On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases
In the paper, the solvability of one nonlinear boundary-value problem arising in kinetic theory of gases is studied. We prove the existence of global solvability of a boundary-value problem in the Sobolev space W¹∞ (R+). The limit of the solution is found by using some a'priori estimations. For...
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| Cite this: | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases / A.Kh. Khachatryan, Kh.A. Khachatryan, T.H. Sardaryan // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 320-327. — Бібліогр.: 6 назв. — англ. |
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| citation_txt | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases / A.Kh. Khachatryan, Kh.A. Khachatryan, T.H. Sardaryan // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 320-327. — Бібліогр.: 6 назв. — англ. |
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| description | In the paper, the solvability of one nonlinear boundary-value problem arising in kinetic theory of gases is studied. We prove the existence of global solvability of a boundary-value problem in the Sobolev space W¹∞ (R+). The limit of the solution is found by using some a'priori estimations. For the case of power nonlinearity, the uniqueness of the solution in a certain class of functions is proved. Some examples illustrating the obtained results are given.
Изучается вопрос глобальной разрешимости одной нелинейной краевой задачи, возникающей в кинетической теории газов. Доказано существование глобальной разрешимости краевой задачи в пространстве Соболева W¹∞ (R+). С использованием априорных оценок найден предел решения в бесконечности. В случае степенной нелинейности доказана единственность решения в определенных классах функций. Приведены примеры, иллюстрирующие полученные результаты.
|
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Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 3, pp. 320–327
On One Nonlinear Boundary-Value Problem
in Kinetic Theory of Gases
A.Kh. Khachatryan
Armenian National Agrarian University
74 Teryan St., Yerevan, 0009, Armenia
E-mail: Aghavard@hotbox.ru
Kh.A. Khachatryan and T.H. Sardaryan
Institute of Mathematics of National Academy of Sciencesof Armenia
24/5 Baghramyan Ave., Yerevan 0019, Armenia
E-mail: Khach82@rambler.ru
Sardaryan.tigran@gmail.com
Received September 9, 2013, revised February 5, 2014
In the paper, the solvability of one nonlinear boundary-value problem
arising in kinetic theory of gases is studied. We prove the existence of global
solvability of a boundary-value problem in the Sobolev space W 1
∞(R+). The
limit of the solution is found by using some a’priori estimations. For the
case of power nonlinearity, the uniqueness of the solution in a certain class
of functions is proved. Some examples illustrating the obtained results are
given.
Key words: boundary-value problem, monotony, nonlinear integral equa-
tion, iteration, limit of solution.
Mathematics Subject Classification 2010: 45G05, 35G55.
1. Introduction. Statement of the Problem
The paper is devoted to the following nonlinear boundary-value problem:
±s
∂ϕ±(x, s)
∂x
+ ϕ±(x, s) = G(U(x)), x > 0, s > 0, (1)
ϕ+(0, s) = G1
∞∫
0
Q(s, p)ϕ−(0, p)dp
, (2)
This work was supported by State Committee of Science MES RA in the frame of the
research project CS 13-1A068.
c© A.Kh. Khachatryan, Kh.A. Khachatryan, and T.H. Sardaryan, 2014
On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases
ϕ−(x, s) = o
(
e
x
s
)
, x → +∞, (3)
where
U(x) =
1√
π
∞∫
0
e−p2
[ϕ+(x, p) + ϕ−(x, p)]dp. (4)
The functions G and G1 describe the nonlinear dependence in the right-hand side
of integro-differential equation (1) and the nonlinear dependence of boundary
condition (2), respectively. The function Q(s, p) describes the general law of
reflection and possesses the substochasticity property
Q(s, p) ≥ 0, (s, p) ∈ R+ × R+,
∞∫
0
Q(s, p)dp ≤ 1. (5)
Boundary-value problem (1)–(4) can be derived from the Boltzmann equation
within the framework of one model suggested in [5] and it has important appli-
cations in kinetic theory of gases (see [1–6] and references therein). By means of
equations (1), (4) with boundary value conditions (2), (3), the flow of a gas with
average mass velocity U(x) in a half space x > 0 bounded by the plate wall x = 0
is described.
