Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model
The interaction between the "accelerating-packing" flows in a gas of rough spheres is studied. A bimodal distribution with the Maxwellian modes of special forms is used. Different sufficient conditions for the minimization of the uniform-integral error between the sides of the Bryan-Pidduk...
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irk-123456789-1067572016-10-05T03:02:11Z Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model Gukalov, A.A. The interaction between the "accelerating-packing" flows in a gas of rough spheres is studied. A bimodal distribution with the Maxwellian modes of special forms is used. Different sufficient conditions for the minimization of the uniform-integral error between the sides of the Bryan-Pidduk equation are obtained. Исследовано взаимодействие двух "ускоряющихся-уплотняющихся" потоков в газе из шероховатых сфер. Использовано бимодальное распределение с максвелловскими модами специального вида. Получены различные условия, достаточные для минимизации равномерно-интегральной невязки между частями уравнения Бриана-Пиддака. 2013 Article Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model / A.A. Gukalov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 316-331. — Бібліогр.: 15 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106757 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The interaction between the "accelerating-packing" flows in a gas of rough spheres is studied. A bimodal distribution with the Maxwellian modes of special forms is used. Different sufficient conditions for the minimization of the uniform-integral error between the sides of the Bryan-Pidduk equation are obtained. |
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| author |
Gukalov, A.A. |
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Gukalov, A.A. Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model Журнал математической физики, анализа, геометрии |
| author_facet |
Gukalov, A.A. |
| author_sort |
Gukalov, A.A. |
| title |
Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model |
| title_short |
Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model |
| title_full |
Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model |
| title_fullStr |
Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model |
| title_full_unstemmed |
Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model |
| title_sort |
interaction between "accelerating-packing" flows for the bryan-pidduck model |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/106757 |
| citation_txt |
Interaction between "Accelerating-Packing" Flows for the Bryan-Pidduck Model / A.A. Gukalov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 316-331. — Бібліогр.: 15 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT gukalovaa interactionbetweenacceleratingpackingflowsforthebryanpidduckmodel |
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2025-07-07T18:57:09Z |
| last_indexed |
2025-07-07T18:57:09Z |
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1837015635752648704 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 3, pp. 316–331
Interaction between ”Accelerating-Packing” Flows for
the Bryan–Pidduck Model
A.A. Gukalov
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61077, Ukraine
E-mail: gukalex@ukr.net
Received March 2, 2012, revised June 20, 2012
The interaction between the ”accelerating-packing” flows in a gas of
rough spheres is studied. A bimodal distribution with the Maxwellian modes
of special forms is used. Different sufficient conditions for the minimization
of the uniform-integral error between the sides of the Bryan–Pidduk equation
are obtained.
Key words: rough spheres, Bryan–Piddack equation, Maxwellian, ”acce-
lerating-packing” flows, error, bimodal distribution.
Mathematics Subject Classification 2000: 76P05, 45K05 (primary); 82C40,
35Q55 (secondary).
1. Introduction
In the paper a model of rough spheres [1], first introduced by Bryan in 1894,
is studied. The methods developed by Chapman and Enskog for general non-
rotating spherical molecules were extended to Bryan’s model by Pidduck in 1922.
The advantage of this model over all other variably rotating models is that no
variables are required to specify its orientation in the space.
These molecules are perfectly elastic and perfectly rough to be interpreted
as follows. When two molecules collide, the velocities at the points of contact
are not the same. It is supposed that the two spheres grip each other without
slipping; first each sphere is strained by the other, and then the strain energy
gets reconverted into kinetic energy of translation and rotation, no energy being
lost. The effect is that the relative velocity of the spheres at the point of contact
is reversed by the impact.
The Boltzmann equation for the model of rough spheres (or the Bryan–
Pidduck equation) has the form [1–4]:
D(f) = Q(f, f); (1)
c© A.A. Gukalov, 2013
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
D(f) ≡ ∂f
∂t
+ V
∂f
∂x
; (2)
Q(f, f) ≡ d2
2
∫
R3
dV1
∫
R3
dω1
∫
Σ
dαB(V − V1, α)
×
[
f(t, V ∗
1 , x, ω∗1)f(t, V ∗, x, ω∗)− f(t, V, x, ω)f(t, V1, x, ω1)
]
.
