Homogenization of Parabolic Equations with Non-self-similar Scales

Abstract

This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in non-self-similar scales. The homogenization problem involves two oscillating scales. We obtain large-scale interior and boundary Lipschitz estimates as well as interior $C^{1, \alpha }$ and $C^{2, \alpha }$ estimates by utilizing higher-order correctors. We also investigate the problem of convergence rates for initial-boundary value problems.

1 Introduction

In this paper we shall be interested in the quantitative homogenization of a parabolic operator with periodic coefficients varying rapidly in space and time, in different scales. More precisely, we consider the parabolic operator

$$\begin{aligned} \partial _t +{\mathcal {L}}_\varepsilon \end{aligned}$$

(1.1)

in ${\mathbb {R}}^{d+1}$, where $\varepsilon >0$ and

$$\begin{aligned} {\mathcal {L}}_\varepsilon =-\text { div} \big ( A(x/\varepsilon , t/\varepsilon ^k)\nabla \big ), \end{aligned}$$

(1.2)

with $0<k<\infty$. We will assume that the coefficient tensor $A=A(y,s)=\big (a_{ij}^{\alpha \beta } (y, s)\big )$, with $1\le i, j \le d$ and $1\le \alpha , \beta \le m$, is real, bounded measurable and satisfies the ellipticity condition

$$\begin{aligned} \Vert A\Vert _\infty \le \mu ^{-1} \quad \text { and }\quad \mu |\xi |^2 \le a^{\alpha \beta }_{ij}(y,s)\xi _i^\alpha \xi ^\beta _j \end{aligned}$$

(1.3)

for any $\xi =(\xi _i^\alpha ) \in {\mathbb {R}}^{m\times d} \text { and almost everywhere } (y,s)\in {\mathbb {R}}^{d+1}$, where $\mu >0$ (the summation convention is used throughout). We also assume that A is 1-periodic in (y, s); that is

$$\begin{aligned} A(y+z,s+t)=A(y,s)~~~\text { for }(z,t)\in {\mathbb {Z}}^{d+1}\text { and almost everywhere }(y,s)\in {\mathbb {R}}^{d+1}.\nonumber \\ \end{aligned}$$

(1.4)

The qualitative homogenization theory for the operator (1.1) has been known since the 1970s (see for example [10]). As $\varepsilon \rightarrow 0$, the weak solution $u_\varepsilon$ of the initial-Dirichlet problem for the parabolic system $(\partial _t + {\mathcal {L}}_\varepsilon )u_\varepsilon =F$ in $\Omega _T = \Omega \times (0, T)$ converges weakly in $L^2(0, T; H^1(\Omega ))$ and strongly in $L^2(\Omega _T)$. Moreover, the limit $u_0$ is a solution of the initial-Dirichlet problem for $(\partial _t +{\mathcal {L}}_0) u_0=F$ in $\Omega _T$, where ${\mathcal {L}}_0$ is a second-order elliptic operator with constant coefficients. Furthermore, the (homogenized) coefficients of ${\mathcal {L}}_0$ as well as the first-order correctors depend on k, but only for three separated cases: $0<k<2$; $k=2$; and $2<k<\infty$.

In recent years there has been a great amount of interest in the quantitative homogenization theory for partial differential equations, where one is concerned with problems related to the large-scale regularity and convergence rates for solutions $u_\varepsilon$. Major progress has been made for elliptic equations and systems in the periodic and non-periodic settings (see [3,4,5,6,7,8,9, 11, 16, 17, 19, 27, 28] and references therein). Some of these works have been extended to parabolic equations and systems in the self-similar case $k=2$. In particular, we established the large-scale Lipschitz and $W^{1, p}$ estimates in [13] and studied the problem of convergence rates in $L^2(\Omega _T)$ as well as error estimates for two-scale expansions in $L^2(0, T; H^1(\Omega ))$ in [14]. Also see related works in [22,23,24, 30]. Most recently, in [15], we have studied the asymptotic behavior of the fundamental solution and its derivatives and established sharp estimates for the remainders. We refer the reader to [2] for quantitative stochastic homogenization of parabolic equations.

If $k\ne 2$, the $\varepsilon$ scaling in the coefficient tensor $A(x/\varepsilon , t/\varepsilon ^k)$ is not consistent with the intrinsic scaling $(x, t) \rightarrow (\delta x, \delta ^2 t)$ of the second-order parabolic equations. To the authors’ best knowledge, very few quantitative results are known in this case. Direct extensions of the existing techniques developed for elliptic equations fail due to the fact that the homogenization problem involves two oscillating scales mentioned above. For more recent work as well as motivations on homogenization problems with more than one oscillating scale, which are referred to as reiterated homogenization, see [1, 12, 18, 26, 29] and references therein.

