Linear Sobolev Type Equations with Relatively <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.38907pt" id="M1" height="13.4115pt" version="1.1" viewBox="-0.0657574 -9.02243 10.4144 13.4115" width="10.4144pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g></svg>-Sectorial Operators in Space of “Noises”

Favini, A.; Sviridyuk, G. A.; Manakova, N. A.

doi:https://doi.org/10.1155/2015/697410

Abstract and Applied Analysis

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Volume 2015 | Article ID 697410 | https://doi.org/10.1155/2015/697410

Linear Sobolev Type Equations with Relatively -Sectorial Operators in Space of “Noises”

A. Favini,¹G. A. Sviridyuk,²and N. A. Manakova²

Academic Editor: Juan C. Cortés

Received04 Apr 2015

Revised01 Sept 2015

Accepted06 Sept 2015

Published23 Nov 2015

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.

1. Introduction

Let and be Banach spaces, the operator (linear and continuous), and the operator (linear, closed, and densely defined). Consider the equation

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 formwhere and are some linear operators; is a deterministic external influence and is a stochastic external influence; is unknown random process. Firstly was understood in the sense of differential of the Wiener process 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 of K-random processes with a.s. continuous trajectories and the space of K-random processes, whose trajectories are a.s. continuously differentiable in the Nelson-Gliklikh sense up to order 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 -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 -sectorial operators is developed; namely, the stochastic Sobolev type equationis considered. Here is the unknown random process, is its Nelson-Gliklikh derivative, is a random process, responsible for external influence; the operators ; moreover the operator is -sectorial, . Add to (3) with a weakened Showalter-Sidorov conditionwhere , . Condition (4) is a natural generalization of conditionwhich is in its turn the generalization of the Cauchy conditionNote 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 and be Banach spaces, and let the operators , . Consider the -resolvent set of , , the L-spectrum of the operator , and the right and the left L-resolvents of the operator , , , respectively. Let The operator-functionsare called the right and the left -resolvents of the operator .

Definition 1. Operator is said to be p-sectorial, with respect to the operator (or shortly -sectorial), if(i), such that the sector(ii) such thatfor all ,

Remark 2. Without loss of generality we can put in Definition 1. Indeed, if we find a resolving semigroup of (1) for , then the semigroup will be resolving when .

Let . Consider two equivalent forms of the linear homogeneous Sobolev type equation (1)as concrete interpretations of the equationdefined on a Banach space , where the operators . Operator is linear bounded on a dense set in and it can be uniquely continued to a bounded operator defined on . For (10) the space and for (11) .

Definition 3. The vector function satisfying (12) on is called a solution of (12).

Definition 4. The mapping is called a semigroup of the resolving operators (a resolving semigroup) of (12), if(i) for all and any ;(ii) is a solution of (12) for any from a dense set in .

The semigroup is called uniformly bounded, ifThe semigroup is called analytic, if it can be extended to some sector containing the ray with fulfillment of properties (i), (ii) in Definition 4.

Theorem 5 (see [2, p. 60]). Let the operator be -sectorial, . Then there exists a uniformly bounded and analytic resolving semigroup of (10) and (11) and it is represented bywhere and contour is such that for , .

For example, contour , where is taken from Definition 1.

Let be the closure of () in the norm of the space . The setis called a kernel [2, p. 61] of the semigrop and the setis called an image [2, p. 61] of the semigrop .

Theorem 6 (see [2, p. 62]). Let the operator be -sectorial. Then ,

Further we assume that the operator is -sectorial. Set , . By denote the restriction of the operator on .

Theorem 7 (see [2, pp. 63, 64]). Let the operator be -sectorial. Then(i)the operator , and the operator ;(ii)there exists the operator ;(iii)the operator is nilpotent with degree less or equal to .

By denote the restriction of the operator on

Consider the following conditions:and there exists the operator

Remark 8. Condition holds, for example, in the case where is strongly -sectorial on the right (left) or when the space is reflexive [2, page 69]. Condition holds in the case when the operator is strongly -sectorial or when it is -sectorial, condition is fulfilled and .

Condition is equivalent to the existence of the projector along on .

