TIME AVERAGING FOR RANDOM NONLINEAR ABSTRACT PARABOLIC EQUATIONS

It is well known that the averaging principle is a powerful tool of investigation of ordinary differential equations, containing high frequency time oscillations, and a vast work was done in this direction (cf. [1]). This principle was extended to many other problems, like ordinary differential equations in Banach spaces, delayed differential equations, and so forth (for the simplest result of such kind we refer to [2]). It seems to be very natural to apply such an approach to the case of parabolic equations, either partial differential, or abstract ones. However, only a few papers deal with such equations. Most of them deal with linear and quasilinear equations in the case when high oscillations in coefficients and/or forcing term are of periodic or almost periodic nature [4, 6, 9, 8, 13, 14, 18]. Moreover, many applications give rise naturally to parabolic equations with highly oscillating random coefficients. For linear equations of such kind the averaging principle was studied in [15, 16, 17]. Note that, in [17] the so-called spatial and space-time averaging (homogenization) is investigated, while the time averaging is also considered. In the present paper, we study the averaging problem for an abstract monotone parabolic equation, the operator coefficient of which is a stationary (operator valued) stochastic process. We prove that in this case the averaging takes place almost surely, that is, with probability 1. As a consequence, we get an averaging result for the case of almost periodic coefficients (almost periodic functions may be regarded as a particular case of a stationary process). This result is, so to speak, individual, in contrast to the main theorem which is statistical in its nature. Our approach differs from those used in the references we pointed out above, except [17], and is based on the theory of G-convergence of abstract parabolic operators. The last theory was developed in [7] in


Introduction
It is well known that the averaging principle is a powerful tool of investigation of ordinary differential equations, containing high frequency time oscillations, and a vast work was done in this direction (cf.[1]).This principle was extended to many other problems, like ordinary differential equations in Banach spaces, delayed differential equations, and so forth (for the simplest result of such kind we refer to [2]).It seems to be very natural to apply such an approach to the case of parabolic equations, either partial differential, or abstract ones.However, only a few papers deal with such equations.Most of them deal with linear and quasilinear equations in the case when high oscillations in coefficients and/or forcing term are of periodic or almost periodic nature [4,6,9,8,13,14,18].Moreover, many applications give rise naturally to parabolic equations with highly oscillating random coefficients.For linear equations of such kind the averaging principle was studied in [15,16,17].Note that, in [17] the so-called spatial and space-time averaging (homogenization) is investigated, while the time averaging is also considered.
In the present paper, we study the averaging problem for an abstract monotone parabolic equation, the operator coefficient of which is a stationary (operator valued) stochastic process.We prove that in this case the averaging takes place almost surely, that is, with probability 1.As a consequence, we get an averaging result for the case of almost periodic coefficients (almost periodic functions may be regarded as a particular case of a stationary process).This result is, so to speak, individual, in contrast to the main theorem which is statistical in its nature.Our approach differs from those used in the references we pointed out above, except [17], and is based on the theory of G-convergence of abstract parabolic operators.The last theory was developed in [7] in

