CC BY
13
MSC 35C15, 60H30, 76B15
DOI: 10.14529/ mmp200203
STOCHASTIC MATHEMATICAL MODEL OF INTERNAL WAVES
E.V. Bychkov1, A.V. Bogomolov2, K.Yu. Kotlovanov1
1 South Ural State University, Chelyabinsk, Russian Federation,
2Saint Petersburg Institute for Informatics and Automation of RAS, Saint Petersburg, Russian Federation
E-mails: [email protected], [email protected], [email protected]
The paper studies a mathematical model of internal gravitational waves with additive "white noise", which models the fluctuations and random heterogeneity of the medium. The mathematical model is based on the Sobolev stochastic equation, Dirichlet boundary conditions and the initial Cauchy condition. The Sobolev equation is obtained from the assumption of the propagation of waves in a uniform incompressible rotation with a constant angular velocity of the fluid. The solution to this problem is called the inertial (gyroscopic) wave, since it arises due to the Archimedes's law and under the influence of inertia forces. By "white noise" we mean the Nelson-Gliklikh derivative of the Wiener process. The study was conducted in the framework of the theory of relatively bounded operators, the theory of stochastic equations of Sobolev type and the theory of (semi) groups of operators. It is shown that the relative spectrum of the operator is bounded, and the solution of the Cauchy-Dirichlet problem for the Sobolev stochastic equation is constructed in the operator form.
Keywords: relatively bounded operator; Sobolev equation; propagators; "white noise"; Nelson-Gliklikh derivative.
Introduction
The internal waves arise at the interface of two fluids with different densities. In internal waves the maximum vertical displacement of particles takes place not on the surface of the fluid, but inside it. For example, this fact can be observed in the ocean at the location of desalinated water over heavier water with greater salinity. In this place a part of power of the ship engine is consumed on excitation of internal waves, resulting in a decrease of sheep's speed. In the simplest case the two-layer fluid model of internal waves is quite similar to the surface waves. They also concentrate near the interface. Assuming that the fluid fills entirely each half space on both sides of the border, the dispersion relation for internal waves is identical to the dispersion relation u2 = gk, where g is the acceleration due to gravity and k is the wavenumber, for gravitational waves but with a different effective value gravity acceleration [1].
The mathematical model of waves in homogeneous incompressible fluid rotating with a constant angular velocity Q is described by the linear system of hydrodynamic equations (the Sobolev system of equations [2])
where v ={u, v, w} is a vector of velocity, p is a pressure is applied perpendicular to the free surface, p0=const is an equilibrium density, and the buoyancy frequency is equal to zero. By directing the Oz axis to be collinear to the vector of Q, we can obtain the equation for the vertical velocity component of the fluid particles (the Sobolev equation [2])
ut + ¿Vp + 2[П x и] = О,
Pt = о, Vv = О,
(1)
A wtt + F2wzz = 0, (2)
where F is the Coriolis parameter. The wave solutions that satisfy (2) are called inertial (or gyroscopic) waves propagating on the surface of a rotating fluid. A solution to equation
(2) in an unbounded domain was obtained in [2] by the Green function method. The paper
[3] describes the behavior of solutions to two-dimensional Hamiltonian systems arising in the theory of small oscillations of rotating ideal fluid and constructs a mathematical model of the incipience of a vortex structure.
In this paper, we study the inhomogeneous stochastic equation
A Wtt +F2wzz =f k (3)
with the initial-boundary conditions
w(x,t) = 0, (x,t) e dD x R, (4)
w(x, 0) = w0(x), wt (x, 0) = wi(x), (5)
where wt (wtt) is the first (second) Nelson-Gliklikh derivative of a random process w, f K (t) is the "white noise", which models heterogeneity of fluid and random fluctuations, D C R3 is a domain with a smooth boundary dD, w0(x) and w1(x) are random K-variables. By "white noise" we mean the Nelson-Gliklikh derivative of the Wiener process.