In this paper we develop a new approach to study homogenization of parabolic equations and systems with non-self-similar scales. This allows us to establish large-scale interior and boundary Lipschitz estimates for the parabolic operator (1.1) with any $0<k<\infty$, under conditions (1.3) and (1.4).

Let $Q_r (x_0, t_0)=B(x_0, r) \times (t_0-r^2, t_0)$ denote a parabolic cylinder. The following is one of the main results of the paper:

Theorem 1.1

Assume $A=A(y, s)$ satisfies (1.3) and (1.4). Let $u_\varepsilon$ be a weak solution to

$$\begin{aligned} (\partial _t +{\mathcal {L}}_\varepsilon )u_\varepsilon =F \quad \text { in } Q_R=Q_R (x_0, t_0), \end{aligned}$$

(1.5)

where $R>\varepsilon +\varepsilon ^{k/2}$ and $F\in L^p(Q_R)$ for some $p> d+2$. Then for any $\varepsilon +\varepsilon ^{k/2}\le r<R$,

(1.6)

where C depends only on d, m, p, and $\mu$.

The inequality (1.6) may be regarded as a large-scale interior Lipschitz estimate. We also obtain large-scale $C^{1, \alpha }$ and $C^{2, \alpha }$ excess-decay estimates, which are new even for $k=2$, for solutions of $\partial _t +{\mathcal {L}}_\varepsilon$ (see Sections 4 and 5). Regarding the condition $R> r\ge \varepsilon +\varepsilon ^{k/2}$, we mention that there exists $u_\varepsilon$ such that $(\partial _t+{\mathcal {L}}_\varepsilon ) u_\varepsilon =0$ in ${\mathbb {R}}^{d+1}$ and $\nabla u_\varepsilon$ is $\varepsilon$-periodic in x and $\varepsilon ^k$-periodic in t (the solution $u_\varepsilon$ is given by $x_j +\varepsilon \chi ^\lambda _j (x/\varepsilon , t/\varepsilon ^2)$ with $\lambda =\varepsilon ^{k-2}$; see Section 2). Note that if the periodic cell $(0, \varepsilon )^d \times (-\varepsilon ^k, 0)$ for $\nabla u_\varepsilon$ is contained in the parabolic cylinder $Q_r (0, 0)$, then $r^2\ge \varepsilon ^k$ and $2r\ge \sqrt{d} \varepsilon$. This implies that $r\ge (\varepsilon +\varepsilon ^{k/2})/4$. As a result, the condition $R> r\ge \varepsilon +\varepsilon ^{k/2}$ for (1.6) is more or less necessary without additional smoothness assumptions on A.

The next theorem gives the large-scale boundary Lipschitz estimate, which is new even in the case $k=2$. Let $\Omega$ be a bounded $C^{1, \alpha }$ domain in ${\mathbb {R}}^d$ for some $\alpha \in (0, 1)$. Define $D_r (x_0, t_0)=\big ( B(x_0, r)\cap \Omega \big ) \times (t_0-r^2, t_0)$ and $\Delta _r (x_0, t_0)=\big ( B(x_0, r)\cap \partial \Omega \big ) \times (t_0-r^2, t_0)$, where $x_0\in \partial \Omega$ and $t_0 \in {\mathbb {R}}$.

Theorem 1.2

Assume $A=A(y, s)$ satisfies (1.3) and (1.4). Suppose that $(\partial _t + {\mathcal {L}}_\varepsilon ) u_\varepsilon =F$ in $D_R=D_R (x_0, t_0)$ and $u_\varepsilon =f$ on $\Delta _R=\Delta _R (x_0, t_0)$, where $\varepsilon +\varepsilon ^{k/2}<R\le 1$, $F\in L^p (D_R)$ for some $p>d+2$, and $f\in C^{1+\alpha }(\Delta _R)$ for some $\alpha \in (0, 1)$. Then for any $\varepsilon +\varepsilon ^{k/2} \le r< R$,

(1.7)

where C depends only on d, m, $\alpha$, p, $\mu$, and $\Omega$.

Under the additional Hölder continuity condition on A, the large-scale estimates in Theorems 1.1 and 1.2 imply the uniform interior and boundary Lipschitz estimates for $|\nabla u_\varepsilon (x_0, t_0)|$. In the case $k=2$, this follows readily from a simple blow-up argument by considering $u_\varepsilon ( \varepsilon x, \varepsilon ^2 t)$ and using the classical Lipschitz estimates for parabolic operators with Hölder continuous coeffcients. If $k\ne 2$, we may consider the function $u_\varepsilon (\delta x, \delta ^2 t)$ with either $\delta =\varepsilon$ or $\varepsilon ^{k/2}$. It leads to the problem of uniform Lipschitz estimates for parabolic operators of forms, $\partial _t -\text { div} \big (A(x/\varepsilon , t)\nabla \big )$ and $\partial _t -\text { div} \big (A(x, t/\varepsilon )\nabla \big )$, with locally periodic coefficients. The details will appear elsewhere.

