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Математика
DOI: 10.14529/mmph210201
INVARIANT SPACES OF OSKOLKOV STOCHASTIC LINEAR EQUATIONS ON THE MANIFOLD
O.G. Kitaeva
South Ural State University, Chelyabinsk, Russian Federation E-mail: kitaevaog@susu.ru
The Oskolkov equation is obtained from the Oskolkov system of equations describing the dynamics of a viscoelastic fluid, after stopping one of the spatial variables and introducing a stream function. The article considers a stochastic analogue of the linear Oskolkov equation for plane-parallel flows in spaces of differential forms defined on a smooth compact oriented manifold without boundary. In these Hilbert spaces, spaces of random K-variables and K-"noises" are constructed, and the question of the stability of solutions of the Oskolkov linear equation in the constructed spaces is solved in terms of stable and unstable invariant spaces and exponential dichotomies of solutions. Oskolkov stochastic linear equation is considered as a special case of a stochastic linear Sobolev-type equation, where the Nelson-Glicklich derivative is taken as the derivative, and a random process acts as the unknown. The existence of stable and unstable invariant spaces is shown for different values of the parameters entering into the Oskolkov equation.
Keywords: Sobolev-type equations; differential forms; Nelson-Glicklich derivative; invariant spaces.
Introduction
Consider the Oskolkov equation
(l-A)Ay nyJJyW . (1)
( Xi, x2)
Equation (1) is a model of the flow of a viscous and elastic incompressible fluid [1], in which the parameter v is responsible for the viscous properties of the fluid. The parameter X that determines the elastic properties of a fluid can take positive and negative values [2]. In [3-5], the solvability of the Cauchy-Dirichlet problem for the Oskolkov equation (1) was considered, and in [6] for the linear Oskolkov equation
(l-A)Ay = nA 2y. (2)
In [7], the problem of stability of solutions to equation (2) was solved in terms of exponential dichotomies, and in paper [8], the problem of stability of solutions in a neighborhood of the zero point of equation (1) was solved in terms of invariant manifolds.
This article discusses the stability of the stochastic linear Oskolkov equation on manifold that has no boundary. To solve this issue, we use equation (2) as an equation of the following form:
Ln = Mh , (3)
where n derivative in the sense [9] of the sought-for random process h = h(t). The number of works devoted to the study of equations of the form (3) is quite large at the present time (see, for example, [1013]), in which this equation was considered in various aspects. The present work is closest to [14] and [15], in which we study the solvability and stability of the Barenblatt-Zheltov-Kochina stochastic equation on a manifold.
The article contains four parts. The first section is the introduction, the fourth section is the bibliography. The second point is dedicated to deals with spaces of q-forms defined on a manifold that has no boundary, recalls the notions of a random variable, stochastic process, Nelson-Gliklikh derivative, constructs spaces of random K -variables and K -"noises". The third point contains a description of the invariant spaces of the Oskolkov stochastic equation.
1. Spaces of "noises" on a manifold
Consider «-dimensional manifold W that has no boundary. Let it have the properties of connectedness, compactness and smoothness. Consider spaces of smooth shapes Eq = Eq (M), 0 < q < n on W, where the scalar products are defined by the following formulas:
(a, b )0 = j aA * b, (a, b )2 = (Aa, Ab )0 + (Aa, b )0 +(a, b )0 ,
W
A = dS + Sd is the Laplace-Beltrami operator, S = (-i)n(q+1)+1 * d *, where * is the Hodge operator associating a differential form Eq with a differential form En-q , d is the outer differentiation operator. The spectrum <r(A) = {ak} of the operator A is positive discrete, and +¥ is the point of its condensation. Denote by Hq and Hq the completions of the lineal Eq with respect to the norms || • ||0 and || • 112 . The basis in Hilbert spaces Hq is the sequence of eigenfunctions {jk} of the operator A or-thonormalized by the norms || • ^ , l = 0,2.
Next, we turn to the construction of spaces of random K - variables and K -"noises" in Hf . Let W = (W,A,P) be a full probability space. We define a random variable as a mapping £: W® R and stochastic process as mapping h: 3 x W ® R (where 3 is a certain interval from R, a function h = h(,w) is a trajectory of the stochastic process). If almost all trajectories of a random process are continuous then such a process is called continuous.
L2 is the set of random variables £ for which the variance D is finite and the mathematical expectation E is zero, and CL2 is the set of continuous stochastic processes n. We fix t e 3, let
Ef = E (| N n), N n the a -algebra generated by the random variable n. By the derivative in the sense
[9] n of the stochastic process he CL2 b te3 we mean a limit (if it exists)
1 i,™ hh(t + At,•)-h(t,•)V hh(t,•)-h(t-At,•)^^
n =— lim Eh
2 ^ Dt ®o+ ^ h
+ lim Eh
Ai®0+
h
Denote by C1L2 the space of stochastic processes whose trajectories are almost sure differentiable in the sense [9] on 3 . The spaces C1L2 are called spaces of differentiable "noises".
