Asymmetrical Bimodal Distributions with Screw Modes
The Boltzmann equation for the model of hard spheres is considered. Approximate bimodal solutions for the Boltzmann equation are built for the case when the Maxwellian modes are screws with di®erent degrees of infinitesimality of angular velocities. Some su±cient conditions for the minimization of t...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1066832016-10-03T03:02:17Z Asymmetrical Bimodal Distributions with Screw Modes Gordevskyy, V.D. Sazonova, E.S. The Boltzmann equation for the model of hard spheres is considered. Approximate bimodal solutions for the Boltzmann equation are built for the case when the Maxwellian modes are screws with di®erent degrees of infinitesimality of angular velocities. Some su±cient conditions for the minimization of the uniform-integral remainder between the sides of the Boltzmann equation are obtained. 2011 Article Asymmetrical Bimodal Distributions with Screw Modes / V.D. Gordevskyy, E.S. Sazonova // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 3. — С. 212-224. — Бібліогр.: 8 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106683 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The Boltzmann equation for the model of hard spheres is considered. Approximate bimodal solutions for the Boltzmann equation are built for the case when the Maxwellian modes are screws with di®erent degrees of infinitesimality of angular velocities. Some su±cient conditions for the minimization of the uniform-integral remainder between the sides of the Boltzmann equation are obtained. |
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Gordevskyy, V.D. Sazonova, E.S. |
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Gordevskyy, V.D. Sazonova, E.S. Asymmetrical Bimodal Distributions with Screw Modes Журнал математической физики, анализа, геометрии |
| author_facet |
Gordevskyy, V.D. Sazonova, E.S. |
| author_sort |
Gordevskyy, V.D. |
| title |
Asymmetrical Bimodal Distributions with Screw Modes |
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Asymmetrical Bimodal Distributions with Screw Modes |
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Asymmetrical Bimodal Distributions with Screw Modes |
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Asymmetrical Bimodal Distributions with Screw Modes |
| title_full_unstemmed |
Asymmetrical Bimodal Distributions with Screw Modes |
| title_sort |
asymmetrical bimodal distributions with screw modes |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/106683 |
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Asymmetrical Bimodal Distributions with Screw Modes / V.D. Gordevskyy, E.S. Sazonova // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 3. — С. 212-224. — Бібліогр.: 8 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT gordevskyyvd asymmetricalbimodaldistributionswithscrewmodes AT sazonovaes asymmetricalbimodaldistributionswithscrewmodes |
| first_indexed |
2025-07-07T18:51:43Z |
| last_indexed |
2025-07-07T18:51:43Z |
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1837015293628514304 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2011, vol. 7, No. 3, pp. 212–224
Asymmetrical Bimodal Distributions with Screw Modes
V.D. Gordevskyy and E.S. Sazonova
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61077, Ukraine
E-mail: gordevskyy2006@yandex.ru
sazonovaes@rambler.ru
Received November 19, 2010
The Boltzmann equation for the model of hard spheres is considered.
Approximate bimodal solutions for the Boltzmann equation are built for the
case when the Maxwellian modes are screws with different degrees of in-
finitesimality of angular velocities. Some sufficient conditions for the mini-
mization of the uniform-integral remainder between the sides of the Boltz-
mann equation are obtained.
Key words: hard spheres, Boltzman equation, Maxwellian, screws,
remainder, bimodal distribution.
Mathematics Subject Classification 2000: 76P05, 45K05 (primary), 82C40,
35Q55 (secondary).
1. Introduction
To describe the interaction between flows of a gas of hard spheres, the kinetic
integro-differential Boltzmann equation [1 – 3] is used
D(f) = Q(f, f), (1)
D(f) =
∂f
∂t
+ v
∂f
∂x
, (2)
Q(f, f) =
d2
2
∫
R3
dv1
∫
Σ
dαB(v − v1, α)[f(t, v′1, x)f(t, v′, x)
− f(t, v1, x)f(t, v, x)],
(3)
where f(t, v, x) is the distribution to be found, ∂f/∂x is its spatial gradient,
t ∈ R1 is the time, x = (x1, x2, x3) ∈ R3 is the position; v = (v1, v2, v3) ∈ R3 is
the velocity of a molecule, d > 0 is its diameter, v and v1 are the velocities of the
molecules before the collision, v′ and v′1 are the velocities of particles after the
c© V.D. Gordevskyy and E.S. Sazonova, 2011
Asymmetrical Bimodal Distributions with Screw Modes
collision, α ∈ Σ, where Σ is the unit sphere in R3. The velocities of the particles
after the collision are defined by the formulae
v′ = v − α(v − v1, α), v′1 = v + α(v − v1, α). (4)
There are the well-known exact solutions of (1)–(4) in the form of global and
local Maxwellians [1–3]. Much progress has been made but only in the special
case of Maxwell molecules and some generalizations.
