A. Favini 1 and G. A. Sviridyuk 2 and N. A. Manakova 2
Academic Editor:Juan C. Cortes
1, Department of Mathematics, University of Bologna, 5 Piazza di Porta San Donato, 40126 Bologna, Italy
2, Department of Mathematics, Mechanics and Computer Sciences, South Ural State University, 76 Lenin Avenue, Chelyabinsk 454080, Russia
Received 4 April 2015; Revised 1 September 2015; Accepted 6 September 2015; 22 November 2015
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be Banach spaces, the operator [figure omitted; refer to PDF] (linear and continuous), and the operator [figure omitted; refer to PDF] (linear, closed, and densely defined). Consider the equation [figure omitted; refer to PDF]
Equations of the form (1) were firstly studied in the works of A. Poincare. Then they appeared in the works of S. V. Oseen, J. V. Boussinesq, S. G. Rossby, and other researchers that were dedicated to the investigation of some hydrodynamics problems. Their systematical study started in the middle of the XX century with the works of S. L. Sobolev. The first monograph [1] devoted to the study of equations of the form (1) appeared in 1999. Nowadays the number of works devoted to such equations is increasing extensively [1-3]. Sometimes such equations are called "equations that are not of Cauchy-Kovalevskaya type," "pseudoparabolic equations," "degenerate equations," or "equations unsolved with respect to the higher derivative." We call equations of the form (1) the Sobolev type equations. This term was firstly proposed in the works of Carroll and Showalter [4]. The Sobolev type equations constitute the vast area in nonclassical equations of mathematical physics [5]. The theory of degenerate semigroups of operators is a suitable mathematical tool for the study of such problems [2].
The right part of (1) can be subjected to random perturbations, such as white noise. Abstract stochastic equations are of great interest nowadays due to the large amount of applications. Linear stochastic differential equation in the simplest case can be represented in the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are some linear operators; [figure omitted; refer to PDF] is a deterministic external influence and [figure omitted; refer to PDF] is a stochastic external influence; [figure omitted; refer to PDF] is unknown random process. Firstly [figure omitted; refer to PDF] was understood in the sense of differential of the Wiener process [figure omitted; refer to PDF] and was traditionally treated as white noise. K. Ito was the first to study ordinary differential equations of the form (2); then R. L. Stratonovich and A. V. Skorokhod developed research. The Ito-Stratonovich-Skorokhod approach in the finite-dimensional case remains popular to this day [6, 7]. Moreover, it was successfully distributed to infinite-dimensional situation [8, 9], and even it was applied to studies of the Sobolev type equations [10, 11]. Another approach was presented in [12], where (2) was considered in the Schwartz spaces and the distributional derivative of the Wiener process makes sense.
A new approach to studying (2), where the noise is defined by the Nelson-Gliklikh derivative of the Wiener process, appeared recently and is actively developing [13, 14]. At first white noise was used in the theory of optimal measurements [15], where a special noises space was constructed [16]. In [17] the concept of white noise was also extended to the infinite-dimensional space [figure omitted; refer to PDF] of K -random processes with a.s. continuous trajectories and the space [figure omitted; refer to PDF] of K -random processes, whose trajectories are a.s. continuously differentiable in the Nelson-Gliklikh sense up to order [figure omitted; refer to PDF] The solvability of Showalter-Sidorov problem for linear stochastic Sobolev type equations with relatively bounded operators was studied in [17]. Our purpose is to study the solvability of weakened (in sense of S. G. Krein) Showalter-Sidorov problem for linear stochastic Sobolev type equation with relatively sectorial operator. The purpose of such extention is the development of the theory of stochastic Sobolev type equations and application of this theory to nonclassical models of mathematical physics of practical value.