Statement of the problem and the main result
Let be a probability space, with a probability measure P .Assume that on it is given an action of a measure preserving dynamical system for every measurable set ᐁ ⊂ .In addition, we always assume the dynamical system T (t) to be ergodic.Recall that T (t) is called ergodic if for each measurable function f (ω) on such that f (T (t)ω) = f (ω) almost everywhere (a.e.) one has f (ω) = const.a.e.In what follows we use standard notations for the Lebesgue spaces, as well as for the space of continuous functions.Moreover, f stands for a mean value of measurable function f on : Let V be a separable reflexive Banach space over the field R of reals, and let V * be its dual space and H a Hilbert space identified with its dual, H * = H.It is assumed that V ⊂ H ⊂ V * and all the embeddings here are dense and compact.We denote by ) ) for every u, u 1 , u 2 ∈ V and almost all (a.a.) ω ∈ , where It is always assumed that m ≥ 2m 2 which implies (u 1 , u 2 ) > 0 provided u 1 + u 2 > 0. Now, we introduce a family A ω (t) (ω ∈ ) of operator valued functions defined by It is not difficult to verify (cf.[11]) that for a.a.ω ∈ the operator function A ω (t) is well defined, and satisfies the Carathéodory condition (on the real line now) and inequalities (3.1), (3.2), (3.3), and (3.4) below which are similar to (2.3), (2.4), (2.5), and (2.6).In particular, the operator A ω (t) is bounded, coercive, and strictly monotone uniformly with respect to ω and t.Therefore, due to standard results on abstract monotone parabolic equations (cf.[10]), for a.a.ω ∈ the following Cauchy problem: ) has a unique solution Here τ > 0 is an arbitrary, but fixed, real number.
We remark that at this point the whole set of assumptions (2.3), (2.4), (2.5), and (2.6) is not needed.We use them only to apply the results on G-convergence [11].
Let Â : V → V * be an operator defined by the mean value of A(ω)u.It is easily seen that Â acts continuously from V into V * and satisfies inequalities (3.1), (3.2), (3.3), and (3.4).By the Birkhoff ergodic theorem (cf.[3]), for a.a.ω ∈ one has The following result justifies in the case we consider the principle of averaging.

G-convergence of abstract parabolic operators
To prove Theorem 2.1, we need certain preliminary results on G-convergence (we refer to [11] for more details).First, we recall some definitions.
Let A k (t), t ∈ [0, τ ], (k = 0, 1, . ..) be operators acting from V into V * .Assume that they satisfy the Carathéodory condition on [0, τ ] and inequalities where Consider parabolic operators acting from the space into L p (0, τ ; V * ).Endowed with the graph norm W 0 becomes a reflexive Banach space.As it was already mentioned, due to our assumptions the operators L k are invertible.One says that L 0 is a G-limit of L k , k = 0, 1, . . ., We have the following results [11].We now point out that, in fact, our parabolic operators act on a larger space consisting of all functions from L p (0, τ ; V ) which have first derivative in L p (0, τ ; V * ).Such functions are not necessarily vanishing at 0. ) where u is a (unique) solution of the Cauchy problem for L with the same initial data u 0 .
Proof.Multiplying (3.9) by u k and integrating, we obtain Now due to assumption (3.2), we see that u k is a bounded sequence in L p (0, τ ; V ) and C([0, τ ]; H ). Using (3.1) and (3.9), we obtain from the last observation the boundedness of u k in L p (0, τ ; V * ).Since L p (0, τ ; V ) and L p (0, τ ; V * ) are reflexive spaces, passing to a subsequence, we can assume that u k → u weakly in L p (0, τ ; V ) and u k → u weakly in L p (0, τ ; V * ).In addition, due to Lemma 1.3.4 of [11], we can also assume that u k → u strongly in C([0, τ ]; H ). (In fact, this lemma is stated in [11] only under a stronger assumption u 0 = 0.However, the proof works equally well if we assume only that u k (0) = u 0 ∈ H .) By Theorem 3.1, u is a solution of Lu = f , while u(0) = u 0 due to convergence of u k in C([0, τ ]; H ). Since such a solution u is unique, the passage to a subsequence above is unnecessary and the proof is complete.