In [4–8], the bimodal distributions, i.e. the linear combinations of two Max-
wellians, in particular, the local Maxwellians of special form describing the screw-
shaped stationary equilibrium states of a gas (in short-screws or spirals), were
considered. They have the form [1,8]:
M(v, x) = ρ0e
βω2r2
(
β
π
) 3
2
e−β(v−v−[ω×x])2 . (5)
Physically, distribution (5) corresponds to the situation when the gas has an
inverse temperature β = 1/2T and rotates in whole as a solid body with the
angular velocity ω ∈ R3 around its axis on which the point x0 ∈ R3 lies,
x0 =
[ω × v]
ω2
. (6)
The square of the distance from the axis of rotation is
r2 =
1
ω2
[ω × (x− x0)]2, (7)
and the density of the gas has the form
ρ = ρ0e
βω2r2
(8)
(ρ0 is the density on the axis, that is with r = 0), v ∈ R3 is the linear mass
velocity for x for which x||ω, and v + [ω×x] is the mass velocity in the arbitrary
point x. It is easy to see that formula (5) gives not only a rotational, but also a
translational movement along the axis with the linear velocity
(ω, v)
ω2
ω.
Thus, it really describes a spiral movement of the gas in general, moreover, (5)
is stationary (independent of t) but inhomogeneous.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3 213
V.D. Gordevskyy and E.S. Sazonova
As, for instance, in [8], we will also consider an inhomogeneous, non-stationary
linear combination of two Maxwellians, i.e. a distribution
f = ϕ1M1 + ϕ2M2 =
2∑
k=1
ϕi(t, x)Mi(v, x), (9)
Mi(v, x) = ρie
βiω
2
i r2
i
(
βi
π
) 3
2
e−βi(v−ṽi)
2
. (10)
ṽi = ṽi(x) = vi + [ωi × x], i = 1, 2. (11)
It is assumed that the coefficient functions ϕi, i = 1, 2, are non-negative and
belong to C1(R4). It is required to find ϕi and the behavior of all parameters so
that the mixed remainder [4–8], i.e. the functional of the form
∆ = sup
(t,x)∈R4
∫
R3
|D(f)−Q(f, f)|dv, (12)
becomes vanishingly small.
We will find the coefficient functions ϕi,
ϕi(t, x) = ψi(t, x)e−βiω
2
i r2
i , i = 1, 2, (13)
and suppose that the angular velocities have the following form:
ωi =
ω0isi
βki
i
, i = 1, 2, (14)
where si > 0 are any constants, ω0i are arbitrary fixed vectors, ki > 0, i = 1, 2
(the other parameters are also arbitrary and fixed so far).
Some approximate solutions of a given kind, for which the Maxwellians with
i = 1 and i = 2 behave in the same way, were obtained in [8]. The two angular
velocities ω1 and ω2 were supposed to tend to zero equally fast with β1, β2 → +∞
(the speeds of tending to zero are different and are equal to the selected degrees
of βi in (14), namely 1, 1
2 or 1
4).
The aim of the present paper is to consider other possible values of ki, i = 1, 2,
and asymmetrical (i.e. for different degrees with different i) behavior of the
angular velocities.
214 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3
Asymmetrical Bimodal Distributions with Screw Modes
2. Main Results
Before formulating and proving the results, we recall some denotations intro-
duced in [8] to be used below:
Ai(u, t, x) = ψiψjρj
d2
√
π
∫
R3
dwe−w2
∣∣∣∣
u√
βi
+ (vi − vj) + [(ωi − ωj)× x]− w√
βj
∣∣∣∣, (15)
Bi(u, t, x) =
∂ψi
∂x
(
u√
βi
+ vi − [ωi × x]
)
+ 2ψi
√
βi{(u, [ωi × vi])
− [ωi × u][ωi × x]}. (16)
Due to the above it is possible to formulate the first theorem.
