term, and
V = v − w,
1
v0 = (v + w + |V|n),
2
1
w0 = (v + w − |V|n), n ∈ S2 ,
2
g(cos θ ) denotes the scattering cross section multiplied by |V| and, as such, it is independent of |V| for Maxwell
molecules. The “true" cross section for Maxwell’s potential (inversely proportional to r−4 ) leads to the asymptotic
20
behavior [7] g(µ ) ∼
= (1 − µ )−5/4 as µ → 1. Therefore we sometimes consider a cutoff collision operator (pseudoMaxwell molecules) with g ∈ L1 (−1, 1) and normalize g(µ ) is such a way that
µ
¶
Z
Z 1
V·n
dng
= 2π
g(µ )d µ = 1
(3)
|V|
S2
−1
by an appropriate scaling of the time variable in (2). The conditions (1), (3) allow us to write Eq. (2) (in the cutoff
case) and the corresponding initial condition as
ft = Q+ ( f , f ) − f ,
f|t=0 = f0 ,
(4)
1 is normalized by Eq. (1). The usual restriction of the type
where f0 ∈ L+
Z
dv f0 (v)(1 + |v|2 ) = 1
ℜ3
(5)
1 (ℜ3 ) is enough. On the other hand, it is well-known
is obviously not necessary for the solution of (4) to exist: f0 ∈ L+
that the restriction (5) guarantees that the solution f (v,t) tends, as t → ∞, to a Maxwellian distribution. Let us assume
1 (ℜ3 ), but
now that f0 ∈ L+
Z
ℜ3
Let
dv f0 (v)|v|2 = ∞.
(6)
Z
fˆ(k,t) =
ℜ3
dv f (v,t)e−ik·v ,
k ∈ ℜ3 ;
(7)
then the normalization in Eq. (1) becomes
fˆ(0,t) = 1,
and the corresponding initial value problem reads
µ
¶
Z
k·n
fˆt =
dng
[ fˆ(k+ ) fˆ(k− ) − fˆ(0) fˆ(k)],
|k|
S2
(8)
fˆ|t=0 = fˆ0 (k),
1
k± = (k ± |k|n),
2
n ∈ S2 .
(9)
In the cutoff case (3) we similarly obtain
fˆt = Q̂+ ( fˆ, fˆ) − fˆ,
fˆ|t=0 = fˆ0 .
(10)
GENERAL RESULTS FOR MAXWELL MOLECULES
We can prove that the Fourier-transformed Boltzmann equation has a solution in the space homogeneous case exists
1,
and this solution is unique in the class of locally bounded functions, for cutoff Maxwell molecules and f0 ∈ L+
without any further assumption. We describe a series representation for solutions with infinite second moments, and
give estimates for the coefficients of the series, which guarantee its convergence. Among these solutions there are
some which are self-similar. Apart from constants which are related to simple invariance properties, these solutions
turn out to depend on a parameter α , taking values between 0 and 1. There is essentially one solution for each value
of this parameter.
A similar representation was considered 25 years ago when one of the authors [8] constructed self-similar solutions
with finite energy. These solutions obey the same equation as our solutions, but with α > 1. Several authors considered
these solutions in more detail and finally it was proved by Barnsley and Cornille [9] (for the simplest case) that all
these solutions, except the so-called “BKW -mode" [10], [11], do not correspond to positive distribution functions.
Thus those solutions do not appear to be useful for applications. On the contrary, all self-similar solutions with infinite
energy constructed in [1] are positive.
We can also prove the existence of the solutions discussed in the previous section and prove that a solution which is
not self-similar is asymptotic (in a very precise sense) to the self-similar one with the same value of α .
21
SEARCH FOR CLOSED FORM SOLUTIONS
In the case of a Maxwell-isotropic cross section we found two self-similar, closed form solutions [1], [13] in the way
we are going to illustrate:
We consider the Boltzmann equation [7] with a cross-section 1/(4π |V|) where V = v − w is the relative velocity of
two colliding molecules, in the space-homogeneous case:
ft =
1
4π
Z
R3 ×S2
dwdn[ f (v0 ) f (w0 ) − f (v) f (w)],
(11)
where
1
1
v0 = (v + w + |V|n), w0 = (v + w − |V|n), n ∈ S2 ,
2
2
and, for simplicity, we do not indicate the time dependence in the collision term.
