CC BY
11
MSC 47D06, 35K70
DOI: 10.14529/ mmpl70212
MULTIPOINT INITIAL-FINAL VALUE PROBLEM FOR THE MODEL OF DEVIS WITH ADDITIVE WHITE NOISE
A.S. Konkina, South Ural State University, Chelyabinsk, Russian Federation, [email protected]
The evolution of the free surface of the filtering fluid in a reservoir of limited power is modeled by the Davis equation with homogeneous Dirichlet conditions. Depending on the nature of the free term describing the internal source of the liquid, the model will be deterministic or stochastic. The deterministic model has been studied in various aspects by many researchers with different initial (initial-final value conditions). The stochastic model is studied for the first time. The main result is the proof of the unique solvability of the evolutionary model with an additive white noise and a multipoint initial-final value condition.
Keywords: white noise; Wiener K-process; Davis model; multipoint initial-final value problem; stochastic model.
Introduction. Let Q C Rra be a bounded domain with boundary dQ of class C. In the cylinder Q x R+ consider the evolutionary Davis model
(A - A)ut = aAu - /3A2u + f, (1)
u(x,t) = Au(x,t) = 0, (x,t) e dQ x R+, (2)
where A e R, a,3 e R+ characterize the environment. Equation (1), together with
f = f(t)
of the liquid, can be reduced to an evolution Sobolev type equation
Lu = Mu + f, (3)
where the operators L e L(U; F) and M e Cl(U; F) act in some Banach spaces U and F-The same problem (1), (2), where the external action f = f (t) is a white noise leads to a stochastic Sobolev type equation
Ldu = Mudt + NdW. (4)
Here U is a Banach space, F is a real separable Hilbert space, the operators L e L(U; F) M e Cl(U; F) and W = W(t) is an F-valued Wiener K -process.
Take To = 0 and Tj e R+, such that rj-\ < Tj for j = 1,n. Equations (3), (4) can be supplemented by a multipoint initial-final value condition
tlim+ Po(u(t) - Co) = 0, Pj(u(Tj) - j) = 0, j = in. (5)
where Pj are relatively spectral projectors [1].
At the end of the statement of the problem we note that three directions of the solution of the problem (4), (5) are actively developing now. White noise is used to solve
K
We also note the approach presented by the school of I.V. Melnikova, in which equation
(4) is considered in Schwartz spaces, where the generalized derivative of a Wiener K-process makes sense [3]. Meanwhile, a new approach is actively developing in the studies of equation (4), where by "white noise" is meant the Nelson - Glickich derivative of the K
1. A Determinate Sobolev Type Equation. Let ^d F be Banach spaces, operators L E F) (that is, linear and continuous) and M E Cl(U; F) (i-e., linear, closed, and densely defined). Let M be (L,p)-sectorial, p E {0}UN. Consider the Sobolev type equation (3) equipped with condition
lim Po(u(t) - uo) = 0, Pj(u(j) - Uj) = 0, j = 1, n. (6)
Moreover, we recall that the condition of an (L,p)-sectoriality of the operator M is a sufficient condition for the existence of degenerate analytic semigroups of operators [5]
U = — [uL - M )-1Lefldp H F1 = — i L(pL - M )-1e^d/j, 2ni Jr 2ni ,/r
defined on spaces ^d F respectively. Introduce the kernels ker U' = U0, ker F' = F0 and the images imU' = U1, imF' = F1 of these semigroups. Suppose that condition
U0 © U1 = U (F0 © F1 = F), (A1)
is fulfilled. It holds either in the case of strong (L,p)-sectorial operator M on the right (left), p E {0} U N or the reflexivity of space U (F).
Denote by Lk (Mk) contraction of op erator L (M) on Uk (domM n U^, k = 0,1. If the operator M is (L,p)-secto rial p E {0} U N, the n Lk E L(Uk ; Fk ) Mk E C l(Uk ; Fk ) k = 0,1, and there exists an
operator M-1 E L(F;U0), as well as a projector P = s - lim Ul, (Q = s - lim Fr), splitting space U (F according to (Al), and U1 = imP (F1 = imQ). Introduce one more condition:
there exists an operator L— E C(Fl; U1), (A2)
( L, p) M p E { 0 } U N
( L, p)
M p E {0} U N (L, p) M
(left), p E {0}UN and if we add condition (A2) to them, we get a strong (L,p)-sectoriality of the operator M, p E {0} U N Then the operator G = M-1L0 E L(U0) is nilpotent of degree p, and the operator S = L-lM1 E Cl(U1) is sectorial.