The paper is organized as follows. In the second section we introduce the definition of a strongly relatively [figure omitted; refer to PDF] -sectorial operator and construct semigroups of the resolving operators. In the third section the Nelson-Gliklikh derivative of K -random process with values in real separable Hilbert spaces is considered. In particular the K -Wiener process is studied. Then the space of such processes, containing the K -Wiener process and its Nelson-Gliklikh derivative (i.e., white noise), is constructed. In the fourth section the theory of stochastic Sobolev type equations with relatively [figure omitted; refer to PDF] -sectorial operators is developed; namely, the stochastic Sobolev type equation [figure omitted; refer to PDF] is considered. Here [figure omitted; refer to PDF] is the unknown random process, [figure omitted; refer to PDF] is its Nelson-Gliklikh derivative, [figure omitted; refer to PDF] is a random process, responsible for external influence; the operators [figure omitted; refer to PDF] ; moreover the operator [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -sectorial, [figure omitted; refer to PDF] . Add to (3) with a weakened Showalter-Sidorov condition [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Condition (4) is a natural generalization of condition [figure omitted; refer to PDF] which is in its turn the generalization of the Cauchy condition [figure omitted; refer to PDF] Note that condition (4) is more natural for the Leontieff type system and for the Sobolev type equations [5] than the traditional Cauchy condition (6). Problem (5) for the deterministic Sobolev type equation was firstly studied in [1]. This investigation formed the basis of the study of problem (5) for linear stochastic Sobolev type equation (3) [11]. The existence and the uniqueness of classical solution for problem (3), (4) are proved in the fourth section of our paper. In the fifth section we apply the abstract scheme to the investigation of the Dzektser model [18], describing free surface evolution of filtered liquid.
2. Holomorphic Degenerate Semigroups of Operators
Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be Banach spaces, and let the operators [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Consider the [figure omitted; refer to PDF] -resolvent set of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the L -spectrum [figure omitted; refer to PDF] of the operator [figure omitted; refer to PDF] , and the right and the left L -resolvents of the operator [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , respectively. Let [figure omitted; refer to PDF] The operator-functions [figure omitted; refer to PDF] are called the right and the left [figure omitted; refer to PDF] -resolvents of the operator [figure omitted; refer to PDF] .
Definition 1.
Operator [figure omitted; refer to PDF] is said to be p-sectorial , [figure omitted; refer to PDF] with respect to the operator [figure omitted; refer to PDF] (or shortly [figure omitted; refer to PDF] -sectorial ), if
(i) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that the sector [figure omitted; refer to PDF]
(ii) [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF]
Remark 2.
Without loss of generality we can put [figure omitted; refer to PDF] in Definition 1. Indeed, if we find a resolving semigroup of (1) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , then the semigroup [figure omitted; refer to PDF] will be resolving when [figure omitted; refer to PDF] .
Let [figure omitted; refer to PDF] . Consider two equivalent forms of the linear homogeneous Sobolev type equation (1) [figure omitted; refer to PDF] as concrete interpretations of the equation [figure omitted; refer to PDF] defined on a Banach space [figure omitted; refer to PDF] , where the operators [figure omitted; refer to PDF] . Operator [figure omitted; refer to PDF] is linear bounded on a dense set in [figure omitted; refer to PDF] and it can be uniquely continued to a bounded operator [figure omitted; refer to PDF] defined on [figure omitted; refer to PDF] . For (10) the space [figure omitted; refer to PDF] and for (11) [figure omitted; refer to PDF] .
Definition 3.
The vector function [figure omitted; refer to PDF] satisfying (12) on [figure omitted; refer to PDF] is called a solution of (12).
Definition 4.
The mapping [figure omitted; refer to PDF] is called a semigroup of the resolving operators (a resolving semigroup ) of (12), if
(i) [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and any [figure omitted; refer to PDF] ;
(ii) [figure omitted; refer to PDF] is a solution of (12) for any [figure omitted; refer to PDF] from a dense set in [figure omitted; refer to PDF] .
The semigroup is called uniformly bounded , if [figure omitted; refer to PDF] The semigroup is called analytic , if it can be extended to some sector containing the ray [figure omitted; refer to PDF] with fulfillment of properties (i), (ii) in Definition 4.
Theorem 5 (see [2, p. 60]).
Let the operator [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -sectorial, [figure omitted; refer to PDF] . Then there exists a uniformly bounded and analytic resolving semigroup of (10) and (11) and it is represented by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and contour [figure omitted; refer to PDF] is such that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
For example, contour [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is taken from Definition 1.
Let [figure omitted; refer to PDF] be the closure of [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) in the norm of the space [figure omitted; refer to PDF] . The set [figure omitted; refer to PDF] is called a kernel [2, p. 61] of the semigrop [figure omitted; refer to PDF] and the set [figure omitted; refer to PDF] is called an image [2, p. 61] of the semigrop [figure omitted; refer to PDF] .
Theorem 6 (see [2, p. 62]).
Let the operator [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -sectorial. Then [figure omitted; refer to PDF] , [figure omitted; refer to PDF]
Further we assume that the operator [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -sectorial. Set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] denote the restriction of the operator [figure omitted; refer to PDF] on [figure omitted; refer to PDF] .
Theorem 7 (see [2, pp. 63, 64]).
