Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
The Maxwellians of a special type, which correspond to inhomogeneous, nonstationary flows and describe the acceleration and packing of gas along some direction, are studied. The approximate description of interaction between these two flows for the model of hard spheres, when the temperatures are su...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
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irk-123456789-1065322016-10-01T03:01:45Z Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas Gordevskyy, V.D. Andriyasheva, N.V. The Maxwellians of a special type, which correspond to inhomogeneous, nonstationary flows and describe the acceleration and packing of gas along some direction, are studied. The approximate description of interaction between these two flows for the model of hard spheres, when the temperatures are sufficiently small, is obtained in a form of bimodal distribution with various coefficient functions. 2009 Article Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas / V.D. Gordevskyy, N.V. Andriyasheva // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 38-53. — Бібліогр.: 8 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106532 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The Maxwellians of a special type, which correspond to inhomogeneous, nonstationary flows and describe the acceleration and packing of gas along some direction, are studied. The approximate description of interaction between these two flows for the model of hard spheres, when the temperatures are sufficiently small, is obtained in a form of bimodal distribution with various coefficient functions. |
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Article |
| author |
Gordevskyy, V.D. Andriyasheva, N.V. |
| spellingShingle |
Gordevskyy, V.D. Andriyasheva, N.V. Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas Журнал математической физики, анализа, геометрии |
| author_facet |
Gordevskyy, V.D. Andriyasheva, N.V. |
| author_sort |
Gordevskyy, V.D. |
| title |
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas |
| title_short |
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas |
| title_full |
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas |
| title_fullStr |
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas |
| title_full_unstemmed |
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas |
| title_sort |
interaction between "accelerating-packing" flows in a low-temperature gas |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106532 |
| citation_txt |
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas / V.D. Gordevskyy, N.V. Andriyasheva // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 38-53. — Бібліогр.: 8 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT gordevskyyvd interactionbetweenacceleratingpackingflowsinalowtemperaturegas AT andriyashevanv interactionbetweenacceleratingpackingflowsinalowtemperaturegas |
| first_indexed |
2025-07-07T18:36:41Z |
| last_indexed |
2025-07-07T18:36:41Z |
| _version_ |
1837014347879022592 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 1, pp. 38�53
Interaction between "Accelerating-Packing" Flows
in a Low-Temperature Gas
V.D. Gordevskyy and N.V. Andriyasheva
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:gordevskyy2006@yandex.ru
Received January 26, 2008
The Maxwellians of a special type, which correspond to inhomogeneous,
nonstationary �ows and describe the acceleration and packing of gas along
some direction, are studied. The approximate description of interaction be-
tween these two �ows for the model of hard spheres, when the temperatures
are su�ciently small, is obtained in a form of bimodal distribution with
various coe�cient functions.
Key words: rare�ed gas, Boltzmann equation, hard spheres, Maxwellian,
bimodal distribution, approximate solution, error, low temperature.
Mathematics Subject Classi�cation 2000: 76P05, 45K05 (primary), 82C40,
35Q55 (secondary).
1. Introduction
In kinetic theory the state of rare�ed gas is described by the distribution
function f(t; v; x), where t 2 R1 is time, v =
�
v1; v2; v3
�
2 R3 is the velocity of
molecule, and x =
�
x1; x2; x3
�
2 R3 is its position in the space. This function is
a solution of non-linear integro-di�erential Boltzmann equation [1�3]
D(f) = Q(f; f); (1)
D(f) =
@f
@t
+ v
@f
@x
; (2)
Q(f; f) =
d2
2
Z
R3
dv1
Z
�
d�j(v � v1; �)j
�[f(t; v0
1; x)f(t; v
0
; x)� f(t; v1; x)f(t; v; x)]; (3)
c
V.D. Gordevskyy and N.V. Andriyasheva, 2009
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
where @f
@x
(or simply f
0
) denotes the spatial gradient of distribution f , d is the
diameter of particles, v; v1, v
0
, v
0
1 are velocities of two molecules before and after
collision, respectively; the vector � belongs to the unit sphere � � R3.