Problem (1)–(4) in a standard way can be reduced to the nonlinear integral
equation
U(x) = µ(x,U) +
∞∫
0
K(x− t)G(U(t))dt, (6)
where
µ(x,U) =
1√
π
∞∫
0
e−
x
s e−s2
G1
∞∫
0
Q(s, p)dp
∞∫
0
e
− t
p G(U(t))
dt
p
ds, (7)
K(x) =
1√
π
∞∫
0
e−
|x|
s e−s2 ds
s
. (8)
In the linear case, where G(x) ≡ x, G1(x) ≡ x, the investigation of the problem
(1)–(4) was carried out in a number of works (see [1, 6] and references therein).
In all the papers mentioned, the average mass velocity possesses asymptotics
O(x) when x tends to +∞.
In the case of the linear law of reflection (i.e., where G1(x) ≡ x), in [5], by
imposing some natural conditions on the function G, it was shown that there
exists qualitative difference between the solutions for the linear (G(x) ≡ x) and
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 321
A.Kh. Khachatryan, Kh.A. Khachatryan, and T.H. Sardaryan
nonlinear cases. In the linear case, the solution has a linear growth away from
the wall, while in the nonlinear case it has a bounded solution with the finite
limit at infinity.
In the present paper, the question of solvability of nonlinear integral equation
(6) is considered. Under certain conditions imposed on the functions G,G1 (see
below), the existence of a positive bounded solution of equation (6) is proved. The
limit of the solution at infinity representing isothermal sliding coefficient is found.
In the case of power nonlinearity (i.e., where G(x) = xα), the uniqueness of the
solution in a certain class of functions is proved. Some examples of nonlinearity
illustrating the obtained results are given.
2. Basic Results
Let G0(z) be a real measurable function defined on the set (−∞, +∞) and
satisfying the following conditions:
a) The numbers η and ξ are assumed to be the first positive roots of the
equations G0(z) = z and G0(z) = 2z, respectively, and besides 2ξ < η,
b) G0 ∈ C[0, η], G0 ↑ on the interval [ξ, η].
Here are the examples of the above function:
1) G0(z) = zp; 0 < p < 1, ξ =
(
1
2
) 1
1−p
; η = 1, (9)
2) G0(z) = ez−1, ξ ≈ 0, 2, η = 1. (10)
Below, assuming that the initial function G(z) is the local majorant for the
function G0(z), η is a fixed point for the function G1(z), and imposing some
natural conditions on the functions G and G1, we will prove global solvability of
equation (6) in the space of essentially bounded functions.
Moreover, by using special a’priori estimations, the limit of the solution at
infinity will be found. In one important particular case, where G(z) = zp, 0 <
p < 1
2 and the function G1 additionally satisfies the Lipschitz condition
|G1(z1)−G1(z2)| ≤ α|z1 − z2|, α ∈ (0; 1], z1; z2 ∈
[(
1
2
) 1
1−p
, 1
]
, (11)
the uniqueness of the solutions in a certain class of functions will be proved.
The following results are true:
Theorem 1. Let the functions G(s) and G1(z) satisfy the following condi-
tions:
i1) G(z) ≥ G0(z), z ∈ [ξ, η], G(η) = G1(η) = η, (12)
322 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases
i2) G,G1 ↑ on the interval [ξ, η] and G1(z) ≥ 0, z ∈ [ξ, η], G;G1 ∈ C[0; η]
(13)
and the function Q(s, p) satisfy condition (5).
Then equation (6) has a positive essentially bounded solution U(x), and be-
sides
lim
x→∞U(x) = η. (14)
Theorem 2. Let G(z) = zp; p ∈ (
0, 1
2
)
, and the function G1 satisfy the
conditions of Theorem 1 and condition (11). Then equation (6) has a unique
solution in the following class of measurable functions:
P = {f(x) :
(
1
2
) 1
1−p
≤ f(x) ≤ 1, x ∈ (0, +∞)}.
3. Proof of the Main Results
P r o o f of Theorem 1. With the help of equation (6), we consider the
auxiliary Hammerstein type nonlinear integral equation
ϕ(x) =
∞∫
0
K(x− t)G0(ϕ(t))dt, x > 0, (15)
with respect to an unknown measurable real function ϕ(x), where kernel K is
given by formula (8).