(3)
Here d is the diameter of the molecule associated with the moment of inertia
I by the relation
I =
bd2
4
,
where b , b ∈ (
0, 2
3
]
, is the parameter characterizing the isotropic distribution
of the matter inside the gas particle; t is the time; x = (x1, x2, x3) ∈ R3 is the
spatial coordinate; V = (V 1, V 2, V 3) and w = (w1, w2, w3) ∈ R3 are the linear
and angular velocities of the molecule, respectively; ∂f
∂x is the gradient of the
function f over the variable x; Σ is the unit sphere in the space R3; α is the
unit vector of R3 directed along the line connecting the centers of the colliding
molecules;
B (V − V1, α) = |(V − V1, α)| − (V − V1, α)
is the collision term.
The linear (V ∗, V ∗
1 ) and angular (w∗, w∗1) molecular velocities after the colli-
sion can be expressed by the appropriate values before the collision:
V ∗ = V − 1
b + 1
(
b(V1 − V )− bd
2
α× (ω + ω1) + α(α, V1 − V )
)
,
V ∗
1 = V1 +
1
b + 1
(
b(V1 − V )− bd
2
α× (ω + ω1) + α(α, V1 − V )
)
,
ω∗ = ω +
2
d(b + 1)
{
α× (V − V1) +
d
2
[α(ω + ω1, α)− ω − ω1]
}
,
ω∗1 = ω1 +
2
d(b + 1)
{
α× (V − V1) +
d
2
[α(ω + ω1, α)− ω − ω1]
}
.
Exact Maxwell solutions of the Boltzmann equation for a more traditional and
simpler model of hard spheres were found and classified in detail
in [5–8]; their descriptions can also be found in [1, 9, 10]. The general form of
Maxwell solutions (i.e., exact solutions of the system D = Q = 0) to the Bryan–
Pidduck equation was firstly given in [11]. In particular, there was obtained the
explicit form of the Maxwellian distribution describing ”accelerating-packing” gas
flow for this model.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 317
A.A. Gukalov
The explicit approximate solutions of kinetic equations, which have a bimodal
structure, were given by several authors. In particular, for the models of interac-
tion between the molecules, we are interested in, they are to be found in [3, 4, 8,
12–15].
In [4], the interaction of two ”screws” (stationary inhomogeneous
Maxwellians) in a gas of rough spheres was studied, and the interaction of two ”ed-
dies” (non-stationary inhomogeneous Maxwellians) for the same Bryan–Pidduck
model was described in [14]. Our goal is to study the interaction of two flows
describing the motion of the ”accelerating-packing” type. It should be noted that
the hard-sphere model was solved and described in [15].
We use the following error firstly proposed in [4]:
∆ = sup
(t,x)∈R4
∫
R3
∫
R3
dV dω
∣∣∣D(f)−Q(f, f)
∣∣∣. (4)
Next, we consider a bimodal distribution
f = ϕ1M1 + ϕ2M2, (5)
where the functions ϕi = ϕi(t, x), (here and below the index i takes only val-
ues 1 and 2), and the Maxwellians Mi correspond to the ”accelerating-packing”
movement and have the form
Mi = ρiI
3/2
(
βi
π
)3
e
−βi
(
(V−V i)2
+Iω2
)
, (6)
where ρi denotes the gas density
ρi = ρ0ie
βi
(
V
2
i +2uix
)
, (7)
and
V i = V̂i − uit (8)
is the mass velocity of molecules, βi = 1
2T denotes the inverse temperature, and
ρ0i, ui, V̂i are arbitrary constants of the spaces R and R3.
The next section contains the results of providing various sufficient conditions
for the minimization of the residual (4) by a suitable choice of the coefficient
functions ϕi and the parameters of the distribution.
2. Main Results
Theorem 1. Let the functions ϕi in distribution (5) have the form
ϕi(t, x) =
Di
(1 + t2)ξi
Ci
x + ui
(
V̂i − uit
)2
2u2
i
, (9)
318 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
where the constants Di, ξi are as follows:
Di > 0, ξi > 1
2
, (10)
and the functions Ci, which are nonnegative and belong to the space C1(R3),
have finite supports (i.e., finite functions) or are fast decreasing at infinity. Also
let the following requirements be fulfilled:
V̂i =
V̂0i
βki
i
, ui =
u0i
βni
i
(11)
with the conditions
ki > 1
2
, ni > 1, ki > 1
2
ni, (12)
and u0i, V̂0i be arbitrary fixed three-dimensional vectors.
