Multipoint initial-final problem for one class of Sobolev type models of higher order with additive "white noise"

Sobolev type equations make up a vast area of nonclassical equations of mathematical physics. Their systematic study started in the middle of the twentieth century with the seminal work of S.L. Sobolev, though many such equations had been studied earlier on; we recall, in particular, the famous Navier-Stokes equation system (see the excellent review in [1]). Recently, there has been a major increase in the research of Sobolev type ecpiations. We should mention several monographs about these problems [2-7]. Different aspects of the incomplete higher-order Sobolev type equations

Au ^{( n} ) = Bu + g,

with the assumption kerA = { 0 } , have been studied [8-11]. Here the operators A, B G L (U; F) (i.e. linear am.1 continuous). U and F are Banach spaces. airsolute term g = g ( t ) models the external force, and n >  2 is a natural number. One of the prototypes of equation

(1) is

( A - A) vtt = a A v + g,

which models, among others, the incompressible fluid free surface perturbation under the assumption of motion potentiality and conservation of mass in a layer [12], longitudinal vibrations of an elastic rod [13] and wave processes in smectic and plasma [14].

The shortcoming of the model (2) with the deterministic absolute term is that in natural experiments the system is exposed to random perturbation, for example in the form

G.A. Sviridyuk, A.A. Zamyshlyaeva, S.A. Zagrebina of white noise. Stochastic ordinary differential equations with different additive random processes (i.e. not only white noise, but more general Markov and diffusion processes) are now actively studied [15]. The traditional Ito-Stratonovich-Skorohod approach is the most widely followed, although new and very promising avenues of research have recently appeared [11,16].

The first results concerning stochastic Sobolev type equations of the first order can be found in [17]. They are based on the extension of the Ito-Stratonovich-Skorokhod method to partial differential equations (see, for example, [18-20]). In this paper, the stochastic higher order Sobolev type equation

An ^{( n} ) = Bn + Nw                            (3)

is considered. Here, w in the right hand side denotes the random process. It is required to find the random process n ( t ) satisfying (in some sense) equation (3) and the multipoint initial-final conditions

Pj ⁽ П ⁽ k ⁾⁽ Tj ) - (k ) = 0 , j VLm.k = 0 ,n - 1 ,                       (4)

where Tj E R with Tj < Tj ₊₁, j = 0 , m, j k = 0 , n — 1, are given random variables, and Pj are the relatively spectral projectors.

At first, w was understood as white noise, which is a generalized derivative of the Wiener process. Later, a new approach to the investigation of equation (3) appeared [15] and is being actively developed [16, 21-23], where "white noise" means the Nelson-Gliklikh [15, 24] derivative of the Wiener process. This "white noise" was first used in optimal measurement theory [25,26], which constructs a special space of "noises". The concept of "white noise" in this theory (that is, only in the finite dimensional spaces) proved to be highly efficient, therefore suggesting to extend the concept to infinite-dimensional spaces [17,27]. The main goal of this extension is to develop a theory of stochastic Sobolev type equations and its applications to nonclassical models of mathematical physics of practical importance [28].

Besides the introduction, the paper consists of three sections. The first one deals with the deterministic inhomogeneous linear Sobolev type equation of higher order. We define a multipoint initial-final problem and state a theorem on the existence of a unique solution. We borrowed results from [10,29] and therefore give them without proofs. The second section extends the deterministic results of the first one to the stochastic setup by analogy with [23]; sketches of proofs complement the results. In the third section we consider the linear stochastic Boussinesq-Love equation. In conclusion, we outline possible directions for further research.

1.    A Deterministic Linear Sobolev Type Equation of Higher Order with Relatively p-Bounded Operators

Let U and F Ire separable Hilbei3 spaces, operators A, B E L(U; F). Following [18]. Introduce an A-resolvent set pA (B) = {ц E C : (цA — B) -1 E L (F; U)} and an A-spectrum. aA(B) = C \ pA(B) of operator B. The operator-functlons (цA — B)-1, RA (B) = (цА — B)-1 A, LA (B) = A(цА — B)-1 with the domain pA(B) are called

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ the A-resolvent, the right and the left A-resolvents of operator B correspondingly. If the set aA(B) is bounded (i.e. there exists a > 0 : |p| < a for all p E aA(B)) then the operator B is ordied (A, a)-bounded.