Theorem 1. Let conditions (13), (14) be valid. Let the functions
ψi,
∂ψi
∂t
,
∣∣∣∣
∂ψi
∂x
∣∣∣∣ , |[ω0i × x]|ψi,
(
[ω0i × x],
∂ψi
∂x
)
, i = 1, 2 (17)
be bounded with respect to t, x on R4.
Then the remainder ∆ in (12) is correctly defined, and there is a value ∆′
that
∆ ≤ ∆′. (18)
If
1
2
< ki ≤ 1, i = 1, 2, (19)
or
1
4
< ki ≤ 1
2
, i = 1, 2, (20)
and
[ω0i × vi] = 0, i = 1, 2, (21)
then there is the finite limit
L = lim
βi→+∞
i=1,2
∆′ =
2∑
i,j=1
i 6=j
ρi sup
(t,x)∈R4
∣∣∣∣
∂ψi
∂t
+ vi
∂ψi
∂x
+ ρjπd2ψ1ψ2|vi − vj |
∣∣∣∣
+ 2πd2ρ1ρ2|v1 − v2| sup
(t,x)∈R4
(ψ1ψ2).
(22)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3 215
V.D. Gordevskyy and E.S. Sazonova
To prove Theorem 1 we need the following lemma [8], which gives a sufficient
condition for the continuity of supremum of the function of special kind of many
variables. The supremum is taken respectively to a part of variables.
L e m m a. Suppose the following conditions:
1) ∀z ∈ Z, the function g(y, z) is bounded on Y ;
2) g(y, z) is continuous by z uniformly with respect to y,i.e.,
∀z0 ∈ Z,∀ε > 0, ∃δ > 0, ∀y ∈ Y, ∀z ∈ Z,
|z − z0| < δ ⇒ |g(y, z)− g(y, z0)| < ε
are valid for the function g(y, z) : Y × Z → R1; Y ∈ Rp;Z ∈ Rq.
Then the function
l(z) = sup
y∈Y
|g(y, z)|
is continuous on the variable z ∈ Z.
P r o o f of Theorem 1. According to (15) and (16), we write the inequality
obtained in [8]
∫
R3
|D(f)−Q(f, f)|dv ≤
2∑
i=1
∫
R3
[∣∣∣∣
∂ψi
∂t
+ Bi(u, t, x) + Ai(u, t, x)
∣∣∣∣
+ Ai(u, t, x)
]
· ρi
π3/2
e−u2
du.
(23)
From (12), (15)–(17), (23) and the properties of supremum, there follows the
existence of the remainder ∆, and the following holds:
∆ ≤ ∆′ =
2∑
i=1
ρi
π3/2
∫
R3
[
sup
(t,x)∈R4
∣∣∣∣
∂ψi
∂t
+ Bi(u, t, x) + Ai(u, t, x)
∣∣∣∣
+ sup
(t,x)∈R4
Ai(u, t, x)
]
e−u2
du.
(24)
If we substitute (14) into (15), (16) and introduce the new denotation
γ = (γ1, γ2) =
(
1√
β1
,
1√
β2
)
, (25)
then
Ai(u, t, x) = ψiψjρj
d2
√
π
∫
R3
dwe−w2 |γiu + (vi − vj) + siγ
2ki
i [ω0i × x]
− sjγ
2kj
j [ω0j × x]− γjw|,
(26)
216 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3
Asymmetrical Bimodal Distributions with Screw Modes
Bi(u, t, x) =
∂ψi
∂x
(
γiu + vi + siγ
2ki
i [ω0i × x]
)
+ 2ψisiγ
2ki−1
i
{
(u, [ω0i × vi])− siγ
2ki
i [ω0i × u][ω0i × x]
}
,
(27)
where i = 1, 2, i 6= j.
Apply the aforesaid lemma to every supremum contained in (24), where y =
(t, x), Y = R4, z = (u, γ), Z = R3 ×R2
+. Fulfillment of the first condition follows
from (17), (26), (27), and the second condition of the lemma can be verified with
the help of (17), polynomial structure of (27) with respect to variables u and γ,
and uniform convergence of the integral (26) by u and γ at any compact, and
with respect to t, x — on the whole space R4.
