CC BY
14
MSC 35S10, 60G99
DOI: 10.14529/ mmp190204
EXPONENTIAL DICHOTOMIES IN BARENBLATT-ZHELTOV-KOCHINA MODEL IN SPACES OF DIFFERENTIAL FORMS WITH "NOISE"
O.G. Kitaeva1, D.E. Shafranov1, G.A. Sviridyuk1
1 South Ural State University, Chelyabinsk, Russian Federation E-mails: [email protected], [email protected], [email protected]
We investigate stability of solutions in linear stochastic Sobolev type models with the relatively bounded operator in spaces of smooth differential forms defined on smooth compact oriented Riemannian manifolds without boundary. To this end, in the space of differential forms, we use the pseudo-differential Laplace-Beltrami operator instead of the usual Laplace operator. The Cauchy condition and the Showalter-Sidorov condition are used as the initial conditions. Since "white noise" of the model is non-differentiable in the usual sense, we use the derivative of stochastic process in the sense of Nelson-Gliklikh. In order to investigate stability of solutions, we establish existence of exponential dichotomies dividing the space of solutions into stable and unstable invariant subspaces. As an example, we use a stochastic version of the Barenblatt-Zheltov-Kochina equation in the space of differential forms defined on a smooth compact oriented Riemannian manifold without boundary.
Keywords: Sobolev type equations; differential forms; stochastic equations; Nelson-Gliklikh derivative.
Dedicated to the 60-th birthday of outstanding mathematician Jacek Banasiak.
Introduction
The Barenblatt-Zheltov-Kochina equation
(A - A)U = aAu + f (1)
simulates dynamics of pressure of a liquid filtered in a fractured-porous medium. Parameters a and A are real and characterize the environment and properties of the liquid, respectively, and function f = f (x) plays the role of an external influence.
The study of solvability of the initial-boundary value problems for equation (1) in Banach spaces with the Cauchy condition is based on the approach described, for example, in [1], where this equation is reduced to abstract linear Sobolev type equation
LU = Mu + f (2)
in suitable function spaces U and F. Paper [2] considers splitting of similar spaces and splitting of the action of elliptic operators in spaces of smooth differential forms defined on smooth Riemannian manifolds without boundary.
Further, paper [3] considers stochastic equations of Sobolev type
L n = Mn + Nu, (3)
where n = n(t) is a stochastic process, V is the Nelson-Gliklikh derivative of process [4], w = w(t) is a stochastic process that corresponds to an external influence; L,M,N G L(U; F) are operators, and operator M is (L,p)-bounded, p G {0} U N. Paper [5] shows the results of studying equation (3) in the case when operator M is (L,p)-sectorial, p G {0}U N. Then, paper [6] considers equation (3) in the case when operator M is (L,p)-radial, p G {0} (J N. Note that all three papers [3,5,6] along with classical Cauchy problem
n(0) = no (4)
consider Showalter-Sidorov problem
P(n(0) - no) = 0. (5)
Paper [7] considers more general initial-finite conditions for equation (3), and paper [8] investigates the Cauchy and Showalter-Sidorov problems posed for the Sobolev type equation of high order.
The stochastic Barenblatt-Zheltov-Kochina model with additive "white noise" given in a bounded domain is considered as a concrete interpretation of abstract stochastic equation (3) in [3] and is transferred to a Riemannian manifold without boundary in [9]. Paper [10] was the first to consider the dichotomies of solutions to abstract homogeneous Sobolev type equation
LU = Mu, (6)
where operator M is (L,p)-bounded, p G {0} (J N.
The paper is devoted to the study of dichotomy of solutions to abstract homogeneous stochastic Sobolev type equation
L n = Mn, (7)
where operator M is (L,p)-bounded, p G {0} (J N. The Barenblatt-Zheltov-Kochina stochastic equation given on a Riemannian manifold without boundary is considered as a specific model.
Note also other approaches to the study of groups and semigroups of stochastic equations, for example, proposed in [11] or in [12-14].