The most general form of the local-equilibrium Maxwell solutions of Boltz-
mann equation (in short, local Maxwellians M =M(t; v; x)), i.e., the exact solu-
tions of the system
D(M) = Q(M;M) = 0; (4)
was studied in [2, 4, 5]. A rather full description of the solutions mentioned
above and a particular analysis of their physical sense can be found, for example,
in [1�3]. The geometrical structure and the physical sense of local Maxwellians
were studied in detail in [6] and the complete analysis of corresponding possible
motions of gas was carried out.
One of these motions was called "accelerating-packing" as it can be described
by the following Maxwellian:
M = �
�
�
�
�3=2
e��(v�ev)
2
; (5)
where
� = � � e�(ev2+2ux) (6)
is the density of the �ow; � = const; � = 1
2T is its inverse temperature (T is the
absolute temperature); ev = v � ut (7)
is its mass velocity (here u; v 2 R3 - arbitrary constant vectors). It is easy to see
from (6), (7) that the vector u has a role of "mass acceleration", and the density �
changes from 0 to +1, besides for any �xed x 2 R3 its minimum value is reached
when t = t0, where
t0 =
1
u2
(u; v); (8)
but for any �xed t 2 R1 it increases only along the vector u.
For the approximate description of interaction between two �ows of "accele-
rating-packing" type, which have su�ciently small temperatures, let us consider,
by the analogy with [6�8], the following bimodal distribution:
f = '1M1 + '2M2; (9)
where the Maxwellians Mi, i = 1; 2, have the form (5)�(7) but with di�erent
hydrodynamical parameters �i, �i, �i, evi, vi, ui, i = 1; 2, and the coe�cient
functions
'i = 'i(t; x); i = 1; 2; (10)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 39
V.D. Gordevskyy and N.V. Andriyasheva
are nonnegative and smooth. The purpose is to �nd such a form of functions (10)
and such a behaviour of parameters vi; ui, i = 1; 2, and so on that together with
the "low-temperature limiting transition"
�i ! +1; i = 1; 2; (11)
make the error, i.e., some norm of di�erence between the sides of Boltzmann
equation (1)�(3), arbitrary small.
In Section 2 the rigorous statement of the problem is formulated and several
possible variants of its solution are presented.
2. Main Results
Following [6�8], consider the "mixed" or "uniform-integral" error between the
values D and Q (see (1)�(3)):
� = sup
(t;x)2R4
Z
R3
jD(f)�Q(f; f)jdv: (12)
The problem is to �nd any possible su�cient conditions for the in�nitesimality
of the value (12) if the distribution f has a bimodal form (9), (10) with the modes
Mi, i = 1; 2, of the type (5)�(7) with the limiting restriction (11).
Now we will prove a number of theorems and corollaries which give various
possibilities for solving this problem.
First, it is convenient to adopt the de�nition.
De�nition 1. Denote as P (Rn) a class of nonnegative functions from C1(Rn)
which have �nite supports (in short, �nite functions) or fast decrease at in�nity.
Theorem 1. Let the functions 'i, i = 1; 2, in distribution (9) have the form
'i(t; x) =
Di
(1 + t2)�i
Ci
x+ ui
(vi � uit)
2
2u2i
!
; i = 1; 2; (13)
where the constants �i; Di are as follows:
Di > 0; �i �
1
2
; i = 1; 2; (14)
and the functions Ci belong to P (R3); i = 1; 2. Let the conditions be ful�lled:
ui =
uoi
�nii
; i = 1; 2 (15)
vi =
voi
�kii
; i = 1; 2; (16)
40 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
where
ni � 1; ki �
1
2
; ki �
1
2
ni; i = 1; 2; (17)
and uoi; voi 2 R3 are arbitrary �xed vectors.
