INVARIANT MANIFOLDS OF THE HOFF MODEL IN "NOISE" SPACES Текст научной статьи по специальности «Математика»

MSC 35S10, 60G99 DOI: 10.14529/mmp210402

INVARIANT MANIFOLDS OF THE HOFF MODEL IN "NOISE" SPACES

O. G. Kitaeva, South Ural State University, Chelyabinsk, Russian Federation, kit aevaog@ susu.ru

The work is devoted to the study the stochastic analogue of the Hoff equation, which is a model of the deviation of an I-beam from the equilibrium position. The stability of the model is shown for some values of the parameters of this model. In the study, the model is considered as a stochastic semilinear Sobolev type equation. The obtained results are transferred to the Hoff equation, considered in specially constructed "noise" spaces. It is proved that, in the vicinity of the zero point, there exist finite-dimensional unstable and infinite-dimensional stable invariant manifolds of the Hoff equation with positive values of parameters characterizing the properties of the beam material and the load on the beam.

Keywords: the Nelson-Gliklikh derivative; stochastic Sobolev type equations; invariant manifolds.

Introduction

The Hoff model

(A + A)u = au + ¡u3, (1)

u(x, 0) = uo, u e E, u(t, 0) = 0, (x, t) G dE x R (2)

is a model of buckling of an I-beam from the equilibrium position. Here E C Rn is a bounded domain with a smooth boundary dE, the parameter A e R+ is the parameter responsible for the load applied to the beam and a, 3 e R are the parameters responsible for the material from which the beam is made. The paper [1] considers the set of valid initial data of problem (1), (2) understood as a phase space. Here equation (1) was reduced to the semilinear Sobolev type equation

Lu = Mu + N (u), (3)

where L, M, N : U ^ F are the operators and U, F are Banach spaces selected in a special way. In [2], the proof of smoothness and simplicity of the phase space of the equation for positive values of the parameters a and 3 is considered. The stability of solutions to equation (1) in a neighborhood of the zero point is described in [3], which shows the existence of stable and unstable invariant manifolds.

The purpose of this paper is to study the stability of the stochastic analogue of equation (1). We consider the Hoff equation as a special case of the stochastic semilinear Sobolev type equation

L n = Mn + N(n). (4)

o

Here, n denotes the Nelson-Gliklikh derivative [4]. Currently, a large number of papers are devoted to the problem on the solvability of a linear (N = O) equation of the form

(4). Let us note only some of them. The paper [5] considers the existence of solutions to the Cauchy problem

lim (n(t) - no) = 0, (5)

the Showalter-Sidorov problem

P(n(0) - no) = 0 (6)

for linear equation (4) (N = O) in the case of the (L,p)-bounded operator M, p G {0} (J N. Investigation of problems (5), (6) for equation (4) if N = O in the case of the relative sectorial operator M is presented in [6], and in the case of the relative radial operator M is considered in [7].

The paper [8] considers the study of the nonlinear stochastic Sobolev type equation

o

L n= N(n). (7)

The paper establishes the conditions for the existence of solutions to equation (7). In our study, the question of the stability of a semilinear equation of the form (7) is solved. In the linear case (N = O), the existence of stable and unstable invariant spaces was shown in [9]. This work is a continuation of [8,9] on the study of local stability of a semilinear stochastic equation.

The paper is organized as follows. Section 1 contains some concepts and statements on the theory of stability of Sobolev type equations. In Section 2, we describe differential forms with coefficients from a specially selected "noises" spaces obtained by Nelson-Gliklikh derivative. In this sections, we research the exponential dichotomy of linear Sobolev type equations and invariant manifolds of semilinear Sobolev type equations. In Section 3, we present an example for the stochastic analogue of the Hoff equation.

1. Invariant Manifolds of Sobolev Type Equations

Let U and F be Banach spaces, L, M G L(U; F) be operators. The set pL(M) = (u G C : GuL - M)-1 G L(F;U)} is called the L-resolvent set and the set aL(M) = C \ pL(M) is called the L-spectrum of the operator M. The operator M is called the (L,a)-bounded operator, if aL(M) is bounded.