Let the operator B be ( A, a )-bourided. p E { 0 } U N. Construe?t the set aA ( B ) = {p E C : pn E a^A ( B ) } ; it is compact in C due to the compactness of the A -spectrum of operator B. Take the contour y = {p E C : |p| = r, rn > a} that bounds the domain containing the points of aA ( B ) and construct the projectors

P = Д [ Pn- 1 RA- ( B ) dp E L (U) , Q = Д [ pn- ¹ LA- ( B ) dp E L (F) . 2 ni         ^                       2 ni         ^

γ

γ

Here. RA- ( B ) = ( pⁿA - B ) ^{- 1} A and LA- ( B ) = A ( pⁿA - B ) ^- 1. Set U 0 (U¹) = ker P (im P ). F0(F1) = ker Q (im Q ). Thus, the spaces U and F can be decomposed into direct sums U = U 0 Ф U¹ and F = F⁰ Ф F¹, whereas U⁰ D ker A. By Ak ( Bk ) define the restriction of operator A ( B ) on to Uk. k = 0 , 1.

Lemma 1. [10] The operators Ak,B_k E L (U ^k ; F ^k ) , k = 0 , 1, moreover, there exist the operators B - ¹ E L (F⁰; U⁰) an id A- ¹ E L (F¹; U¹).

Construct the operators H = B0 ¹ A ₀ E L (U⁰), S = A- ¹ B 1 E L (U¹) .

The ( A, a )-bounded operator B is ordied ( A,p)-bounded. p E { 0 } U N. if to is a, removable singular point (i.e. H = O , p = 0) or a pole of order p E N (i.e. H^p = O, H^{p +1} = O) of the Aresol vent ( pA — B ) ^{- 1} of operator B.

Introduce the following condition:

a^A ( B ) = [J a^A ( B ) , for m E N; moreover , a^A ( B ) = 0, j =0

there exists a closed contour Yj C C , bounding a domain

( A )

Dj D ajA ( B ) , such that Dj П a A ( B ) = 0 and

D_k П Dl = 0 for all j, k,l = 1 , m with k = l.

Then we have

Lemma 2. [10] If the operator B is ( A, a ^-bounded and condition (A) is fulfilled then (i) there exist relatively spectral projectors

Pj =

2 ni J

pn^{- 1} RA- ( B ) dp E L (U) ,

^j = ^{1 ,m},

γj

Qj =

A" [ p"^{- 1} LA- ( B ) dp E L (F) , 2 ni          ^

j = 1 ,m.

γj

Moreover,

(u) PjP = PPj = P₃. QjQ = QQj = Qj

(in) P_kPl = PlPk = O for all k. l = 1 , m with k = l.

G.A. Sviridyuk, A.A. Zamyshlyaeva, S.A. Zagrebina mm

Put P о = P — ^^ Pj E L (U), Q о = Q — ^^ Qj E L (F). Due to Lemma 3 operators j =1                             j =1                                    ____

P о , Q о are projector^-s. Moreover. PjP 0 = P 0 Pj = O. QjQ о = Q о Qj = O ft>r j = 1 , m.

Thus, assume that condition (A) is fulfilled. Fix Tj E R with Tj < Tj ₊₁, vectors uj E U for j = 0 , m , and vector-function f E C^ю (R; F). Consider the linear inhomogeneous Sobolev type ecpiation

Au ⁽ n ) = Bu + f.                                (5)

Refer to a vector-function u E C^ю (R; U) satisfying (5) as a solution to (5). Refer to a solution u = u ( t ) to (5) satisfying the conditions

Pj ( u ⁽ k )( Tj) — ujk ) = 0 , j = 0 , m, k = 0 ,n — 1 ,                        (6)

as a solution to the multipoint initial-Jin al value problem (6) for (5). Introduce the following operator families uk (t) =

^n-k- 1( ^nA — в ) - ¹ A^ntd^,

k = 0 , 1 ,..., n — 1 , j = 1 ,..., m.