Then the error � in (12) is correctly de�ned, and there exists such a value �
0
that
� � �
0
; (18)
besides it has the limit
lim
�i!+1;i=1;2
�
0
= K(�1; �2)
2X
i=1
�iDi sup
x2R3
f�i(x)Ci(x+ ai)g ; (19)
where K(�1; �2) is some constant, the functions �i(x) are as follows:
�i(x) =
24 1; ni > 1; ki >
1
2
;
expf2uoixg; ni = 1; ki >
1
2
;
expfv2oi + 2uoixg; ni = 1; ki =
1
2
;
(20)
and the vector constants ai, i = 1; 2, are equal to
uoiv
2
oi
2u2oi
if ki =
1
2
ni and they are
equal to zero if ki 6= 1
2
ni.
P r o o f. The substitution of (9) into equations (1)�(3), by taking into
account (5)�(7) (with indexes i = 1; 2 for all values, respectively, see after (9))
and the fact that for each of Mi, i = 1; 2, the relation (4) is valid after some
evident estimations, the changes of variables and transformations analogous to
those done in [7,8] with the utilization of the technique developed in [6], leads to
the following inequality:
� � �
0
= sup
(t;x)2R4
2X
i;j=1;i6=j
24 Z
R3
����@'i@t +
�
up
�i
+ vi � uit
�
@'i
@x
+ '1'2�j(t; x)
d2p
�
Z
R3
Fije
�w2
dw
������ �i(t; x)��3=2e�u2du
+ '1'2
�1(t; x)�2(t; x)
�2
d2
Z
R3
e�w
2
�u2Fijdwdu
35 ; (21)
where
Fij = Fij(u; t; w) =
����� up
�i
+ vi � vj + (uj � ui)t�
wp
�j
����� ; i 6= j; (22)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 41
V.D. Gordevskyy and N.V. Andriyasheva
and
�i(t; x) = �i expf�i((vi � uit)
2 + 2uix)g; i = 1; 2: (23)
From (21)�(23) it can be easily seen that for veri�cation of the existence of
values �, �
0
it is su�cient to check that the products of functions (23) on the
values
'i;
@'i
@t
;
����@'i@x
���� ; 'it; t�ui@'i@x
�
; i = 1; 2; (24)
are bounded with respect to t; x on R4 for any �xed �i, i = 1; 2, if the functions
'i, i = 1; 2, have the form (13). Let us consider every one of the mentioned above
products separately. The �rst one, 'i�i(t; x), as a result of denotation
y = x+ ui
(vi � uit)
2
2u2i
; (25)
evidently, will have the form
�i expf2�iuiyg �
Di
(1 + t2)�i
� Ci(y): (26)
It follows from (26) that not only this expression itself, but also its product
on t is bounded with respect to t; y on R4 because of (14) and the properties of
functions Ci(y); i = 1; 2. The analogous conclusion is also true for other three
products because it follows from (13) that (once more after changing (25))
@'i
@t
= � Di
(1 + t2)�i
�
2�it
1 + t2
Ci(y) + (ui; C
0
i)
(ui; vo)� tu2i
u2i
�
; (27)
@'i
@x
=
Di
(1 + t2)�i
C
0
i(y); i = 1; 2: (28)
Further, the suppositions (15)�(17), as it can be seen from (23), guarantee
that for any (t; x) 2 R4 there exists the limit
lim
�i!+1;i=1;2
�i(t; x) = �i�i(x); i = 1; 2; (29)
with functions �i(x) of the form (20). Expressions (22) for every �xed u; t; w (and
on every compact in R4, even uniformly) tend to zero
lim
�i!+1;i=1;2
Fij = 0; i 6= j: (30)
Besides, from (25) it follows that the value y � x due to (15), (16) has the
form
uoi
2u2oi
0@ voi
�
ki�
1
2
ni
i
� uoit
�
1
2
ni
i
1A2
(31)
42 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
and, consequently, by conditions (17) has the following �nite limit:
lim
�i!+1;i=1;2
(y � x) =
"
0; if ki >
1
2
ni
uoiv
2
oi
2u2oi
; if ki =
1
2
ni
= ai; i = 1; 2: (32)
Thus, because of the supposition of smoothness (see Def.1), the functions Ci(y)
and C
0
i(y) with �i ! +1, i = 1; 2, tend to their values at point x+ ai; i = 1; 2.