Theorem 9 (see [2, pp. 69, 71, 73]). Let the operator be -sectorial and let conditions , be fulfilled. Then(i)the projector () can be represented as(ii)the operator and the operator ;(iii)the operator is sectorial.

The solution to (12) is called a solution to a Cauchy problem if it also satisfies the condition

Definition 10. The set is called a phase space of (12), if(i)any solution of (12) lies in ; that is, for all ;(ii)for any there exists a unique solution of problem (12), (18).

Theorem 11 (see [2, p. 67]). Let the operator be -sectorial. Then phase space of (10) and (11) coincides with the image of semigroup .

3. The Spaces of “Noises”

Let be a complete probability space and let be the set of real numbers endowed with Boreal -algebra. The measurable mapping is called a random variable. The set of random variables with zero mean and finite variances forms a Hilbert space with the scalar product , where denotes the mathematical expectation. This Hilbert space will be denoted by . The random variables , with normal (Gaussian) distribution, will be very important later on; they are called Gaussian random variables. Let be a -subalgebra of -algebra A. Construct the space of random variables, measurable with respect to . Obviously, is a subset of ; denote by the orthoprojector. Let , then is called conditional expectation of the random variable and is denoted by . It is easy to see that , if ; and , if . Finally, the minimal -subalgebra , regarding which random variable is measurable, is called the -algebra generated by

Let be some interval. Consider two mappings: the first one , which maps each to a random variable , and the second one , which maps every pair to the point . The composition , , is called a (one-dimensional) random process. Thus, for every fixed the random process is a random variable; that is, , and for every fixed the random process is called the (sample) trajectory. The random process is called continuous if almost surely (a.s.) all its trajectories are continuous; that is, for almost every (a.e.) the trajectories are continuous. The set of continuous random processes form a Banach space, which will be denoted by . The continuous random process, whose random variables are Gaussian, is called Gaussian.

The (one-dimensional) Wiener process , 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. ; a.s. all its trajectories are continuous, and for all the random variable is Gaussian;(W2)the mathematical expectation and autocorrelation function for all ;(W3)the trajectories are nondifferentiable at any point and have unbounded variation on any small interval.

Example 12. There exists a random process , satisfying properties (W1), (W2); moreover, it can be represented in the formwhere are independent Gaussian variables, , and , where denotes the dispersion.

The random process , satisfying properties (W1)-(W2), will be called Brownian motion.

Now fix and and by denote the -algebra, generated by the random variable . For the sake of brevity, we introduce the notation .

Definition 13. Let , and the random variableis called a forward (a backward ) mean derivative of the random process at the point if the limit exists in the sense of uniform metric on . The random process is called forward (backward) mean differentiable on , if for every point there exists the forward (backward) mean derivative.

Now let the random process be forward (backward) mean differentiable on . Its forward (backward) mean derivative is also a random process; we denote it by (). If the random process is forward (backward) mean differentiable on , then the symmetric mean derivativecan 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 or the random process will henceforth be called the Nelson-Gliklikh derivative for brevity and will be denoted by ; that is, . By , denote the th Nelson-Gliklikh derivative of the random process . Note that if the trajectories of the random process are a.s. continuously differentiable in a “common sense” on , then the Nelson-Gliklikh derivative of coincides with the “regular” derivative.

Theorem 14 (see [14]). Let for all and .

Now let be a real separable Hilbert space; consider the operator with spectrum whose elements are nonnegative, discrete, with finite multiplicity tending only to zero. By denote the sequence of eigenvalues of operator , numbered in decreasing order according to their multiplicity. Note that the linear span of related orthonormal eigenfunctions of operator is dense in . Suppose that the operator is nuclear (i.e., its trace ).

Take the sequence of independent random processes and define the -random processprovided that the series (22) converges uniformly on any compact subset of . Note that if and the -random process exists, then a.s. its trajectories are continuous. Denote the space of such processes by the symbol . Consider in the subspace of random processes, whose random variables belong to ; that is, , if for each . Note that the space contains, in particular, those -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 -random processprovided that the derivatives up to degree of in the right hand side of (23) exist and the series uniformly converges on any compact subset of .