Proof of Theorem 2.1
Consider parabolic operators L ω,ε and L generated by the left-hand sides of (2.9) and (2.14), respectively.First, we point out that for a.a.ω ∈ the operators L ω,ε satisfy all the assumptions of Section 3. To prove Theorem 4.1 we need to introduce an operator of "differentiation" along trajectories of our dynamical system T (t) (see [11,Section 3.1], for more details).Associated to T (t), there exists a one-parameter groups of operators G(t) acting in all the spaces L r ( , E), where The operator G(t) is defined by It is easily seen that G(t) is an isometric operator in each space under consideration.Moreover, Now G(t) is considered as an operator in L r ( ; E) (1 < r < ∞), hence, G * (t) acts in L r ( ; E * ).In particular, G(t) is a group of unitary operator in L 2 ( ; H ). The group G(t) is strongly continuous in L r ( ; E), with 1 ≤ r < ∞.The generator ∂ of this group is a closed linear operator in L r ( ; E).Due to (4.3), ∂ is skew-symmetric: where However, for our purpose we need to consider ∂ as an (unbounded) operator from L p ( ; V ) into L p ( ; V * ).Denote by ᐃ( ) the completion of with respect to the norm This is a reflexive Banach space densely embedded into L p ( ; V ).Now the action of ∂ can be extended to ᐃ( ) and we get the desired operator from L p ( ; V ) into L p ( ; V * ), with the domain ᐃ( ).Making use of the same smoothing arguments in [11, Section 3.1], we see that this operator, still denoted by ∂, is skew-symmetric: We also remark that, due to ergodicity assumption, the kernel ker ∂ consists of constant functions on .

G. Bruno et al. 7
Proof of Theorem 4.1.Independently of τ, for a.a.ω ∈ the operators L ω,ε satisfy the assumptions of Theorem 3.1.Hence, for any sequence of ε's converging to 0, there exists a subsequence, still denoted by ε, and a parabolic operator where 0 is a set of measure 1.To prove the theorem it suffices now to show that A 0 (t) = Â for a.a.t ∈ [0, τ ].In particular, this means that the passage to a subsequence above is superfluous.

Almost periodic averaging
We now consider the averaging problem for the equation We assume that the operator function A(t) : V → V * satisfies inequalities (3.1), (3.2), (3.3), and (3.4), and the function is almost periodic, in the sense of Bohr, in t ∈ R uniformly with respect to v ∈ V [12].More precisely, continuous operators from V into V * , having power growth of order p − 1, form a metric space, with the metric To apply Theorem 2.1, we recall the notion of Bohr compactification R B of R [12].There exist a compact abelian group R B and a dense continuous embedding R ⊂ R B of abelian groups such that every almost periodic function on R is, in fact, a restriction to R of a continuous function on R B .Moreover, each continuous function on R B restricted to R gives rise to an almost periodic function.We refer to [12] for detailed presentation of the theory of almost periodic functions from this point of view.
Now we set = R B and denote by P the normalized Haar measure on R B .We define the dynamical system T (t) by ( (5.7) Due to Proposition 3.4, we have L 0,ε G − − → L. Applying Proposition 3.3, we obtain the result.

An example
Now we consider a simple example.Let Q ⊂ R n be a bounded open set and a(ω, t) a stationary stochastic process a.a.realizations of which are contained between two positive constants.The last assumption may be expressed as follows: a(ω, t) = a(T (t)ω), where a(ω) ∈ L ∞ ( ) and a(ω) ≥ α 0 > 0. The equation where â is the mean value of the process a.

Some generalizations
First of all, we note that in (2.9) we can consider the forcing term f of the form f 0 (t) + f 1 (T (t/ε)ω), where f 0 ∈ L p (0, τ ; V * ) and f 1 ∈ L p ( ; V * ).This situation reduces immediately to the case of Theorem 2.1 if we replace the operator A(ω) by a new operator Ã(ω) = A(ω) − f 1 (ω).It is easily seen that Ã(ω) satisfies all the assumptions of Section 2 whenever A(ω) does.Moreover, one can extend Theorem 2.1 to the case when the equation under consideration contains the slow variable t as well as the fast one t/ε, that is, is of the form u + A ω t, t ε u = f, ( where f = f (t), or even f = f ω (t, t/ε).To do this we need only to consider instead of A(ω) an operator function A(t, ω) defined on [0, τ ] × and satisfying the same assumption as in Section 2, with replaced by [0, τ ] × .Certainly, in this case A similar remark concerns with f = f ω (t, t/ε).
• , | • |, and • * the norms in V ,H , and V * * (ω ∈ ) of operators satisfying the Carathéodory condition (C) for almost all ω ∈ the operator A(ω) : V → V * is continuous, while A(ω)u is a measurable V * -valued function for every u ∈ V , and the following inequalities Corollary 1.3.1]).