At the same time the factor near C
0
i in (27) obviously tends to zero, therefore in
(27) the second summand vanishes. Since the parentheses near @'i
@x
in (21) also
tend to zero, we can, applying Lem. 1 from [7] (its conditions can be easily checked
from (27)�(32)) and using the boundness and continuity of all expressions as well
as good convergence of all integrals in (21), pass to the limit under the signs of
supremums and integrals entering into (21). Finally, the trivial integration with
respect to w and u yields (19), where the constant K(�1; �2) is as follows:
K(�1; �2) = 2max
i=1;2
�
�i sup
t2R1
jtj
(1 + t2)�i+1
�
: (33)
The theorem is proved.
Corollary 1. Let all suppositions of Theorem 1 be ful�lled. Then
8" > 0; 9Æ > 0; 8D1;D2 : 0 < D1;D2 < Æ;
9�o > 0; 8�i > �o; i = 1; 2;
� < ":
(34)
P r o o f is evident because of (18), (19) and the fact that Ci(x+ai) 2 P (R3),
i = 1; 2, for any ai from (32), so the products �i(x)Ci(x+ai) are bounded on R3.
Under the conditions of Th. 1, as it can be seen from (13), (23), the explicit
dependence of the �ows on the temperatures (i.e. on �i; i = 1; 2) is present at the
densities �i(t; x), but not at the desired coe�cient functions 'i(t; x) (they depend
on �i only through (15), (16), and this dependence does not play an essential role
(see (19)).
Let us now consider a result based on some other assumptions which give the
possibility to compensate the increase of �i(t; x) with �i ! +1, i = 1; 2.
Theorem 2. Let
'i(t; x) = i(t; x) exp
�
��i((vi � uit)
2 + 2uix)
; i = 1; 2; (35)
where the smooth functions i � 0 are such that the values (24) with the substi-
tution of i for 'i, i = 1; 2, are bounded with respect to t; x on R4, and (15) is
valid but now for
ni �
1
2
: (36)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 43
V.D. Gordevskyy and N.V. Andriyasheva
Then the inequality (18) holds true, besides for ni >
1
2
lim
�i!+1;i=1;2
�
0
=
2X
i=1
�i sup
(t;x)2R4
����@ i@t + vi
@ i
@x
+ 1 2�d
2�jjv1 � v2j
����
+2�d2�1�2jv1 � v2j sup
(t;x)2R4
( 1 2) = L; (37)
and for ni =
1
2
in addition to (37) a new summand arises
lim
�i!+1;i=1;2
�
0
= L+
4p
�
2X
i=1
�ijuoij sup
(t;x)2R4
i: (38)
P r o o f. From (35) instead of (27), (28) we have
@'i
@t
= exp
�
��i((vi � uit)
2 + 2uix)
�@ i
@t
+ 2�i i((vi; ui)� tu2i )
�
; (39)
@'i
@x
= exp
�
��i((vi � uit)
2 + 2uix)
�@ i
@x
� 2�i iui
�
; i = 1; 2: (40)
The formulas (21), (22) evidently remain true. Thus the substitution of (35),
(39), (40) in (21), taking into account (23), yields (18) with
�
0
= sup
(t;x)2R4
2X
i;j=1;i 6=j
24 Z
R3
����@ i@t + 2�i i((vi; ui)� tu2i )
+
�
up
�i
+ vi � uit
��
@ i
@x
� 2�i iui
�
+ 1 2�j
d2p
�
Z
R3
Fije
�w2
dw
������ �i��3=2e�u2du
+ 1 2
d2
�2
�1�2
Z
R6
e�w
2
�u2Fijdwdu
35 ; (41)
(the existence of the values � and �
0
follows from the conditions of Th. 2) or, in
short, after some obvious simpli�cations,
�
0
= ��3=2 sup
(t;x)2R4
2X
i=1
�i
Z
R3
�����@ i@t +Ai +Bi
����+Ai
�
e�u
2
du; (42)
44 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
where
Ai = Ai(u; t) = 1 2
d2p
�
�j
Z
R3
e�w
2
Fijdw; i 6= j; (43)
Bi = Bi(u; t) =
@ i
@x
�
up
�i
+ vi � uit
�
� 2 i
p
�i(u; ui): (44)
The limiting transition in (42) can be done in the same way as in the proof of
Th. 1, but the result will be di�erent. Indeed it follows from (22) and (15) that
lim
�i!+1;i=1;2
Fij = jvi � vj j; i 6= j; (45)
whence, from (43) after trivial integration with respect to w,
lim
�i!+1;i=1;2
Ai = 1 2�d
2�j jv1 � v2j; i 6= j: (46)
The limit of Bi depends on the quantity of parameter ni
lim
�i!+1;i=1;2
Bi = vi
@ i
@x
+ 2 iHi; i = 1; 2; (47)
where
Hi =
�
0; ni >
1
2
�(u; uoi); ni =
1
2
:
(48)
Passing to the limit in (42) with the use of (46)�(48), after integration with
respect to u (in the second case from (48) the value (42) must be bounded from
above once more when one chooses a "supplementary" summand 2 ij(u; uoi)j
whose integration together with the factor e�u
2
yields the second term in (38)),
we obtain (37), (38). The theorem is proved.