Then the following assertion is true:
∀ε > 0, ∃δ > 0, ∀D1, D2 : 0 < D1, D2 < δ,
∃β0, ∀βi > β0,
∆ < ε.
(13)
P r o o f. First we will show that there exists a value ∆
′
such that
∆ 6 ∆
′
, (14)
and we have
lim
βi→+∞
∆
′
= K(ξ1, ξ2)
2∑
i=1
ρ0iDi sup
x∈R3
[
ηi(x)Ci(x + ai)
]
, (15)
where the functions ηi(x) are following:
ηi(x) =
1, ni > 1; ki > 1
2 ,
e2u0ix, ni = 1; ki > 1
2 ,
eV̂ 2
0i+2u0ix, ni = 1; ki = 1
2 .
(16)
K(ξ1, ξ2) is a constant, and the vector constants ai are equal to u0iV̂
2
0i
2u2
0i
if ki = 1
2ni,
and they are equal to zero if ki 6= 1
2ni.
It is easy to show that for the function f of form (5) the following relations
take place:
D(f) = M1D(ϕ1) + M2D(ϕ2)
= M1
(
∂ϕ1
∂t
+ V
∂ϕ1
∂x
)
+ M2
(
∂ϕ2
∂t
+ V
∂ϕ2
∂x
)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 319
A.A. Gukalov
and
Q(f, f) = ϕ1ϕ2
[
Q
(
M1,M2
)
+Q
(
M2,M1
)]
.
It is well known that the right-hand side of the Bryan–Pidduck equation (3),
considered as a bilinear operator on any two functions f, g, can be decomposed
as follows:
Q(f, g) = G(f, g)− fL(g),
where
G(f, g) =
d2
2
∫
R3
dV1
∫
R3
dω1
∫
∑
dαB(V − V1, α)f(t, x, V ∗
1 , ω∗1)g(t, x, V ∗, ω∗),
and
L(g) =
d2
2
∫
R3
dV1
∫
R3
dω1
∫
∑
dαB(V − V1, α)g(t, x, V1, ω1).
In [14], it was shown that
∫
R3
dV
∫
R3
dωQ(Mi,Mj) = 0, j = 1, 2,
and hence we have the relationship
∫
R3
dV
∫
R3
dωG(Mi,Mj) =
∫
R3
dV
∫
R3
dωMiL(Mj),
i.e., we can get the inequality
∣∣∣D(f)−Q(f, f)
∣∣∣ 6 M1
(
|D(ϕ1)|+ ϕ1ϕ2L(M2)
)
+M2
(
|D(ϕ2)|+ ϕ1ϕ2L(M1)
)
+ ϕ1ϕ2
(
G(M1,M2) + G(M2,M1)
)
.
Integrating the last estimation over the entire space of the linear and angular
velocities, we obtain
∫
R3
dV
∫
R3
dω
∣∣∣D(f)−Q(f, f)
∣∣∣
6
2∑
i,j=1
i 6=j
∫
R3
dV
∫
R3
dω
(
|D(ϕi)|+ ϕiϕjL(Mj)
)
Mi + 2ϕ1ϕ2
∫
R3
dV
∫
R3
dωG(M1,M2)
6
2∑
i=1
∫
R3
dV
∫
R3
dω|D(ϕi)|Mi + 4ϕ1ϕ2
∫
R3
dV
∫
R3
dωG(M1,M2).
320 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
Then we use the formula, the proof of which is given in detail in [4],
∫
R3
dV
∫
R3
dωG(M1,M2)
=
d2ρ1ρ2
π2
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣
q√
β1
− q1√
β2
+ V 1 − V 2
∣∣∣∣ .
