Convergence rates in homogenization of higher order parabolic systems

This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp $O(\va)$ convergence rate in the space $L^2(0,T; H^{m-1}(\Om))$ is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by \cite{suslinaD2013} is used here.

The proof of Theorem 1.1 is mainly based on the duality argument initiated in [25]. To adapt the ideas, we first provide the existence results for the matrix of correctors χ(y, s) and flux correctors B(y, s) (also referred as dual correctors) for operators ∂ t + L ε (ε > 0) in Section 2. Recall that flux correctors play an essential role in the investigation on sharp convergence rate in the homogenization of second order elliptic or parabolic systems [10,11,12,21,7]. In [7], the flux correctors are obtained by considering a harmonic system with periodic boundary conditions in R d+1 (see Lemma 2.1 therein), which is a modification of the approach for second order elliptic systems. The process however seems not applicable to higher order parabolic systems. Indeed, following the process we will obtain a degenerate elliptic system in R d+1 , which is hard to cope with. Instead, we will modify the construction of flux correctors for elliptic systems in another manner to construct the flux correctors for high order parabolic systems, see Lemmas 2.1 and 2.2. This approach also provides us further regularity results on the flux correctors (see (2.5) 2 ). As we can see from the definition of w ε in (1.12), the higher regularity (H 2m−1 ) on B γ(d+1)β (or B γ(d+1)β ) are essential, which however is trivial for second order parabolic systems (m = 1) [7].
Since we consider the systems with coarse coefficients, the correctors χ(y, s) and the flux correctors B(y, s) may be unbounded. Therefore, similar to [7], in Section 3 we introduce the smoothing operator S ε with respect to the space and time variables x, t and establish proper estimates for the smoothing operator. However, to deal with the higher order operators, more general estimates are proved by using an approach quite different from [7].
With preparations in Sections 2 and 3, in Section 4 we introduce the function where ρ ε , ̺ ε are proper cut-off functions, S 2 ε = S ε • S ε , see (4.6) for the details. Then through some delicate analysis and proper use of preparations aforementioned, we prove the following O √ ε The above estimate should be comparable to (3.17) or (3.20) in [7]. Yet, we point out that the auxiliary function w ε is much more complicated than that for second order parabolic systems constructed in [7]. And compared to the proof of (3.17) in [7], the proof of (1.13) needs more delicate analysis. Whence (1.13) is obtained, the desired estimate (1.7) follows directly by the duality argument motivated by [26,7]. The proof of Theorem 1.2 is completely parallel, and is sketched in Section 5.
As the end of the introduction, let us provide a brief review on the background of convergence rates in quantitative homogenization, which is currently a quite active area of research. Sharp convergence rates for second order elliptic equations (systems) has been studied extensively in various circumstances in the past years. To name but a few, in [8,25,26] the optimal O(ε) convergence rate was obtained for second order elliptic equations with Dirichlet or Neumann boundary conditions in C 1,1 domains. In [11,21], the optimal O(ε) and suboptimal convergence rates (like O(ε ln 1 ε )) were derived for second order elliptic systems with Dirichlet or Neumann boundary conditions in Lipschitz domains. See also [2,13,9,22,23] and references therein for more related results.
For second order parabolic equations with time-independent coefficients, the sharp convergence rate has also been studied widely, see [31,24] for the Cauchy problems on the whole space, and [15,16] for the initial boundary value problems in C 1,1 cylinders. Quantitative estimates for parabolic equations with time dependent coefficients are a bit more intricate and little progress was made until very recently [6,7,4,29,1]. In [7] the optimal O(ε) convergence rate in L 2 (0, T ; L 2 (Ω)) was obtained in homogenization of second order parabolic systems in C 1,1 cylinders, while in [29] the suboptimal O(ε ln(1/ε)) convergence rate in L 2 (0, T ; L 2 (Ω)) was obtained for parabolic systems of elasticity in Lipschitz cylinders. More recently, in [1] the convergence rate and uniform regularity estimates in homogenization of second order stochastic parabolic equations were deeply studied. See also [6,4] for more results on the uniform regularity estimates in the periodic setting.
Homogenization of higher order elliptic equations arises in the study of linear elasticity [3,10,20], for which the qualitative results have been obtained for many years [3,10]. Few quantitative results were known in the homogenization of higher order elliptic or parabolic equations until very recently. In [19,20,14], the optimal O(ε) convergence rate was established in homogenization of higher-order elliptic equations in the whole space. In [27,28], some O(ε) two-parameter resolvent estimates were obtained for more general higher order elliptic systems with homogeneous Dirichlet or Neumann boundary data in bounded C 2m domains. Shortly, the sharp convergence rate and uniform regularity estimates in the homogenization of higher order elliptic systems with symmetric or nonsymmetric coefficients were further studied in [17,18], see also [30] for the results in the almost-periodic setting.
As far as we know, quantitative estimates in homogenization of higher order parabolic equations have not been studied, especially for those with time dependent coefficients. The present paper seems to be the first attempt in this direction. Our results in Theorems 1.1 and 1.2 extend the convergence results for higher order elliptic equations in [19,20,27,28] to parabolic systems on the one hand, and on the other hand they extend the results for second order parabolic systems in [7] to higher order parabolic systems.
Furthermore, there exists a constant C depending only on d, n, m, µ such that Proof. For simplicity of presentations, let us suppress the subscripts i, j. Since for |β| = m, B ıβ (y, s) are 1-periodic in R d+1 , and for any s ∈ R, where ∆ d denotes the Laplacian in R d . We define for |β| = m, By the Liouville property for ∆ m d and the periodicity of f , we know that |α|=m D α f αβ (·, s) + ∂ s f (d+1) β (·, s) is a constant. Consequently, it's not hard to verify that Moreover, note that for any |α| = |β| = |γ| = m, which implies the first estimate in (2.5). Similarly, the second part of (2.5) follows from The proof is complete.
for some positive constant C.
Proof. Integrating equation (2.6) over Y d , we get ∂ s B (d+1) β (s) = 0, which together with the fact Let L * ε be the adjoint operators of L ε , i.e., where A * = (A * αβ ij ) = (A βα ji ). Parallel to (2.1), we can introduce the matrix of correctors is the solution to the following cell problem, (2.8) We can also introduce B * ıβ (y, s) and B * αβ (s) as Lemmas 2.1 and 2.2. It is not difficult to see that χ * γ , B * ıβ and B * αβ (s) satisfy the same properties as χ γ , B ıβ and B αβ (s) respectively, since A * satisfies the same conditions as A.
Taking χ * α i and χ γ j as test functions in (2.1) and (2.8) respectively, we get In view of (2.2), this provides another expression ofĀ in terms of A * and χ * ,