Corollary 2. Let the requirements (35), (15), (36) be ful�lled and the func-
tions i be of the form
i = DiCi(t); i = 1; 2; (49)
where Di > 0, and smooth, nonnegative functions Ci are such that the expressions
tCi and tC
0
i are bounded on R1.
Then:
a) For C1, C2, v1, v2 which satisfy the following conditions:
suppC1 \ suppC2 = � (50)
or
v1 = v2; (51)
the statement (34) holds true.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 45
V.D. Gordevskyy and N.V. Andriyasheva
b) For arbitrary C1, C2, v1, v2 the statement
8" > 0; 9Æ > 0; 8D1;D2; d : 0 < D1;D2; d < Æ;
9�o > 0; 8�1; �2 > �o
� < "
(52)
is valid.
P r o o f. Requirement (49) under the mentioned conditions imposed on
functions Ci(t) ensures the ful�llment of suppositions of Th. 2. Further, by virtue
of (49) and (50) or (51), or with
d! 0 (53)
(the last condition is the only new fact in (52) in comparison with (34)), the
nonzero terms retained in (37) or (38), will be only
Di sup
t2R1
jC 0
i(t)j; Di sup
t2R1
jCi(t)j; i = 1; 2: (54)
These two supremums due to the smoothness of functions Ci(t) are �nite. So, we
have (34) and (52) for situations a), b) of Corollary 2, respectively. The corollary
is proved.
There also exist two possible variants when the exponent in (35) contains not
two summands but only one. Now we will describe these variants representing
the following statements.
Theorem 3. Let the conditions of Theorem 2 be valid, but now instead of
(35) and (36) it is supposed that
'i(t; x) = i(t; x) expf��i(vi � uit)
2g; i = 1; 2; (55)
ni � 1; i = 1; 2; (56)
and the functions i, i = 1; 2, are such that the products of the values (24) (with
i instead of 'i) on the factors expf2�iuixg, i = 1; 2, are bounded with respect to
t; x on R4. Then the inequality (18) holds true, where for ni > 1
lim
�i!+1;i=1;2
�
0
=
2X
t=1
�i sup
(t;x)2R4
�����i(x)�@ i@t + vi
@ i
@x
�
+ 1 2�1(x)�2(x)�d
2�jjv1 � v2j
��+ 2�d2�1�2jv1 � v2j
� sup
(t;x)2R4
[�1(x)�2(x) 1(t; x) 2(t; x)] = N (57)
46 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
with �i(x), i = 1; 2, which correspond to the �rst and second cases from (20), and
for ni = 1
lim
�i!+1;i=1;2
�
0
= N + 2
2X
i=1
�ij(uoi; vi)j sup
(t;x)2R4
f�i(x) i(x)g: (58)
P r o o f. By di�erentiating (55), we obtain
@'i
@t
= exp
�
��i(vi � uit)
2
�
�
@ i
@t
+ 2�i i
�
(vi; ui)� tu2i
��
; (59)
@'i
@x
=
@ i
@x
exp
�
��i(vi � uit)
2
: (60)
Thus, remembering (23), from (21) and the conditions of Th. 3 we will have the
value �
0
for (18)
�
0
= sup
(t;x)2R4
2X
i�j=1;i 6=j
24 Z
R3
����@ i@t + 2�i i
�
(ui; vi)� tu2i
�
+
�
up
�i
+ vi � uit
�
@ i
@x
+ 1 2�je
2�jujx
d2p
�
Z
R3
Fije
�w2
dw
������
��ie2�iuix��3=2e�u
2
du+ 1 2
�
d
�
�2
�1�2e
2x(�1u1+�2u2) �
Z
R6
e�w
2
�u2Fijdwdu
35 ;
(61)
where Fij again has the form (22) but now (42)�(44) will be somewhat compli-
cated:
�
0
= ��3=2 sup
(t;x)2R4
2X
i=1
�ie
2�iuix
Z
R3
�����@ i@t +Ai +Bi
����+Ai
�
e�u
2
du; (62)
Ai = 1 2
d2p
�
�je
2�jujx
Z
R3
e�w
2
Fijdw; i 6= j; (63)
Bi =
@ i
@x
�
up
�i
+ vi � uit
�
+ 2�i i
�
(ui; vi)� tu2i
�
: (64)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 47
V.D. Gordevskyy and N.V. Andriyasheva