(17)
We extend the estimation taking into account (8) and (17),
∫
R3
dV
∫
R3
dω
∣∣∣D(f)−Q(f, f)
∣∣∣ 6
2∑
i=1
∫
R3
dV
∫
R3
dω|D(ϕi)|Mi + Y
=
2∑
i=1
∫
R3
dV
∫
R3
dω
∣∣∣∣
∂ϕi
∂t
+ V · ∂ϕi
∂x
∣∣∣∣ ρiI
3/2
(
βi
π
)3
e−βi(V−V i)2−βiIω2
+ Y,
where the value Y is determined by the expression
4d2ρ1ρ2ϕ1ϕ2
π2
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 + (u2 − u1)t
∣∣∣∣ .
Thus, we can integrate over the space of angular velocities ω (three-dimensional
Euler–Poisson integral) to get
∫
R3
dV
∫
R3
dω
∣∣∣D(f)−Q(f, f)
∣∣∣
6
2∑
i=1
∫
R3
dV
∣∣∣∣
∂ϕi
∂t
+ V · ∂ϕi
∂x
∣∣∣∣ ρi
(
βi
π
)3/2
e−βi(V−V i)2
+ Y.
By changing the variable
V =
p√
βi
+ V i,
whose Jacobian is β
−3/2
i , we can obtain the inequality, which will be often used
in our further calculations,
∫
R3
dV
∫
R3
dω
∣∣∣D(f)−Q(f, f)
∣∣∣
6
2∑
i=1
ρi
π3/2
∫
R3
dp
∣∣∣∣
∂ϕi
∂t
+
(
p√
βi
+ V̂i − uit
)
· ∂ϕi
∂x
∣∣∣∣ e−p2
+ Y.
(18)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 321
A.A. Gukalov
For the existence of value (4) and the validity of inequality (14), as seen from
estimation (18), it is sufficient to verify that the products of gas density (7) on
the functions
ϕi;
∂ϕi
∂t
;
∣∣∣∣
∂ϕi
∂x
∣∣∣∣ ; ϕit;
(
ui,
∂ϕi
∂x
)
t (19)
are bounded for any (t, x) from R4.
In the representation of functions (9) let us introduce a redesignation
l = x + ui
(
V̂i − uit
)2
2u2
i
,
whence (
V̂i − uit
)2
= 2uil − 2uix,
and consequently we have
ϕiρi = ρ0ie
2uilβi
Di
(1 + t2)ξi
Ci(l). (20)
The product (20) is a bounded function on (t, x) ∈ R4 due to the properties of
the function Ci(l). It should be noted that the boundedness will remain true even
after multiplying the value (20) by a variable t due to the expression (1 + t2)ξi ,
contained in the denominator, and condition (10). Similarly, we can prove the
boundedness of the last three products by using the equations, obtained by direct
differentiation of the function ϕi of the form (9) with respect to the time t and
the position in space x,
∂ϕi
∂t
= − Di
(1 + t2)ξi
2tξi
1 + t2
Ci(l) +
(
C
′
i(l), ui
)
(
V̂i, ui
)
− tu2
i
u2
i
, (21)
∂ϕi
∂x
=
Di
(1 + t2)ξi
C
′
i(l). (22)
Taking into account assumptions (11), we have a low-temperature limit (βi → +∞)
of the density (7) depending on the numbers ni and ki
lim
βi→+∞
ρi = ρ0i ·
1, ni > 1, ki > 1
2 ;
e2u0ix ni = 1, ki > 1
2 ;
eV̂ 2
0i+2u0ix ni = 1, ki = 1
2 .
322 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
Taking into account equalities (11) and conditions (12), the following equation
is evident:
lim
βi→+∞
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 + (u2 − u1)t
∣∣∣∣ = 0. (23)
As a result, substituting the calculated derivatives (21), (22) into (18), taking
the supremum on both sides of (18) and performing a low-temperature limit by
using the technique of [3, 4, 14, 15], we get the equality
lim
βi→+∞
∆
′
=
2∑
i=1
ρ0iDi sup
(t,x)∈R4
{
ηi(x) lim
βi→+∞
2|t|ξiCi(l)
(1 + t2)ξi+1
}
,
where the variable l is the sum of the variables x and ri, the latter of which can
be represented as follows:
ri =
u0i
2u2
0i
V̂0i
β
ki− 1
2
ni
i
− u0it
β
1
2
ni
i
2
.