Finally, introduce one more important condition on the relative spectrum of an M
aL(M) = У af(M), n E N, with af(M) = 0 contained in a bounded
rf(M), n E N^th af' j=0
domain Dj С С with a piecewise smooth boundary dDj = Г С C.
(A3)
Moreover Dj П ) = 0 Mid Dk П Di = 0 for all j,k,l = l,n,k = l. ,
Construct relatively spectral projectors [6] Pj E Qj E L(F), j = l,n, of the form
Pj = 2- i (»L - MQj = 2-1 L(^L - mu, j = i,n. (7)
*J P j *J P j
It turns out that under the condition of (L, p)-sectoriality of the operator M and conditions (Al), (A2), PjP = PPj = Pj and QjQ = QQj = Qj, j = l,n. Hence, in this case there
n
exists a projector Po = P — Pj> Po E L(U).
j=i
Suppose that conditions (Al) - (A3) are satisfied. Fix t0 = 0 Tj E R+ (rj-1 < Tj), Uj E U, j = 0,n, and consider the multipoint initial-final value condition (6) for a linear Sobolev type equation (3). A vector function u E C 1([T0,Tn]; U) fl C([T0,Tn]; U), satisfying (3), is called a solution of (3); solution u = u(t) of equation (3) satisfying (6) is called a solution of a multipoint initial-final value problem (3), (6).
Theorem 1. [6] Let the operator M be (L,p)-sectorial, (Al) - (A3) hold. Then for any vector-function f0 E Cp([To,T,n]; F0) f Cp+1((To,Tn); F0), f1 E C([To.Tn]; F1) there exists a unique solution of (3), (6), given by
p n / t \
u(t) = — £ GqM-1f °(q)(t) + £ U— Uj W U^L-^Qj f (s)ds q=1 j=1 \ jTj '
2. The Spaces of Noises. Let Q = (Q, A, P) ^e ^^^^^^^e probability space and let R be the set of real nimbers endowed Borel A measurable mapping £ : Q ^ R
is called a random variable. Note that all random variables, having a normal distribution (i.e. gaussian), are contained in space L2 see [4].
Consider two mappings f : I ^ matching each t E I to a random variable £ E L2, and g : L2 x Q ^ R matching each pair (£, u) ^o a point £(u) E R, where I C R is some interval. By a stochastic process we call a mapping n : I x Q ^ R, having the form n = n(t,u) = g(f (t), u). Note that the stochastic process n = n(t, •)> if t E I is fixed, is a random variable, and the stochastic process n = n(^,u), if u E Q is fixed, will
n
u E Q a trajectory n(t, u) is continuous on I.
Denote by P = P(I x Q; U) a space of stochastic processes. The space of continuous
nE
if n(t, •) E L2 for all t E I. Note that CL2 is a subspace of P. Note that the space CL2 in particular contains those stochastic processes, whose all trajectories are continuous, and all (independent) random variables are Gaussian.
Consider the operator K E L(R), whose spectrum a(K) is positive, i.e. a(K) E R+. This is possible when K positive and self-adjoint. The sequence of eigenvalues of the operator K is denoted by {Xk}■ Suppose that the spectrum a(K) is discrete, finite and is condensed only to the zero point. Enumerate in {Xk} nonincreasing order, taking into account their multiplicities.
(X
Operator K is called nuclear, if Tr K = < We note that the linear span of
k=1
the set {<^k} of the corresponding eigenfunctions of the operator K is dense in R. Let us define the Brownian motions, take the sequence {£k}, of independent random variables.
Definition 1. A stochastic process
/3(t) = /(t, = & sin 2(2k + l),t E R+, (8)
k=0
is called a Brownian motion, if
(Wl) /(0) = 0 almost everywhere in Q, and the trajectories of stochastic process /(t) are almost surely continuous on R+;
3(t)
any point t E R+ and on any interval I C R+ have unbounded variation. As an example consider the Wiener K-process [4]
<x
W(t) = ^ (t)Vk, (9)
k=0
where /k(t) are Brownian motions, k E N.
Theorem 2. [2] For any nuclear operator K E L(U) and arbitrary sequence of Brownian motions } a Wiener K-process W E CL2.
For the solvability of problem (4), (5) we need another condition
QN = N. (A4)
Then the formal solution u = u(t) of the multipoint initial-final problem (5) for equation (4) has the form
u(t) = Ufo +
j=l
U t-sL-1Qj NdW (s) + Ut-Tj j + L-Q3 NW (t)
(10)
Theorem 3. Let the operator M be (L,p)-sectorial and conditions (Al)-(A4) be fulfilled . Then for arbitrary U1 -valued Gaussian random variables j j = 0, n independent on W(t) and satisfying (9), there exists a unique strong solution of (4), (5), given by (10).