Lemma 3. [10] If the operator B is ( A,p^-bounded and condition (A) is fulfilled then

fl) Uk ( t ) k = 0 , 1 ,... ,n — 1 , j = 1 ,... ,m are propagators of homogeneous (f = 0J equation (5);

(U) ( Uk ( t )) (l ) = Uj^k—1 ( t ) for k = 0 , 1 ,..., n — 1, j = 0 , 1 ,..., m, l = 0 , 1 ,..., k;

(Uk ( Uk ( t )) t' ⁾ in ( Uk ( t )) t^kk

t =о

= O for k = l;

t =о

= j

Introduce the subspaces U1 j = im Pj a nd F1 j = im Qj for j = 0, m. By construction, mm

U¹ = ф U¹ j ^aild F¹ = ф F¹ j.

j =о                   j =о

Denote by A 1 j the restiaction of A tо U^{1 j} ашI by B 1 j the restriction of B tо U^{1 j} ft>r j = 0 , m.

Theorem 1. [29] (generalized spectral theorem) Suppose that A, B E L (U; F), the operator B is ( A, a ^-bounded, and condition (A) is satisfied then

(i) A 1 j E L (U^{1 j} ; F^{1 j} ) an id B 1 j E L (U^{1 j} ; F^{1 j} ) fo r 3 = 0 ,m:

fit) the. operators A— E L (F^{1 j} ; U^{1 j} ) exist. for j = 0 , m.

Theorem 2. [10] If the operator B is (A,p)-bounded for p E {0}U N and condition (A) holds then for- all f E Cpn+n (R; F) as id ujk E U. for j = 0, m, k = 0, n — 1. there exists a unique solution to (5), (6) given by u (t ) = — ib Hq B о 1(I — Q) f(qn)(t)+ q=о

m n— 1

+E E u f ( t—Tj ) uk + j =о k =о

t m j=о J

τj

U^n— 1( t — s ) A —j¹ Qj f ( s ) ds.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

2.    A Stochastic Linear Sobolev Type Equation of Higher Order with Relatively p-Bounded Operators

For a real separable Hilbert space U = (U , (•, •) ), take an operator K E L (U) whose spectrum a ( K ) is nonnegative, discrete, finite, and accumulates only to zero. Denote by {A,} the sequence оf eigenvalues of K enumerated in the non-increasing order taking into account the multiplicities. The linear span of the set {ф,} of corresponding orthonormal eigenfunctions of K is dense in U. Assume also that K is a nuclear operator, that is, its ^

trace Tr K = ^ A j <  + w. j =1

Take a sequence {n,} of independent stnj : D x I ^ R. a, complete probability space D. and an interval I C R. Equip R with the Borel a-algebra. Assume that the random variables nj (w,t) E L₂ are Gaussian for all ш E A arid t E I. where A is a a-algebra on D. In addition, the sample trajectory nj(ш, •) is almost surely continuous, that is. nj E CL₂ (for a detailed desciiption of the spaces C^lL₂ fc>r l E {0} U N. see [23]). Define the U-valued stocliastic K-process

^

e к ( t ) = £ j, n ( t ) ф ,                         (S)

j =1

assuming that series (8) converges uniformly on every compact subset of I. Observe that if {n,} C CL ₂ then the existence of a stochastic K -process e _K implies that its trajectories are almost surely (a.s.) continuous. Introduce the Nelson-Gliklikh derivatives

^

eK ⁾( t ) = V_x Aⁿ- )( t ) ф ,                                (9)

j =1

of the stochastic K -process assuming that the derivatives in the right-hand side up to order l exist and all series converge miifcirmly on every compact subiset of I. (For a detailed description of the Nelson-Gliklikh derivative, see [15,23]). Introduce [23] the space of differentiable "noises" C K L ₂ of stochastic K -processes whose trajectories are a.s. continuously differentiable on I in the sense of Nelson Gliklikh up to order l E { 0 } U N.