Let M be a (L, a)-bounded operator. Then there exist a splitting of the spaces U0 (U1) = kerP (imP), F0 (F1) = kerQ (imQ), the operators Lk (Mk) G L(Uk;Fk)

Here

(k = 0,1), and the operators M0-1 G L(F0; U0) and L-1 G L(F1; U1) (see, for example, [10]).

P=— [UL - M)~lLdn G £(il), Q = — [ LUL - M)~ldn G C($) 2ni J 2ni J

Y Y

are projectors, and the closed contour 7 C C bounds a domain containing aL(M). Consider the operators H = L-1 M0 G L(U0) and S = L-1 M1 G L(U1). If the operator M is (L, a)-bounded operator and H = O, p = 0 or Hp = O, Hp+1 = O, then the operator M is called (L,p)-bounded operator.

The vector function u G Ck((-t, t); U), k G N U satisfying equation (3) for some t G R+ is called a solution to this equation. The solution u = u(t) to equation (3) is called a solution to the Cauchy problem,

u(0) = u0 (8)

for equation (3), if equality (8) is satisfied for some u0 G U.

Definition 1. The set P e U is called the phase space of equation (3) if

(i) any solution u = u(t) to equation (3) belongs to P, i.e. u(t) e P for every t e ( t, t);

(ii) for any u0 e P, there exists a unique solution u e Ck((-t,t); U), k e N U {to} to Cauchy problem (8) for equation (3).

Cauchy problem (8) for equation (3) can be either unsolvable in general, or solvable, but not uniquely, even in the case when to is a pole of the order p e {0} U N of the L-resolvent of the operator M. Starting from the paper [1], in order to study the solvability of Cauchy problem (8) for equation (3), it is proposed to limit to quasi-stationary trajectories, i.e. such solutions to equation (3) for which Hu0(t) = O. These solutions belong to the set

M = {u e U : (I - Q)(Mu + N(u)) = 0}.

Note that if the operator N = O, then the set M = U1.

Let u e M. The set M is called the Cl-manifold at the point u, if there exist neighborhoods O C M and O1 C U1 of the points u e M and u1 = Pu e U1, respectively, and the Cl-diffeomorphism D : O1 ^ O such that D-1 is a restriction of the projector P by M, l e N U {to}. The pair (D, O) is called a map of the set M. The set M is called a Banach Cl-manifold if it is such at each of its points. A connected Banach Cl-manifold is called a simple manifold, if any of its atlases is equivalent to an atlas containing a single map.

Theorem 1. [1] Let M be a (L,p)-bounded operator, p e {0} U N, the operator N e Ck(U,F), and the set M be a simple Banach Cl-manifold at the point u0. Then, for some t e R+, there exists a unique solution u e Cm((-t, t); M), m = min{k,l}, to equation (3) passing through the point u0.

Remark 1. If the operator N = O, then the set M = U1 and the phase space of the equation

Lu = Mu (9)

is a subspace of U1.

Definition 2. If, for any solution u0 e J C U to problem (9), (8) is u e C 1(R; 1), then the space J is called an invariant space of equation (9).

Remark 2. For the existence of invariant spaces, it is sufficient to fulfill the condition

aL (M ) = af(M )U 4(M), ^L (M ) = 0, af(M) is a closed set.

Remark 3. Any invariant space J of the equation (7) is a subspace of its phase space. Definition 3. If there exist constants N1(2), v1(2) e R+ and

II^COHu < N1 e-vl(s-i)|u1(s)|u for s > t (u1 e 1+) (||u2(t)||u < N2e-V2(t-s)||u2(s)|u for t > s (u2 e I-)),

then invariant space 1+(-) C P is called a stable (unstable) invariant space of equation

(9).

(10)

Remark 4. (i) If J+ = P (J- = P), then we talk about the stability (unstability) of the stationary solution to equation (9).