Then we have that
ai = lim
βi→+∞
(l − x) =
[
0, ki > 1
2ni;
u0i
V̂ 2
0i
2u2
0i
, ki = 1
2ni.
As a result, we obtain
lim
βi→+∞
∆
′
=
2∑
i=1
ρ0iDi sup
(t,x)∈R4
{
ηi(x)
2|t|ξi
(1 + t2)ξi+1
Ci(x + ai)
}
6 K(ξ1, ξ2)
2∑
i=1
ρ0iDi sup
x∈R3
[
ηi(x)Ci(x + ai)
]
,
where the constant K(ξ1, ξ2) is defined as follows:
K(ξ1, ξ2) = 2 max
i
{
ξi sup
t∈R
|t|
(1 + t2)ξi+1
}
.
Thus, we have shown that equality (15) is fulfilled, from which (with
(14) being taken into account) the statement of our theorem, i.e. (13), follows.
The theorem is proved. ¥
Theorem 2. Assume that the functions ϕi(t, x) have the representation
ϕi(t, x) = ψi(t, x)e−βi
(
(V̂i−uit)2
+2uix
)
, (24)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 323
A.A. Gukalov
where
ψi = DiCi(t),
here Di > 0, and Ci are finite functions.
Let the condition
ui =
u0i
βni
i
, ni > 1
2
, (25)
hold.
Then:
a) if the equality
suppC1 ∩ suppC2 = ∅
takes place, or
V̂1 = V̂2,
then assertion (13) remains true.
b) In the case of arbitrary supports of the functions (C1 and C2) and the
velocities V̂1, V̂2, Theorem 1 still holds if being complemented by the condition of
the infinite smallness of the diameter of the gas particles (d < δ).
P r o o f. First we introduce and prove an auxiliary assertion that there
exists a value of ∆
′
, and the inequality (14) is fulfilled, but if ni > 1
2 , then
lim
βi→+∞
∆
′
=
2∑
i=1
ρ0i sup
(t,x)∈R4
∣∣∣∣
∂ψi
∂t
+ V̂i
∂ψi
∂x
∣∣∣∣
+4πd2ρ01ρ02
∣∣∣V̂1 − V̂2
∣∣∣ sup
(t,x)∈R4
(ψ1ψ2) = Z,
(26)
while for ni = 1
2 we have
lim
βi→+∞
∆
′
= Z +
4√
π
2∑
i=1
ρ0i|u0i| sup
(t,x)∈R4
ψi. (27)
If the assertion is true, then items (a) and (b) are also true.
Note that inequality (18) remains true, therefore it is necessary to compute
the derivatives of (24) by t and x
∂ϕi
∂t
= e
−βi
(
(V̂i−uit)2
+2uix
) {
∂ψi
∂t
+ 2βiψi
((
V̂i, ui
)
− tu2
i
)}
, (28)
∂ϕi
∂x
= e
−βi
(
(V̂i−uit)2
+2uix
) {
∂ψi
∂x
− 2βiuiψi
}
. (29)
324 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
Taking into account that the functions ψi(t, x) are smooth and nonnegative,
and basing on their type, it follows that the above expressions in (19) remain
bounded. After replacing ϕi(t, x) on ψi(t, x), we can pass to the supremum in
inequality (18), primarily substituting in it the expressions for derivatives (28)
and (29). Thus we have
∆ 6 ∆
′
= sup
(t,x)∈R4
2∑
i=1
ρi
π3/2
∫
R3
dp
∣∣∣∣e
−βi
(
(V̂i−uit)2
+2uix
) (
∂ψi
∂t
+ 2βiψi
((
V̂i, ui
)
− tu2
i
))
+
(
p√
βi
+ V̂i − uit
)
e
−βi
(
(V̂i−uit)2
+2uix
) (
∂ψi
∂x
− 2βiuiψi
)∣∣∣∣ e−p2
+4 sup
(t,x)∈R4
ψ1ψ2e
−β1
(
(V̂1−u1t)2
+2u1x
)
−β2
(
(V̂2−u2t)2
+2u2x
)
d2ρ1ρ2
π2
×
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 + (u2 − u1)t
∣∣∣∣ .