The concept of the Nelson-Gliklikh derivative was introduced in the monograph [4]; the first derivative of an arbitrary random process was also found there. Later, derivatives of higher-order random processes were calculated, and the first mathematical models with "white noise", were investigated [5]. The Nelson-Gliklikh derivative is based on the concept of the average derivative introduced by Nelson [6]. In addition to the approach to "white noise", as the Nelson-Gliklikh derivative, the Ito-Stratonovich-Skorokhod approach is used by Kovac and Larson [7]. Also, in [8], a equation of the parabolic type was considered in the form of Ito differentials with uniform initial conditions, and a solution to the problem was found by reducing the equation to a first-order system. I.V. Melnikova takes a similar approach. The paper [8] introduces the spaces of generalized H-valued random variables, in which the H-valued white noise turns out to be smooth with respect to the variable t. It was shown in [9] that the Nelson-Gliklikh derivative of the Wiener process is in good agreement with the predictions of the Einstein - Smoluchowski theory of Brownian motion; therefore, the stochastic process was called "white noise". This approach is successfully applied to the study of equation of the Sobolev type, mathematical model based on one [10-15], the dichotomies of the stochastic equation defined on the manifold [16], and to the study of mathematical models of measuring devices [9].
The paper is organized as follows. In Section 1, we define the space of random K-variables, the space of random K-"process", and the space of K-"noises". In Section 2, we introduce the basic definitions and concepts of the relatively bounded operators theory and findings the solution of abstract equation. In Section 3, the mathematical model is reduced to the Cauchy problem for an abstract operator-differential equation and the propagators for stochastic Sobolev equation (3) are constructed when D is a parallelepiped.
1. Space of Differentiable K-"noises"
Currently, the theory of Sobolev type equations is transferred to the spaces of K-"noises" [9-16]. In this section, for completeness, we give only the necessary information about the spaces of differentiable K-"noises", which are considered in [9, 16]. Denote by
Q = (Q, A, P) the total probability space. A measurable map £ : Q ^ R is called a random, variable. The set of random variables whose expectations are equal to zero (i.e., E£ = 0) and the variances are finite (i.e., D£ < forms the Hilbert space L2 with the inner
product (£^£2) = E£1 £2. Denote by A0 a a-subalgebra of the a-algebra A and construct the space L2 of random variables measurable with respect to A0, then L0 is a subspace of the space L2. Let £ e L2, then n : L2 ^ L0 is an orthoprojector, and n£ is called the conditional expectation of a random variable £ and denoted by E(£|A0).
Let I = (0,T), T e R+. Consider two mappings: f : I ^ L2, which associates each t e I with a random variable £ e L2, and g : L2 x Q ^ R, which associates each pair (£, u) with a point £(u) e R. The map n : I x Q ^ R of the form n = n(t, u) = g(f (t), u) is called (one-dimensional) random process. If all the trajectories of a random process are almost surely (a.s.) continuous, then this process is called continuous. The set of continuous random processes forms a Banach space, which we denote by CL2. An example of a continuous random process is the one-dimensional Wiener process ^ = ^ (t), which can be represented as
0(t) = J>s in(|(2fc + l)i), (6)
k=0
where £k are uncorrelated Gaussian random variables such that E£k = 0, D£k = [f + 1)] ~2 .
Now we fix an arbitrary continuous random process n e CL2 and t e I. Let Ntn be the a-algebra generated by the random process n(t), and Ef = E(-IN7) be the conditional expectation.
Let n e CL2, then by the average derivative from the right Dn(t, ■) (left D*n(t, ■)) of the random process n at the point t e (e, t) we mean a random variable
if the limit exists in the sense of the uniform metric on R. A random process n is called average differentiate from the right (left) on I if there exists an average derivative from the right (left) at each point t e I. Let n e CL2 be a random process, which is average differentiable from the right and left on I. Then the average symmetric derivative is defined
O 1
as V= Dsr] = j (D + D*) rj. Further, we refer to the average symmetric derivative as the
Nelson-Gliklikh derivative. By n(1), l e N, we denote the l-th Nelson-Glicklikh derivative of the random process n. Note that if n(t) is a deterministic function, then the Nelson-Gliklikh derivative coincides with the classical derivative. In the case of the one-dimensional Wiener process ^ = ^ (t), the following statements are true:
(i) = ^ for all t e E+;
(ii)^(0(i) = (-l),"1nt1i(2i-l)®> ¿GN, l> 2.
We construct the noise space C'L2, l e N, as the space of random processes from CL2, whose trajectories are almost sure differentiable in the sense of the Nelson-Gliklikh derivative on I up to the l-th order inclusive.