(ii) If P = J+ © J-, then there exists an exponential dichotomy of solutions to equation

(9).

(11)

Let the following condition be fulfilled:

aL(M) = a+ (M) U aL(M) and (M) = (u G aL(M) : Reu > (<)0}, a+(_)(M) = 0

Then we can construct the projectors

P«r) = ¿t J Ri(M)dta, Ql(r) = ¿t y

Yl(r) Yl(r)

where the contour 7^) belongs to the left (right) half-plane and bounds the domain containing the part of the L-spectrum of the operator M which belongs to this half-plane.

Theorem 2. [4] Let M be the (L,p)-bounded operator and condition (11) be fulfilled. Then there exist the stable J+ = imp and unstable J- = imPr invariant spaces of equation (9).

Definition 4. The set

M+(-) = (u0 G U : ||PWu0||u < R1, ||u(t,u0)|u < R2, t G R+(-}

is such that

(i) M+(-) is diffeomorphic to a closed ball in 1+(-);

(ii) M+(-) touches 1+(-) at the zero point;

(iii) for any u0 G M+(-) and for t ^ + (-||u(t, u0)||u ^ 0 is called a stable (unstable) invariant manifold of equation (3).

Here u(t, u0) is a quasi-stationary trajectory of equation (3) passing through the point u0 G M.

Theorem 3. [3] Let M be the (L,p)-bounded operator, p G (0} U N, condition (11) be fulfilled, and the operator N G C^(U,F) be such that N(0) = 0, N0 = O. Then for some Rj, j = 1, 2 there exist the stable and unstable invariant manifolds of equation (3). Moreover, if for some u0 G M, there exist |P1(r)u0|U < R1 and |u(t,u0)|U < R2 for t ^ + (-)ro, then u0 G M+(-).

2. Stable and Unstable Invariant Manifolds in "Noise" Spaces

Let Q = (Q, A, P) be a complete probability space and L2 be a set of random variables £ : Q ^ R, whose mathematical expectation is zero (E£ = 0) and the variance (D) is finite. In L2, we define the scalar product (£1,£2) = E£1£2. Denote by L2 C L2 the subspace of random variables measurable with respect to A0, where A0 is a a-subalgebra of the a-algebra A. The orthoprojector n : L2 ^ L0 is called a conditional mathematical expectation and is denoted by E(£|A0).

The mapping n : R x Q ^ R is called a stochastic process. If we fix t G J C R, then the stochastic process n = n(t, ■) is a random variable. If we fix u G Q, then the stochastic

process n = n(',u) is called a trajectory. If almost certainly all the trajectories of the stochastic process n are continuous (i.e. for almost all u G H the trajectories of n(',u) are continuous), then n is called a continuous process. Denote by CL2 the set of continuous process. Fix n G CL2 and t G J, and denote by Nf the a-algebra generated by a random variable n(t) and E? = E(-|N7).

Definition 5. [4] Let n G CL2. If there exists the limit

5= I ( Um El + + lim ^ ^(t.O-^-At.QNN

2 yAt—s-o+ f \ At J Ai^o+ f \ At J J '

o

then n is called the Nelson-Gliklikh derivative of the stochastic process n at the point t G J.

Denote by C1 L2, l G N the space of stochastic processes whose trajectories are almost certainly differentiable by Nelson-Gliklikh on the interval J up to the order l inclusively. The spaces C1 L2 are called the spaces of differentiable "noises".

Let U (F) be a real separable Hilbert space with a basis {^k} ({^k}) orthonormal with respect to the scalar product < ■, ■ >U (< ■, ■ >f). Choose the sequence K = {Ak} C R

ro

such that A2 < to, and the sequence {£k} C L2 ({(k} C L2) of uniformly bounded k=i

random variables. Next, we construct the random K-value

ro / ro

£ = S Afc^ ( Z = Afc Zfc

fc=i V fc=i /

The completion of the linear shell with the random K-values according to the norm ll£IIukL2 = £ Aklie||FKL2 = £ AkDZk

is a Hilbert space, which we denote by UKL2 (FKL2) and call the space of random K-values.