Let the operator [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -sectorial. Then
(i) the operator [figure omitted; refer to PDF] , and the operator [figure omitted; refer to PDF] ;
(ii) there exists the operator [figure omitted; refer to PDF] ;
(iii): the operator [figure omitted; refer to PDF] is nilpotent with degree less or equal to [figure omitted; refer to PDF] .
By [figure omitted; refer to PDF] denote the restriction of the operator [figure omitted; refer to PDF] on [figure omitted; refer to PDF]
Consider the following conditions: [figure omitted; refer to PDF] and there exists the operator [figure omitted; refer to PDF]
Remark 8.
Condition [figure omitted; refer to PDF] holds, for example, in the case where [figure omitted; refer to PDF] is strongly [figure omitted; refer to PDF] -sectorial on the right (left) or when the space [figure omitted; refer to PDF] is reflexive [2, page 69]. Condition [figure omitted; refer to PDF] holds in the case when the operator [figure omitted; refer to PDF] is strongly [figure omitted; refer to PDF] -sectorial or when it is [figure omitted; refer to PDF] -sectorial, condition [figure omitted; refer to PDF] is fulfilled and [figure omitted; refer to PDF] .
Condition [figure omitted; refer to PDF] is equivalent to the existence of the projector [figure omitted; refer to PDF] along [figure omitted; refer to PDF] on [figure omitted; refer to PDF] .
Theorem 9 (see [2, pp. 69, 71, 73]).
Let the operator [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -sectorial and let conditions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be fulfilled. Then
(i) the projector [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) can be represented as [figure omitted; refer to PDF]
(ii) the operator [figure omitted; refer to PDF] and the operator [figure omitted; refer to PDF] ;
(iii): the operator [figure omitted; refer to PDF] is sectorial.
The solution to (12) is called a solution to a Cauchy problem if it also satisfies the condition [figure omitted; refer to PDF]
Definition 10.
The set [figure omitted; refer to PDF] is called a phase space of (12), if
(i) any solution [figure omitted; refer to PDF] of (12) lies in [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;
(ii) for any [figure omitted; refer to PDF] there exists a unique solution of problem (12), (18).
Theorem 11 (see [2, p. 67]).
Let the operator [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -sectorial. Then phase space of (10) and (11) coincides with the image of semigroup [figure omitted; refer to PDF] .
3. The Spaces of "Noises"
Let [figure omitted; refer to PDF] be a complete probability space and let [figure omitted; refer to PDF] be the set of real numbers endowed with Boreal [figure omitted; refer to PDF] -algebra. The measurable mapping [figure omitted; refer to PDF] is called a random variable . The set of random variables with zero mean and finite variances forms a Hilbert space with the scalar product [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the mathematical expectation. This Hilbert space will be denoted by [figure omitted; refer to PDF] . The random variables [figure omitted; refer to PDF] , with normal (Gaussian) distribution, will be very important later on; they are called Gaussian random variables. Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] -subalgebra of [figure omitted; refer to PDF] -algebra A . Construct the space [figure omitted; refer to PDF] of random variables, measurable with respect to [figure omitted; refer to PDF] . Obviously, [figure omitted; refer to PDF] is a subset of [figure omitted; refer to PDF] ; denote by [figure omitted; refer to PDF] the orthoprojector. Let [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is called conditional expectation of the random variable [figure omitted; refer to PDF] and is denoted by [figure omitted; refer to PDF] . It is easy to see that [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] ; and [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] . Finally, the minimal [figure omitted; refer to PDF] -subalgebra [figure omitted; refer to PDF] , regarding which random variable [figure omitted; refer to PDF] is measurable, is called the [figure omitted; refer to PDF] -algebra generated by [figure omitted; refer to PDF]
Let [figure omitted; refer to PDF] be some interval. Consider two mappings: the first one [figure omitted; refer to PDF] , which maps each [figure omitted; refer to PDF] to a random variable [figure omitted; refer to PDF] , and the second one [figure omitted; refer to PDF] , which maps every pair [figure omitted; refer to PDF] to the point [figure omitted; refer to PDF] . The composition [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , is called a (one-dimensional) random process . Thus, for every fixed [figure omitted; refer to PDF] the random process [figure omitted; refer to PDF] is a random variable; that is, [figure omitted; refer to PDF] , and for every fixed [figure omitted; refer to PDF] the random process [figure omitted; refer to PDF] is called the (sample) trajectory . The random process [figure omitted; refer to PDF] is called continuous if almost surely (a.s.) all its trajectories are continuous; that is, for almost every (a.e.) [figure omitted; refer to PDF] the trajectories [figure omitted; refer to PDF] are continuous. The set of continuous random processes form a Banach space, which will be denoted by [figure omitted; refer to PDF] . The continuous random process, whose random variables are Gaussian, is called Gaussian.