It is easy to see that (45) is preserved, but in (46) the factor �j(x) arises and it
is equal to 1 if nj > 1 and to expf2uojxg if nj = 1. As for (47), only the value
(48) will be changed
Hi =
�
0; ni > 1
(uoi; vi); ni = 1:
(65)
That is why the limiting transition in (62) leads to the expressions (57), (58).
The theorem is proved.
Corollary 3. Let the conditions of Theorem 3 be ful�lled and the functions
i be of the form:
i =
Di
(1 + t2)�i
Ci ([x� vi]) ; i = 1; 2; (66)
if
(vi; ui) = 0; i = 1; 2; (67)
and
i =
Di
(1 + t2)�i
Ci (x) ; i = 1; 2; (68)
for arbitrary v1; v2, where (14) is valid and the functions Ci belong to P (R3).
Then the statements a), b) of Cor. 2 remain true.
P r o o f. The expressions (68) under the indicated conditions evidently
are in concord with the requirements of Th. 3. Let us check whether the same
statement is true for the functions of the form (66) if (67) is ful�lled. Decompose
an arbitrary vector x 2 R3 by the orthogonal (because of (67)) basis
ui; vi; [ui � vi]; (69)
i.e.,
x = x1ui + x2vi + x3[ui � vi]; (70)
then
ie
2�iuix =
Di
(1 + t2)�i
Ci([x� vi])e
2�iuix
=
Di
(1 + t2)�i
Ci(x1[ui � vi]� x3[vi � [ui � vi]])e
2�ix1u
2
i
=
Di
(1 + t2)�i
Ci(x1[ui � vi]� x3uiv
2
i )e
2�ix1u
2
i ; (71)
but with the increasing of x1 when the exponent in (71) also increases, the ar-
gument of the function Ci obviously increases too without any connection with
the behaviour of x3 (the component x2 is not present in (71) at all) because of
48 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
the perpendicularity of the components of this argument. Thus, the function Ci
either will vanish (if it is �nite) or will compensate the increasing exponent in
a view of supposition of its fast decrease. So, the expression (71) in whole turns
out to be a bounded one, whose behaviour with x3 !1 is evident. The product
of (71) by t is also bounded because of (14). The derivative @ i
@t
behaves itself
completely in the same way (see (33)). Further, from (66) we will �nd
@ i
@x
=
Di
(1 + t2)�i
h
vi � C
0
i
i
; i = 1; 2; (72)
i.e., the values ����@ i@x
���� e2�iuix; t
�
ui
@ i
@x
�
e2�iuix (73)
are bounded by the same reasons as (71), because C
0
i([x � vi]) is �nite or fast-
decreasing, too. Therefore, for the expressions (66), (68) all conditions of Th. 3
are ful�lled. Consequently, (57) or (58) holds true. If (67) is valid, then the
second summand in (58) vanishes, i.e., it is essential only when ni = 1 and (68)
is ful�lled. But in the cases a), b) of Cor. 2 the only nonzero expression in (57)
remains
�i(x)
�
@ i
@t
+ vi
@ i
@x
�
: (74)
And in (58) an "additional" summand to the value N may remain. However, as
it can be seen from (72), under the supposition (66) we have
vi
@ i
@x
= 0; i = 1; 2; (75)
and for (68)
vi
@ i
@x
=
Di
(1 + t2)�i
�
vi; C
0
i(x)
�
; i = 1; 2; (76)
that is, for all possible cases the expressions (57), (58) up to some constant factors
reduce to the values of type (19), where there are either functions Ci themselves
or their derivatives C
0
i , i = 1; 2. This fact, obviously, yields to (34) and (52) in
the same way as in the proofs of previous corollaries.
