Analysis of the Wentzell stochastic system composed of the equations of unpressurised filtration in the hemisphere and at its boundary

Let Q E R n , n >  2 , be a region with boundary Г of the class C “ . On a compact Q U Г we consider a system of two Dzekzer equations [1], modelling the evolution of the free surface of the filtering fluid

(A — A)ut = a0Au — e0A2u - Y0u, u = u(t, x), (t, x) E R x Q,

∂u

(A — A)vt = a1 Av — в1 A2v + dY1v, v = v(t, x), (t, x) E R x Г,

∂u dv =0, (t,x) E R x Г, tr u = v, на R x Г.

The symbol A in (1) denotes the Laplace operator in the region Q , and in (2) the same symbol denotes the Laplace - Beltrami operator on a smooth Riemannian manifold Г . The symbol v = v ( t,x ) , (t,x) E R x Г stands for the normal R x Г external to R x Q .

The parameters a ₀ , a 1 , A, в ₀ , в 1 , Y 0 , Y 1 E R describe the medium.

The condition of the form (2) and initial conditions (4) have been studied previously in various situations [2, 3], so we will only give a brief history. It first appeared in [4] when constructing the Feller semigroup generator [5] for multidimensional diffusion processes in the bounded Q region. In [6] it was shown for the first time that (2) arises naturally in biophysics to describe diffusion inside a cell and on its membrane. This approach to the study of problems where boundary conditions are treated not as limit values of the desired function and its derivatives, but as a description of some processes on the boundary, possibly only partially depending on the processes inside the region, led to the construction of a new direction in potential theory [7, 8], where solutions of one-phase and two-phase Wentzell problems with the use of repeated double and simple layer potentials were obtained. Another approach is based on the ideas and methods of semigroup operator theory. In [9] it was first shown that the operator including the Laplace operator A inside the region Q and the Laplace - Beltrami operator A on its boundary dQ is a generator of a C0-semigroup. In [10] this result was used in solving a number of applied problems. The first results of research in this direction were summarised in [11]. Moreover, in [12–15] analyticity conditions for solving C0-continuous semigroups of operators were found. Finally, in [16] the case when the operator A s replaced by A2 in Q region, while on the boundary the Laplace - Beltrami operator A remains the same.

Our approach to the study of the problem (6) – (9) is unconventional – intending in the future to consider different cases of the domain Q and the boundary Г (for instance, Q is a bounded connected Riemannian manifold with edge Г ) we consider it necessary to call (1), (2) a system of equations, albeit defined on sets of different geometric dimension. This approach is supported by the fact that equations (1) and (2) describe the same physical process of moisture filtration. The term “boundary conditions” should be reserved for equations defined on the boundary (edge) of a region (manifold) and having a lower order of derivatives on spatial variables (see the classical treatise [17]).

In the simplest case, we will study the solvability of system (1) - (3): Q = { ( r, Ө, ^ ) : r E [0 ,R ) ,Ө E [0 , ПП ] ,A E [0 , 2 n ] } in R ³ , but Г = { ( Ө,^ ) : Ө Е [0 , f ) ,^ Е [0 , 2 п ) } is a hemisphere with boundary. In this case, (1) – (3) is transformed to the form

( A - A r^ u t = a o A r^ u - в о А^ ,^ и — Y o u, u = u(t, r, Ө , a ) , ( t, r, Ө, ^ ) E R x Q , (5)

(A - Aө,^)vt = a1Aө,^v - в1A2Iө,^v + dRu - Yiv, v = v(t, Ө, a), (t, Ө, ^) E R x Г, dRu = 0, v = v(t, Ө, ^), (t, Ө, ^) E R x Г, where

A„ = (r - R) A Ал - r) I) + AA + EL, ∂r         ∂r    ∂θ2

d2    d2_ d

ө, ^ = дӨ2 + d^2 ’    R = dr r=R ’

Л _ д2    д2_ д

= дӨ ^{2 +} ^ 2 ’  ^{д R} = dr _r = _R ■

To the given system we add the matching condition (4) and equip it with initial conditions

u(0,r,Ө,^) = uo(r,Ө,^), v(0^,^) = Vo (Ө,^).