Performing the Fourier transform
Z
fˆ(k) =
we obtain [12]
1
fˆt =
4π
ℜ3
dv f (v)e−ik·v ,
k ∈ ℜ3 ,
(12)
Z
S2
dn[ fˆ(k+ ) fˆ(k− ) − fˆ(0) fˆ(k)],
(13)
where
1
k± = (k ± |k|n),
2
We consider the simplest class of solutions
n ∈ S2 .
fˆ(v,t) = φ (x,t), x = |k|2 /2,
(14)
corresponding to isotropic distribution functions f (t, |v|) in (11). Then the equation for φ (x,t) reads
φt =
Z 1
0
ds[φ (sx)φ ((1 − s)x) − φ (0)φ (x)].
(15)
Our aim here is to look for self-similar solutions of the form:
φ (x,t) = ψ (xe−at ),
ψ (0) = 1.
(16)
Then the equation for ψ (x) reads:
1
−axψx =
x
Z x
0
dyψ (y)ψ (x − y) − ψ (x),
(17)
where we let y = sx.
USE OF THE LAPLACE TRANSFORM
Equation (17) can be obviously simplified by using the Laplace transform
u(p) = L [ψ ] =
satisfying:
Z ∞
0
dx ψ (x)e−px ,
−a(pu)00 − u0 = u2 ,
f = L −1 [F],
pu(p) → p→∞ 1,
(18)
or, equivalently, in terms of y = pu(p):
−ap2 y00 − py0 + y(1 − y) = 0,
22
y(p) → p→∞ 1.
(19)
These equations were established in [13], where two solutions were found, by means of a transformation leading
from a (known) solution with a > 0 to solutions with a < 0. Both solutions have the form
y = Yα (p) =
1
.
(1 + p−α )2
(20)
Here we proceed to checking that there are exactly two solutions of this form, corresponding to α = 1/2, a = −2/3
or α = 1/3, a = −3/2.
In fact, we have (20)
y0 = 2α
1
p−α −1 ,
(1 + p−α )3
y00 = 6α 2
1
1
p−2α −2 − 2α (α + 1)
p−α −2
−
α
4
(1 + p )
(1 + p−α )3
and, consequently,
ap2 y00 + py0 = 6aα 2
1
1
p−2α − 2α (aα + a − 1)
p−α ,
−
α
4
(1 + p )
(1 + p−α )3
y(1 − y) =
1
1
−
;
(1 + p−α )2 (1 + p−α )4
These two relations yield:
ap2 y00 + py0 − y(1 − y) =
6aα 2 p−2α + 1 ([2aα 2 + 2(a − 1)α ] + 1)p−α + 1
−
.
(1 + p−α )4
(1 + p−α )3
If aα 2 = −1/6 and (a − 1)α = −5/6 the two fractions become equal and the equation is satisfied. The conditions
imposed on a and α give α = 1/2, a = −2/3 or α = 1/3, a = −3/2 for Yα to be a solution.
Thus we have found the Laplace transforms of two solutions of the Boltzmann equation:
u(p) =
1
1
,
p (1 + p−α )2
α = 1/2, 1/3.
We need to invert them and this we do by using the following Lemma [14]:
Lemma Let F(p) be an analytic function having no singularities in the cut plane C/R− . We assume that F(p) =
F(p) and that the limiting value
F ± (t) = lim F(te±iφ ),
φ →π −0
F + (t) = F − (t)
exist for almost all t > 0. Let the following also be true:
(A) F(p) = o(1) for |p| → ∞, F(p) = o(|p|−1 ) for |p| → 0, uniformly in any sector |argp| < π − η , π > η > 0;
(B) there exists ε > 0 such that for every π − ε < φ ≤ π
F(re±iφ )
∈ L1 (R+ ), |F(re±iφ )| ≤ a(r),
1+r
where a(r) does not depend on φ and a(r)e−δ r ∈ L1 (R+ ) for any δ > 0. Then the inverse Laplace transform of F is
given by:,
Z
1 ∞
L −1 [F] =
dt e−xt ℑF − (t).