The paper, in addition to the introduction, conclusion and bibliography, contains three sections. In the first section, we introduce the terminology of linear Sobolev type equations, including the terminology related to stability, and the main theorems proved in other papers. The second section describes a stochastic analogue of the Barenblatt-Zheltov-Kochina equation in specially selected spaces. The third section gives the main result on existence of exponential dichotomies of solutions to the equations. The conclusion presents several directions for further research. Note also that the bibliography does not pretend to be complete, but reflects only the personal preferences of the authors.
1. Dichotomies of Solutions to Equations in Banach Spaces
Let U and F be Banach spaces, and operators L, M G L(U; F). Consider L-resolvent set pL(M) = {u G C : (uL - M)-1 G L(F; U)} and L-spectrum aL(M) = C \ pL(M) of operator M. If L-spectrum aL(M) of operator M is bounded, then operator M is called (L, a)-bounded. If operator M is (L, a)-bounded, then there exist projectors
Р=1Ь / е Q = b! е
Here RL(M) = (^L - M)-1 L and (M) = L(^L - M)-1 are the right and the left
L-resolvents of operator M, respectively, and closed loop 7 C C bounds the domain
containing aL(M). Set U0 (U1) = kerP (imP), F0 (F1) = kerQ (imQ). Denote the restriction of operator L (M) to Uk by Lk (Mk), k = 0,1.
Theorem 1. [1, Ch.3] Let M be (L, a)-bounded operator, then
(i) operators Lk (Mfc) G L(Uk;Fk), k = 0,1;
(ii) there exist operators M-1 G L(F0; U0) and L-1 G ^(F1; U1).
Corollary 1. Let M be (L, a)-bounded operator, then
<x <x
OuL - M)-1 = Sk-1L-1 Q + £HkM0-1 (I - Q),
k=0 k=1
where operator H = L-1 M0 G L(U0), S = L-1M1 G ¿(U1).
Further, (L, a)-bounded operator M is called (L,p)-bounded, p G {0} U N, if to is a removable singular point (H = O, p = 0) or a pole of order p G N (i.e. Hp = O, Hp+1 = O) of L-resolvents (^L - M)-1 of operator M. We consider vector function u G C 1(R; U) as a solution of equation (6), if u substituted to equation (6) makes (6) true. Solution u = u(t) of equation (6) is called a solution of Cauchy problem,
u(0) = u0 (8)
for equation (6), if equality (8) holds for some u0 G U.
Definition 1. Set P C U is called a phase space of equation (6), if
(i) any solution u = u(t) to equation (6) belongs to P pointwise, i.e. u(t) G P for all t G R;
(ii) for any u0 G P there exists unique solution u G C 1(R; U) of Cauchy problem (8) of equation (6).
Theorem 2. [1, Ch.3] Let M be (L,p)-bounded operator, p G {0} U N. Then phase space of equation (6) is the subspace U1.
Note that if there exists operator L-1 G L(F; U), then phase space of equation (6) is space U.
Definition 2. [10] Subspace J C U is called an invariant space of equation (6), if for any u0 G J the solution of problem (6), (8) is u G C 1(R; I).
Note that if equation (6) has phase space P, then any its invariant space J C P.
Definition 3. [10] Solution u = u(t) of equation (6) has exponential dichotomy , if
(i) phase space P of equation (6) splits into a direct sum of two invariant spaces (i.e. P = J+ © J-), and
(ii) there exist constants Nk G R+, vk G R+, k =1, 2, such that
Y
Y
llu^Hu < N1e-Vl(s-i)|u1(s)|u for s > t, ||u2(t)HU < N2e-V2(s-i)|u2(s)!u for t > s,
where u1 = u1(t) G J+ and u2 = u2(t) G J- for any t G R. Space J+(J-) is called stable (unstable) invariant space of equation (6). And if J+ = P (J- = P), then stationary solution of equation (6) is stable (unstable).
