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MSC 35G15, 65N30
DOI: 10.14529/mmph240403
STOCHASTIC WENTZEL SYSTEM OF FREE FLUID FILTRATION EQUATIONS ON A HEMISPHERE AND ON ITS EDGE
N.S. Goncharov, G.A. Sviridyuk
South Ural State University, Chelyabinsk, Russian Federation
E-mail: [email protected], [email protected]
Abstract. Deterministic and stochastic Wentzell systems of the Dzekzer equations describing the evolution of the free surface of a filtering fluid in a hemisphere and at its edge are studied. In the deterministic case, the unambiguous solvability of the initial problem for the Wentzell system in a particular constructed Hilbert space is established. In the case of the stochastic system, the theory of Nelson-Glicklich derivatives is used and a stochastic solution is constructed to quantify the change in the free filtration of the fluid.
Keywords: stochastic Dzekzer equation; system of Wentzell equations; the Nelson-Glicklich derivative.
Introduction
Let Qcl", n> 2, be a manifold with an edge Y. In particular, ,pe[0,2;r]} be a hemisphere in M3,and r = {<p: <p e [0,2^-]}} be a edge ofhem-
Q = {(0,p) :0<
isphere. The system of two Dzekzer equations [1], which describing free fluid filtration is defined on the compact Qlj T
(A-A9,<p)ut =aoA9,q,u-fioA9,<pu-you,u = u{t,@,(P),{t>0>(P)^M+xn W
(/I - A^v, = a^u - + dRu - yxu,v = v(t,R,<p),(t,R,<p) e M+ xT, (2)
where the Laplace-Beltrami operator A0 p on the hemisphere and the Laplace-Beltrami operator Ap on the edge of the hemisphere have the following form
i a f. „ a ^ 1 a2 . 1 a2 a
^ = sin0 dd I^ dd J + sin2 0 dp2' Ap = sin2 0 dp2 '=dd
(3)
n v '
e=—
2
Here, the symbol v = v(i,#,p),(V,#,p)eR+xr, denotes the external normal to M+xQ. The parameters a(),a},A, f3{),/?,,Yo,Y\ e^ characterize the medium. To this system we add the matching condition
trw = v Ha M+xr, (4)
and equip it will initial conditions
u(0,0,p) = u^ (0,p), v(0, p) = v0 (p). (5)
Let us call the solution of the problem (1)-(5) the deterministic solution of the Wentzell system. If we replace u and v, defined by Q and r respectively, on r = r(t) and K = x(t) are stochastic processes on the interval (0,r), we obtain stochastic Wentzell system, where the derivative of stochastic
processes is understand by the Nelson-Gliklikh derivative of the process. It associated with correct definition of "white noise" as one-dimensional Wiener process (see, for example, [2-7]). Let us call the solution of the corresponding problem the stochastic solution of the Wentzell system.
The paper, in addition to the introduction and the list of references, consists of two parts. The first part considers the existence and uniqueness of the deterministic Wentzell system of equations of free filtration of fluid on a hemisphere and at its edge. The second part contains the proof of existence and uniqueness of the stochastic system of Wentzell equations of free fluid filtration on a hemisphere and at its edge.
The deterministic Wentzell system of free fluid filtration equations
If 0k = k (k +1) eigenvalues of the Laplace-Beltrami operator A0 p , then
Goncharov N.S., Sviridyuk G.A.
Stochastic Wentzel System of Free Fluid Filtration Equations
on a Hemisphere and On Its Edge
Pjm (cos#)cosmф, m = 0,...,k; Pkm (cos#)sin | m | ф, m = -k,...,-1,
are the corresponding eigenfunctions orthonormalized with respect to the scalar product. Here,
1 dk Pk (t )=ds
(t2 - i)k
is a Lejandre polynomial of degree k , and (t) = (1 -12)' ^ d—pk (t) is the attached Lejandre polynomial. The scalar product is calculated using the following formula
{Yk?Yk7) = 7'cos — yos m2ydy\ P— (t) P— (t) dt.
o -1
Consider the following series
œ k
=ZZexp
k=1 m=0
Jok4 -«0k2 -Го Л + k2
(am,k COs^ + bm,k sinmV)Pk (COs^),
(6)
where
2ж
n/ 2 2n n/ 2
amk = j u0 (O,y)cosmydy j Pkm (0)sin OdO, bm k = j u0 (O,y)sinmydy j Pkm (0)sin OdO. 0 0 0 0 It is easy to see that the series constructed above is a formal solution of the equation (1). Moreover, if the series in (6) converge uniformly, then we have a solution to the problem (1), (5), where deu = 0. Given this, we can construct a solution to the problem (2), (5)
/k4 - a1k2 - y1
= Z exP
k=1
Л + k2
(ck cos kф + dk sin kф),
(7)
where
2 ж
2 ж
ck = j v0 (ф) coskфdф, dk = j v0 (ф)sinkфdф.