Let us call the solution of the problem (4) – (9) a deterministic solution of the Wentzell system. We note that by transforming the operator (17) to Cartesian coordinates we obtain

∂∂ A ■ = x + ydy	+ z| + (z2 + y 2j    + 1 dz \      J dx 2	xy ( x 2 + y 2 + z 2 ) x 2 + y 2	2 xy^	d 2 ∂x∂y
+ ^x	2 , 2 x 2 + y 2 + z 2 A d 2 + У   x 2 + y 2 / dy 2	(x 2 + y 2 +P ) 'Р ■

We shall transfer the consideration of the Laplace operator in standard spherical coordinates to our future studies.

In addition to the introduction and the list of references, the paper contains two parts. The first part considers the existence and uniqueness of a deterministic system of Wentzell equations in a hemisphere and on its boundary. The second part contains abstract reasoning consisting in the construction of the space of ( H -valued ) K -“noises” and the proof of existence and uniqueness of the stochastic system of Wentzell equations in the hemisphere and on its boundary.

1.    Deterministic Wentzell System

Let us consider the following series

∞ u = £ exp(t —ek-A—ak-k=2    \

+ cos kp) \ + £ exp ( t— J   k =i    \

“^ (RRr) a^as sin кӨ (sin kp + cos kp ) + b k cos kӨ (sin kp +

β 0 k ⁴ - α 0 k ² - γ 0

A + k ²

+d k cos kӨ (sin kp + cos kp)^ ,

^ ^ck sin kӨ (sin kp + cos kp ) +

where

2n     Rk ak = /dp/ uo(r,Ө,p) ^Rk   sin kӨ(sin kp + cos kp)rdr,

2n     Rk bk = fdP,/ u0(r^,P)~—Rk   cos kӨ(sin kp + cos kp)rdr,

π 2n ck = /dp/ v0(Ө, p) sin kӨ(sin kp + cos kp)dӨ, 00 π 2n      2

d k = /dp/ V 0 ^p) cos kӨ (sin kp + cos kp)d Ө .

It is not complicated to notice that the constructed series above is a formal solution of (5). Furthermore, if the series in (10) converges uniformly, then we have a solution of the problem (5), (9), where d R u = 0 , A u = 0 . Taking this into account, we can construct a solution of problem (6), (9)

v

\    (l - ^{в 1 k} 4 - ^a 1 k ² - ^{Y 1} A(          , л

=    exp ^t ^ + k 2 ------J \ c_k,n cos kp + dk.

sin kp ,

where in the case a ₀ = a 1 , в 0 = в 1 , Y o = Y 1 the solutions of the problem (6) — (9) will satisfy the matching condition (4).

The closure of the lineal span { ( R ^k ) - 1 ( R — r ) ^k sin kӨ (sin kp + cos kp), (R ^k ) - 1 ( R — r) ^k cos kO - • (sin kp + cos kp ) : k G N \{ 1 } ,r G (0 ,R ) ,Ө G [0 , П ] ,p G [0 , 2 n ) } generated by the scalar product

π

2 π 2

⁽ p^{,V )}

dr dϕ

p(r, Ө, p ) V ( r, Ө, p ) r ² sin Ө d Ө ,

we denote by the symbol A (Q) . Then, the closure of the lineal span { sin kӨ (sin kp + cos kp ) , cos kӨ (sin kp + cos kp ) : k G N , Ө G [0 , ⁿ ],p G [0 , 2 n ) } by the norm generated by the scalar product

π

2 π 2

(p^

dϕ

P ^{( r,ө,}p' ) ^{V ( r,ө,}p' ) ^dӨ,

we denote with the symbol А (Г) .

Thus, the following theorem occurs.

Theorem 1. For any u ₀ G A (Q) and v ₀ G А (Г) such that (4) is satisfied, and for any coefficients α ₀ , α ₁ , λ, β ₀ , β ₁ , γ ₀ , γ ₁ ∈ R , such that the following condition is satisfied a ₀ = a 1 , в ₀ = в 1 , Y ₀ = Y 1 , а ^ = k ² , where k G N , cthere exists a single solution ( u,v ) G C “ A (Q) ф А (Г)) of the problem (4) - (9) .

2.    Stochastic Wentzell System

Let Q = (Q , A , P ) be a complete probability space with probability measure P , associated with the a -algebra A of subsets of the set Q , and let R be the set of real numbers endowed with a Borel a -algebra. A measurable mapping ( : Q G R is called a random variable . The set of random variables with zero expectation and finite variance forms a Hilbert space L ₂ with scalar product ( ( 1 ,( 2 ) = E ( 1 ( ₂ .