Theorem 3. [10] Let M be (L,p)-bounded operator, p G {0}U N, and aL(M) f|{iR} = 0. Then solution u = u(t) of equation (6) has exponential dichotomy.
2. Dichotomies of Solutions of Equations in Spaces of Differentiable "Noises"
Let Q = (Q, A, P) be a full probability space, R be set of real numbers endowed with the Borel a-algebra. Measurable mapping £ : Q ^ R is called a random variable. A set of random variables having zero expectation (E£ = 0) and finite dispersion forms Hilbert space L2 with scalar product (£^£2) = E£1£2. Let Ao be a a-subalgebra of a-algebra A. Construct subspace L0 C L2 of random variables measurable with respect to A0. Denote the orthoprojector by n : L2 ^ L2. Let £ G L2, then n£ is called a conditional expectation of the random variable £, and is denoted by E(£|A0).
Consider set I C R and the following two mappings. First, f : I ^ L2, associates each t G I with random variable £ G L2. Second, g : L2 x Q ^ R, associates each pair (£,u) with point £(u) G R. The mapping n : R x Q ^ R having form n = n(t, u) = g(f (t), u) is called (one-dimensional) stochastic process. Therefore, stochastic process n = n(t, ■) is a random variable for each fixed t G I, i.e. n(t, ■) G L2, and stochastic process n = n(',u) is called a (sample) path for each fixed u G Q. Stochastic process n is called continuous, if all its paths are almost sure continuous (i.e. at almost all u G Q paths n(',u) are continuous). The set of continuous stochastic processes form a Banach space, which we denote by CL2. Fix n G CL2 and t G I, and denote by Nf7 the a-algebra generated by a random variable n(t). For brevity, Evt = E(-|Ni'?).
Definition 4. Let n G CL2. A random variable
0= I(iim E;+ + lim ^(«(*.■)- f-At-)
2 VAi^o+ * \ At ) At^o+ 1 \ At
o
is called Nelson-Gliklikh derivative n of the stochastic process n at point t G I, if the limits exist in the sense of the uniform metric on R.
Let C'L2, l G N, be a space of stochastic processes whose paths are almost sure differentiable in the sense of the Nelson-Gliklikh derivative on I up to order l inclusively. Spaces C1 L2 are called the spaces of differentiable "noises". Let I = {0} U R+, then a well-known example [3, 5] of a vector of space C1 L2 is given by a stochastic process that describes the Brownian motion in Einstein-Smoluchowski model
2
k=0
-2
where independent random variables G L2 are such that dispersion D^k = [f (2fc +1)]
k G {0} U N. As shown in [4], ¡3 (t) = ^,t G R+.
Now let U (F) be a real separable Hilbert space with orthonormal basis }).
Denote by UlL2 (FmL2) the Hilbert space, which is a completion of the linear span of random L-variables
П = ^fcUfcCfc^fc = ^fc/*Cfc"0fc) (9)
k=i k=i
by the norm
lu = £ DCk (NIF = £^k/DC*). k=i k=i
Here sequence L = {Ak} C R+ (M = {^k} C R+) is such that A| < (J2 +
fc=i fc=i
{uk} ({/}) is a sequence of coefficients of vector u G U (/ G F) expansion by basis {^fc} ({^fc}), and {£k} C L2 ({Zk} C L2) is a sequence of random variables. Note that for existence of a random L-variable n G ULL2 (w G FML2) it is enough to consider a sequence of random variables {£k} C L2 ({(fc} C L2) having uniformly bounded dispersions, i.e. D£k < const, k G N (D(fc < const, k G N).