(21)
π 0
If we take:
¸
·
1
1
ψ = L −1
, α = 1/2, 1/3.
p (1 + p−α )2
Eq. (21) gives immediately:
ψα (x) = 2
sin απ
απ
Z ∞
0
ds (1 + s cos απ ) −xs−1/α
e
,
(1 + s2 + 2s cos απ )2
23
α = 1/2, 1/3.
Then we remark that x = |k|2 /2 and that
2 /2
e−Θ|k|
Z
= F [MΘ ] =
Rd
dv MΘ (|v|) exp(−ik · v),
where MΘ (|v|) denotes the Maxwellian distribution
2 /(2Θ)
MΘ (|v|) = (2π Θ)−3/2 e−|v|
(22)
with “temperature" Θ > 0. Then we finally obtain:
fα (|v|) = 2
sin απ
απ
Z ∞
0
ds (1 + s cos απ )
M (|v|),
(1 + s2 + 2s cos απ )2 Θ(s)
α = 1/2, 1/3
(23)
where Θ(s) = s−1/α .
We remark that we have done all our calculations in the three-dimensional case, but one can check that the solution
holds in any dimension d ≥ 2, provided only that we replace the three-dimensional Maxwellian MΘ(s) with the ddimensional Maxwellian (the only change in the definition is that the exponent -3/2 is replaced by -d/2).
NON-MAXWELL MOLECULES
In this section we study the case of power-like potentials U(r) = α /rn (α = 0, n > 1) with or without angular cutoff.
Then the collision kernel of the Boltzmann equation reads
g(|u|, cos θ ) = |u|γ gγ (cos θ ),
γ = 1−
4
n
(24)
The asymptotic case (n → ∞)
g(|u|) = |u|
d2
4
(25)
corresponds to hard spheres with diameter d.
We assume that the initial distribution function f0 (v) ≥ 0 does not posses a second moment (energy) finite. Then
we have:
Z
dv f (v,t){1, v, |v|2 } = {1, 0, ∞}
(26)
ℜ3
provided these conditions are fulfilled at t = 0.
We want to discuss briefly the following
Question. Can the results for Maxwell molecules be generalized to the case γ 6= 0, i. e. to non-Maxwell power
potentials?
Following the same scheme as in [1] we consider Eqs. (11)(24) with arbitrary γ 6= 0 and transform f (v,t) to a scaling
form
f (v,t) = s3 (t)F[vs(t),t]
(27)
with an unknown scaling function s(t). Assuming that s(t) > 0, F(v,t) ≥ 0, we obtain the following equation for
F(v,t):
∂F
sγ
+ (sγ −1 st )divv (vF) = Qγ (F, F)
(28)
∂t
where Qγ (F, F) = Q(F, F) with the collision kernel (24). The self-similar solution
leads to the equation:
f∗ (v,t) = s3 (t)F∗ [vs(t)]
(29)
sγ −1 st = −λ ⇒ sγ = sγ (0) − λ γ t
(30)
24
Moreover λ γ > 0 since
Z
H( f∗ ) =
R3
Z
dv f∗ (v,t) log f∗ (v,t) =
R3
dvF∗ (v) log F∗ (v) + 3 log s(t)
(31)
must be a decreasing function of t.
Now we consider separately two cases :
(A) soft potentials with γ < 0;
(B) hard potentials with γ > 0.
In the first case (γ < 0) we obtain:
s(t) = s0 (1 + λ |γ |s|0γ |t)−1/|γ | , s0 = s(0),
If we introduce the notation
Z t
dt 0
τ (t) =
0
s(t 0 )
,
(32)
(33)
then Eq. (28) for F̃(v, τ ) = F(v,t) reads
∂ F̃
− λ divv (vF̃) = Qγ (F̃, F̃)
∂τ
(34)
Moreover τ (t) → ∞ with t. Eq. (34) is very similar (in both cases of positive and negative values of γ ) to that with γ = 0.