Similarly, introduce the space of -random processes with a.s. continuous Nelson-Gliklikh derivatives up to order , whose random variables belong to .

As an example consider the -Wiener processwhich is defined on .

Corollary 15. Let for all , , and nuclear operator

Moreover, the -Wiener process (24) satisfies conditions (W1) a.s. , a.s. all its trajectories are continuous, and for all the random variable is Gaussian; (W2) the mathematical expectation and autocorrelation function

4. The Stochastic Sobolev Type Equation with Relatively -Sectorial Operator

Let , , be real separable Hilbert spaces. Let the operator be -sectorial, let and conditions , be fulfilled, and let the operator Let . Let the operator be nuclear with eigenvalues . 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:

Definition 17. The -random process is called a (classical) solution of (3), if a.s. all its trajectories satisfy (3) with some -random process for all The solution of (3) is called a solution of weakened Showalter-Sidorov problem (3), (4), if it also satisfies condition (4).

Suppose that the -random process , satisfies condition

Theorem 18. Let the operator be -sectorial, let and conditions , be fulfilled, and let operator . For any -random process satisfying (27) and for any -valued random variable , independent of w, there exists a unique solution to problem (3), (4), given by

Proof. Proof of the theorem is analogous to the deterministic case [2]. Acting on (3) and condition (4) by projectors and and using Theorems 7 and 9, reduce it to the equivalent system of two independent problemswhere , , . Since the operator is nilpotent, it follows from (29) that necessarilySince the operator the solution of problem (30) exists and can be represented in the form

Consider the weakened Showalter-Sidorov problem (4) for equationhere the right hand side includes the Nelson-Gliklikh derivative of the K-Wiener process The white noise 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:By virtue of the definition of Nelson-Gliklikh derivative for all , , we can make integration by parts. Letting in (34) we get

Theorem 19. Let the operator M be -sectorial and let and conditions , be fulfilled. For any and for any -valued random variable , independent of , there exists a unique solution of problem (4), (33), given by

5. Dzektser Stochastic Model

Let be a bounded domain with a boundary of class . Consider a boundary value and initial valueproblems for the stochastic equationHere the parameters , . This model describes evolution of free surface of filtered liquid.

Define the space and the space with the scalar productDenote by the sequence of eigenvalues of the homogeneous Direchlet problem for the operator , numbered in nonincreasing order with regard to multiplicities and tending to . By denote the orthonormal (in the sense of ) family of corresponding eigenfunctions , . Introduce the F-valued -random process. Define the operator with the domainIt is rather easy to find such a number according to fixed number (which is the dimension of the domain ) that the mentioned series converges. For example, can be equal to . Note that the operator has the same eigenfunctions , as the Laplace operator, but its spectrum consists of eigenvalues . Since their asymptotic , , we take such number that the series converges. Then the operator is continuously invertable on , whereas the inverse operator (i.e., the Green operator) has the spectrum consisting of eigenvalues . We take this operator as the nuclear operator for F-valued -random process.

Fix , , and define the operators ,. The operator , and the operator with

Lemma 20 (see [2, p. 198]). For any , the operator is -sectorial.

The -spectrum of the operator consists of all points of the formBy Theorem 5 there exists a holomorphic resolving semigroup for (39) in the form

Lemma 21. For any , conditions , are fulfilled.

Construct subsetsObviously . Thus the projector has the formThe projector is constructed analogously.

Moreover there exists the operatorConditions (3) and (38) take the formThus, we have reduced problem (37)–(39) to problem (3), (4). From Theorem 18 we have the following assertion.

Theorem 22. For any , and for any -random process satisfying (27) and for any -valued random variable , independent of , there exists a unique solution for problem (37)–(39), given byHere

Consider the initial-boundary value problem (37), (38) for equationwhere the right part includes the Nelson-Gliklikh derivative of the K-Wiener process From Theorem 19 we have the following assertion.

Theorem 23. For any , , and -valued random variable , independent of there exists a unique solution of problem (37), (38), (51) given byHere

Conflict of Interests

The autors declare that they have no conflict of interests regarding the publication of this paper.

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Copyright © 2015 A. Favini et al. 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.

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