In this paper we also investigate the rate of convergence in $L^2(\Omega _T)$ for the initial-Dirichlet problem

$$\begin{aligned} (\partial _t +{\mathcal {L}}_\varepsilon )u_\varepsilon =F \quad \text { in } \Omega _T \quad \text { and } \quad u_\varepsilon =f \quad \text { on } \partial _p \Omega _T, \end{aligned}$$

(1.8)

where $\partial _p \Omega _T$ denotes the parabolic boundary of $\Omega _T$.

Theorem 1.3

Let $\Omega$ be a bounded $C^{1, 1}$ domain in ${\mathbb {R}}^d$ and $0<T<\infty$. Assume $A=A(y, s)$ satisfies (1.3) and (1.4). Also assume that $\Vert \partial _s A\Vert _\infty <\infty$ for $0<k<2$ and $\Vert \nabla ^2 A\Vert _\infty <\infty$ for $k>2$. Let $u_\varepsilon$ be a weak solution of (1.8) and $u_0$ its homogenized solution (with the same data F and f). Suppose that $u_0\in L^2(0, T; H^2(\Omega ))$ and $\partial _t u_0\in L^2(\Omega _T)$. Then

$$\begin{aligned} \begin{aligned}&\Vert u_\varepsilon -u_0\Vert _{L^2(\Omega _T)}\\&\le C \Big \{ \Vert u_0\Vert _{L^2(0, T; H^2(\Omega ))} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \Big \} \cdot \left\{ \begin{aligned}&\varepsilon ^{k/2}&\quad&\text {if } \ 0<k\le 4/3,\\&\varepsilon ^{2-k}&\quad&\text {if } \ 4/3< k< 2,\\&\varepsilon ^{k-2}&\quad&\text {if } \ 2<k<3,\\&\varepsilon&\quad&\text {if } \ k=2 \text { or} \ 3\le k<\infty ,\\ \end{aligned} \right. \end{aligned} \end{aligned}$$

(1.9)

for any $0<\varepsilon <1$, where C depends only on d, m, k, A, $\Omega$, and T.

The convergence rates $\varepsilon ^\gamma$ for different k’s in (1.9) are obtained as a result of the formula $\gamma =\min ( k/2, 2-k)$ for $0<k<2$; $\gamma =1$ for $k=2$; and $\gamma =\min (1, k-2)$ for $k>2$. In Theorem 1.3 we do not specify conditions on F and f, but rather require that $u_0\in L^2(0, T; H^2(\Omega ))$ and $\partial _t u_0\in L^2(\Omega _T)$. Notice that if $\Omega$ is $C^{1,1}$, $F\in L^2(\Omega _T)$ and $f=0$, then $\Vert \nabla ^2 u_0\Vert _{L^2(\Omega _T)} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \le C \Vert F\Vert _{L^2(\Omega _T)}$. It follows that

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{L^2(\Omega _T)} \le C \varepsilon ^\gamma \Vert F\Vert _{L^2(\Omega _T)}, \end{aligned}$$

(1.10)

where $\gamma \in (0, 1]$ is given above.

Remark 1.4

Whether the convergence rate in (1.10) is sharp for $0<k<2$ and $2<k<3$ remains open. We point out that even though the homogenized equation does not depend on k for $0<k<2$ and for $2<k<\infty$, the sharp convergence rate for (1.10) may depend on k. This is already clear in the trivial case $A^\varepsilon =A(t/\varepsilon ^k)$ for k close to 0. We also note that the convergence rate given by Theorem 1.3 is not continuous in k at $k=2$, and that the non-self-similar case $k\ne 2$ requires additional smoothness conditions. These seem to be consistent with the known results in reiterated homogenization. In particular, in the elliptic case with coefficient tensor $A^\varepsilon =A(x/\varepsilon , x/\varepsilon ^k)$ for $0<k<1$, the convergence rate obtained in [25] for $\Omega ={\mathbb {R}}^d$ is $\varepsilon ^\gamma$ with $\gamma =\min (k, 1-k)$ under the assumption that A(y, z) is Lipschitz continuous in z.

