Stochastic Wentzel system of free fluid filtration equations on a hemisphere and on its edge

Let    Q с I ^й , n >  2, be a manifold with an edge Г . In particular,

Q = { ( Ө , p ) : Ө e

0,П ,p e [0,2n]} be a hemisphere in R3 , and Г = {p: pe [0,2n]} } be a edge of hem- isphere. The system of two Dzekzer equations [1], which describing free fluid filtration is defined on the compact Qu Г

(л-A^p ) Ut = a0 AӨpU - e0 ^pU - /0U,U = u (t °p) , (t Ap)^^^

(Л - Ap)Vt = axkpU -ftApU + drU - Y1u, v = v(t,R ,p),(t,R ,p) ей+хГ,(2)

where the Laplace-Beltrami operator A _Ө , _p on the hemisphere and the Laplace-Beltrami operator A _p on the edge of the hemisphere have the following form

*        1 d f .  _ d >     1 d2 л       1    d2d

A_Op =--1 sln Ө — 1 + —:--7, A _p =— --—,д _R =—

п

Ө =—

sin O дӨ ^     дӨ J sin² Ө dp^{2   v}  sin² Ө dp²      дӨ

Here, the symbol v = v(t,O,p),(t,Ө,p)eЖ+xГ, denotes the external normal to I+xQ . The parameters a0,a1,Л,в0,P1,Y0,Y1 eR characterize the medium. To this system we add the matching condition tr u = v на Ж+хГ,

and equip it will initial conditions

U ( 0, Ө , p ) = U 0 ( Ө , p ) , V ( 0, p ) = V 0 ( p ) .

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 Г respectively, on n = n(t) and к = к(t) are stochastic pro- cesses 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 Ө _к = k ( k + 1 ) eigenvalues of the Laplace-Beltrami operator A _Ө , _p , then

үт ^ө )=

P m ( cos Ө ) cos m p , m = о,

...

, k ;

P_k ^{m |} ( cos Ө ) sin| m | p , m = - k ,

...,

- 1,

are the corresponding eigenfunctions orthonormalized with respect to the scalar product. Here,

Pk (t )=^ £(t2 - ’)‘ is a Lejandre polynomial of degree к , and P|ml (t) = (1 -12)'   ~~^Pk (t) is the attached Lejandre pol ynomial. The scalar product is calculated using the following formula

2 п

(Y m , Y m ² )= j cos m ^ cos m 2⁽ P d⁽ P j P m ( t ) P mm ² ( t ) dt .

-1

Consider the following series

Г u=XX exp t k=1 m=о     I

в о k ⁴ - « о k ² - / о

A + k2

( a m , k ^cos m p ⁺ b m , k ^sin m p ) P m ( ^{COs Ө} ) ,

where

2 n

П 2

2 n

П 2

a m , k

= j u _о ( Ө , p ) cos m p d p j P km ( о ) sin Ө d Ө , b_m , _k = j u _о ( Ө , p ) sin m p d p j P^¹ ( о ) sin Ө d Ө .

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 д₀u = о. Given this, we can construct a solution to the problem (2), (5)

»

u = X exp t k=1     I

в 1 k ⁴ - « 1 k ² - ү 1 A + k ²

( c_k cos k p + d_k sin k p ) ,

where

2 n

2n

c k = j v o ( p ) cos k p d p , d_k = j v _о ( p ) sin k p d p .

In the case of the matching condition (4) we obtain the following equation

Г XX exp t k=1 m=о     I

в о k ⁴ - « 0 k ² - Ү о

A + k2

x

=X exp t k=1     V

( a m , k ^cos m p ⁺ b m , k ^sin m p ) P m ( ^{COs Ө} )

Ө=П 2

в 1 k ⁴ - « 1 k ² - ү 1 A + k ²

( c_k cos k p + d_k sin k p ) .