From (8), it follows that
K(−x) = K(x), x ≥ 0 and
+∞∫
−∞
K(x)dx = 1. (16)
In [4], not only the existence of positive solution ϕ(x) for equations (15), (16)
was proved, but also the following properties were established:
lim
x→∞ϕ(x) = η; ϕ(x) ≥ ξ, x ≥ 0. (17)
Let us consider the iteration for basic equation (6) taking into account (7),
Un+1(x) =
1√
π
∞∫
0
e−s2
e−
x
s G1
∞∫
0
Q(s, p)
dp
p
∞∫
0
e
− t
p G(Un(t))dt
ds
+
∞∫
0
K(x− t)G(Un(t))dt,
(18)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 323
A.Kh. Khachatryan, Kh.A. Khachatryan, and T.H. Sardaryan
U0(x) = ϕ(x), n = 0; 1; 2; . . . , x ≥ 0. (19)
Due to monotony of the functions G and G1 on the interval [ξ, η], it is easy to
check that
a) Un(x) ↑ in n,
b) the functions Un(x) are measurable on the set R+; n = 0; 1; 2, . . . .
Below we prove that
Un(x) ≤ η, n = 0; 1; 2, . . . . (20)
In fact, in the case n = 0, inequality (20) is obvious because of U0(x) = ϕ(x), ϕ(x) ↑
on R+ and lim
x→∞ϕ(x) = η.
We assume that inequality (20) takes place for some n ∈ N. Then in view of
(12) and (5), from (18) we get
Un+1(x) ≤ 1√
π
∞∫
0
e−s2
e−
x
s G1
η
∞∫
0
Q(s, p)dp
ds + η
∞∫
0
K(x− t)dt
≤ 1√
π
∞∫
0
e−s2
e−
x
s G1(η)ds + η
x∫
−∞
K(y)dy
= η
1√
π
∞∫
0
e−s2
e−
x
s ds +
x∫
−∞
K(y)dy
≡ J.
It is easy to verify that
∞∫
x
K(y)dy =
1√
π
∞∫
0
e−s2
e−
x
s ds,
therefore J = η, and hence Un+1 ≤ η.
Thus, the sequence of measurable functions {Un(x)}∞n=0 has a pointwise limit
as n → +∞. By B. Levi’s theorem, the function U(x) = lim
n→∞Un(x) satisfies
equation (6). From (18)–(20), it also follows that
ϕ(x) ≤ U(x) ≤ η, x ∈ R+. (21)
As lim
x→∞ϕ(x) = η, then in view of (21), we immediately get
lim
x→∞U(x) = η. (22)
The theorem is proved.
324 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases
P r o o f of Theorem 2. We assume the opposite. Let equation (6) have two
solutions from P. We denote their difference by ∆U = U1−U2; U j ∈ P, j = 1, 2.
Then from (6), taking into account (11) and G(z) = zp, p ∈ (
0, 1
2
)
, we have
∆U(x) =
1√
π
∞∫
0
e−s2
e−
x
s
×
G1
∞∫
0
Q(s, p)
∞∫
0
e
− t
p G(U1(t))
dtdp
p
−G1
∞∫
0
Q(s, p)
∞∫
0
e
− t
p G(U2(t))
dtdp
p
ds
+
∞∫
0
K(x− t)
[
G(U1(t))−G(U2(t))
]
dt. (23)
By the Lagrange theorem, it is easy to verity that
if
(
1
2
) 1
1−p
≤ x1, x2 ∈ 1, then |xp
1 − xp
2| ≤ 2p |x1 − x2|. (24)
Using (24) and (5), from (23) we obtain
|∆U(x)| ≤ α√
π
∞∫
0
e−s2
e−
x
s
∞∫
0
Q(s, p)
∞∫
0
e
− t
p |G(U1(t))−G(U2(t))|1
p
dtdpds
+
∞∫
0
K(x− t)|G(U1(t))−G(U2(t))|dt
≤
α√
π
2p
∞∫
0
e−s2
e−
x
s
∞∫
0
Q(s, p)dpds + 2p
x∫
−∞
K(y)dy
sup
t≥0
|U1(t)− U2(t)|
≤
2pα
∞∫
x
K(y)dy + 2p
x∫
−∞
K(y)dy
sup
t≥0
|∆U(t)| ≤ 2p sup
t≥0
|∆U(t)|.
Hence,
(1− 2p) sup
t≥0
|∆U(t)| ≤ 0. (25)
As p ∈ (
0, 1
2
)
, then due to (25) we obtain that ∆U(t) = 0 almost everywhere on
(0;+∞), therefore U1(t) = U2(t) almost everywhere on (0; +∞). The theorem is
proved.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 325
A.Kh. Khachatryan, Kh.A. Khachatryan, and T.H. Sardaryan
E x a m p l e s. As the function G(z) can be chosen, for the examples of
the functions G0 see (9) and (10). However, we also give an example different
from G0,
G(z) = G0(z) +
η
π
sin2 G0(z)π
η
.