3. The Davis Model. Consider problem (1), (2), where white noise appears as an external action of f = f (t). Let F = L2(Q) and U = {u E W22(Q) : u(x) = 0,x E dQ}. Operators L and M are given by formulas L = A — A and M = a A — / A2,
domM = U n {u E W24(Q) : Au(x) = 0,x E dQ}.
Obviously, for all A E R operator L E L(U; F), and for all a E R, / E R \ {0} operator M E Cl(U; F).
Lemma 1. For all A E R,a,/ E R+ operator M is (L, 0)-sectorial.
Let {^k} be an orthonormal set of eigenfunctions of the homogeneous Dirichlet problem
AQ
{Ak}
oo
(^L - M)u = ^(^A - (^ + a)Xk + px2k) <и,фк > фк
к=1
t
j
for any u E dom M, v E C, then
<x
(vL - M)-1 = £ </' +k> . ) Vk (11)
fXk - aXk + V(X - Xk)
C
points
Vk = Xk^ ,k E N. (12)
Xk - X
Since the spectrum a(A) negative, discrete, finite and condensed only to then
it follows from (12) that the L-spectrum aL(M) operator M it real, discrete, finite and condensed only to In considering the problem, we confine ourselves to the values
of the parameter X lying in the spectrum of the operator A. Therefore, from the set of numbers (12) it is necessary to delete the numbers vk with numbers k for which X = Xk. So, let X E &(A), then we get
oo
( L _ M)-1 = ' < ''Vk > Vk__y^ < -,Vk > Vk
(V ) ¿-r PXk - aXk + v(X - Xk)+ ^ f3X2 - aX ''
k—1 Ak=X
RL(M) = £ '(v + Xk- < ->Vk >Vk = LL(M k—1 k
o* - m )-iLHM) = £ -0» + Xk s^ )-1 x P)i _<£+>*_ Xk)
k—1
k
X = Xk. Hence it is not difficult to obtain the strong L-spectoriality of the operator M. For the sake of simplicity, we take the operator N = Q, then condition (9) is obviously satisfied. Denote by {vk} a sequence of eigenvalues of the Laplace operator A with condition (2) in a domain Q, numbered in nonincreasing order with taking into account their multiplicities, and by {vk} a sequence of eigenfunctions. Then
u(t) = T,T, -eVkt&,Vk )vk + Y, Y'^f^dMV, (13)
j—1 k—1 j:vjeaf(M) k—1 Vk
where vk = —v-v— are the points of the L-spectrum of the operator M, {Xk} are
(a - Vk)
the eigenvalues of a special constructed nuclear operator K. A prime at a sum sign means the absence of terms such that X = vk■
Corollary 1. Let X E R a,f E R+ and conditions (Al) - (A4) be fulfilled. Let satisfy (9). Then there exists a unique strong solution of (1), (2), (5), given by (13).
Acknowledgements. The work was supported by Act 211 Government of the Russian Federation, contract No. 02.A03.21.0011.
References
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Received March 25, 2017
УДК 517.9 DOI: 10.14529/mmpl70212
МНОГОТОЧЕЧНАЯ НАЧАЛЬНО-КОНЕЧНАЯ ЗАДАЧА ДЛЯ МОДЕЛИ ДЕВИСА С АДДИТИВНЫМ БЕЛЫМ ШУМОМ
A.C. Конкина, Южно-Уральский государственный университет, г. Челябинск
Эволюция свободной поверхности фильтрующейся жидкости в пласте ограниченной мощности моделируется уравнением Девиса с однородными условиями Дирихле. В зависимости от характера свободного члена,описывающего внутренний источник жидкости, модель будет детерминированной или стохастической. Детерминированная модель изучалась в разных аспектах многими исследователями с различными начальными (начально-конечными) условиями. Стохастическая модель изучается впервые. Основной результат-докозательство однозначной разрешимости эволюционной модели с аддитивным белым шумом и многоточечным начально-конечным условием.
Ключевые слова: белый шум; винеровский K-процесс; модель Девиса; многоточечная начально-конечная задача; стохастическая модель.
Александра Сергеевна Конкина, ассистент кафедры «Уравнения математической физики:», аспирант кафедры «Математическое и компьютерное моделирование:», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Поступила в редакцию 25 марта 2011 г.