Next, consider interval I = (e, t) C R. Mapping n : (e, t) ^ UlL2 given by formula
^
n(t) = AfcUfc£fc(t)^fc, (10)
fc=i
where {£k} C CL2 is a sequence, is called U-valued continuous stochastic L-process, if the series on the right-hand side converges uniformly on any compact in I by norm || • ||u, and paths of process n = n(t) are almost sure continuous. Continuous stochastic L-process n = n(t) is called continuously Nelson-Gliklikh differentiate on I, if series
n (t) = J] Afcufc £fc (t)^fc (11)
fc=1
oo
converges uniformly on any compact in I by norm || • ||u, and paths of process n=n (t) are almost sure continuous. Let C(1, UlL2) be a space of continuous stochastic L-processes, and C1 (I, ULL2) be a space of continuously differentiable up to order l G N stochastic L-processes. An example of a stochastic L-process, which is continuously differentiable up to any order l G N inclusively, is Wiener L-process [3,5]
WL(t) = J] Afc& (t)^fc, fc=i
where } C C'L2 is a sequence of Brownian motions on R+. Similarly, spaces C(1, FML2) and C1 (I, FML2), l G N, are constructed. Note also that spaces QL2, C(1, UlL2) and Cz(1, FML2), l G N, are called the spaces of differentiable L-"noises" [3].
Consider an operator A G L(U; F). It is clear that the same operator A G L(UlL2; FML2). Moreover, there exists the following lemma holds.
Lemma 1. Let operators L, M G L(U; F), where operator M is (L,p)-bounded, p G {0} U N. Then operator M G L(UlL2; FmL2) is also (L,p)-bounded,
ЭО
p G {0} U N, where operator L G L(UlL2; FmL2). Moreover, L-spectrum of operator M G L(U; F) coincides with L-spectrum of operator M G L(UlL2; FmL2).
The interested reader is encouraged to prove this statement. According to Lemma 2.1, all results of section 1 are transferred from Banach spaces to spaces of differentiable L-"noises".
Let operators L, M G L(UlL2; FmL2). Consider equation
L n = Mn. (12)
Stochastic L-process n G C1(R; UlL2) is called a solution of equation (12), if n substituted to equation (12) makes (12) true. Solution n = n(t) of equation (12) is called a solution of Cauchy problem,
n(0) = no, (13)
for equation (12), if equality (13) holds for some random L-variable n0 G ULL2.
Definition 5. Set PlL2 C UlL2 is called a stochastic phase space of equation (12), if
(i) almost surely each path of solution n = n(t) of equation (12) belong to PlL2, i.e. n(t) G PlL2,t G R for almost all paths;
(ii) there exists the unique solution of problem (12), (13) for almost all paths n0 G PlL2 .
Since the solution of problem (12), (13) is a stochastic L-process, and only one of its paths is observed in reality, we consider necessary to make an explanation. Recall [5-7] that stochastic L-processes n = n(t) and Z = Z(t) are considered equal, if almost surely each path of one of them coincides with any path of the other. Next, extend projector P of section 1 from Banach space U to the space of random L-variables UlL2 . It is easy to show that operator P G L(UlL2) is also a projector. Set ULL2 = ker P, ULL2 = imP such that ULL2 = ULL2 © ULL2.
Theorem 4. Let operators L, M G L(UlL2; FmL2), where operator M is (L,p)-bounded. Then phase space of equation (12) is space ULL2.
Note that if there exists operator L-1 G L(FML2; UlL2), then ULL2 = UlL2.
Definition 6. Subspace ILL2 C UlL2 is called an invariant space of equation (12), if for any n0 G IL L2 the solution of problem (12), (13) is n G C1(R; IlL2).
Note that if equation (12) has phase space PlL2 and invariant space IlL2, then IlL2 C PlL2 .
Definition 7. Solution n = n(t) of equation (12) has exponential dichotomy, if
(i) phase space PlL2 of equation (12) splits into a direct sum of two invariant spaces (i.e. PlL2 = I+L2 © I-L2), and
(ii) there exist constants Nk G R+, vk G R+, k = 1, 2, such that
11n1 (t)||u < N1 e-vi(s-t)||n1(s)||u for s > t, ||n2(t)||u < N2e-V2(s-t)||n2(s)||u for t > s,
where n1 = n1(t) G I+L2 and n2 = n2(t) G I-L2 for all t G R. Space I+L2 (I-L2) is called the stable (unstable) invariant space of equation (12).