Here are the examples of the functions G1(z):
a) G1(z) = ez−1; η = 1; α = 1,
b) G1(z) = η − βG̃(η − z); β ∈
0, min
1;
1
max
ξ≤z≤η
G̃′(z)
,
where G̃(η) = η, G̃ ↑ on [ξ, η], max
ξ≤z≤η
G̃′(z) < +∞, G̃(z) ≥ 0, z ∈ [ξ, η].
For example, if G̃(z) = z2, then η = 1 and G1(z) = 1− β(1− z)2, β ∈ (0, 1
2 ].
R e m a r k. It should be noted that the solution of initial boundary value
problem (1)–(4) belongs to the space W 1∞(R+) in x.
Thus, from (1)–(3), we get
ϕ+(x, s) = C(s)e−
x
s +
x∫
0
e−
(x−t)
s G(U(t))
dt
s
, (26)
ϕ−(x, s) =
∞∫
x
e−
(t−x)
s G(U(t))
dt
s
, (27)
where
C(s) = G1
∞∫
0
Q(s, p)
dp
p
∞∫
0
e
− t
p G(U(t))dt
.
As U ∈ L∞(0, +∞), then from (26), (27) it follows that for each fixed s ∈ (0, +∞),
ϕ±(x, s) ∈ W 1
∞(R+).
References
[1] C. Cercignani, The Boltzmann Equation and its Applications. Springer–Verlag,
New York, 1988.
[2] M.M.R. Williams, Mathematical Methods in Particle Transport Theory. Butter-
worth, London, 1971.
326 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases
[3] C. Villani, Cereignami’s Conjecture is Sometimes True and Always Almost True
Communications in Mathematical Physics. — Israel J. Math. 234 (2003), No. 3,
455–490.
[4] A.Kh. Khachatryan and Kh.A. Khachatryan, On an Integral Equation with Mono-
tonic Nonlinearity. — Memoirs on Differential Equations and Mathematical Physics
51 (2010), 59–72.
[5] A.Kh. Khachatryan and Kh.A. Khachatryan, Qualitative Difference between Solu-
tions for a Model of the Boltzmann Equation in the Linear and Nonlinear Cases. —
J. Theor. Math. Phys. 172 (2012), No 3, 1315–1320.
[6] N.B. Engibarian and A.Kh. Khachatryan, Exact Linearization of the Sliding Prob-
lem for a Dilute Gas in the Bhatnagar–Gross–Krook model. — J. Theor. Math.
Phys. 125 (2000), No 2, 239–342.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 327
|
| id | nasplib_isofts_kiev_ua-123456789-106801 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T15:29:03Z |
| publishDate | 2014 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Khachatryan, A.Kh. Khachatryan, Kh.A. Sardaryan, T.H. 2016-10-05T19:47:02Z 2016-10-05T19:47:02Z 2014 On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases / A.Kh. Khachatryan, Kh.A. Khachatryan, T.H. Sardaryan // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 320-327. — Бібліогр.: 6 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106801 In the paper, the solvability of one nonlinear boundary-value problem arising in kinetic theory of gases is studied. We prove the existence of global solvability of a boundary-value problem in the Sobolev space W¹∞ (R+). The limit of the solution is found by using some a'priori estimations. For the case of power nonlinearity, the uniqueness of the solution in a certain class of functions is proved. Some examples illustrating the obtained results are given. Изучается вопрос глобальной разрешимости одной нелинейной краевой задачи, возникающей в кинетической теории газов. Доказано существование глобальной разрешимости краевой задачи в пространстве Соболева W¹∞ (R+). С использованием априорных оценок найден предел решения в бесконечности. В случае степенной нелинейности доказана единственность решения в определенных классах функций. Приведены примеры, иллюстрирующие полученные результаты. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases Article published earlier |
| spellingShingle | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases Khachatryan, A.Kh. Khachatryan, Kh.A. Sardaryan, T.H. |
| title | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases |
| title_full | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases |
| title_fullStr | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases |
| title_full_unstemmed | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases |
| title_short | On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases |
| title_sort | on one nonlinear boundary-value problem in kinetic theory of gases |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106801 |
| work_keys_str_mv | AT khachatryanakh ononenonlinearboundaryvalueprobleminkinetictheoryofgases AT khachatryankha ononenonlinearboundaryvalueprobleminkinetictheoryofgases AT sardaryanth ononenonlinearboundaryvalueprobleminkinetictheoryofgases |