Note on quantitative homogenization results for~parabolic systems in $\mathbb{R}^d$

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a semigroup $e^{-tA_\varepsilon}$, $t\geqslant 0$, generated by a matrix elliptic second order differential operator $A_\varepsilon \geqslant 0$. Coefficients of $A_\varepsilon$ depend on $\mathbf{x}/\varepsilon$ and oscillate rapidly as $\varepsilon \rightarrow 0$. Approximations for $e^{-tA_\varepsilon}$ were obtained by T. A. Suslina (2004, 2010) via the spectral method and by V. V. Zhikov and S. E. Pastukhova (2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by T. A. Suslina (2015).


Introduction
The subject of this note is quantitative estimates in periodic homogenization, i. e., approximations for the corresponding resolving operator in the uniform operator topology. There are several approaches to obtaining results of such type, see [BSu,CDaGr,Sh,ZhPas3].
In introduction, let us consider the simplest periodic operator A ε = −divg(ε −1 x)∇, ε > 0, acting in L 2 (R d ). Here g is a periodic positive definite matrix valued function such that g, g −1 ∈ L ∞ . Let u ε be the solution of the equation A ε u ε + u ε = F , where F ∈ L 2 (R d ). The homogenization problem is to describe the behavior of the solution u ε in the small period limit ε → 0. The classical answer is that u ε → u 0 in the L 2 -norm, where the limit function u 0 is the solution of the equation of the same type A 0 u 0 + u 0 = F but with the so-called effective operator A 0 = −div g 0 ∇ with the constant matrix g 0 .
By using the spectral method, M. Sh. Birman and T. A. Suslina [BSu] proven that u ε − u 0 L 2 Cε F L 2 . This estimate can be rewritten as approximation for the resolvent (A ε + I) −1 in the uniform operator topology. Approximations for the semigroup e −tAε , t 0, were obtained in [Su1,Su2], and [ZhPas2] via the spectral and shift methods, respectively: Since the point zero is the lower edge of the spectra for A ε and A 0 , estimate (1) can be treated as a stabilization result for t → ∞. Later in [MSu], it was observed that the quantitative results for parabolic problems can be derived from the corresponding elliptic results with the help of the identity e −tAε = − 1 2πi γ e −ζt (A ε − ζI) −1 dζ, where γ ⊂ C is a contour enclosing the spectrum of A ε in a positive direction. But in [MSu] only problems in a bounded domain O ⊂ R d were studied and operators under consideration were positive definite. In the case of the Dirichlet boundary condition, it was obtained that The unique conceptual difference between (1) and (2) is the behaviour at t → 0. While (2) contains exponentially decaying factor e −ct , we can not speak on stabilization for A D,ε . Indeed, for the difference of the exponentials we have the rough estimate e −tA D,ε −e −tA 0 D 2e −c * t , where c * > 0 is a common lower bound for A D,ε and A 0 D . In (2), the constant c is such that 0 < c < c * and the constant C depends on our choice of c and grows as c → c * . This caused by the used Cauchy integral representation and the behaviour of the error estimate in approximation the resolvent (A D,ε − ζI) −1 for small fixed |ζ|. According to the results of [Su4], the estimate for the resolvent (A ε − ζI) −1 has different behaviour with respect to ζ compared to the one for (A D,ε − ζI) −1 .
The difference between methods of the present paper and [MSu] consists of choosing the contour γ depending on time t. This idea is inspired by the proof of [INZ,Lemma 1].

Preliminaries. Known results
Let Γ ⊂ R d be a lattice, and let Ω be the cell of the lattice Γ. By H 1 per (Ω) we denote the subspace of matrix valued functions from H 1 (Ω) whose Γ-periodic extension belongs to H 1 loc (R d ). For any Γ-periodic matrix valued function f we use the notation we denote the operator of multiplication by the matrix valued function f ε (x).
In L 2 (R; C n ), we consider matrix elliptic second order differential operator A ε , ε > 0, formally given by the expression is a first order differential operator with constant (m × n)-matrix valued coefficients b l , l = 1, . . . , d. The coefficients of matrices g(x) and b l , l = 1, . . . , d, are in general complex. Suppose that m n and that the symbol b(ξ) = d l=1 b l ξ l satisfies the full rank condition rank b(ξ) = n, 0 = ξ ∈ R d . Or, equivalently, there exist constants α 0 , α 1 such that Under the above assumptions, the operator A ε is self adjoint, non-negative and strongly elliptic. The precise definition is given via the corresponding quadratic form on H 1 (R d ; C n ).
The simplest example of the operator under consideration it the acoustics operator A ε = −div g ε (x)∇. The operator of elasticity theory also can be written as b(D) * g ε (x)b(D), see details in [BSu,Chapter 5].
The coefficients of the operator A ε oscillate rapidly as ε → 0. The limit behaviour of its resolvent or the semigroup e −tAε is given by the corresponding function of the so-called effective operator A 0 = b(D) * g 0 b(D) with the constant matrix g 0 . The definition of g 0 is given in terms of the Γ-periodic (n × m)-matrix valued function Λ: where Λ ∈ H 1 per (Ω) is the weak solution of the cell problem By S ε we denote the Steklov smoothing operator acting in L 2 (R d ; C m ) by the rule

According to [ZhPas1, Lemma 1.2], for any Γ-periodic function
. By using this statement and inclusion Λ ∈ H 1 per (Ω), one can show that the so-called corrector acts continuously from L 2 (R d ; C n ) to H 1 (R d ; C n ), and K(ε; ζ) L 2 →H 1 = O(ε −1 ) for fixed ζ ∈ C \ R + . The (L 2 → H 1 )-continuity of the operator (10) below can be checked by using the same arguments.
The following result was obtained in [Su4, Theorems 2.2 and 2.4].
Theorem 1 ( [Su4]). Let the above assumptions be satisfied. Let ζ ∈ C \ R + , φ = arg ζ. Denote Then for ε > 0 we have Let K(ε; ζ) be the corrector (4). Then for ε > 0 we have The constant C 1 depends only on α 0 , α 1 , g L∞ , g −1 L∞ , and parameters of the lattice Γ. The constants C 2 and C 3 depend on the same parameters and also on m and d.
The aim of the present paper is to give another proof for the following theorem.
Theorem 2 ( [Su1], [ZhPas2], [Su3]). Under the above assumptions, for ε > 0 and t 0 we have Then for ε > 0 and t > 0 we have The constant C 4 depends only on α 0 , α 1 , g L∞ , g −1 L∞ , and parameters of the lattice Γ. The constants C 5 and C 6 depend on the same parameters and also on m and d.

Discussion
Since we derive the parabolic estimates from the elliptic ones, the achievement of the present paper can be interpreted as a quantitative Trotter-Kato like result in homogenization context. For derivation of hyperbolic results from elliptic ones, see the preprint [M].