Theorem 5. Let M be (L,p)-bounded operator, p G {0} U N, and aL(M) f|{iR} = 0. Then solution n = n(t) of equation (12) has exponential dichotomy.
Let us give an idea of the proof. In order to define projectors in the space of random L-variables ULL2, we use formulas
Pl = 1tii e £(UlL2)> P2 = ¿T J Ri(M)dn G £(UlL2),
ri r2
where contour r (r2) belongs to the left (right) half-plane of the complex plane and bounds a part of L-spectrum of operator M
aL(M) p|{^ : Re^ < 0} (aL(M) Q{^ : Re^ > 0}).
Set I+L2 = imP1 and I-L2 = imP2. Obviously, U[L2 = I+L2 © I-L2. Let n1 G I+L2. If s > t, then
ll(t)U<e-v^±- I \R^M)\ \dr\ ЫШ^ < N^is)^,
where r = ^ + vb and Rer > 0, r G Г1. The estimate for n2 G I- L2 is obtained similarly.
3. Exponential Dichotomies of the Barenblatt-Zheltov-Kochina Stochastic Equation in Spaces of Differential Forms
Let Пп be a n-dimensional smooth compact oriented connected Riemannian manifold without boundary, and Eq = Eq (Qn), 0 < q < n be a space of differential q-forms on Qn. In particular, E0(Rn) is a space of functions of n variables. Consider Laplace-Beltrami operator A : Eq ^ Eq, defined by equality A = id + d$, where d : Eq ^ Eq+1 is the operator of external differential from differential forms, and $ : Eq ^ Eq-i can be presented as linear Hodge operator $ = (—l)n(q+1)+1 * d*, * : Eq ^ En-q, which associates a q-form on Qn with (n — q)-form. Denote the space of harmonic q-forms by Hq = (w G Eq : Aw = 0}.
It follows from the Hodge decomposition (see, for example, in [9])
Eq = A(Eq) ф Hq = d$(Eq) 0 $d(Eq) 0 Hq (14)
that equation Aw = a has solution w G Eq, if a q-form a is orthogonal to space of harmonic forms Hq.
Define scalar product in space Eq, q = 0,1,..., n, by formula
(£,n)o = J £ A*n, £,П G Eq, (15)
where * is the Hodge operator, and denote the corresponding norm by || • ||0. Continue
n
scalar product (15) to direct sum ф Eq, such that different spaces Eq are orthogonal. Let
q=0
H0 be a completion of space Eq by norm || ■ ||0, and PqA be an orthoprojector on HA-Introduce the scalar product on Eq by formulas
(£,n)1 = (A£,n)o + (£a ,nA)0, (16)
(e,n)2 = (A£, An)0 + (e,n)1, (17)
where = PqAw. Let H? and H2 be completions of lineal Eq by corresponding norms || ■ ||1 and || ■ ||2, respectively. In fact, sub index means how many times q-forms are differentiable in generalized sense in the corresponding spaces. Spaces Hq, l = 1, 2, are Banach spaces (their Hilbert structure does not interest us further), moreover, there exist continuous and dense embeddings H2 C H1 C H0, and for any q = 0,1,..., n their exist the splittings of spaces
h' = h'a © HA,
where HqA = (I - Pa)[H'], l = 0,1, 2.