The stochastic process n : (e, t) ^ UKL2 is defined by the formula

ro

n(t) = }_^ Ak £k (t)^k, (12)

k=1

where {£k} is some sequence from CL2 and J = (e, t) C R, which is called a stochastic continuous K-process, if the number on the right side converges uniformly on any compact set in J with the norm || ■ ||ukL2 , and the trajectory of the process n = n(t) is almost surely continuous. A continuous stochastic K-process n = n(t) is called a process continuously differentiable by Nelson-Gliklikh on J, if the series

ro

oo

n (t) = £ Ak £k (%k (13)

k=1

converges on any compact in J according to the norm || ■ ||u and the trajectories of

oo

the process n=n (t) are almost certainly continuous. The symbol C( J, UKL2) denotes the

space of continuous stochastic K-processes and the symbol Cl( J, UKL2) denotes the space of the stochastic K-processes continuously differentiable up to the order l G N. Examples of the vector space Cl L2 and the stochastic K-process continuously differentiable up to any order l G N are given by the stochastic process describing Brownian motion in the Einstein-Smolukhovsky model

ro k=0

where ^ G L2, = [f(2k + l)]"2, k G {0} U N, /3 (t) = t G R+, and the Wiener's K-process

ro

WL(t) = J] Ak& (t)^k, k=1

where } C Cl L2 is a sequence of Brownian motion on R+ [4,5].

The following lemma gives the opportunity to transfer all the considerations of Section 1 to the spaces of the random K-values.

Lemma 1. The operator A : U ^ F is a linear and continuous operator (A G L(U; F)) if and only if the same operator A : UKL2 ^ FKL2 is a linear and continuous operator (A gL(UkL2; FKL2)).

Remark 5. Let af(M) be the L-spectrum of the operator M, where the operators L, M : U ^ F, and af(M) be the L-operator spectrum of the operator M, where the operators L, M : UKL2 ^ FKL2. Then af (M) = a^(M).

Assume that the operators L, M G L(UKL2; FKL2), consider the equation

L n = Mn + N(n). (14)

Let J = {0} U R+. A stochastic K-process n G C 1 (J; UKL2) is called a solution to equation (14), if all its trajectories satisfy equation (14) for all t G J .A solution n = n(t) to equation (14) is called a solution to the Cauchy problem,

lim (n(t) - no) = 0, (15)

i^+ro

if equality (15) holds for some random K-value n0 G UKL2.

Definition 6. The set PKL2 C UKL2 is called a stochastic phase space of equation (14), if

(i) probably almost every solution path n = n(t) of equation (14) belongs to PKL2, i.e. n(t) G PKL2,t G R, for almost all trajectories;

(ii) for almost all n0 G PKL2, there exists a solution to problem (14), (15).

Let M be the (L,p)-bounded operator. Then we can extend the projector P considered in Section 1 from the Banach space U to the space of the random K-values UKL2. If condition (11) is satisfied, then we extend the projectors Pi and Pr by UKL2. Denote UKL2 = imP, UKL2 = imPi and UKL2 = imPr. Along with semilinear equation (14), we consider the linear equation

L n= Mn (16)

with the initial condition

n(0) = no- (17)

Theorem 4. Let the operators L, M G L(UKL2; FKL2) and M be the (L/p)-bounded operator. Then the phase space of equation (16) is the space UKL2.

Remark 6. Under the conditions of Theorem 4, if there exists an operator L-1 G L(FkL2; UkL2), then UKL2 = UkL2.

Definition 7. The subspace IKL2 C UKL2 is called the invariant space of equation (16), if, for any n0 G IkL2, solution to problem (16), (17) n G C1(R; IKL2).

Remark 7. If equation (16) has a phase space PKL2 and an invariant space IKL2, then

IKL2 C PKL2.