Let U be a separable Hilbert space with the orthonormal basis {pk}• Each element u G U can be written as
u = ^ uk (k. k=l
Let K = {vk} be a monotonically decreasing numerical sequence such that (=1 ul < Choose a sequence of random variables {£k} C L2 such that (=1 v|D£k < Then a Hilbert space UKL2 is called a random K-variable space, moreover £ = (=1 vk£kpk.
Choose a sequence {rqk} from the space CL2 and define the U-valued continuous random, K-process by the formula
£(t) = Yl Vk£k(t)Pk (7)
k=1
provided that series (7) converges uniformly on any compact set from I in the norm of
o
UkL2. We introduce the Nelson-Gliklikh derivatives of the random K-process £(t) =
o
S(=1 vk £k(1)(t)pk provided that the Nelson-Gliklikh derivatives from the right exist up to the l-th order inclusive, and all the series converge uniformly on any compact set of I in the norm of UKL2. Therefore, we define the space Cl(I; UKL2) of continuous random K-processes whose trajectories are a.s. continuously differentiable with respect to Nelson-Gliklikh up to the l-th order inclusive. For shortness, the space Cl(I; UKL2) is called the space (of differentiable) K-"noises".
2. Cauchy Problem for Abstract Stochastic Equation
Let consider two spaces of random K-variable U = UKL2, F = FKL2, the operators L, M, N G L(U; F), and L be a Fredholm operator. Consider the Sobolev type equation of the second order in the space of differentiable K-"noises"
o
L wtt = Mw + Ng (8)
with the initial conditions oo
w (0) = w0, wt (0) = w1. (9)
Using the following lemma, we can transfer the theory of relatively bounded operators to the spaces of random K-variables
Lemma 1. Let U, F be separable Hilbert spaces and A G L(U; F). Then A G L(UkL2; FKL2).
The sets
pL(M) = G C : (uL - M)-1 G L(F; U)}
and aL(M) = C \ pL(M) are called the L-resolvent set and L-spectrum of the operator M, respectively.
The operator M is said to be a-bounded with respect to the operator L (or (L,a)-bounded) if 3 a G R+ : V u G C (|u| > a) ^ (pL - M)-1 G L(U; F)).
The operator-functions (pL - M)-1, R^(M) = (pL - M)-1L and LL(M) = L(pL - M)-1 with the domain pL(M) are called the L-resolvent, the right L-resolvent and the left L-resolvent of the operator M, respectively. If the set aL(M) is bounded, then the operator M is called (L, a)-bounded.
If the operator M is (L, a)-bounded, then the operators
Y Y
exist. Here 7 C C is a closed contour which bounds a domain containing aL(M). Let U0 (U1) = ker P (imP), F0 (F1) = ker Q (imQ). Denote by Lk (Mk) the restriction of the operator L (M) on the subspaces Uk, k = 0,1.
Theorem 1. (The splitting theorem [10]) Let the operator M be (L, a)-bounded. Then
(i) the operators Lk , Mk £ L(Uk, Fk), k = 0,1;
(ii) there exist the operators M—1 £ L(F0,U0) and L-1 £ ^(F1,U1).
Construct the operators H = M0-1Lo £ L(U0), S = L—1 Mx £ ¿(U1). The operator M is called (L,p)-bounded, p £ {0} U N, if to is a removable singularity of the L-resolvent of the operator M (i.e. H = O,p = 0) or a pole of the order p (p £ N) (i.e. Hp = O, Hp+1 = O) of the L-resolvent of the operator M.
Definition 1. A random K-process w £ C2(I; UKL2) is called a solution to equation (8), if a.s. all its trajectories satisfy equation (8) for all t £ I .A solution w = w(t) of equation (8) is called a solution to problem (8), (9), if moreover condition (9) is fulfilled.
Note that, generally speaking, problem (8), (9) is unsolvable for arbitrary w0,w1 £ U. Construct the set aL(M) = {^ £ C : ^n £ aL(M)}. This set is compact in C since the L-spectrum aL(M) of the operator M is compact. Consider a closed contour r = {|^| = R : Rn > a} and construct the operator-functions
Vt = — [ ¡in~k-\iinL - My'Le^dii, 2nt J r
where m = 0,1, and the integral is understood in the Riemann sense.
Definition 2. The map V• £ C^(R; L(U)) is called a propagator of homogeneous equation (8), if the vector-function w(t) = Vlv is a solution to (8) for all v £ U.