As an example, let us present "black noise", a stochastic K -process whose trajectories a.s. coincide with zero (that is, absolute silence), as well as "white noise"

w к ( t ) = WKit),                            ⁽¹⁰⁾

which is the Nelson-Gliklikh derivative of the Wiener K -process

W k ( t ) = £ V A-в P, ( t ) Фз, t E R+ • j =1

Here Pj = Pj (t) is the Brownian motion of the form n x            • n(2k + 1)

Pj(t) = ^L^jk sin----2---" t, t E R+, k=1

G.A. Sviridyuk, A.A. Zamyshlyaeva, S.A. Zagrebina where j are pairwise independent Gaussian random variables such that Ej = 0 arid

d j =

n (2 k + 1) 2

- 2

that is. ^j_k E L₂.

Having considered the deterministic ecpiation (5) in the previous section, we now proceed to the stochastic equation (3). Assume that the operator B is (A,p)-bounded, with p E {0} U N, and condition (A) is satisfied. Consider the linear stochastic Sobolev type equation

о ( n )

(П)

An = Bn + Nw,

0 ^{( n} )

where n = n ( t ) ^1S the required stochastic K -process, n is its Nelson-Gliklikh derivative of the n -th order, w = w ( t ) is a, known stochastic K -process. arid the operator N is defined below.

Take t ₀ = 0 and t , E R₊ with t , _ 1 <  t , for j = 1 , m . Complement (11) with the multipoint initial-final conditions

о ^{( k} )                  ,                           ________ ________________

Pj (n (Tj) - ej) = 0, j = 0,m, k = 0,n - 1,(12)

where Pj are the relatively spectral projectors from Lemma 3. In view of (10), we also have to consider the weak (in the sense of S. Krein) multipoint initial-final conditions lim Po(n(k) (t) - e0k) = 0, Pj(n(k)(Tj) - ej) = 0, j = 1 ,m, k = 0,n - 1.(13)

t^T o +

Here

^

ek = EVkерфк,j = 0,m, k = 0,n -1,(w)

i =1

ejki E L ₂ is a Gaussian random variable such that series (14) converges. (For instance Dj A Cjk- i E N. j = 0 ,m. k = 0 ,n - 1). Call a, stochastic K -process n E C K L ₂ a (classical) solution to (11) whenever a.s. all its trajectories satisfy (11) for some stochastic K -process w E C _K L ₂, some о perator N E L ( U ; F ), am1 all t El. (Here and henceforth I = (0 , + to )). Call a solution n = n ( t ) to (H) ^a (classical) solution to problem (11), (12) (problem (11), (13)) whenever in addition condition (12) (condition (13)) is satisfied.

Theorem 3. For p E {0} U N take an (A,p^-bounded operator B and assume that condition (A) holds. Given Tj E R+ Jor j = 1, m, an operator N E L(U; F), a nuclear operator K E L(U) with real spectrum a(K). a stoehastie K-process w = w(t) such that (I - Q)Nw E CK+nL2 aird QNw E CKL2- anal random variables ej E L2. for j = 0,m k = 0, n - 1. such that Щ) is fuljitled. there exists a unique, solution n E CKL2 to problem (11), (12) given by n (t) = - ^^ hq b- 1(I - Q) w(qn)(t)+ q=0

m

+E

j =0

n- 1

E Uk (t k=0

-

Tj ) ej + t Un^- 1( t - s ) A^-1Q, w ( s ) ds

τj

t ∈ I.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Let us sketch the proof. It is straight forward to verify that (17) is a solution to problem (11), (12). To establish the uniqueness, reduce the problem to the equivalent system о (n)                    о (k)                          _______ _____________

An = Bn, Pj П   ( Tj ) = 0 , j = 0 ,m, k = 0 , n — 1 .