Definition 8. Solutions n = n(t) of equation (16) have an exponential dichotomy if

(i) the phase space PKL2 of equation (16) splits into a direct sum of two invariant spaces (i.e. PkL2 = IKL2 © IKL2);

(ii) there exist constants Nk G R+, vk G R+, k = 1, 2 such that

ln1(t)|uKL2 < Nie-vi(s-i)11n1 (s)||ukL2 for s > t, lln2(t)|uKL2 < N2e-V2(i-s) ||n2(s)|uKL2 for t > s,

where n1 = n1(t) G IKL2 and n2 = n2(t) G IKL2 for all t G R. The space IKL2 (IkL2) is called the stable (unstable) invariant space of equation (16).

Theorem 5. [9] Let M be the (L,p)-bounded operator and condition (11) be fulfilled. Then the solutions of equation (16) have an exponential dichotomy and the spaces UKL2 and UKK2 are stable and unstable invariant spaces of equation (16).

Next, we arrive at questions about the solvability and stability of stochastic semilinear equation (3). If, for some fixed u G H, there exists a solution n = n(t) to equation (14), then n belongs to the set

M L = I {n G UkL2 :(I - Q)(Mn + N(n)) = 0}, if ker L = {0}; MkL2 1 UkL2 , if ker L = {0},

and the following theorem is true.

Theorem 6. [8] Let M be the (L,p)-bounded operator, the operator N G C1 (UkL2, FKL2), and the set MKL2 be a simple Banach C1-manifold at the point n0 G UkL2. Then the set MKL2 is the phase space of equation (14).

Definition 9. The set

MK(-)L2 = {no G UkL2 : ||Pi(r)no||uKL2 < R, lln(t, no)||uLL2 < R, t G R+(-} is such that

(i) MK(-)L2 is diffeomorphic to a closed ball in IK(-)L2;

(ii) MK( )L2 concerns IK( )L2 at the zero point;

(iii) for any n0 G M+( )L2, for t ^ +(-||n(t, n0) I|ulL2 ^ 0 is called a stable (unstable) invariant manifold of equation (14).

From Lemma 1 and Theorem 3 we obtain the following result.

Theorem 7. Let M be the (L,p)-bounded operator, condition (11) be fulfilled, and the operator N G Ck(U, F) be such that N(0) = 0, N0 = O. Then there exist stable and unstable invariant manifolds of equation (14) in the neighborhood of the zero point.

3. Hoff Stochastic Equation

◦1 -i

Consider the stochastic analogue of equation (1). Let U = W2, F = W2 (functional spaces are defined on the domain £). The space U is a real separable Hilbert space densely and continuously nested in F. In the space U, basis is orthonormal in the sense of U of consecutive eigenfunctions } of the Laplace operator A corresponding to {vk}. Here {vk} is a sequence of eigenvalues of the Laplace operator numbered in nondecreasing order taking into account multiplicity. By analogy with Section 2, we construct the spaces of the random K-values UKL2, FKL2 and the spaces of differentiable "noise" C'UKL2,

ro

l G {0}U N. Let K = {Ak} be a sequence such that A| < For example (see [5]),

k=1

as K = {Ak}, we can choose a sequence of eigenvalues of the Green operator Ak = |vk |-m

ro

(here m G N is chosen in such a way that the series Y1 |vk|-m converges).

k=1

The operators L, M and N are defined by formulas L : x ^ (A + A)x, X G UwkL2, M : x ^ aAx, N : n ^ ftx3, X G UKL2. (18) Then the stochastic analogue of Hoff equation (1) is represented as the equation

L X = Mx + N(x). (19)

Lemma 2. For any A G R+ and a, ft G R \ {0},

(i) the operators L, M G L(UKL2; FKL2);

(ii) the operator M is (L, 0)-bounded operator;

(iii) the operator N G Cro(UkL2; UkL2), N(0) = 0 and N0 = O.

Proof. (i) According to Lemma 1, the operators L, M G L(UkL2; FkL2). (ii) The L-spectrum of the operator M has the form

L/— 1 a

aL(M) = Uk = —-, (20)

[ л + J

therefore it is bounded. The kernel of the operator L has the form

ker L = span{w : v = — Л}.