R e m a r k 1. The expressions (66) and (68) are similar but they do not
reduce to each other. Really, Ci([x � vi]) describes a function on x, which is
constant along the vector vi, i.e. (because of (67)) in a direction perpendicular to
the direction of acceleration and packing of i-th �ow, i = 1; 2, and �nite or fast-
decrease particularly along the vector ui that is, at the direction of the increasing
of factor �i(x)). However, (68) corresponds to some "clot" of a gas concentrated
on a bounded in R3 support, and the factors depending on t, which are common
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 49
V.D. Gordevskyy and N.V. Andriyasheva
for (66) and (68), mean that the interaction between two �ows is weakened with
t! �1 but not too quickly.
Theorem 4. Let us suppose that instead of (35) or (55) the following equality
is ful�lled:
'i(t; x) = i(t; x) expf�2�iuixg; (77)
and the requirements (15), (16), (36) are valid with
ki �
1
2
; i = 1; 2; (78)
besides the smooth, nonnegative functions i ensure boundness on R4 of the same
expressions as in the conditions of Th. 3, but with substitution of the factor
expf�i(vi � uit)
2g for expf2�iuixg, i = 1; 2. Then the inequality (18) holds
true once more, where:
i. If ni >
1
2
; ki >
1
2
, then
lim
�i!+1;i=1;2
�
0
=
2X
i=1
�i sup
(t;x)2R4
����@ i@t
���� : (79)
ii. If ni >
1
2
; ki =
1
2
, then
lim
�i!+1;i=1;2
�
0
=
2X
i=1
�ie
v2oi sup
(t;x)2R4
����@ i@t
���� : (80)
iii. If ni =
1
2
; ki >
1
2
, then
lim
�i!+1;i=1;2
�
0
=
2X
i=1
�i sup
(t;x)2R4
�
et
2u2oi
����@ i@t + 2 itu
2
oi
�����
+
2p
�
2X
i=1
�i juoij sup
(t;x)2R4
n
et
2u2oi i
o
: (81)
iiii. If ni = ki =
1
2
, then
lim
�i!+1;i=1;2
�
0
=
2X
i=1
�i sup
(t;x)2R4
�
e(voi�uoit)
2
����@ i@t + 2 itu
2
oi
�����
+2
2X
i=1
�i
� juoijp
�
+ j(uoi; voi)j
�
sup
(t;x)2R4
n
e(voi�uoit)
2
i
o
: (82)
50 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Interaction between "Accelerating-Packing" Flows in a Low-Temperature Gas
P r o o f. It is evident that in our situation the analogues of formulae (59)�(61)
will be the following:
@'i
@t
=
@ i
@t
expf�2�iuixg; (83)
@'i
@x
= expf�2�iuixg �
�
@ i
@x
� 2�i iui
�
; (84)
�
0
= sup
(t;x)2R4
2X
i�j=1;i6=j
24 Z
R3
����@ i@t +
�
up
�i
+ vi � uit
��
@ i
@x
� 2�i iui
�
+ 1 2�je
�j(vj�ujt)
2 d2p
�
Z
R3
Fije
�w2
dw
������ � �ie�i(vi�uit)2��3=2e�u2du
+ 1 2
�
d
�
�2
�1�2 exp
�
�1(v1 � u1t)
2 + �2(v2 � u2t)
2
Z
R6
e�w
2
�u2Fijdwdu
35 :
(85)
Thus, instead of (62)�(64) we will have:
�
0
= ��3=2 sup
(t;x)2R4
2X
i=1
�ie
�i(vi�uit)
2
Z
R3
�����@ i@t +Ai +Bi
����+Ai
�
e�u
2
du; (86)
Ai = 1 2
d2p
�
�je
�j(vj�ujt)
2
Z
R3
e�w
2
Fijdw; i 6= j; (87)
Bi =
@ i
@x
�
up
�i
+ vi � uit
�
+ 2�i i
�
� 1p
�i
(u; ui)� (ui; vi) + tu2i
�
: (88)
The limit of exponents presented in�
0
