Invariant spaces of Oskolkov stochastic linear equations on the manifold

Consider the Oskolkov equation

( X - А ) А ^ = v A² ^ — E ^ i A ^ ) .                          (1)

( x, x 2 )

Equation (1) is a model of the flow of a viscous and elastic incompressible fluid [1], in which the parameter ν is responsible for the viscous properties of the fluid. The parameter λ that determines the elastic properties of a fluid can take positive and negative values [2]. In [3–5], the solvability of the Cauchy-Dirichlet problem for the Oskolkov equation (1) was considered, and in [6] for the linear Oskolkov equation

( X -A ) A ^ = v A ² ^ .                                 (2)

In [7], the problem of stability of solutions to equation (2) was solved in terms of exponential dichotomies, and in paper [8], the problem of stability of solutions in a neighborhood of the zero point of equation (1) was solved in terms of invariant manifolds.

This article discusses the stability of the stochastic linear Oskolkov equation on manifold that has no boundary. To solve this issue, we use equation (2) as an equation of the following form:

L^о η = M η ,                                      (3)

where on derivative in the sense [9] of the sought-for random process n = n ( t ). The number of works devoted to the study of equations of the form (3) is quite large at the present time (see, for example, [10– 13]), in which this equation was considered in various aspects. The present work is closest to [14] and [15], in which we study the solvability and stability of the Barenblatt–Zheltov–Kochina stochastic equation on a manifold.

The article contains four parts. The first section is the introduction, the fourth section is the bibliography. The second point is dedicated to deals with spaces of q-forms defined on a manifold that has no boundary, recalls the notions of a random variable, stochastic process, Nelson–Gliklikh derivative, constructs spaces of random K -variables and K -“noises”. The third point contains a description of the invariant spaces of the Oskolkov stochastic equation.

1. Spaces of “noises” on a manifold

Consider n -dimensional manifold Q that has no boundary. Let it have the properties of connectedness, compactness and smoothness. Consider spaces of smooth shapes E^q = E^q ( M ), 0 <  q <  n on Q , where the scalar products are defined by the following formulas:

(a, b )0 = J aЛ* b, (a, b )2 = (Aa, Ab )0 + (Aa, b )0 +( a, b )0,

Q

A = dS + Sd is the Laplace-Beltrami operator, 5 = (-1)n(q+1)+1 * d *, where * is the Hodge operator associating a differential form Eq with a differential form En q , d is the outer differentiation operator. The spectrum ^(A) = {ak} of the operator A is positive discrete, and +да is the point of its condensation. Denote by Hq and Hq the completions of the lineal Eq with respect to the norms || • || and || -|| . The basis in Hilbert spaces Hf is the sequence of eigenfunctions {^k} of the operator A orthonormalized by the norms || • ||, . l = 0,2 .

Next, we turn to the construction of spaces of random K - variables and K -“noises” in H ^q . Let

Q = ( Q , А , Р ) be a full probability space. We define a random variable as a mapping £ : Q^ R and stochastic process as mapping ? : 3 x Q ^ R (where 3 is a certain interval from R , a function ? = ? ( - , ^ ) is a trajectory of the stochastic process). If almost all trajectories of a random process are continuous then such a process is called continuous.

• L ₂ is the set of random variables ^ for which the variance D is finite and the mathematical expectation E is zero, and CL ₂ is the set of continuous stochastic processes n . We fix * e 3 , let E * =E ( iy n ) , N n the σ -algebra generated by the random variable η . By the derivative in the sense

• [9] n of the stochastic process ? e CL₂ в * e 3 we mean a limit (if it exists)

n = ¹ f Um E ? ( ? * ^{+ A t} ’ ° - ^{n (} * ’ ■ )  . lim E ? ( ? * ’ - ^{n (} * -^A * ’° 1\

• 2 ^ A * ^ 0 +    ^         n        J   A * ^ 0 +    ^         n        J,

Denote by C ¹ L ₂ the space of stochastic processes whose trajectories are almost sure differentiable in the sense [9] on 3 . The spaces C ¹ L ₂ are called spaces of differentiable “noises”.