Using the representation for density (7) and opening the parenthesis, after
collecting similar terms, we can find that the value of ∆
′
is equal to the expression
sup
(t,x)∈R4
2∑
i=1
ρ0i
π3/2
∫
R3
dp
∣∣∣∣
∂ψi
∂t
+
(
p√
βi
+ V̂i − uit
)
∂ψi
∂x
− 2βiuiψi√
βi
p
∣∣∣∣ e−p2
+4 sup
(t,x)∈R4
d2ψ1ψ2ρ01ρ02
π2
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 + (u2 − u1)t
∣∣∣∣ .
Next, by using condition (25), the value ∆
′
can be converted to the form
sup
(t,x)∈R4
2∑
i=1
ρ0i
π3/2
∫
R3
dp
∣∣∣∣
∂ψi
∂t
+
(
p√
βi
+ V̂i − u0i
βni
i
t
)
· ∂ψi
∂x
− 2β
1
2
−ni
i u0iψip
∣∣∣∣ e−p2
+4 sup
(t,x)∈R4
d2ψ1ψ2ρ01ρ02
π2
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
×
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 +
(
u02
βn2
2
− u01
βn1
1
)
t
∣∣∣∣ ,
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 325
A.A. Gukalov
which can be estimated by the following sum:
2∑
i=1
ρ0i
π3/2
∫
R3
dp sup
(t,x)∈R4
∣∣∣∣
∂ψi
∂t
+
(
p√
βi
+ V̂i − u0i
βni
i
t
)
· ∂ψi
∂x
∣∣∣∣ e−p2
+
4ρ01ρ02d
2
π2
sup
(t,x)∈R4
ψ1ψ2
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
×
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 +
(
u02
βn2
2
− u01
βn1
1
)
t
∣∣∣∣
+2
2∑
i=1
ρ0i
π3/2
∫
R3
dp sup
(t,x)∈R4
β
1
2
−ni
i |u0iψip| e−p2
.
Now, performing the limiting passage (βi → +∞) under the sign of inequality
and supremum, as in the proof of Theorem 1, we obtain equality (26) for ni > 1
2 ,
and the validity of (27) can be proved by using the equality
∫
R3
|p|e−p2
dp = 2π
obtained by direct integration in the spherical coordinates. Thus, we have shown
the validity of the assertions of Theorem 2. ¥
Theorem 3. Let the functions ϕi in distribution (5) take the form
ϕi(t, x) = ψi(t, x) · e−βi(V̂i−uit)2
, (30)
and condition (25) hold true, but now for ni > 1.
Then the assertion of Theorem 2 remains true if:
a) the functions ψi have the form
ψi(t, x) =
Di
(1 + t2)ξi
Ci
([
x× V̂i
])
, (31)
(10) holds and, in addition to that, V̂i⊥u0i;
b) there holds the representation
ψi(t, x) =
Di
(1 + t2)ξi
Ci (x) , (32)
and on the functions Ci, used here, the same restrictions as in Theorem 1 are
imposed.
326 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
P r o o f. Before proving Theorem 3, we will prove a proposition which
states that there exists a value of ∆
′
such that (14) is fulfilled and there holds
the equality
lim
βi→+∞
∆
′
=
2∑
i=1
ρ0i sup
(t,x)∈R4
µi(x)
∣∣∣∣
∂ψi
∂t
+ V̂i
∂ψi
∂x
∣∣∣∣
+ 4πd2ρ01ρ02
∣∣∣V̂1 − V̂2
∣∣∣ sup
(t,x)∈R4
[µ1(x)µ2(x)ψ1(t, x)ψ2(t, x)]
+ 2θ
2∑
i=1
ρ0i
∣∣∣
(
u0i, V̂i
)∣∣∣ sup
(t,x)∈R4
[µi(x)ψi(t, x)] ,
(33)
where at ni > 1 : θ = 0, and µi = 1; in the case ni = 1 : θ = 1, µi(x) = e2u0ix.