Let us introduce into consideration the space Hq, l = 0,2 whose elements are random K -variables
h=Y.\xkVk.
k=1
The norm in this space is defined by the following formula:
N1 Hq = Z12 D4, k=1
where {£k} is a sequence of random variables with bounded variance, {jk} are the eigenfunctions of the operator A, orthonormalized by (•, )l, l = 0,2, and K = {Xk} is a monotone sequence such that
Zlk < +¥ . Let C(3;Hq) be the set of continuous stochastic K -processes
k=i
h(t) = Z 1hk (t)jk , hk e CL2 , (4)
k=1
and C1(3;Hq) be the set of continuously Nelson-Gliklikh differentiable K -processes
'(t) = Zl nk (t)jk , nk e ClL2, (5)
k=1
Kitaeva O.G. Invariant Spaces of Oskolkov Stochastic Linear Equations
on the Manifold
if series (4) and (5) converge uniformly on 3 c R (3 is compact set in R).
2. Stable and unstable invariant spaces
Let us define the operators
L = -(X + A)A, M =nA2 and the equation (2) in the space Hq can be considered in the form
L°n =Mn . (6)
The operators L, M: Hq ® Hq have the properties of linearity and continuity, and the operator M is (L, 0) -bounded operator.
By a solution to equation (6) we mean a stochastic K -process he C1(3; Hq) if, after substituting it into equation (6), we obtain the identity.
Definition 1. A set P e Hq such that the following conditions are satisfied:
(i) almost sure each trajectory of the solution h = h(t) to equation (6) belongs to P;
(ii) for almost all h0 e P, there exists a solution to equation (6) satisfying the condition
h(0) =h0 (7)
is called a phase space of equation (6).
It was shown (see, for example, [14]) that the phase space of equation (3) is the image of the resolving group Ut I" (mL -M)-1 Lemdm . Therefore, the following theorem is true.
2piJ
g
Theorem 1. The set of the following form:
P = jHq, Ae[ak }, (8)
[he Hq : (h, jn)o = 0,X = an
is the phase space of equation (6)
If the solution to problem (6), (7) is he C1(3;I) for any n0 e L, I c P, then the set I called a invariant space of equation (6).
Definition 2. A set I+ such that the following conditions are satisfied
(i) I+ is an invariant space;
(ii) h1(t) q £ N1e~m>1^s t) h(s) q , s > t, where positive constants N1, m1, h e I+ for all t e R
H0 H 0 is called a stable invariant space of equation (6). A set I_ such that the following conditions are satisfied
(i) I+ is an invariant space;
(ii) h2(t) нq £ N2e hl(s) q , t > s ,where positive constants N2, m2, h2 e I_ for all
,2
'2' m2 '
'0
te R
is called an unstable invariant space of equation (6).
Due to the fact that the relative spectrum of the operator M S (M) = s+L (M) + S (M), where
(M) = in-, A>_at}, (M) = jn-, A<_at},
[X + ak J [1 + ak J
and the results presented in [15] we obtain
Theorem 2. (i) The stable invariant space is set of the form (8) for n > 0 and X < 0. (ii) The stable invariant space is set of the form
I+={he Hq : (h, jk )0 = 0, X>_ak }
and the unstable invariant space is the set of the form
I-={he H q : (h, jk )0 = 0,1<-ak }
for n > 0 and 1 < 0.
Remark 1. For n > 0 and 1 < 0 there is an exponentially dichotomous behavior of solutions to the equation (6).
Conclusion
In the future, we intend to study the question on the solvability and stability of the stochastic analogue of semilinear equation (1). In addition, we intend to transfer all results for equation (1) to spaces of q-forms defined on a manifold with border.
References
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11. Favini A., Sviridiuk G.A., Sagadeeva M.A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of "Noises". Mediterranean Journal of Mathematics, 2016, Vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x
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13. Favini A., Zagrebina S.A., Sviridiuk G.A. Multipoint Initial-Final Value Problems for Dynamical Sobolev-type Equations in the Space of Noises. Electronic Journal of Differential Equations, 2018, Vol. 2018, p. 128.
Kitaeva O.G.
Invariant Spaces of Oskolkov Stochastic Linear Equations
on the Manifold
14. Shafranov D.E., Kitaeva O.G. The Barenblatt-Zheltov-Kochina Model with the Showalter-Sidorov Condition and Additive "White Noise" in Spaces of Differential Forms on Riemannian Manifolds without Boundary. Global and Stochastic Analysis, 2018, Vol. 5, no. 2, pp. 145-159.