Let I C R be an internal. We call the measurable mapping n : I x Q G R , a stochastic process, for each fixed w G Q the function nG ^w ) : I G R is its trajectory, and for each fixed t G I the random variable n ( t, • ) : Q G R is its cross section. We call a stochastic process n = n ( t ) , t G I , continuous stochastic process if almost probably all its trajectories are continuous (i.e. if almost all w G A the trajectories nt,w ) are continuous functions). A multitude of continuous stochastic processes forms a Banach space, which we denote by the symbol C (I; L 2 ) with norm

H n H c L 2 = sup⁽ D n ⁽ t,w ^{)) 1 / 2} . t ∈ I

Let A ₀ be an σ -subalgebra of σ -algebra A . Let us construct a subspace L ⁰ ₂ ⊂ L ₂ of random variables. of random variables measurable with respect to A ₀ . We denote by П : L ₂ G L 2 the orthoprojector. Let ( G L ₂ , then П ( is called the conditional mathematical expectation of the random variable ( is denoted by E ( ( |A ₀ ) . We fix n G C (I; L ₂ ) and t G I , denote by N t the a -algebra, generated by the random variable n(t) , and define E ⁿ = E^ N ⁴ ) .

Definition 1. Let n £ C (I; L 2 ) • The derivative of the Nelson–Glicklich process η of stochastic process n at a point t £ I is a random variable

n^lflta En (rttiAt-WL)V

^,        2 \л / - o ■ t                At

+ lim En (n(t, ) - \t - At,') jj , At^0+ t \        At        J J if the limit exists in the sense of a uniform metric on R.

о

If the Nelson - Glickich derivatives n (t, • ) of the stochastic process n(t, • ) exist at all (or

О p.c.) points of the interval I, then we say that the Nelson-Glicklich derivative n (t, •) on I (p.c. on I). The set of continuous stochastic processes having continuous Nelson–Glicklich derivative n orm a Banach function C1(I; L2) space with norm

◦

.

llnllc^ = sup Dn(t w) + D n (t, w) teI v7

Let us further define by induction the Banach spaces C l (I; L ₂ ) , l £ N , of stochastic processes whose trajectories are Nelson-Glicklich differentiable on I up to order l £ { 0 }U N inclusive. Their norms are given by the formulas

/ l оV/ llnllciL2 = sup £D n (k\t,w)

.

^{te I} \Z0

Here we will _o consider the zero-order Nelson-Glicklich derivative as the initial random process, e.g. ⁿ (⁰⁾ = n . Note also that the spaces C l (I; L ₂ ) , l £ { 0 } U N , for the sake of for brevity we will call the spaces of “noise” .

Let us proceed to the construction of the space of random K -values. Let H be a real separable Hilbert space with orthonormalised basis { ^ _k } , monotone sequence K = { A k } C M

R+, such that £ Ak < +^, and also the sequence {£k} = £k(w) C L2 random variables k=i such that ||(k|L2 < C, for some constant C £ R+ and for all k £ N. Let us construct an H-valued random K-values

M

£ ( w ) = £A k C k ( w ) ^ k .

k =i

Completion of the linear envelope of the set { A k £ _k ^ _k } by norm

^{M            1 / 2}

IlnU L 2 = IE A k D&)

is called the space of ( H -valued) random K -values and is denoted by the symbol H _K L ₂ . It should be obvious that the space H K L ₂ is Hilbertian, with the random K -value £ = £ ( w ) £ H K L ₂ . Equivalently, we define the Banach space of ( H -valued) K -“noises” C^l (I; H K L ₂ ) , l £ { 0 } U N , to be an enlargement of the linear envelope of the set { A k n k ^ k } by norm

IMIc l H K L 2

= sup t e I

( M      l

E Ai E Dn k=1 m=1

( m ) k

у/ 2

where the sequence of “noises” { n k } ^ C l (I; L ₂ ) , l ^G { 0 } U N . Obviously, the vector

∞

n(t, w ) = У^ A k n k ( t, и)^ k k =1

lies in the space C l ( I ; H K L ₂ ) , if the sequence of vectors {n k } C C l ( I ; L ₂ ) and all their Nelson-Glicklich derivatives up to and including order l G { 0 } U N are uniformly bounded in norm || • ||c l L 2 .