In the case of this theorem, estimation (18) also holds, so we again begin with
calculation of the derivatives of (30) contained in the inequality. The derivative
with respect to time t is expressed as follows:
∂ϕi
∂t
= e−βi(V̂i−uit)2
{
∂ψi
∂t
+ 2βiψi
((
ui, V̂i
)
− u2
i t
)}
, (34)
and on the spatial coordinate x, it has the form
∂ϕi
∂x
=
∂ψi
∂x
e−βi(V̂i−uit)2
. (35)
Taking into account the conditions imposed on the functions ψi(t, x) and
derivatives (34), (35), we will pass to the supremum in inequality (18). Its exis-
tence follows from the conditions imposed on the functions Ci in the statement
of the theorem. Then we transform the resulting expression using representation
(7). The value ∆
′
gets the form
sup
(t,x)∈R4
2∑
i=1
ρ0ie
2βiuix
π3/2
∫
R3
dp
∣∣∣∣
∂ψi
∂t
+ 2βiψi
((
ui, V̂i
)
− u2
i t
)
+
(
p√
βi
+ V̂i − uit
)
∂ψi
∂x
∣∣∣∣ e−p2
+ sup
(t,x)∈R4
4d2ρ01ρ02ψ1ψ2e
2x(β1u1+β2u2)
π2
×
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 + (u2 − u1)t
∣∣∣∣ ,
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 327
A.A. Gukalov
which admits an upper bound by the following sum:
sup
(t,x)∈R4
2∑
i=1
ρ0ie
2β
1−ni
i u0ix
π3/2
∫
R3
dp
∣∣∣∣
∂ψi
∂t
+
(
p√
βi
+ V̂i − uit
)
∂ψi
∂x
∣∣∣∣ e−p2
+
4d2ρ01ρ02
π2
sup
(t,x)∈R4
ψ1ψ2e
2x
(
β
1−n1
1 u01+β
1−n2
2 u02
)
×
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 +
(
u02
βn2
2
− u01
βn1
1
)
t
∣∣∣∣
+2
2∑
i=1
ρ0ie
2β
1−ni
i u0ix
π3/2
∫
R3
dp sup
(t,x)∈R4
ψi
∣∣∣∣∣
(
u0i
βni
i
, V̂i
)
− u2
0i
β2ni
i
t
∣∣∣∣∣ e−p2
.
Thus, if we now pass to the low-temperature limit, we will obtain assertion
(33) with the corresponding values of θ and µi(x).
The verification of item (b), i.e., the functions of the form (32), is obvious
enough, but (31) should be considered in more detail.
To begin with, we introduce a new orthogonal (due to the conditions of item
(a)) basis consisting of the vectors ui, V̂i and
[
ui × V̂i
]
. In this basis, we expand
an arbitrary vector x
x = x1ui + x2V̂i + x3
[
ui × V̂i
]
.
Then, for the product ψie
2βiuix, we get the representation
Di
(1 + t2)ξi
Ci
(
x1
[
ui × V̂i
]
− x3uiV̂
2
i
)
e2βiu
2
i x1
,
which is constant on the second component of x2. It is easy to see that on the
remaining components of
(
x1, x3
)
, as well as on βi, it is also bounded if we take
into account the compact support property of Ci and requirement (25).
Next, let us compute the derivative with respect to x,
∂ψi
∂x
=
Di
(1 + t2)ξi
[
V̂i × C ′
i
([
x× V̂i
])]
,
and scalar multiply it by the vector V̂i. It is obvious that the resulting product
is equal to zero.
Taking this fact into account and using (33), it is easy to show the validity of
the assertions for the functions of the form (31). The theorem is proved. ¥
328 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
Theorem 4. Assume that the representation
ϕi(t, x) = ψi(t, x) · e−2βiuix (36)
takes place and requirement (11) is retained, but with
ni > 1
2
, ki > 1
2
. (37)
Let the function ψi(t, x) have the form
ψi(t, x) = DiCi(t)Ei(x), (38)
where Di > 0, Ci(t) has the same properties as in the previous theorems, and the
function Ei(x) is nonnegative, finite or rapidly decreasing at infinity and bounded
together with its gradient on x. Then (13) holds true.