The (one-dimensional) Wiener process [figure omitted; refer to PDF] , modeling Brownian motion on the line in Einstein-Smolukhovsky theory, is one of the most important examples of the continuous Gaussian random processes. It has the following properties:
(W1) a.s. [figure omitted; refer to PDF] ; a.s. all its trajectories [figure omitted; refer to PDF] are continuous, and for all [figure omitted; refer to PDF] the random variable [figure omitted; refer to PDF] is Gaussian;
(W2) the mathematical expectation [figure omitted; refer to PDF] and autocorrelation function [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;
(W3) the trajectories [figure omitted; refer to PDF] are nondifferentiable at any point [figure omitted; refer to PDF] and have unbounded variation on any small interval.
Example 12.
There exists a random process [figure omitted; refer to PDF] , satisfying properties (W1), (W2); moreover, it can be represented in the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are independent Gaussian variables, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the dispersion.
The random process [figure omitted; refer to PDF] , satisfying properties (W1)-(W2), will be called Brownian motion.
Now fix [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and by [figure omitted; refer to PDF] denote the [figure omitted; refer to PDF] -algebra, generated by the random variable [figure omitted; refer to PDF] . For the sake of brevity, we introduce the notation [figure omitted; refer to PDF] .
Definition 13.
Let [figure omitted; refer to PDF] , and the random variable [figure omitted; refer to PDF] is called a forward [figure omitted; refer to PDF] (a backward [figure omitted; refer to PDF] ) mean derivative of the random process [figure omitted; refer to PDF] at the point [figure omitted; refer to PDF] if the limit exists in the sense of uniform metric on [figure omitted; refer to PDF] . The random process [figure omitted; refer to PDF] is called forward (backward) mean differentiable on [figure omitted; refer to PDF] , if for every point [figure omitted; refer to PDF] there exists the forward (backward) mean derivative.
Now let the random process [figure omitted; refer to PDF] be forward (backward) mean differentiable on [figure omitted; refer to PDF] . Its forward (backward) mean derivative is also a random process; we denote it by [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). If the random process [figure omitted; refer to PDF] is forward (backward) mean differentiable on [figure omitted; refer to PDF] , then the symmetric mean derivative [figure omitted; refer to PDF] can be defined. Since the mean derivatives were introduced by Nelson [19], and the theory of these derivatives was developed by Gliklikh [7], the symmetric mean derivative [figure omitted; refer to PDF] or the random process [figure omitted; refer to PDF] will henceforth be called the Nelson -Gliklikh derivative for brevity and will be denoted by [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denote the [figure omitted; refer to PDF] th Nelson-Gliklikh derivative of the random process [figure omitted; refer to PDF] . Note that if the trajectories of the random process [figure omitted; refer to PDF] are a.s. continuously differentiable in a "common sense" on [figure omitted; refer to PDF] , then the Nelson-Gliklikh derivative of [figure omitted; refer to PDF] coincides with the "regular" derivative.
Theorem 14 (see [14]).
Let [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .
Now let [figure omitted; refer to PDF] be a real separable Hilbert space; consider the operator [figure omitted; refer to PDF] with spectrum [figure omitted; refer to PDF] whose elements are nonnegative, discrete, with finite multiplicity tending only to zero. By [figure omitted; refer to PDF] denote the sequence of eigenvalues of operator [figure omitted; refer to PDF] , numbered in decreasing order according to their multiplicity. Note that the linear span of related orthonormal eigenfunctions [figure omitted; refer to PDF] of operator [figure omitted; refer to PDF] is dense in [figure omitted; refer to PDF] . Suppose that the operator [figure omitted; refer to PDF] is nuclear (i.e., its trace [figure omitted; refer to PDF] ).
Take the sequence of independent random processes [figure omitted; refer to PDF] and define the [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] provided that the series (22) converges uniformly on any compact subset of [figure omitted; refer to PDF] . Note that if [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] exists, then a.s. its trajectories are continuous. Denote the space of such processes by the symbol [figure omitted; refer to PDF] . Consider in [figure omitted; refer to PDF] the subspace [figure omitted; refer to PDF] of random processes, whose random variables belong to [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] . Note that the space [figure omitted; refer to PDF] contains, in particular, those [figure omitted; refer to PDF] -random processes for which almost surely all trajectories are continuous, and all (independent) random variables are Gaussian.