Define spaces HqL2 of smooth differential q-forms
x1 , x2 , xn) ^ ^ \il,i2,...,iq (t, X1 , x2 , xn)dxil ^ dxi2 ^ --- ^ dxin ,
|il,i2,...,iq |=q
where coefficients xil)i2,...,iq(t,x1 ,x2, ...,xn) G C1(R; ULL2) are stochastic L-processes, and xi are one-dimensional Brownian processes. We can separate time and local coordinates at non-relativistic speeds. Here time t is the same at all points of the manifold and impacts only on the coefficients of differential forms that are stochastic continuous L-processes differentiable in the sense of Nelson-Gliklikh. For fixed a G R, A G R introduce operators
L = (A + A), M = aA, (18)
where A is the Laplace-Beltrami operator. Consider a stochastic equation with differential forms
L n= Mn (19)
with Cauchy condition
n(0) = n0- (20)
Paper [8] establishes solvability of problem (19), (20). Introduce
I+L2 = {n G HqL2 : = 0, v > A} , (21)
and
I-L2 = {n G HqL2 : (■, <A)c= 0, vi < A} . (22)
The following theorem is true.
Theorem 6. For any a G R, A G R+, no G ULL2, solution n = n(t) of problem, (19), (20) has exponential dichotomies, and I+L2 and I-L2 of form (21), (22) are infinite-dimensional stable and finite-dimensional unstable invariant spaces of equation
(19), respectively.
Remark 1. For any a G R_, A G R+ and no G UlL2, solution n = n(t) of problem (19),
(20) has exponential dichotomies, and there exist finite-dimensional stable and infinite-dimensional unstable invariant spaces of equation (19). For any a G R+ and A G R- we can only talk about exponential stability of solutions of problem (19), (20), and the solutions of problem (19), (20) are exponentially unstable for a, A G R-.
Conclusion
Further, we plan to continue the results of the paper in several directions. Namely, to generalize the results for the case of sectorial operator [5] and even more general case of radial operator [6]. Also, to investigate generalized Showalter-Sidorov problem, and multipoint problem [22]. Moreover, the theory of degenerate resolving groups and semigroups of operators, as well as numerical methods [23], require such a research.
Acknowledgements. The work was supported by Act 211 Government of the Russian Federation, contract no. 02.A03.21.0011.
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Received December 24, 2018
УДК 517.9
БОТ: 10.14529/ шшр190204
ЭКСПОНЕНЦИАЛЬНЫЕ ДИХОТОМИИ В МОДЕЛИ БАРЕНБЛАТТА - ЖЕЛТОВА - КОЧИНОЙ В ПРОСТРАНСТВАХ ДИФФЕРЕНЦИАЛЬНЫХ ФОРМ С «ШУМАМИ»
О.Г. Китаева1, Д.Е. Шафранов1, Г.А. Свиридюк1
1Южно-Уральский государственный университет, г. Челябинск, Российская Федерация
Исследована устойчивость решений в линейных стохастических моделях соболевского типа с относительно ограниченным оператором в пространствах гладких дифференциальных форм, определенных на гладких компактных ориентированных рима-новых многообразиях без края. Для этого в пространстве дифференциальных форм используем вместо обычного оператора Лапласа псевдодифференциальный оператор Лапласа - Бельтрами. В качестве начальных использованы условие Коши и условие Шоуолтера - Сидорова. В связи с недифферинцируемостью, в обычном понимании, имеющегося в модели «белого шума» используем производную стохастического процесса в смысле Нельсона - Гликлиха. Для исследования устойчивости решений устанавливаем наличие экспоненциальных дихотомий разделяющих пространство решений на устойчивое и неустойчивое инвариантные подпространства. В качестве примера используется стохастический вариант уравнения Баренблатта - Желтова - Кочиной в пространстве дифференциальных форм, определенных на гладком компактном ориентированном римановом многообразии без края.
Ключевые слова: уравнения соболевского типа; дифференциальные формы; стохастические уравнения; производная Нельсона - Гликлиха.
Ольга Геннадьевна Китаева, кандидат физико-математических наук, доцент, кафедра «Уравнения математической физики», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Дмитрий Евгеньевич Шафранов, кандидат физико-математических наук, доцент, кафедра «Уравнения математической физики», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Георгий Анатольевич Свиридюк, доктор физико-математических наук, профессор, кафедра «Уравнения математической физики», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Поступила в редакцию 24 декабря 2018 г.