Theorem 2. [13] Let the operator M be (L,p)-bounded. Then the operator-functions Vk, k = 0,1 are propagators of homogeneous equation (8).
Theorem 3. [13] Let the operator M be (L,p)-bounded. Then for any N £ L(U, F) and a random process w such that (I — Q)Ng £ Cp+2(R, FKL2) and QNg £ C(R, FKL2), and for arbitrary independent random variables w0,w1 £ UKL2, which are independent of g for all fixed t £ [0,t], there exists a unique solution w to problem (8), (9) given by
p
w(t) = ^2 Vtwk + I VtsL-1QNg (s)dsHq M0-1 (I - Q)N 9(q) (t). 10
k=0 0 q=0
3. Propagators of the Sobolev Stochastic Equation
We consider a case when the domain D is a parallelepiped. There are domains with
_ d2
an analytic boundary for which the operator А where it has continuous spectrum.
_qzr_
1
Consider a space over the domain D such that the operator A 1 is a compact operator and the operator is bounded, and therefore their composition is a compact operator.
_ d2
Therefore, the spectrum of A is bounded. Later we will show that the L-spectrum
dz2
_ $2
of the operator M coincides with the spectrum of the operator A Let the domain
dz2
D be the parallelepiped [0,a] x [0,6] x [0,c]. Fix p G N and define the spaces U = {u G W2p+2(D) : u(x,y,z,t) = 0 (x,y,z,t) G x R+} and F = W2p(D). The space U is a separable Hilbert space by construction. Denote by — Xf = — — (-f-)2 — the eigenvalues of the Dirichlet problem for the Laplace operator A. Denote by <pi>m>n = sin sin (^y^) sin ) the eigenfunctions corresponding to — Xf .
We introduce the U-valued random K-processes. The sequence K is defined as follows: K = {vl>m>n : vl,m,n = A_m„}. By formula (7), we obtain the F-valued Wiener K-process in the form
,m,ni
,,m,nyl,m,n\ l,m,n=1
where ^l)TO,n(i) is a product of three independent one-dimensional Wiener processes (6). Define the operators
d 2
L = A, M = — F2——, N = 1 dz2
as elements of the space L(UKL0; FKL2) by virtue of Lemma 1. Also, define the inhomogeneity function as the Nelson-Gliklikh derivative of the Wiener process
o , .
g =WK (t).
Therefore, we reduce mathematical model (3) - (5) to Cauchy problem (9) for abstract equation (8).
Since {<£l)m>n} C (D), then
-Alm~ F2 (—i
V c /
L - M =
l,m,n=1
where < •, • > is the inner product in L2(D). The equation
^y + F2 (™)2 = 0
determines the relative spectrum of the operator M
Fm
c\rtfrnn
± Fim
m,n ~ / „ 1'
Therefore, aL(M) = is the L-spectrum of the operator M and is bounded.
Construct the propagators according to the Theorem 2. Since the relative spectrum of the operator M is discrete, we obtain
V0W0 = E cos ( f™ t ) < <Pltmtn,W0 > <Pltmtn,
l,m,n=1
<x /
v>i = E sin ) < > ^
_l.m.n=l Cv I,m,n \CVAi,m,n
Due to the fact that "white noise" wK (t) is not differentiable with t = 0, the integral in formula (10) does not make sense and conditions of the Theorem 3 are not satisfied. In order to overcome this difficulty, we use the method proposed in [10]. In order to use this
method, we transform the second term from the right-hand side of (10) as follows:
t t
J Vts w°K (t)(s)ds = -Vt£wK(t) - J js(y*-8)wK{s)ds =
£ £
t
= -Vt£WK(e) + J Vot-sWK(s)ds.
£
In this case, integration by parts makes sense for any e G (0,t), t G R+, by virtue of the definition of the Nelson-Gliklikh derivative. If e ^ 0, then we obtain
t t
J Vf-s wk (s)ds = J V0t-swk(s)ds. 0 0
Therefore, the conditions of Theorem 3 are fulfilled. Hence, there exists the unique solution to problem (2) - (5) given by
w(x,t) = Vftwo + VtW\ + Vt swK(s)ds.
o,
Acknowledgments. This work was supported by a grant from the President of the Russian Federation for state support of leading scientific schools of the Russian Federation (SSh-2553.2020.8).