Let us introduce into consideration the space Hf, l = 0,2 whose elements are random K -variables к=1

The norm in this space is defined by the following formula:

да

I ? l H q = £ 4 ² D ^ k , ^l     к = 1

where [ck} is a sequence of random variables with bounded variance, {^k} are the eigenfunctions of the operator A , orthonormalized by (•,-)z, l = 0,2 , and K = {Xk} is a monotone sequence such that да

£ Xk < +да . Let C(3;Hq ) be the set of continuous stochastic K -processes к=1

n ( * ) = £ ^ k ? k ( *)Фк , ? k e CL 2 ,                                (4)

k = 1

and C 1 ( 3 ; H^q ) be the set of continuously Nelson-Gliklikh differentiable K -processes

ⁿ ( * ) = £ ^ k ⁿ к ( *)ф_к , ⁿ к e C^lL 2 ,                                  (5)

к = 1

Kitaeva O.G.

if series (4) and (5) converge uniformly on 3 c R ( 3 is compact set in R ).

• 2.    Stable and unstable invariant spaces Let us define the operators

and the equation (2) in the space H q can be considered in the form o Lη = Mη .

The operators L , M : H q ^ H q have the properties of linearity and continuity, and the operator M is ( L , 0) -bounded operator.

By a solution to equation (6) we mean a stochastic K -process n g С ¹ ( 3 ; H q ) if, after substituting it into equation (6), we obtain the identity.

Definition 1. A set P g H q such that the following conditions are satisfied:

• (i)    almost sure each trajectory of the solution n = n ( t ) to equation (6) belongs to P ;

• (ii)    for almost all n ₀ g P , there exists a solution to equation (6) satisfying the condition

  
  

7(0) = По is called a phase space of equation (6).

It was shown (see, for example, [14]) that the phase space of equation (3) is the image of the resolv ing group Ut = —— f (цL - M)-1 Le^dц . Therefore, the following theorem is true.

2eH

Y

Theorem 1. The set of the following form: P =f Hq, Xg^}, nG H0 :(n, Фп )0 =

0, X = a n

is the phase space of equation (6)

If the solution to problem (6), (7) is n g С 1 ( 3 ; I ) for any П о g L , I c P , then the set I called a invariant space of equation (6).

Definition 2. A set I such that the following conditions are satisfied

• (i)    I₊ is an invariant space;

• (ii)    | |n 1 ( t )|| ^_q <  N 1 e m ^{1( 5} t ) m 1 ( ⁵ )||_яу , ⁵ - 1 , where positive constants N 1 , m 1 , n 1 g I ₊ for all t g R

is called a stable invariant space of equation (6). A set I such that the following conditions are satisfied

• (i)    I₊ is an invariant space;

• (ii)  || n ² ( t )|| _Hq <  N ₂ e m ² ( t ⁵ ) Ш 1 ( 5 )|| _Hq , t - 5 ,where positive constants   N ₂, m ₂ ,   n ² g I -   for all

t G R is called an unstable invariant space of equation (6).

Due to the fact that the relative spectrum of the operator M T L ( M ) = T L ( M ) + T L ( M ), where

-va_k   i         I

----—, X <-a k ) ,

X + a _k

t ₊ ^l ( M ) =

and the results presented in [15] we obtain

Theorem 2. (i) The stable invariant space is set of the form (8) for v >  0 and X <  0.

• (ii)    The stable invariant space is set of the form

and the unstable invariant space is the set of the form

• I -    = { n e H 0 : ( ПФ к ) o = О, ^X<~^a_k }

for v >  0 and X <  0.

Remark 1. For v >  0 and X <  0 there is an exponentially dichotomous behavior of solutions to the equation (6).

Conclusion

In the future, we intend to study the question on the solvability and stability of the stochastic analogue of semilinear equation (1). In addition, we intend to transfer all results for equation (1) to spaces of q -forms defined on a manifold with border.