Its physical meaning is obvious: it is the Boltzmann equation for a space-homogeneous, rarefied gas in a dissipative
medium with a constant coefficient of friction λ . By an analogy with the case γ = 0 [1] we can expect that there exists
a stationary solution F∗ (v) with infinite energy and that this solution is an attractor for certain classes of initial data.
Hence, we conjecture that this also holds for γ 6= 0 , where the time dependence is now given by Eqs. (29)(32), is also
valid for soft potentials (with the exception of the statement indicating that they are eternal, see below).
Let us now consider the case (B) of hard potentials (γ > 0). Then Eq. (30) results in
γ 1/γ
s(t) = s0 (1 − λ γ s−
t) , s0 = s(0),
0
0<γ ≤1
(35)
By using the same substitution (33) we obtain Eq. (34) again, and expect that there exists a stationary solution F∗ (v)
and that F̃(v, τ ) → F∗ (v) as τ → ∞ for a certain class of initial data. There is, however, an essential difference from
the case γ ≤ 0: the limit τ → ∞ does not correspond to the limit t → ∞, but to the finite value
s0γ
t = t∗ (γ ) =
λγ
,
γ > 0.
On the other hand, τ (t) → ∞ as t → t∗ (γ ) for all 0 < γ ≤ 1. Thus we expect that, for some initial data satisfying the
Boltzmann equation, the solution f (v,t) is defined for 0 < t < T∗ (γ ) and
f∗ (v,t) ∼
=t→t∗ s3γ (t)F∗ [vsγ (t)],
sγ (t) = const.[t∗ (γ ) − t]1/γ ,
0, γ < 1.
Then the collision frequency behaves asymptotically (as t → tγ ) as
Z
dw f (v,t)|w − w|γ ∼
=
Z
dw f (w,t)|w|γ ∼
=
1
t∗ (γ ) − t
and therefore the solution cannot be extended to larger values, t ≥ t∗ (γ ).
This means that global in time solutions with infinite energy do not exist for power potentials of the hard type (in
particular, rigid spheres). On the other hand, the possible self-similar solution (29) for γ > 0 is well defined for all
t < 0, whereas, as we have seen, it becomes singular at a certain positive time. The solutions for soft-potentials exist
for all positive times, but become singular for a certain negative time. Hence it seems probable that eternal solutions
(defined for all t ∈ R) exist only for Maxwell molecules (among other power-like potentials)
In spite of these results, a self-similar solution may still be used to describe the asymptotics of the solution describing
the structure of an infinitely strong shock wave, in the manner previously indicated.
25
CONCLUDING REMARKS
We have examined a class of self-similar solutions of the Boltzmann equation with infinite energy. The problem has
been treated to a satisfactory level for Maxwell molecules, but has been only the subject of a preliminary investigation
for other power law potentials (including hard spheres). The only difference which can be clearly proved is that,
whereas the solutions for Maxwell molecules are eternal, those for other interactions (if they exist) exist only for
−t < t < ∞ (soft potentials), or −t < t < ∞ (soft potentials), or for−∞ < t < t (hard potentials).
What happens in the case of finite energy? We mention here briefly some very recent results [16] related to pseudoMaxwell models (elastic and inelastic):
1. Eternal self-similar solutions of (2) having the form (29) with s(t) = exp(at), a > 0, exist for any 0 < a < amax .
They have a zero energy (second moment) and therefore cannot be positive. Their generalizations (obtained by
convolution with the Maxwellian (22)) have positive energy but it seems very doubtful there there is any new positive
solution among them.
2. The most interesting application of the self-similar solutions relates to inelastic Maxwell models proposed in
[17]. In this case there exists a special self-similar solution with finite energy which is asymptotic (for large positive
times) to a wide class of initial date (including Maxwellians). This property was first conjectured by Ernst and Brito
[18][19] and recently proved by the present authors [16].
Acknowledgments
The research described in the paper was performed in the frame of European TMR (contract n. ERBFMRXCT97O157) and was also partially supported by MURST of Italy and NFR of Sweden.
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