We now introduce the Nelson-Gliklikh derivatives of a [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] provided that the derivatives up to degree of [figure omitted; refer to PDF] in the right hand side of (23) exist and the series uniformly converges on any compact subset of [figure omitted; refer to PDF] .
Similarly, introduce the space [figure omitted; refer to PDF] of [figure omitted; refer to PDF] -random processes with a.s. continuous Nelson-Gliklikh derivatives up to order [figure omitted; refer to PDF] , whose random variables belong to [figure omitted; refer to PDF] .
As an example consider the [figure omitted; refer to PDF] -Wiener process [figure omitted; refer to PDF] which is defined on [figure omitted; refer to PDF] .
Corollary 15.
Let [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and nuclear operator [figure omitted; refer to PDF]
Moreover, the [figure omitted; refer to PDF] -Wiener process (24) satisfies conditions (W1) a.s. [figure omitted; refer to PDF] , a.s. all its trajectories [figure omitted; refer to PDF] are continuous, and for all [figure omitted; refer to PDF] the random variable [figure omitted; refer to PDF] is Gaussian; (W2) the mathematical expectation [figure omitted; refer to PDF] and autocorrelation function [figure omitted; refer to PDF]
4. The Stochastic Sobolev Type Equation with Relatively [figure omitted; refer to PDF] -Sectorial Operator
Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be real separable Hilbert spaces. Let the operator [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -sectorial, let [figure omitted; refer to PDF] and conditions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be fulfilled, and let the operator [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] . Let the operator [figure omitted; refer to PDF] be nuclear with eigenvalues [figure omitted; refer to PDF] . Consider the linear stochastic Sobolev type equation (3) with condition (4).
Remark 16.
Due to Theorem 6 condition (4) is equivalent to the following condition: [figure omitted; refer to PDF]
Definition 17.
The [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] is called a (classical ) solution of (3), if a.s. all its trajectories satisfy (3) with some [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] The solution of (3) is called a solution of weakened Showalter -Sidorov problem (3), (4), if it also satisfies condition (4).
Suppose that the [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] , satisfies condition [figure omitted; refer to PDF]
Theorem 18.
Let the operator [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -sectorial, let [figure omitted; refer to PDF] and conditions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be fulfilled, and let operator [figure omitted; refer to PDF] . For any [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] satisfying (27) and for any [figure omitted; refer to PDF] -valued random variable [figure omitted; refer to PDF] , independent of w, there exists a unique solution [figure omitted; refer to PDF] to problem (3), (4), given by [figure omitted; refer to PDF]
Proof.
Proof of the theorem is analogous to the deterministic case [2]. Acting on (3) and condition (4) by projectors [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and using Theorems 7 and 9, reduce it to the equivalent system of two independent problems [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Since the operator [figure omitted; refer to PDF] is nilpotent, it follows from (29) that necessarily [figure omitted; refer to PDF] Since the operator [figure omitted; refer to PDF] the solution of problem (30) exists and can be represented in the form [figure omitted; refer to PDF]
Consider the weakened Showalter -Sidorov problem (4) for equation [figure omitted; refer to PDF] here the right hand side includes the Nelson-Gliklikh derivative of the K -Wiener process [figure omitted; refer to PDF] The white noise [figure omitted; refer to PDF] does not satisfy condition (27). One of feasible approaches to overcome this difficulty was proposed in [10, 11]. The advantage of this approach comes from transformation of the second summand in the right hand side of (28) as follows: [figure omitted; refer to PDF] By virtue of the definition of Nelson-Gliklikh derivative for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we can make integration by parts. Letting [figure omitted; refer to PDF] in (34) we get [figure omitted; refer to PDF]
Theorem 19.
Let the operator M be [figure omitted; refer to PDF] -sectorial and let [figure omitted; refer to PDF] and conditions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be fulfilled. For any [figure omitted; refer to PDF] and for any [figure omitted; refer to PDF] -valued random variable [figure omitted; refer to PDF] , independent of [figure omitted; refer to PDF] , there exists a unique solution [figure omitted; refer to PDF] of problem (4), (33), given by [figure omitted; refer to PDF]
5. Dzektser Stochastic Model
Let [figure omitted; refer to PDF] be a bounded domain with a boundary [figure omitted; refer to PDF] of class [figure omitted; refer to PDF] . Consider a boundary value [figure omitted; refer to PDF] and initial value [figure omitted; refer to PDF] problems for the stochastic equation [figure omitted; refer to PDF] Here the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . This model describes evolution of free surface of filtered liquid.