Example. Vector lying in all spaces C l (R ₊ ; H K L ₂ ) , l G { 0 } U N ,

∞

W k ( t,w ) = ^ A k в к (t,w ) ^ k , k =1

where { в _к } C C l (I; L ₂ ) is a sequence of Brownian motions, is called an ( H -valued) Wiener K -process .

Let U ( F ) now be a real separable Hilbert space with orthonormalised basis Wk }

∞

(W k } ). Let us introduce a monotone sequence K = { A _k } C { 0 }U R such that £ A k <  + to .

k =1

By the symbol U _K L ₂ ( F k L ₂ ) we denote the Hilbert space. which is a replenishment of the linear envelope of random K -values

∞

€ = У ^A k ⁽ k ^ k , i k ^G L 2 , k =1

( M

Z = E k=1

M k C k ^ k , C k ^G L 2

,

by norm

∞

I IMI F = E M k D Zd ■

MU = E^A k ⁿ '

k =1

Note that in different spaces ( U _k L ₂ и F k L 2 ) the sequence K can be different ( K = { A _k } and U _K L ₂ и K = { M _k } в F _K L ₂ ), but all sequences marked by K , must be monotone and summable with square. All results will generally be true for different sequences { A _k } and { M _k } , but for the sake of simplicity we will restrict ourselves to the case A _k = M _k •

Let A : U G F be a linear operator. By the formula

∞

A( = £A k ( k A^ k

k =1

we define a linear operator A : U _K L ₂ G F k L 2 , and if the series in the right-hand side of (12) converges (in the F _k L ₂ metric), then ( G dom A , and if diverges, then ( / dom A .Traditionally the spaces of linear continuous operators L ( U _K L ₂ ; F _k L ₂ ) and linear closed densely defined operators are traditionally defined. The following holds

Lemma 1. (i) Operator A G L (U; F) is exactly and only if A G L ( U _K L ₂ ; F _k L ₂ ) .

Since it is clear to see,

∞∞

IK ||f <  E ^A k ^{D (} k IKkIlF <  const E ^A k ^{D (} k = const IK ||u.

k =1                           k =1

• (ii)    Operator A E Cl (U; F) is exactly and only if A E Cl ( UKL 2 ; FKL 2 ) .

For reasons of simplicity, let U = {u E W22(Q) Ф W22(Г) : dRu = 0}, F = L2(Q) ФL2(r). Following the algorithm outlined above, we then construct the spaces of random K-values. A random K-value £ E UKL2 has the following form e = £ Ak' ^,                            (13)

k =1

where { ^ _k } is the family of eigenfunctions of the modified Laplace operator А _г,_ө,^ E L (U; F) orthonormalised in the sense of the scalar product ( • , • ) from L ₂ (Q) . Let us consider the linear stochastic Wentzell system of the moisture filtration equation in the balloon and at its boundary. In this case (1), (2) is transformed to the form

(A - Ar,e,v)nt = аоАг,ө,^п - eoАг,ө,^п - үоn,n E C”(R+; UKL2),

• (A - Аө,^)nt = а1АӨ,^n - в1 АГ,өЕІ + dRn - ү1п, n E C№(R+i UKL2),

dRn = 0,n E C“(R+; UkL2), where

Ar,„ = (r - R) A ((R - r)d-^ + A + d, ∂r         ∂r    ∂θ2

A _ d2 , d2A дӨ2 + dp2 ’ dR   dr r=R

A _ d2    d2

= д Ө ^{2 +} d^ 2 ’ ^d R = dr _{r = R}

For this system we add a matching condition and equip it with initial conditions

П(0) = По

The solution of the problem (14) – (18) we call the stochastic solution of the Wentzell system.

Theorem 2. For any n 0 E UKL 2 (Q) and for coefficients a ₀ , a 1 , A, в о , в 1 , Y 0 , Y 1 E R , such that the following condition a ₀ = a 1 , в 0 = в 1 , Y 0 = Y 1 , аnd A = k ² is satisfied, where k E N , there exists a single solution n E C “ (R ₊ ; UKL 2 ) of the stochastic Wentzell system (14) – (18) .

Proof. The existence and singularity of the solution are proved by analogy with the deterministic case due to the validity of Lemma 1.

□

Conclusion

We constructed the resolution group in the Cauchy–Wentzell system in the hemisphere and its biundary. Further, we plan to continue the results of the paper by applying the Wentzell conditions in directions related to [18–20].

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