15. Kitaeva O.G., Shafranov D.E., Sviridiuk G.A. Exponential Dichotomies in the Barenblatt-Zheltov-Kochina Model in Spaces of Differential Forms with "Noise". Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming and Computer Software (Bulletin SUSU MMCS), 2019, Vol. 2, no. 12, pp. 47-57. DOI: 10.14529/mmp190204
Received January 16, 2021
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2021, vol. 13, no. 2, pp. 5-10
УДК 517.9 DOI: 10.14529/mmph210201
ИНВАРИАНТНЫЕ ПРОСТРАНСТВА СТОХАСТИЧЕСКОГО ЛИНЕЙНОГО УРАВНЕНИЯ ОСКОЛКОВА НА МНОГООБРАЗИИ
О.Г. Китаева
Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: kitaevaog@susu.ru
Уравнение Осколкова получается из системы уравнений Осколкова, описывающей динамику вязкоупругой жидкости, после купирования одной из пространственных переменных и введения функции тока. В статье рассматривается стохастический аналог линейного уравнения Осколкова плоскопараллельных течений в пространствах дифференциальных форм, определенных на гладком компактном ориентированном многообразии без края. В данных гильбертовых пространствах строятся пространства случайных K-величин и К-«шумов» и решается вопрос об устойчивости решений линейного уравнения Осколкова в построенных пространствах в терминах устойчивого и неустойчивого инвариантных пространств и экспоненциальных дихотомий решений. Стохастическое линейное уравнение Осколкова рассматривается как частный случай стохастического линейного уравнения соболевского типа, где в качестве производной берется производная Нельсона-Гликлиха, а в качестве неизвестного выступает случайный процесс. При различных значения параметров, входящих в уравнение Осколкова, показано существование устойчивого и неустойчивого инвариантных пространств.
Ключевые слова: уравнения соболевского типа; дифференциальные формы; производная Нельсона-Гликлиха; инвариантные пространства.
Литература
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2. Амфилохиев, В.Б. Течения полимерных растворов при наличии конвективных ускорений / В.Б. Амфилохиев, Я.И. Войткунский, Н.П. Мазаева, Я.И. Ходорковский // Труды Лениниградско-го кораблестроительного института. - 1975. - Т. 96. - С. 3-9.
3. Свиридюк, Г. А. Квазистационарные траектории полулинейных динамических уравнений типа Соболева / Г.А. Свиридюк // Изв. РАН, сер. матем. - 1993. - Т. 57, № 3. - С. 192-202.
4. Свиридюк, Г.А. Фазовое пространство начально-краевой задачи для системы Осколкова / Г.А. Свиридюк, М.М. Якупов // Дифференц. уравнения. - 1996. - Т. 32, № 11. - С. 1538-1543.
5. Свиридюк, Г.А. Задача Коши для одного класса полулинейных уравнений типа Соболева / Г.А. Свиридюк, Т.Г. Сукачева // Сиб. матем. журн. - 1990. -Т. 31, № 5. - С. 109-119.
6. Свиридюк, Г.А. Задача Коши для линейного уравнения Осколкова на гладком многообразии / Г.А. Свиридюк, Д.Е. Шафранов // Вестник Челябинского государственного университета. Серия 3. Математика, Механика, Информатика. - 2003.- № 1(7). - С. 146-153.
7. Свиридюк, Г.А. Инвариантные пространства и дихотомии решений одного класса линейных уравнений типа Соболева / Г.А. Свиридюк, А.В. Келлер // Изв. вуз. Матем. - 1997. - № 5. -С.60-68.
8. Китаева, О.Г. Устойчивое и неустойчивое инвариантные многообразия уравнения Оскол-кова / О.Г. Китаева, Г.А. Свиридюк // Труды международного семинара «Неклассические уравнения математической физики», посвященного 60-летию со дня рождения профессора В.Н.Врагова, Новосибирск, 3-5 октября 2005 г. - Новосибирск: Изд-во Ин-та математики. - 2005. - С. 160-166.
9. Gliklikh, Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics / Yu.E. Gliklikh. - Springer, London, Dordrecht, Heidelberg, N.-Y. - 2011. - 436 p.
10. Favini, A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "noises" / A. Favini, G.A. Sviridyuk, N.A. Manakova // Abstract and Applied Analysis. - 2015. -Vol. 2015. - Article ID 697410.
11. Favini, A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of "Noises" / G.A. Sviridiuk, M.A. Sagadeeva // Mediterranean Journal of Mathematics. - 2016. - Vol. 13, no.6. - P. 4607-4621.
12. Favini, A. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise" / A. Favini, G.A. Sviridyuk, A.A. Zamyshlyaeva // Communications on Pure and Applied Analysis. -Springer, 2016. - Vol. 15, no. 1. - P. 185-196.
13. Favini, A. Multipoint Initial-Final Value Problems for Dynamical Sobolev-type Equations in the space of noises / A. Favini, S.A. Zagrebina, G.A. Sviridyuk // Electronic Journal of Differential Equations. - 2018. - Vol. 2018. - P. 128.
14. Shafranov, D.E. The Barenblatt-Zheltov-Kochina Model with the Showalter-Sidorov Condition and Additive "White Noise" in Spaces of Differential Forms on Riemannian Manifolds without Boundary / D.E. Shafranov, O.G., Kitaeva // Global and Stochastic Analysis. - 2018. - Vol. 5, no. 2. -P.145-159.
15. Kitaeva, O.G. Exponential Dichotomies in the Barenblatt-Zheltov-Kochina Model in Spaces of Differential Forms with "Noise" / O.G. Kitaeva, D.E. Shafranov, G.A. Sviridiuk // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2019. - Т. 2, № 12. - С. 47-57.
Поступила в редакцию 16 января 2021 г.