Considering, that в о = в і , « о = « , Ү о = Ү і we obtain equalivent system of equations

k

X ( a m , k cos m p + b_m , _k sin m p ) P ^m ( о ) = c_k cos k p + d_k sin k p , where m + n = 2 k . m =о

Substituting the integral coefficients we obtain an equivalent system

k 2 n

я/ 2

2 n

П 2

^ ( j u _о ( Ө , р ) cos m p d p j P m ( о ) sin Ө d Ө cos m p + j u _о ( Ө , р ) sin m p d p j P m ( о ) sin Ө d Ө sin m p P km ( о )

m =о о

2 n                          2 n

= j v о ( p ) cos k p d p cos k p + j v _о ( p ) sin k p d p sin k p .

Here the auxiliary integrals are calculated by the formula л/ 2                       П 2

j Pkm (о)sinӨdӨ = Pkm (о) j sinӨdӨ = Pm (о), and system has the following form

Математика

k s m =0

^2n                               2nA j u0 (Ө, p)cosmpdpcosmp + j u0 (Ө, p)sinmpdpsinmp (Pm(0))2

к 0                                 0

2n j v0 (p)coskpdpcoskp + j v0 (p)sinkpdpsinkp.

Thus in the case в ₀ = в ₁ , a ₀ = a ₁, ү ₀ = ү and the obtained condition (8) the solutions to the problem (1)–(5) will satisfy the matching condition (4).

Lineal       closure       of      the       span{ P m ( cos 0 ) sin m p ,       P m ( cos Ө ) cos m p :

m , k eN \{1}, Ө e 0,— , p e [0,2 n )} generated by the scalar product

2лл/ 2

(p,^) = j j p(3,p^(3,p)sin3d3dp,

0  0

we denote by the symbol A ( Q ) . Next, the closure of the span{sin k p , cos k p : k e N, p e [0,2 n )} by the norm, generated by the scalar product

2 n

(%,P = j %(p)v(p)dp, we denote by the symbol A (Г).

Thus, the following theorem holds.

Theorem 2.1 For any u₀ e A ( Q ) and v ₀ e A ( Г ) , and any coefficients a ₀, a ₁, в ₀, в ₁, Y o , Y ₁, ^ ^G® , such, that the conditions a ₀ = a ₁ , в ₀ = в і , Y ₀ = Y ₁, A ^ k² are satisfied, where k e N , and the system (8) is solvable, then there exists a unique solution ( u , v ) e C ^ ( I; A ( Q ) + A ( Г ) ) of problem (1)-(5).

The stochastic Wentzell system of free fluid filtration equations

For simplicity's sake, let A = {u e W2 (Q) + W22 (Г): 5Ru = 0}, F = L2 (Q) + L2 (Г) . Next, following the algorithm above, construct the spaces of random K -values. The random K -values n,K e UKL2 ж                да has the form n = S Anp, к = SAKPk, where {pk} is the family of eigenfunctions of the Laplace -i=1                 k=1

Beltrami operator Д^ e L ( U_KL₂;F_KL₂ ) orthonormalized in the sense of the scalar product ( • , • ) of L ₂ ( Q ) ; { ^ _k } is the family of eigenfunctions of the Laplace-Beltrami operator Д _p e L ( U_KL₂;F_KL₂ ) orthonormalized in the sense of the scalar product ( • , • ) of L ₂ ( 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-Дө,p)n = «0Дөpn-ДДӨ,pn-Ү0П, nec“(>+;UKL2),(9)

(A-Дp )к = a1ДpK - в1ДpK + d Rn - YK, Ke C ” (K+; UKL2)

To the system (9), (10) we add the corresponding matching condition (8) and initial condition

n( 0) = П0,к(0 ) = K0,

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 n0,K0 e UKL2 (Q) and any coefficients a0,a1,в0,в1,Y0,Y1,Ae®, such, that the conditions a0 = a1, в0 = в1, Y0 = Y1, A ^ k2 are satisfied, where k e N, and the system (8) is solvable, then there exists a unique solution n e Cж (К.+;UKD of problems (9)-(11).

The research was funded by the Russian Science Foundation (project No. 23-21-10056).