Then we can see that every integral in (24) is taken from the function which
is continuous by u, γ and converges uniformly with respect to the variables γ at
any compact, because there is an integrating majorant. Hence, in general, the
value ∆′ is continuous by γ on R2
+. So, in (24) we can pass to the limit with
β1, β2 → +∞, that is equivalent to the tending of γi, i = 1, 2 to zero in (26),
(27).
Thus,
lim
βi→+∞
i=1,2
∆′ =
2∑
i,j=1
i6=j
ρi
π3/2
∫
R3
[
sup
(t,x)∈R4
∣∣∣∣
∂ψi
∂t
+ vi
∂ψi
∂x
+ ψiψjρj
d2
√
π
π3/2|vi − vj |
∣∣∣∣
+ sup
(t,x)∈R4
ψiψjρjπd2|vi − vj |
]
e−u2
du.
(28)
From (28), as a result of integration by u, there follows (22). The theorem is
proved.
In this theorem the behavior of the angular velocity with i = 1 is identical to
that with i = 2. Now we will obtain possible results for the case of asymmetrical
behavior of ω1 and ω2.
Theorem 2. Let condition (14) be valid with
k1 = 1, k2 =
1
2
. (29)
Thereby, if conditions (17) are valid, then inequality (18) holds true. More-
over,
lim
βi→+∞
i=1,2
∆′ = L +
4√
π
ρ2s2|[ω02 × v2]| sup
(t,x)∈R4
ψ2. (30)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3 217
V.D. Gordevskyy and E.S. Sazonova
P r o o f. Let us use estimation (24) without introducing the denotation for
its right-hand side yet. Using again the denotations of (25), we will substitute
(14) into (15) and (16).
Then, instead of (26), (27) we will obtain
A1(u, t, x) = ψ1ψ2ρ2
d2
√
π
∫
R3
dwe−w2 |γ1u + (v1 − v2) + s1γ
2
1 [ω01 × x]
− s2γ
2
2 [ω02 × x]− γ2w|, (31)
B1(u, t, x) =
∂ψ1
∂x
(
γ1u + v1 + s1γ
2
1 [ω01 × x]
)
+ 2ψ1s1γ1
{
(u, [ω01 × v1])− s1γ
2
1 [ω01 × u][ω01 × x]
}
, (32)
A2(u, t, x) = ψ1ψ2ρ1
d2
√
π
∫
R3
dwe−w2 |γ2u + (v2 − v1) + s2γ2[ω02 × x]
− s1γ1[ω01 × x]− γ1w|, (33)
B2(u, t, x) =
∂ψ2
∂x
(γ2u + v2 + s2γ2[ω02 × x])
+ 2ψ2s2 {(u, [ω02 × v2])− s2γ2[ω02 × u][ω02 × x]} . (34)
By substituting (31)–(34) into (24), we can obtain the estimation
∆ ≤ ∆′ =
ρ1
π3/2
∫
R3
[
sup
(t,x)∈R4
∣∣∣∂ψ1
∂t
+ A1(u, t, x) +
∂ψ1
∂x
(γ1u + v1 + s1γ
2
1 [ω01 × x])
+ 2ψ1γ1s1
{
(u, [ω01 × v1])− s1γ
2
1 [ω01 × u][ω01 × x]
}∣∣∣
+ sup
(t,x)∈R4
A1(u, t, x)
]
e−u2
du +
ρ2
π3/2
∫
R3
[
sup
(t,x)∈R4
∣∣∣∂ψ2
∂t
+ A2(u, t, x)
+
∂ψ2
∂x
(γ2u + v2 + s2γ2[ω02 × x])− 2ψ2s
2
2γ2[ω02 × u][ω02 × x]
∣∣∣
+ sup
(t,x)∈R4
A2(u, t, x) + sup
(t,x)∈R4
2ψ2s2|u| |[ω02 × v2]|
]
e−u2
du. (35)
The application of the lemma to each of the supremums in (35) (similarly
as in the proof of Theorem 1, where the last summand under integral in (35)
does not depend on β2, namely γ2 at all) allows to pass to the limit in (35) with
βi → +∞, i = 1, 2. Then the result will differ from (22) by the only mentioned
last addendum calculation of which leads to the following integral:
I =
∫
R3
|u|e−u2
du.
218 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3
Asymmetrical Bimodal Distributions with Screw Modes
The integral can be easily calculated by passing to spherical system of coordinates.
All the above leads to (30). The theorem is proved.