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Received January 20, 2020
УДК 517.9 Б01: 10.14529/шшр200203
СТОХАСТИЧЕСКАЯ МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ВНУТРЕННИХ ВОЛН
Е.В. Бычков1, А.В. Богомолов2, К.Ю. Котлованов1
1 Южно-Уральский государственный университет, г. Челябинск, Российская Федерация
2 Санкт-Петербургский институт информатики и автоматизации Российской академии наук, г. Санкт-Петербург, Российская Федерация
В работе проведено исследование математической модели внутренних гравитационных волн с аддитивным «белым шумом», который моделирует случайные неоднородности среды и флуктуации. Математическая модель строится на стохастическом
уравнении Соболева, краевых условиях Дирихле и начальном условии Коши. Математическая модель строится на стохастическом уравнении Соболева, краевых условиях Дирихле и начальном условии Коши. Уравнение Соболева получено из предположения о распространении волн в однородной несжимаемой вращающейся с постоянной угловой скоростью жидкости. Решение этой задачи называется инерционной (гироскопической) волной, поскольку она возникает в силу закона Архимеда и под воздействием сил инерции. Под «белым шумом» мы подразумеваем производную Нельсона - Гликлиха винеровского процесса. Исследование проведено в рамках теории относительно ограниченных операторов и теории стохастических уравнений соболевского типа и теории (полу)групп операторов. Показано, что относительный спектр оператора ограничен, и построено решение в операторном виде.
Ключевые слова: относительно ограниченные операторов; уравнение Соболева; пропагаторы; «белый шум»; производная Нельсона - Гликлиха.
Литература
1. Бреховских, Л.М. Введение в механику сплошных сред / Л.М. Бреховских, В.В. Гончаров. - М.: Наука, 1982.
2. Соболев, С.Л. Об одной новой задаче математической физики / С.Л. Соболев // Известия АН СССР. Серия: Математическая. - 1954. - Т. 18. - С. 3-5.
3. Фокин, М.В. Гамильтоновы системы в теории малых колебаний вращающейся идеальной жидкости. I / М.В. Фокин // Математические труды. - 2001. - Т. 4, № 2. - C. 155-206.
4. Gliklikh, Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics / Yu.E. Gliklikh. - London; Dordrecht; Heidelberg; N.Y.: Springer, 2011.
5. Гликлих, Ю.Е. Изучение уравнений леонтьевского типа с белым шумом методами производных в среднем случайных процессов / Ю.Е. Гликлих // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2012. - № 27 (286), вып. 13. -С. 24-34.
6. Nelson, E. Dynamical Theory of Brownian Motion / E. Nelson. - Princeton: Princeton University Press, 1967.
7. Kovacs, M. Introduction to Stochastic Partial Differential Equations / M. Kovacs, S. Larsson // Proceedings of New Directions in the Mathematical and Computer Sciences. - 2007. -V. 4. - P. 159-232.
8. Мельникова, И.В. Обобщенная корректность задачи Коши для абстрактного стохастического уравнения с мультипликативным шумом / И.В. Мельникова, М.А. Альшанский // Труды института математики и механики УрО РАН. - 2012. - Т. 18, № 1. - С. 251-267.
9. Сагадеева, М.А. Построение наблюдения для задачи оптимального динамического измерения по искаженным данным / М.А. Сагадеева // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2019. - Т. 12, № 2. - С. 82-96.
10. Sviridyuk, G.A. Динамические модели соболевского типа с условием Шоуолтера - Сидорова аддитивными шумами / Г.А. Свиридюк, Н.А. Манакова // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2014. - Т. 7, № 1. -С. 90-103.
11. 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. - V. 2015. - P. 1-8.
12. Zamyshlyaeva, A.A. The Linearized Benney - Luke Mathematical Model with Additive White Noise / A.A. Zamyshlyaeva, G.A. Sviridyuk // Semigroups of Operators. Theory and Applications. - 2015. - V. 113. - P. 327-337.
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Евгений Викторович Бычков, кандидат физико-математических наук, кафедра «Уравнения математической физики>, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Алексей Валерьевич Богомолов, доктор технических наук, профессор, лаборатория автоматизации научных исследований, Санкт-Петербургский институт информатики и автоматизации Российской академии наук (г. Санкт-Петербург, Российская Федерация), [email protected].
Константин Юрьевич Котлованов, аспирант, кафедра «Уравнения математической физики>, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Поступила в редакцию 20 января 2020 г.