P r o o f. As in the previous theorems, we begin with the introduction of
the auxiliary assertions. Again, we prove that there exists a value ∆
′
such that
inequality (14) is fulfilled and its low-temperature limit is equal to:
a) at ni > 1
2 , ki > 1
2 :
2∑
i=1
ρ0i sup
(t,x)∈R4
∣∣∣∣
∂ψi
∂t
∣∣∣∣ ; (39)
b) in the case ni > 1
2 , ki = 1
2 :
2∑
i=1
ρ0ie
V̂ 2
0i sup
(t,x)∈R4
∣∣∣∣
∂ψi
∂t
∣∣∣∣ ; (40)
c) if ni = 1
2 , ki > 1
2 :
2∑
i=1
ρ0i sup
(t,x)∈R4
{
et2u2
0i
∣∣∣∣
∂ψi
∂t
+ 2ψitu
2
0i
∣∣∣∣
}
+
4√
π
2∑
i=1
ρ0i|u0i| sup
(t,x)∈R4
(
et2u2
0iψi
)
;
(41)
d) and with ki = ni = 1
2 :
2∑
i=1
ρ0i sup
(t,x)∈R4
{
e(V̂0i−u0it)2
∣∣∣∣
∂ψi
∂t
+ 2ψitu
2
0i
∣∣∣∣
}
+ 2
2∑
i=1
ρ0i
(
2|u0i|√
π
+
∣∣∣
(
u0i, V̂0i
)∣∣∣
)
sup
(t,x)∈R4
{
e(V̂0i−u0it)2
ψi
}
.
(42)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 329
A.A. Gukalov
Using inequality (18), remaining true, given that only the form of the function
ϕi is changed, let us find the derivatives of function (36) with respect to t and x
∂ϕi
∂t
= e−2βiuix
∂ψi
∂t
, (43)
∂ϕi
∂x
= e−2βiuix
(
∂ψi
∂x
− 2βiuiψi
)
. (44)
Now, calculating the supremum of both sides of (18) and using the bounded-
ness of all the terms, we substitute the expressions for the derivatives (43), (44).
As a result, for the value ∆
′
, we will have the expression
sup
(t,x)∈R4
2∑
i=1
ρ0ie
βi(V̂i−uit)2
π3/2
∫
R3
dp
∣∣∣∣
∂ψi
∂t
+
(
p√
βi
+ V̂i − uit
)(
∂ψi
∂x
− 2βiuiψi
)∣∣∣∣ e−p2
+
4d2ρ01ρ02
π2
sup
(t,x)∈R4
ψ1ψ2e
β1(V̂1−u1t)2
+β2(V̂2−u2t)2
×
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣
q√
β1
− q1√
β2
+ V̂1 − V̂2 + (u2 − u1)t
∣∣∣∣ .
This sum under conditions (11) and (37) can be rewritten as follows:
sup
(t,x)∈R4
2∑
i=1
ρ0ie
βi
(
V̂0i
β
ki
i
− u0i
β
ni
i
t
)2
π3/2
∫
R3
dp
∣∣∣∣∣
∂ψi
∂t
+
(
p√
βi
+
V̂0i
βki
i
− u0i
βni
i
t
)
×
(
∂ψi
∂x
− 2β1−ni
i u0iψi
)∣∣∣∣ e−p2
+
4d2ρ01ρ02
π2
sup
(t,x)∈R4
ψ1ψ2e
β1
(
V̂01
β
k1
1
− u01
β
n1
1
t
)2
+β2
(
V̂02
β
k2
2
− u02
β
n2
2
t
)2
×
∫
R3
dq
∫
R3
dq1e
−q2−q2
1
∣∣∣∣∣
q√
β1
− q1√
β2
+
V̂01
βk1
1
− V̂02
βk2
2
+
(
u02
βn2
2
− u01
βn1
1
)
t
∣∣∣∣∣ .
Now, performing, as before, the limiting passage and some simple
transformations, we can obtain the values of the low-temperature limits of the
value ∆
′
specified in (39)–(42).
Then, calculating the derivative of (38) with respect to t and substituting
it into (39)–(42), we can see that assertion (13) is true. Hence the theorem is
proved. ¥
330 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Interaction between ”Accelerating-Packing” Flows for the Bryan–Pidduck Model
In the present paper, we obtained the results similar to those described in [15]
for the hard-sphere model. Thus, under the assumptions mentioned in Theorems
1–4, we can extend the results, previously known for the hard-sphere model, on
the model of rough spheres.
Acknowledgement. The author thanks Prof. V.D. Gordevskyy for his
useful comments.
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Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 331
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