By Theorem 1 the first equation here is equivalent to the system

H ( П ⁰)^{( n} ) = n ⁰ , ( П 1)^{( n} ) = Sn ¹ ,

where n ⁰ = ( I — P ) n ^and n ¹ = PH- Taking now the n -th Nelson-Gliklikh derivative of the first equation and acting on the left by H we obtain in succession

= h ( n ⁰)^{( n} ) = n ⁰ .

о = h p ⁺¹( n ⁰)⁽ np ^{+ n} ) =    = H ²( n ⁰)^{(2 n} ) =

By Theorem 2 and the initial-final conditions (12), the second equation of (16) yields m n— 1

n ¹ = E Т.Ч ( t — Tj )0 = 0

j =0 k =0

In view of (10), problem (11), (12) is not solvable when the right-hand side of (11) is the "white noise" w ( t ) = WK ( t ). In this case instead of conditions (12) we should consider conditions (13).

Corollary 1. If all conditions of Theorem 3 hold and w ( t ) = WK ( t ) then, for random variables f— E L ₂ given by (If) there exists a unique solution to problem (11), (13) given by

p

n ( t ) = — 52 HqB0 1(I — Q ) WK ^{( qn +1)}( t ) + q =0

m n— 1

+ E E U ( t — Tj ) j — Un- ¹( t — Tj ) A —1 Qj NW k ( Tj )+ j =0 k =0

+ / Un- ²( t — s ) A-1 Qj NW k ( s ) d^ , t El.

The proof of Corollary 1 is similar to that of Theorem 3. The difference in the additive terms is caused by an application of integration "by parts",

t j Ujn- 1(t — s) A-j1 Qj N Wk (s) ds =

τj

t

—Un- ¹ ( t — Tj ) A-1 Qj NW k ( Tj ) + J Ujn- ²( t — s ) A-}Qj NW k ( s ) ds,

τj which follows from the properties of the Nelson-Gliklikh derivative.

G.A. Sviridyuk, A.A. Zamyshlyaeva, S.A. Zagrebina

3.    The Multipoint Initial-Final Problem for the Stochastic Boussinesq-Love Equation with Additive "White Noise"

Let D C R d be a bounded domaiп with the boundary dD of class C^ю. Flx l G { 0 }U N and set F = Wl ( D ) , U = {u G W 2⁺²( D ) : u ( x ) = 0 ,x G dD}. Obviously, U is a real separable Hilbert space densely and continuously embedded in F .

Let {vj} be the sequence of eigenvalues of the Laplace operator with homogenous Dirichlet boundary conditions, numbered in nondecreasing order according to multiplicity, and by {фj} denote the set of corresponding eigenfunctions, orthonormal in the sense of F

Introduce the U-valued random K -processes.   Construct the   operator

Л = (—1)m-1Am with domain domЛ = {W2+2(m+1)(D) : Aku(x) = 0,x G dD, k = 0, 1, ...,m — 1 },m G N. Note that the operator Л has the same eigenfunctions {ф,}. as the Laplace operator, but its spectrum consists of eigenvalues |v,|m. Since their asymptotic |v,|m ~ j2m ^ ^, j ^ ж. we consider such number m G N for a, fixed d G N that the ∞ series ^2 |vj |-m converges (in particular m = d). Then the operator Л Is continuously j=1

invertible on U, whereas the inverse operator (i.e. the Green operator) has a spectrum consisting of eigenvalues X, = |vj|^-m . We take that very operator as the nuclear operator K and consider the Wiener K -process

∞

W k ( t )= £ V X в, ( t ) Vj.

• j ; ^vj = X

In the cylinder D x [0 , T ], T G R₊ consider the Dirichlet problem

£ ( x,t ) = 0 ,   ( x,t ) G dD x [0 , T ]                           (18)

for the equation

( X — A x ) °_tt = a A x( + W k .                         (19)

Put A = X — A x,B = a A x, N = I

Lemma 4. [10] For all a G R₊, X G R the operator B is ( A, 0)-bounded.