Theorem 3. Let conditions (14) and (17) of Theorem 1 be valid with
k1 =
1
2
, k2 =
1
4
. (36)
Moreover,
[ω02 × v2] = 0. (37)
Then the analogue of (22) below holds true
lim
βi→+∞
i=1,2
∆′ = L +
4√
π
ρ1s1|[ω01 × v1]| sup
(t,x)∈R4
ψ1
+
4√
π
ρ2s
2
2|ω02| sup
(t,x)∈R4
(|[ω02 × x]|ψ2).
(38)
P r o o f. Let us again use estimation (24) without introducing the denotation
∆′ for its right-hand side, and in (15), (16), using the denotations from (25),
substitute (14) with k1 = 1
2 , k2 = 1
4 .
Then we obtain:
A1(u, t, x) = ψ1ψ2ρ2
d2
√
π
∫
R3
dwe−w2 |γ1u + (v1 − v2) + s1γ1[ω01 × x]
− s2γ2[ω02 × x]− γ2w|, (39)
B1(u, t, x) =
∂ψ1
∂x
(γ1u + v1 + s1γ1[ω01 × x])
+ 2ψ1s1 {(u, [ω01 × v1])− s1γ1[ω01 × u][ω01 × x]} , (40)
A2(u, t, x) = ψ1ψ2ρ1
d2
√
π
∫
R3
dwe−w2 |γ2u + (v2 − v1) + s2
√
γ2[ω02 × x]
− s1
√
γ1[ω01 × x]− γ1w|, (41)
B2(u, t, x) =
∂ψ2
∂x
(γ2u + v2 + s2
√
γ2[ω02 × x])
− 2ψ2s
2
2[ω02 × u][ω02 × x]. (42)
By substituting (39)–(42) into (24,) we will obtain the following estimation:
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3 219
V.D. Gordevskyy and E.S. Sazonova
∆ ≤ ∆′ =
ρ1
π3/2
∫
R3
[
sup
(t,x)∈R4
∣∣∣∂ψ1
∂t
+ A1(u, t, x) +
∂ψ1
∂x
(γ1u + v1 + s1γ1[ω01 × x])
− 2ψ1s
2
1[ω01 × u][ω01 × x]
∣∣∣ + sup
(t,x)∈R4
A1(u, t, x)
+ sup
(t,x)∈R4
2ψ1s1|u|
∣∣[ω01 × v1]
∣∣
]
e−u2
du +
ρ2
π3/2
∫
R3
[
sup
(t,x)∈R4
∣∣∣∂ψ2
∂t
+ A2(u, t, x) +
∂ψ2
∂x
(γ2u + v2 + s2
√
γ2[ω02 × x])
∣∣∣
+ sup
(t,x)∈R4
A2(u, t, x) + sup
(t,x)∈R4
∣∣2ψ2s
2
2[ω02 × u][ω01 × x]
∣∣
]
e−u2
du.
(43)
From (39) and (41) we can see that the limit of the value A2(u, t, x) with γ → 0
is the same as the limit of the value A1(u, t, x), and the estimation for the modulus
having Bi(u, t, x), contained in (23), leads to the separation of two summands
independent of γ (i.e of βi, i = 1, 2), in which there appear the expressions
2s1
∫
R3
sup
(t,x)∈R4
[
ψ1|u|
∣∣[ω01 × v1]
∣∣
] ρ1
π3/2
e−u2
du, (44)
2s2
2
∫
R3
sup
(t,x)∈R4
[
ψ2|ω02||u|
∣∣[ω02 × x]
∣∣
] ρ2
π3/2
e−u2
du. (45)
Further calculation of (44) and (45), analogously as in the proof of Theorem
2, leads to (38). The theorem is proved.
Theorem 4. Let condition (14) be valid with
k1 = 1, k2 =
1
4
. (46)
Thereby, if conditions (17) and (37) are valid, then inequality (18) holds true,
where
lim
βi→+∞
i=1,2
∆′ = L +
4√
π
ρ2s
2
2|ω02| sup
(t,x)∈R4
(|[ω02 × x]|ψ2). (47)
P r o o f is the same as that of Theorem 2. But now, owing to (14) with
k1 = 1, k2 = 1
4 and (37), instead of (33), (34), we get (41), (42) from (15), (16).