depends on the behaviour of the parameters
ni, ki as follows:
lim
�i!+1;i=1;2
e�i(vi�uit)
2
= �i(t) =
26664
1; if ni >
1
2
; ki >
1
2
;
ev
2
oi ; if ni >
1
2
; ki =
1
2
;
et
2u2oi ; if ni =
1
2
; ki >
1
2
;
e(voi�uoit)
2
; if ni = ki =
1
2
:
(89)
Therefore, with passing the limit the analogous exponent in Ai (with index j 6= i)
always has a �nite limit �j(t), i.e., Ai in whole tends to zero because of (15), (16)
which yield (30). Finally, since vi ! 0; i = 1; 2, in (47) there will be maintained
only the second summand:
lim
�i!+1;i=1;2
Bi = 2 iHi; (90)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 51
V.D. Gordevskyy and N.V. Andriyasheva
where
Hi =
24 0; if ni >
1
2
; ki � 1
2
;
tu2oi � (u; uoi); if ni =
1
2
; ki >
1
2
;
tu2oi � (u; uoi)� (uoi; voi); if ni = ki =
1
2
:
(91)
Taking into account all these facts, the equality (86) leads to (79)�(82).
The theorem is proved.
Corollary 4. Let all conditions of Theorem 4 be ful�lled. Then the statement
(34) holds true if the functions i have the form
i(t; x) = DiCi(t)Ei(x); i = 1; 2; (92)
where Di > 0; Ci(t) 2 P (R1) and Ei(x) � 0 are smooth and bounded together
with E
0
i(x) functions on x 2 R3.
P r o o f. The functions of the form of (92) by the conditions imposed here
ensure the boundless of all expressions mentioned in Th. 4. Moreover, directly
from (79)�(82) it can be seen that the supremums entering into these formulas
are �nite for any possible values of the constants ni, ki � 1
2
, i = 1; 2, and the
presence of factors Di, i = 1; 2, in (92) leads to (34).
R e m a r k 2. It would be of interest to try to minimize the expressions (81),
(82) by solving the following di�erential equation (for each i = 1; 2):
@ i
@t
+ 2 itu
2
oi = 0: (93)
However, it can be easily seen that its solution
i = e�t
2u2oiEi(x); (94)
with Ei(x) being the same as in Cor. 4, in spite of ensuring the existence of both
supremums in (81) and under the additional condition (67) also in (82), does
not satisfy the conditions of Th. 4, because it cannot guarantee the boundness of
indicated there expressions before the limiting transition �i ! +1, i = 1; 2.
References
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1988.
[2] T. Carleman, Problemes Mathematiques dans la Theorie Cinetique des Gas.
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[3] M.N. Kogan, The dynamics of a Rare�ed Gas. Nauka, Moscow, 1967.
[4] H. Grad, On the Kinetic Theory of Rare�ed Gases. � Comm. Pure and Appl. Math.
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52 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
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[5] O.G. Fridlender, Local Maxwell Solutions of the Boltzmann Equation. � Prikl.
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