In order to state initial-final conditions, we need relatively spectral projectors. In this example for the sake of simplicity we confine to three initial-final conditions. First of all present the projectors

'         Iu (If) if X = vj for all j G N, p (Q ) =

^E ^•’Ф, ^ U Фз

I U

-

I F - _λ     ⟨ ·, φj ⟩ _F φj

Furthermore, choose h 1 , h ₂ G R₊ such that h 1 < h ₂ and the sets a A ( B ) = { ц , G aA ( B ) : |Цj | < h 1 }• a A ( B ) = { ц , G a^A ( B ) : h 1 <  | ц , | < h 2 }. aiid a A ( B ) = { ц , G aA ( B ) : | ц , | > h ₂ } are not empty: hence, aA ( B ) П aA ( B ) П aA ( B ) = 0 and coiidltlon ( A ) holds.

no

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Construct the projectors

P 1 = IU “ 52 ⁽‘’Ф^ и ^фj’P 2 = IU — 52 ⁽*’^ф3) U ^фj’ h 1 <\^j\2                                  h2<\m \

Q 1 = IF ^— 52 ⁽’Ф) F ^фj’Q 2 = IF ^— 52 ⁽"’Ф^ F ^фj’         ^

h 1 <\^j\ 2                                     h 2 <\^j \

P 0 = P ^— P 1 ^— P 2 , Q 0 = Q ^— Q 1 ^— Q 2 .

Finally, choose т1 G (0, T) as well as random variables £0, £ 1 and £2 independent of each О other and of stochastic ^-processes WK and pose the initial-final conditions tlim Pо(n (t) —

P1(n(т1)— £1) = 0, P2(n(T) — £2) = 0, where

00                        TOTO

£0 = E лМФ £1 = E V^ 1 ф £2 = £ VAi£2iфг.^

i=1                     i=1

Applying the results of Section 2 to problem (18), (19), (21), we obtain the following theorem.

Theorem 4. For all numbers A G R, a G R \ {0} a nd т1 G (0, T), as well as random variables £₀i. £ 1 i atul £₂i such that Dj < Cj for i G N, j = 0,1, 2 for some Cj G R₊there exists a unique solution n = n(t) /^or t G R₊, to problem (18), (19), (21) given by

О n (t) = B 0 1(1 — Q) WK (t) + Uo (t)£ 0 + Uo1 (t)£ 0 + t

r

⁺J ^U0 ⁽t ^— s ) A10 Q0 WK^{( s} ) ^{ds + U}1 ⁽t ^{— т}1)^£ 1 ^{+ U}1 ⁽t ^{— т}1) ^£ 1 ^—

t

—U1¹(t

-

т1) A Г1¹Q1 Wk

(т1) +

U0( t — s) A n¹Q1 Wk (s) ds+

τ1

^{+ U}2 ⁽t ^{— T}) ^{£ 0 + U}2 ⁽t ^{— T}) ^£ 2 ^{— U}2 ⁽t ^{— T} ) A12 ^Q 2 WK^{(T) +} t

+1U00(t — S)AГ2¹Q2Wk(S)ds, t G R₊.

T

Here

B0 = aA52к‘^,фк^W, νk=λ

^     (•’Фк )фк cos. / ^aVk t’

νk - λ

Цк ^{G CT}A⁽B) ’

λ > νk

^U0(t )=     ^     (•’Фк )фк^Ch\/"T^V^" t⁺

λ - νk

Цк ^G CT A⁽B ) ’

λ < νk u0■(t)=   e   (;Фк>ФкХwkssh awt +   e   (,Фкфsin wtt,

⁰                          ανk     λ-νk                         ανk     νk-λ

Цк ^{G CT}A⁽B) ’                                   Цк ^{G CT}A⁽B) ’

λ < νk                                λ > νk

G.A. Sviridyuk, A.A. Zamyshlyaeva, S.A. Zagrebina

A101 = E ⁽^ - ^к)¹(•’фк^фк, µk ∈σ₀^A (B)

^U0( t ) =     E     (•’фк ^фк ch\R

Цк G a A (B), λ < νk

avL t₊

-

νk