ANALYSIS OF THE STOCHASTIC WENTZELL SYSTEM OF FLUID FILTRATION EQUATIONS IN A CIRCLE AND ON ITS BOUNDARY

. Wentzell boundary condition problems for linear elliptic equations of second order have been studied by various methods. Over time, the condition has come to be understood as a description of a process occurring on the boundary of a domain and affected by processes inside the domain. Since Wentzell boundary conditions in the mathematical literature have been considered from two points of view (in the classical and neoclassical cases), the aim of this paper is to analyse the stochastic Wentzell system of filtration equations in a circle and on its boundary in the space of differentiable K-“noise”. In particular, we prove the existence and uniqueness of the solution that determines quantitative predictions of changes in the geochemical regime of groundwater in the case of non-pressure filtration at the boundary of two media (in the region and on its boundary).


Introduction
Liquid filtration as well as its flow, diffusion, falling, etc. is one of the moisture transfer processes.The study of these processes begins with the study of their mathematical models.Let us consider one of the mathematical models of filtration.Let n  , 2 n  be a connected bounded region with boundary  of class C  .The system of Barenblatt-Zheltov-Kochina equations [1], modelling the process of fluid filtration is defined on the compact          , , , , ,       , , , , , tr .u v on    (3) Here the symbol Δ in (1) denotes the Laplace operator in the region Ω , and in (2) the same symbol denotes the Laplace -Beltrami operator on the smooth Riemannian manifold Γ .The symbol     , , , t x t x      , denotes the normal to  external to .The parameters , , , ,       characterise the medium.
Previously [2], following the tradition of [3][4][5][6][7], we called the condition of the form (2), in which the order of derivatives on spatial variables is not lower than the order on the same ones in (1), the Wentzell boundary condition.However, intending in the future to consider different cases of  and  (for instance,  is a bounded connected Riemannian manifold with edge Γ) we consider it necessary to call (1), (2) a system of equations, albeit defined on sets of different geometric dimension.This is supported by the fact that equations (1), (2) describe the same physical process of fluid filtration.The term "boundary conditions" should be reserved for the equations defined on the boundary (edge) of a region (manifold) and having a lower order of derivatives on spatial variables (see the classical treatise [8]).The name of the system of Wentzell equations emphasises the merits of the discoverer [9] of a new section of mathematical physics.
We will study the solvability of the system (1), (2) in the simplest case: , , , , , , , To the system (4), (5) we add the matching condition (3) and provide it with initial conditions Resorting to the stochastic interpretation of partial derivative equations, in this paper we are going to touch upon the studies of nondeterministic problems in the interpretation necessary for us, the distinguishing feature of which is a different notion of "white noise" in the sense of the Nelson-Glicklich derivative of the Wiener process.The term Nelson-Glicklich derivative was originally introduced in the monograph [10], where the first derivative of a random process was found.This paradigm not only justified the consistency with the Einstein-Smoluchowski theory, which allows us to understand by Brownian motion the sought stochastic process, and by the derivative from this process -"white noise", but also prompted the emergence of a new direction of study of stochastic equations of the Sobolev type.This is reflected in the studies of: dichotomies of a stochastic equation defined on a manifold [11]; application of the phase space method in the case of (L, p)-bounded operator M in [12]; stochastic equations of Sobolev type of high order in [12].
The paper consists of two parts.In the first part the existence and uniqueness of the system of Wentzell equations in a circle and on its boundary are considered.The second part contains abstract reasoning consisting in the construction of space and proof of existence and uniqueness of the stochastic system of Wentzell equations in a circle and on its boundary.

Deterministic case
Since it is not difficult to notice, is the formal solution of equation ( 4).If the series in (7) converge uniformly, then we have a solution of the problem (4), ( 6), with 0. R u  Taking this into account, we can construct the solution of problem (5), ( 6) and in the case α = γ,β = δ the solutions of the problem ( 4)-( 6) will satisfy the matching condition (3).Further, the closure of the linear by the norm generated by the scalar product of we denote by the symbol  .
A  Closure of the linear  by the norm generated by the scalar product of we denote by the symbol  .  3)- (5).
The existence of the solution is proved by formulas ( 7)-( 8), the proof of the singularity of the solution is trivial.

 
,, AP    be a complete probability space with probability measure P associated to the σalgebra A of subsets of the set  , and let R be the set of real numbers endowed with a Borel σalgebra.A measurable mapping : R   is called a random variable.The set of random variables whose expectation E is zero and variance D is finite forms a Hilbert space 2 { : with scalar product   lN  , will be called "noise" spaces for simplicity.(see, example, [3][4][5][6][7]).
Let us proceed to the construction of the space of random K -values.We consider H a real separable Hilbert space with orthonormalised basis {} k  , a monotone sequence Completion of the linear envelope of the set HL is Hilbertian, and the above constructed random Similarly, the Banach space of ( lN  let us define it as an enlargement of the linear envelope of the set {}


where the sequence of "noises" As we can clearly see, the vector It should be noted that in different spaces ( FL), but all sequences marked by K must be monotone and summable with square.All results will, in general, be true for different sequences {}  , but for simplicity's sake we will restrict ourselves to the case kk   .
Let : A A F  be a linear operator.By the formula we define a linear operator 22 : , and if the series in the right-hand side of ( 9) converges (in the metric the prescriptions above, we construct the spaces of random K -values In which {}  10)- (12).The existence and singleness of the solution are proved by analogy with the deterministic case by virtue of Lemmas 3.1 and 3.2.
The research was funded by the Russian Science Foundation (project No. 23-21-10056).

2 : 2 ;
gL   , which maps a point    to each pair   ,ω = g f t ,ω , we will call a (one-dimensional) stochastic process.Considering T  an interval, we call a stochastic process  η = η t , tT,  continuous if a.b. (almost probably) all its trajectories are continuous (i.e., if a.a.(almost all) ωA  trajectories   η ,ω  are continuous functions).The set of continuous stochastic processes forms a Banach space, which we denote by the symbol   the properties of the Nelson-Glicklich derivative η of a stochastic process η at the point tT  (for a detailed description, see for example in monograph [9]).If the Nelson-Glicklich derivatives   η t, of a stochastic process   η t, exist at all (or almost all) points of an interval T , then we refer to the existence of a Nelson-Glicklich derivative   η t, on T (almost probably on T ).The set of stochastic processes, whose trajectories are Nelson-Glicklich differentiable on I up to order {0} lN  inclusively form a Banach space Here we will consider the zero-order Nelson-Glicklich derivative as the initial random process, i.e.   0 η η  .Let us also note that the spaces

values and is denoted by the symbol 2 K
HL.As we can clearly observe, the space 2 K and all their Nelson-Glicklich derivatives up to and including order {0} lN  are uniformly bounded by the norm 2 denote the Hilbert space which is a replenishment of the linear envelope of random K

k 2 L
is the family of eigenfunctions of the Laplace operator , :  .Consider the linear stochastic Wentzel system of the fluid filtration equation in a circle and at its boundary.In this case (1), (2) is transformed to the form solution of the problem (11)-(12)  we will call a stochastic solution.