Define the space [figure omitted; refer to PDF] and the space [figure omitted; refer to PDF] with the scalar product [figure omitted; refer to PDF] Denote by [figure omitted; refer to PDF] the sequence of eigenvalues of the homogeneous Direchlet problem for the operator [figure omitted; refer to PDF] , numbered in nonincreasing order with regard to multiplicities and tending to [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] denote the orthonormal (in the sense of [figure omitted; refer to PDF] ) family of corresponding eigenfunctions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Introduce the F -valued [figure omitted; refer to PDF] -random process. Define the operator [figure omitted; refer to PDF] with the domain [figure omitted; refer to PDF] It is rather easy to find such a number [figure omitted; refer to PDF] according to fixed number [figure omitted; refer to PDF] (which is the dimension of the domain [figure omitted; refer to PDF] ) that the mentioned series converges. For example, [figure omitted; refer to PDF] can be equal to [figure omitted; refer to PDF] . Note that the operator [figure omitted; refer to PDF] has the same eigenfunctions [figure omitted; refer to PDF] , as the Laplace operator, but its spectrum consists of eigenvalues [figure omitted; refer to PDF] . Since their asymptotic [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we take such number [figure omitted; refer to PDF] that the series [figure omitted; refer to PDF] converges. Then the operator [figure omitted; refer to PDF] is continuously invertable on [figure omitted; refer to PDF] , whereas the inverse operator (i.e., the Green operator) has the spectrum consisting of eigenvalues [figure omitted; refer to PDF] . We take this operator as the nuclear operator [figure omitted; refer to PDF] for F -valued [figure omitted; refer to PDF] -random process.
Fix [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and define the operators [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The operator [figure omitted; refer to PDF] , and the operator [figure omitted; refer to PDF] with [figure omitted; refer to PDF]
Lemma 20 (see [2, p. 198]).
For any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] the operator [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -sectorial.
The [figure omitted; refer to PDF] -spectrum of the operator [figure omitted; refer to PDF] consists of all points of the form [figure omitted; refer to PDF] By Theorem 5 there exists a holomorphic resolving semigroup for (39) in the form [figure omitted; refer to PDF]
Lemma 21.
For any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] conditions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are fulfilled.
Construct subsets [figure omitted; refer to PDF] Obviously [figure omitted; refer to PDF] . Thus the projector [figure omitted; refer to PDF] has the form [figure omitted; refer to PDF] The projector [figure omitted; refer to PDF] is constructed analogously.
Moreover there exists the operator [figure omitted; refer to PDF] Conditions (3) and (38) take the form [figure omitted; refer to PDF] Thus, we have reduced problem (37)-(39) to problem (3), (4). From Theorem 18 we have the following assertion.
Theorem 22.
For any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and for any [figure omitted; refer to PDF] -random process [figure omitted; refer to PDF] satisfying (27) and for any [figure omitted; refer to PDF] -valued random variable [figure omitted; refer to PDF] , independent of [figure omitted; refer to PDF] , there exists a unique solution [figure omitted; refer to PDF] for problem (37)-(39), given by [figure omitted; refer to PDF] Here [figure omitted; refer to PDF]
Consider the initial-boundary value problem (37), (38) for equation [figure omitted; refer to PDF] where the right part includes the Nelson-Gliklikh derivative of the K -Wiener process [figure omitted; refer to PDF] From Theorem 19 we have the following assertion.
Theorem 23.
For any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] -valued random variable [figure omitted; refer to PDF] , independent of [figure omitted; refer to PDF] there exists a unique solution [figure omitted; refer to PDF] of problem (37), (38), (51) given by [figure omitted; refer to PDF] Here [figure omitted; refer to PDF]
Conflict of Interests
The autors declare that they have no conflict of interests regarding the publication of this paper.
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Abstract
The concept of "white noise," initially established in finite-dimensional spaces, is transferred to infinite-dimensional case. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical interest. To reach this goal the Nelson-Gliklikh derivative is introduced and the spaces of "noises" are developed. The Sobolev type equations with relatively sectorial operators are considered in the spaces of differentiable "noises." The existence and uniqueness of classical solutions are proved. The stochastic Dzektser equation in a bounded domain with homogeneous boundary condition and the weakened Showalter-Sidorov initial condition is considered as an application.
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