Thus, A1(u, t, x), B1(u, t, x) have the form of (31), (32). From (41), (42) we can
220 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3
Asymmetrical Bimodal Distributions with Screw Modes
see that the limit of the value A2(u, t, x) with γ → 0 is the same as the limit
in Theorem 3 and similarly to A1(u, t, x) from Theorem 2. Furthermore, the
estimation for the modulus having B2(u, t, x), contained in (23), was obtained
in the proof of Theorem 3. So, the result will differ from (22) only by the last
addendum defined in (45). Further calculation leads to (47). The theorem is
proved.
Now, by means of the expressions obtained for the limits with βi → +∞,
i = 1, 2, we can find the sufficient conditions under which the remainder ∆ tends
to zero. For more convenience, the conditions are formulated in the form of
corollaries of Theorems 1–4. In what follows it is assumed that the passage to
the limit β1, β2 → +∞ is performed.
Corollary 1. Let all the assumptions of Theorem 1 be valid. Then the state-
ment
∆ → 0 (48)
holds true if at least one of the conditions fulfils the following:
1) v1 = v2 = 0,
ψi = ψi(x), i = 1, 2, (49)
where ψi(x) are any functions satisfying the requirement (17);
2) v1 = v2 6= 0, condition (21) is valid, and
ψi = Ci ([x× vi]) , i = 1, 2, (50)
where Ci ≥ 0 are any smooth and finite (i.e. with finite support) or fast decreasing
functions on its vector arguments;
3) v1 = v2 6= 0, condition (21) is valid, and
ψi = Ci (x− vit) , i = 1, 2, (51)
where Ci ≥ 0 are the same as in item 2);
4) v1 = 0, the vectors v2, ω01, ω02 are collinear,
ψ1 = ψ1(t, x) = h ([x× v2])
{
λ + C ([x× v2])
×
[
−πd2|v2|h ([x× v2])
(
x1
v1
2
(
ρ2
µ
+
ρ1
λ
)
− ρ2
µ
t
)]}−1
, (52)
ψ2 = ψ2(t, x) =
1
µ
(
h ([x× v2])− λψ1(t, x)
)
, (53)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3 221
V.D. Gordevskyy and E.S. Sazonova
where λ, µ > 0 are any constants, the functions h and C have the same properties
as Ci, i = 1, 2 in (50), and, in addition,
d → 0; (54)
5) v1 6= 0, v2 6= 0 are arbitrary, equality (21) is valid, the functions ψi, i = 1, 2,
have the form of (50) or (51), and
suppψ1 ∩ suppψ2 = ∅; (55)
6) v1 6= 0, v2 6= 0 are arbitrary, the functions ψi, i = 1, 2, have the form of
(50) or (51) under the condition of (21), and (54) is valid.
P r o o f of all items is based on (21) and (22).
1) All summands of the expressions having vi, i = 1, 2 become equal to zero,
and ∂ψi
∂t is equal to zero too as ψi = ψi(x), i = 1, 2. Consequently, the whole
expression (22) is equal to zero.
2) The summands having the difference |vi − vj |, i, j = 1, 2, i 6= j, become
equal to zero due to the given conditions. Let us consider the addendum having
∂ψi
∂x . Since
(
C
(
[x×a]
))′
x
= [a×C ′], where a is an arbitrary constant vector, then
(
vi,
[
vi, C
′
i]
)
= 0.
The condition (21) means that ω0i||vi. So in (50) we can write ω0i instead
of vi. Thus we obtain the boundedness in (17). The same is true for all the
following items of this corollary.
3) In (22) all summands but one instantly become equal to zero due to the
given conditions. However, this summand also disappears, because the function
of the form of (51) satisfies the following system of equations:
∂ψi
∂t
+ vi
∂ψi
∂x
= 0. (56)
4) Under the conditions mentioned above, the functions (52) and (53) are the
solutions of the following system of differential equations (as shown in [6]):
∂ψi
∂t
+ vi
∂ψi
∂x
= −ρjπd2ψ1ψ2|v2|, i, j = 1, 2, i 6= j. (57)
All other summands tend to zero due to condition (54).
5) Under condition (55), every function ψi 6= 0, i = 1, 2, but the product
ψ1ψ2 = 0 identically.
The function of the form of (50) or (51) satisfies the system of equations (56).
Thus, expression (22) is evidently equal to zero.