If ф G ker L \ {0} then

ф = aiw' lail > 0

VI=—A v;=-A

and

Мф = a аг^г / imL.

VI=—A

(iii) Let x £ UKL2. Then the Frechet derivatives N^, N"X of the operator N at the point n has the form

NX : x ^ 3^x2, NX' : x ^ 6£x-

All other Frechet derivatives of the operator N at the point u are zero. Obviously, the operator N(0) = 0 and the Frechet derivative N0 = O.

□

Theorem 8. Let a,5 > 0. Then the phase space of equation (19) is the set

fix £ Uk L2 : E < (1 - a - £x2)x, ^ > ^ = 0, if - A = MK L2 == < -A=v;

{ UkL2 , if A = Vj.

Proof. The statements of the theorem follow from Lemma 1, Lemma 19, Theorem 6 and the results of [1].

□

Remark 8. The subspace UKL2 = {x £ UKL2 :< x, >= 0, —A = v^j is a model of the manifold MKL2 for —A = vj.

Theorem 9. Let a, A £ R+.

(i) If A < —v1; then equation (19) has only a stable invariant manifold that coincides with MkL2 .

(ii) If — v1 < A, then there exist a finite-dimensional unstable invariant manifold MKL2 and an infinite-dimensional stable invariant manifold MKL2 of equation (19) in the neighborhood of the zero point.

Proof. (i) Let A < v1, then the L-spectrum of the operator M has the form

aL{M) = aL_{M) = (t-^— : A < —U\ [A + vfc

Following Theorem 5, the linear part of equation (19) has only a stable invariant space that coincides with the subspace UKL2. The existence of an unstable invariant manifold that coincides with MKL2 follows from Theorem 7 and Theorem 8.

(ii) If A > — v1, then the L-spectrum of the operator M consists of two parts aL(M) = aL (M)U (M), where

cL-{M) = : A < -uk\, arL(M) = : A > -uk

I A + vfc J [ A + vfc

The existence of a stable manifold MKL2 of equation (19) follows from Theorem 5, Theorem 7 and Theorem 8. The model of the set MKL2 is an infinite-dimensional subspace I^-L2 = span{ui : —A < u{\. The finite-dimensional space IkL2 = span{ui : —ui < A} is a model of an unstable manifold MKL2.

□

Conclusion

In the future, following [12-14], it is proposed to conduct numerical experiments on the study of invariant manifolds of the stochastic analogue of the Hoff model.

References

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Received August 3, 2021

УДК 517.9 DOI: 10.14529/mmp210402

ИНВАРИАНТНЫЕ МНОГООБРАЗИЯ МОДЕЛИ ХОФФА В ПРОСТРАНСТВАХ «ШУМОВ»

О.Г. Китаева, Южно-Уральский государственный университет, г. Челябинск,

Российская Федерация

В данной работе изучается стохастический аналог уравнения Хоффа, который является моделью отклонения двутавровой балки от положения равновесия. Показана устойчивость модели при некоторых значениях параметров данной модели. При исследовании модель рассматривается как стохастическое полулинейное уравнение соболевского типа, где стохастический процесс выступает в качестве искомой величины. Установлены достаточные условия существования инвариантных многообразий полулинейного стохастического уравнения соболевского типа. Полученные результаты перенесены на уравнение Хоффа, рассматриваемого в специально построенных пространствах «шумов». Доказано, что в окрестности точки нуль существуют конечномерное неустойчивое и бесконечномерное устойчивое инвариантные многообразия уравнения Хоффа при положительных значениях параметров, которые определяют свойства материала балки и нагрузку на балку.

Ключевые слова: производная Нельсона - Гликлиха; стохастические уравнения соболевского типа; инвариантные многообразия.

Литература

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Ольга Геннадьевна Китаева, кандидат физико-математических наук, доцент, кафедра «Уравнения математической физики>, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), kitaevaog@susu.ru.

Поступила в редакцию 3 августа 2021 г.