In the case of the matching condition (4) we obtain the following equation
œ k
HexP
k=1 m=0
4 -«0k2 -Г0 ^
Л + k1
(am,k cosmФ + bm,k sinmф)Pk" (c°s#)
в=ж! 2
= Z eXP
k=1
J1k -«1k Л + k 2
( ck cos kф + dk sin kф) .
Considering, that // = //, a0 = a1, y0 = we obtain equalivent system of equations
k
^ (am k cos my + bm k sin my) P— (0) = ck cos ky + dk sin ky, where m + n = 2k.
m=0
Substituting the integral coefficients we obtain an equivalent system
k 2n n/ 2 2n n/2
^ (| u0 (O,y) cos mydy j P— (0) sin Odd cos my + J u0 (O,y) sin mydy j P— (0) sin Odd sin myP^^ (0)
m=0 0 0 0 0
2 n 2 n
= | v0 (y)coskydycosky + j v0 (y)sinkydysinky. 0 0 Here the auxiliary integrals are calculated by the formula
n/2 n/2
J Pkm (0)sin OdO = Pkm (0) j sin OdO = P;^ (0), 0 0 and system has the following form
( pm (0))2
k (2n 2n N
y | u0 (#,p)cosmpdpcosmp+ j" u0 (0,p)sinmpdpsinmp
m=0 ^ o 0 y
2 n 2 n
= | v0 (p)coskydpcoskp+ j v0 (p)sinkpdpsinkp. (8)
0 0 Thus in the case Po = ^, «o = «1, To = Ti and the obtained condition (8) the solutions to the problem (1)-(5) will satisfy the matching condition (4).
Lineal closure of the span{Pkm (cos#)sin mp, P^ (cos0)cosmp:
m,k \ {!},# e
of
[0,2f )} generated by the scalar product
2xxj 2
(p,^) = | | p(d,p)y(Q,p)smOdddp, o o
we denote by the symbol A (Q). Next, the closure of the span {sin/cy. cos kcp\ k eN, ye [0,2tt)} by the norm, generated by the scalar product
w)= j £(p)v(p)dP,
0
we denote by the symbol A( r).
Thus, the following theorem holds.
Theorem 2.1 For any u0 and v0 e^4(T), and any coefficients cCq,^,/^,/^,^,^,^ e№,
5mc/2, that the conditions a(j=al, J3() = /3} , y(j=yl, X^k2 are satisfied, where k e N, and the system (8) is solvable, then there exists a unique solution (u,
v)eC°°(K;^(Q) + ^(r)) of problem (l)-(5).
The stochastic Wentzell system of free fluid filtration equations
For simplicity's sake, let A = {u e W22 (Q) + W2 (r) : dRu = 0}, F = L2 (Q) + L2 (r) . Next, following the algorithm above, construct the spaces of random K -values. The random K -values ?/, k e UKL2
has the form ^ = , k = , where {pk} is the family of eigenfunctions of the Laplace -
i=l k=1
Beltrami operator A0peL(UKL2;FKL2) orthonormalized in the sense of the scalar product (•,•) of L2 (Q); {yk} is the family of eigenfunctions of the Laplace-Beltrami operator Ap eL(UKL2;FKL2)
orthonormalized in the sense of the scalar product (•,•) of L2 (Q). Consider the linear stochastic
Wentzel system of free fluid filtration equations in a hemisphere and at its edge. In this case (1)-(5) is transformed to the form
=a^e,vn-P^20,(pri-yori, r] e Cm(M+;UKL2), (9)
(X-A^K^a^K-frAlK + d^-y.K, K ^Cm(M+,UKL2) (10)
To the system (9), (10) we add the corresponding matching condition (8) and initial condition
;7(0) = 70,K( 0) = K0, (11)
The solution of the problem (9)-(11) will be called a stochastic solution. Thus, using the idea inherent in the results obtained earlier (see, for example [8]), the following theorem holds.
Theorem 3.1 For any ?/0,K"0 e IIK /.2 (Q) and any coefficients a0,ar,p^y^^^A & , such, that
the conditions a0 =. /?0 = /?, . yQ=yl, A, ^k2 are satisfied, where k e N, and the system (8) is solvable, then there exists a unique solution rj e Cm(W+',UKL2) of problems (9)-(ll).
The research was funded by the Russian Science Foundation (project No. 23-21-10056).
Goncharov N.S., Sviridyuk G.A.
Stochastic Wentzel System of Free Fluid Filtration Equations
on a Hemisphere and On Its Edge
References
1. Dzektser E.S. Generalization of the Equation of Motion of Ground Waters with Free Surface. Dokl. Akad. NaukSSSR, 1972, Vol. 202, no. 5, pp. 1031-1033.
2. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "Noises". Abstract and Applied Analysis, 2015, vol. 2015, no. 697410. DOI: 10.1155/2015/697410
3. Favini A., Sviridyuk G.A., Zamyshlyaeva A.A. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure and Applied Analysis, 2016, Vol. 15, no. 1, pp. 185-196. DOI: 10.3934/cpaa.2016.15.185
4. Favini A., Sviridyuk G.A., Sagadeeva M. 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
5. Favini A., Zagrebina S.A., Sviridyuk 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, no. 128, pp. 1-10.
6. Favini A., Zagrebina S.A., Sviridyuk G.A. The Multipoint Initial-Final Value Condition for the Hoff Equations in Geometrical Graph in Spaces of K-"noises". Mediterr. J. Math., 2022, Vol. 19, Iss. 2, Article no. 53. DOI: 10.1007/s00009-021-01940-0
7. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. Springer, London, Dordrecht, Heidelberg, N.-Y., 2011, 436 p. DOI: 10.1007/978-0-85729-163-9
8. Goncharov N.S., Zagrebina S.A., Sviridyuk G.A. Non-Uniqueness of Solutions to Boundary Value Problems with Wentzell Condition. Bulletin of the South Ural State University. Series: MathematimathcalModeling, Programming and Computer Software, 2021, Vol.14, Iss. 4, pp. 102-105. DOI: 10.14529/mmp210408
Received September 15, 2024
Information about the authors
Goncharov Nikita Sergeevich is Post-graduate Student, Equations of Mathematical Physics Department, South Ural State University, Chelyabinsk, Russian Federation, e-mail: [email protected].
Sviridyuk Georgiy Anatol'evich is Professor, Dr. Sc. (Physics and Mathematics), Head of Mathematical Physics Non-Classical Equations Research Laboratory, South Ural State University, Chelyabinsk, Russian Federation, e-mail: [email protected], ORCID iD: https://orcid.org/0000-0003-0795-2277.
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2024, vol. 16, no. 4, pp. 24-28
УДК 517.9, 519.216.2 DOI: 10.14529/mmph240403
СТОХАСТИЧЕСКАЯ СИСТЕМА ВЕНТЦЕЛЯ УРАВНЕНИЙ СВОБОДНОЙ ФИЛЬТРАЦИИ ЖИДКОСТИ НА ПОЛУСФЕРЕ И НА ЕЕ КРАЕ
Н.С. Гончаров, Г.А. Свиридюк
Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: [email protected], [email protected]
Аннотация. Исследуются детерминированные и стохастические системы Вентцеля уравнений Дзекцера, описывающие эволюцию свободной поверхности фильтрующейся жидкости на полусфере и на ее краю. В детерминированном случае установлена однозначная разрешимость начальной задачи для системы Вентцеля в конкретном построенном гильбертовом пространстве. В случае стохастической системы используется теория производных Нельсона-Гликлиха и строится стохастическое решение, позволяющее определить количественное изменение свободной фильтрации жидкости.
Ключевые слова: стохастическое уравнение Дзекцера; система уравнений Вентцеля; производная Нельсона-Гликлиха.
Литература
1. Дзекцер, Е.С. Обобщение уравнения движения грунтовых вод со свободной поверхностью / Е.С. Дзекцер // Доклады Академии наук СССР. - 1972. - Т. 202, № 5. - С. 1031-1033.
2. 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. - P. 697410.
3. 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. -2016. - Т. 15, № 1. - P. 185-196.
4. Favini, A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of «Noises» / A. Favini, G.A. Sviridyuk, M. Sagadeeva // Mediterranean Journal of Mathematics. - 2016. - Vol. 13, no. 6. - P. 4607-4621.
5. 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, no. 128. - P. 1-10.
6. Favini, A. The Multipoint Initial-Final Value Condition for the Hoff Equations in Geometrical Graph in Spaces of K-"noises" / A. Favini, S.A. Zagrebina, G.A. Sviridyuk // Mediterr. J. Math. - 2022. - Vol. 19, Iss. 2. - Article no. 53.
7. Gliklikh, Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics / Yu.E. Gliklikh. - Springer, London, Dordrecht, Heidelberg, N.-Y. - 2011. - 436 p.
8. Goncharov, N.S. Non-Uniqueness of Solutions to Boundary Value Problems with Wentzell Condition / N.S. Goncharov, S.A. Zagrebina, G.A. Sviridyuk // Bulletin of the South Ural State University. Series: Mathematimathcal Modeling, Programming and Computer Software. - 2021. - Vol.14, Iss. 4. -P.102-105.
Поступила в редакцию 15 сентября 2024 г.
Сведения об авторах
Гончаров Никита Сергеевич - ассистент, кафедра уравнений математической физики, ЮжноУральский государственный университет, г. Челябинск, Российская Федерация, e-mail: [email protected].
Свиридюк Георгий Анатольевич - доктор физико-математических наук, профессор, научно-исследовательская лаборатория неклассических уравнений математической физики, ЮжноУральский государственный университет, г. Челябинск, Российская Федерация, e-mail: [email protected], ORCID iD: https://orcid.org/0000-0003-0795-2277.