222 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3
Asymmetrical Bimodal Distributions with Screw Modes
6) Conditions (54) and (55) lead to the fact that the following summands are
equal or tend to zero:
ρjπd2ψ1ψ2|vi − vj |, i, j = 1, 2, i 6= j,
2πd2ρ1ρ2|v1 − v2| sup
(t,x)∈R4
(ψ1ψ2).
For the function ψi of the form of (50) or (51), the statement (56) is fulfilled.
So, the equality of the expression (22) to zero is proved.
Corollary 2. Let all the assumptions of Theorem 2 be valid. Then the state-
ment (48) holds true if one of the conditions of Corollary 1 takes place and, in
addition, at least one of the following requirements is fulfilled :
1. s2 → 0;
2. Condition (21) with i = 2.
P r o o f is evident and leans on (30).
Corollary 3. Let all the assumptions of Theorem 3 be valid. Then the state-
ment (48) holds true if one of the conditions of Corollary 1 takes place and, in
addition, at least one of the following requirements is fulfilled:
1. si → 0, i = 1, 2;
2. s2 → 0, condition (21) with i = 1.
P r o o f is evident and leans on (38).
Corollary 4. Let all the assumptions of Theorem 4 be valid. Then (48) holds
true if at least one of the conditions of Corollary 1 takes place and, in addition,
the first requirement of Corollary 2 is satisfied.
P r o o f can be easily done by means of (47).
R e m a r k 1. It should be noted that the exact solutions of the Boltz-
man equation, which have the well-known physical sense, are the spiral Maxwell
distributions (10), (11) themselves. However, the bimodal distributions (9), con-
sidered in the paper, give only an approximate description of interaction of such
spirals in the sense of minimization of the remainder ∆. Nevertheless, they can
be reasonably interpreted physically. In fact, the common property of all ob-
tained distributions is that they describe the non-uniform cooling gas (βi → +∞,
i = 1, 2). Besides, the rotation of both spirals decelerates (ωi → 0, i = 1, 2),
although in different degrees in accordance with (14) and under the conditions
of (19), (20), (29), (36), and (46) (in some cases, condition 1) of Corollary 3 is
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 3 223
V.D. Gordevskyy and E.S. Sazonova
also assumed). At the same time, the distribution f itself does not tend to any
of Maxwellians (i.e. to the known exact solution of the Boltzmann equations).
R e m a r k 2. Theorem 1 is devoted to the case of equal degrees of
infinitesimality of both angular velocities ωi in the most general case, namely,
when ki ∈ (1
4 , 1]. It is worth being noted that, as we can see from Theorems
2–4, the consideration of different degrees of βi with i = 1 and i = 2 leads to the
appearance of ”mixed” expressions of the limiting values for the estimation ∆′.
These expressions include various summands mentioned in the paper.
References
[1] C. Cercignani, The Boltzman Equation and its Applications. Springer, New York,
1988.
[2] M.N. Kogan, The Dynamics of a Rarefied Gas. Nauka, Moscow, 1967.
[3] T. Carleman, Problems Mathematiques dans la Theorie Cinetique des Gas.
Almqvist & Wiksells, Uppsala, 1957.
[4] V.D. Gordevskyy, An Approximate Bimodal Solution of the Nonlinear Boltzman
Equation for Hard Spheres. — Mat. Fiz., Anal., Geom. 2 (1995), No. 2, 168–176.
(Russian)
[5] V.D. Gordevskyy, A Criterium of Smallness of Difference for a Bimodal Solution
of the Boltzmann Equation. — Mat. Fiz., Anal., Geom. 4 (1997), No. 1/2, 46–58.
(Russian)
[6] V.D. Gordevskyy, Some Classes of the Approximate Bimodal Solutions of the Non-
linear Boltzman Equation. — In: Integral Transforms and its Application for Bound-
ary Problems (M.P. Lenuk, Ed.). — Kyev, In-t of Mathematics of Ukrainian Acad.
Sci. (1997), No. 16, 54. (Ukrainian)
[7] V.D. Gordevskyy, An Approximate Two-Flow Solution to the Boltzmann Equation.
— Teor. Mat. Fiz. 114 (1998), No. 1, 126–136. (Russian)
[8] V.D. Gordevskyy, Biflow Distributions with Screw Modes. — Teor. Mat. Fiz. 126
(2001), No. 2, 283–300. (Russian)
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