Effective operators and homogenized systems
In this part, we prove that the effective operator for ∂ t + L ε is ∂ t + L 0 , where L 0 is defined as in (1.6). LetĀ = (Ā αβ ij ) be defined as in (2.2). In view of (2.1), we havē Using (1.3) and integration by parts we obtain that, which, combined with (2.2), implies thatĀ satisfies the ellipticity condition (1.3) with 1/µ replaced by some constant µ 0 , depending only on d, n, m and µ. Therefore, for any f ∈ L 2 (0, . ) be the unique weak solutions to initial-Dirichlet problems (1.1) and (1.5), respectively. Then as ε −→ 0, Proof. The proof is adapted from Theorem 2.1 in [3, p.140], where similar results was proved for second order parabolic equations. Note that u ε , ∂ t u ε are uniformly (in ε) bounded in L 2 (0, T ; H m 0 (Ω)) and L 2 (0, T ; H −m (Ω)), respectively, |β|=m A αβ ε D β u ε is uniformly bounded in L 2 (0, T ; L 2 (Ω)) for all |α| = m, where A αβ ε (x, t) = A αβ (x/ε, t/ε 2m ). Up to subsequences, we may assume that there exists a function u 0 such that where the last convergence result in (2.12) follows from the well-known Aubin-Simon type compactness result. Moreover, where Ω T = Ω × (0, T ). Hereafter, let us denote the product of H −m (Ω) and H m 0 (Ω) as , for short. On the other hand, taking such a φ as a test function in (1.1) and passing to the limits, we obtain that which, combined with (2.13), implies that u 0 (0) = h. Therefore, to verify that u 0 is a weak solution of (1.5), it remains to prove (2.14) For positive integer k, let P k = (P 1 k , P 2 k , ..., P n k ) | P i k are homogeneous polynomials of y of degree k .
For P m ∈ P m , let ω be the weak solution to the cell problem ω(y, s) is 1-periodic in (y, s) andˆY ω(y, s) dy ds = 0.
Similar to Theorem 2.1, we can prove that the homogenized operator for −∂ t + L * ε is given by whereĀ * = (Ā * αβ ij ) = (Ā βα ji ). Furthermore, the same argument also gives the homogenized system for the initial value problem (1.1) with homogeneous Neumann boundary data.

5)
where C depends only on d, q, Ω, T.
Proof. Note that Taking (3.7) into (3.6) and using Fubini's theorem, it yields which is exactly (3.3). The proof of (3.4) is similar and we therefore pass to (3.5).
Recall that (3.5) was essentially proved in [7, Lemma 3.2] for the case m = ℓ = 1, q = 2 by the Plancherel Theorem, which is not applicable for general q. By (3.2),

(3.9)
And also, which, together with (3.8) and (3.9), gives (3.5). For the last step of (3.10), we have used the following observation To see this, we note that for any g ∈ C ∞ c (Ω T \Ω T,2ε ) and d-dimensional multi-index η with |η| = ℓ, where the integration by parts and Fubini's theorem have been used for the second and the third equalities respectively. Therefore, using estimates similar to (3.4) we may deduce that which implies (3.11) directly. The proof is complete.
In view of the definition of w ε , to prove (1.7) it is sufficient to prove the following estimates, which implies (4.28). In a similar way, we can also get (4.29) and (4.30). To prove (4.31), it suffices to verify Similar to (4.6), we define Using (4.26) and (4.34), we deduce that = I 1 + I 2 + · · · + I 6 .
(4.35) By Lemma 4.2, we have where, for the last step, we have used estimates (4.14), (4.23) and (4.25) for v 0 . The proof of these estimates are completely the same as those for u 0 , since A * (y, T − s) satisfies conditions as A(y, s).  Note that (4.33) follows directly from (4.35), (4.38), (4.39) and (4.41). The proof is thus complete.

2)
where C depends only on d, n, m, µ, T and Ω.
Proof. The proof is almost the same as that of Lemma 4.2, let us omit the details.