We now describe our general approach to Theorems 1.1–1.3. The key insight is to introduce a new scale $\lambda \in (0, \infty )$ and consider the operator

$$\begin{aligned} {\mathcal {L}}_{\varepsilon , \lambda } =-\text { div} \big ( A_\lambda (x/\varepsilon , t/\varepsilon ^2)\nabla \big ), \end{aligned}$$

(1.11)

where $A_\lambda (y, s) = A(y, s/\lambda )$. Observe that the coefficient tensor $A_\lambda$ is 1-periodic in y and $\lambda$-periodic in s. Moreover, for each $\lambda$ fixed, the scaling of the parameter $\varepsilon$ in $A_\lambda (x/\varepsilon , t/\varepsilon ^2)$ is consistent with the intrinsic scaling of the second-order parabolic operator $\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }$. As a result, we may extend some of recently developed techniques for elliptic equations to the parabolic equation $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda } =F$, as in the case $k=2$. We point out that for the results to be useful, it is crucial that the bounding constants C in the estimates of solutions $u_{\varepsilon , \lambda }$ do not depend on $\lambda$ (and $\varepsilon$). This allows us to use the observation ${\mathcal {L}}_\varepsilon = {\mathcal {L}}_{\varepsilon , \lambda }$ for $\lambda =\varepsilon ^{k-2}$ and prove Theorems 1.1 and 1.2. The approach also leads to large-scale $C^{1, \alpha }$ and $C^{2, \alpha }$ excess-decay estimates as well as a Liouville property, expressed in terms of correctors for $\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }$.

The approach described above works equally well for the problem of convergence rates. In addition to the observation ${\mathcal {L}}_{\varepsilon , \lambda }={\mathcal {L}}_\varepsilon$ for $\lambda =\varepsilon ^{k-2}$, we also use the fact that as $\lambda \rightarrow \infty$, the homogenized coefficient matrix $\widehat{A_\lambda }$ for $\partial _t+{\mathcal {L}}_{\varepsilon , \lambda }$ converges to $\widehat{A_\infty }$, the homogenized coefficient matrix for $\partial _t + {\mathcal {L}}_\varepsilon$ in the case $0<k<2$. If $\lambda \rightarrow 0$, then $\widehat{A_\lambda } \rightarrow \widehat{A_0}$, the homogenized coefficient matrix for $\partial _t + {\mathcal {L}}_\varepsilon$ in the case $2<k<\infty$.

The paper is organized as follows: in Section 2 we introduce the first-order correctors $\chi ^\lambda$ and homogenized coefficients for $\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }$, with $\lambda >0$ fixed, as well as correctors and homogenized coefficients for ${\mathcal {L}}_\varepsilon$ in (1.1) with $0<k<\infty$. We also establish precise estimates of $|\widehat{A_\lambda } -\widehat{A_\infty }|$ for $\lambda >1$, and of $|{\widehat{A}}_\lambda -\widehat{A_0}|$ for $0< \lambda <1$, under additional regularity assumptions on A. These estimates are used in the proof of Theorem 1.3. In Section 3 we prove an approximation result for solutions of $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda }=F$ in a parabolic cylinder. This is done by using $\varepsilon$-smoothing and dual correctors. The proof follows the approach used in [14] by the present authors for the case $\lambda =1$. The proof of Theorem 1.1 is given in Section 4, where we also establish a large-scale $C^{1, \alpha }$ estimate. In Section 5 we introduce second-order correctors for the operator $\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }$ and prove a large-scale $C^{2, \alpha }$ estimate. The large-scale boundary Lipschitz estimate in Theorem 1.2 is proved in Section 6. We remark that the approaches used in Sections 4–6 are motivated by recently developed techniques for studying the large-scale regularity in the homogenization theory for elliptic equations and systems [3, 4, 6,7,8, 11, 16, 17]. Finally, we give the proof of Theorem 1.3 in Section 7, where we also obtain error estimates for a two-scale expansion in $L^2(0, T; H^1(\Omega ))$.

The summation convention is used throughout. We will use to denote the $L^1$ average of u over the set E; that is . For notational simplicity we will assume that $m=1$ in the rest of the paper. However, no particular fact pertaining to the scalar case is ever used. All results and proofs extend readily to the case $m>1$—the case of parabolic systems.

2 Correctors and Homogenized Coefficients

Let $A=A(y, s)$ be a matrix satisfying conditions (1.3) and (1.4). For $\lambda >0$, define

$$\begin{aligned} A_\lambda =A_\lambda (y, s)=A(y, s/\lambda ) \quad \text { for } (y, s)\in {\mathbb {R}}^{d+1}. \end{aligned}$$

(2.1)

The matrix $A_\lambda$ is $(1, \lambda )$-periodic in (y, s); that is

$$\begin{aligned} A_\lambda (y+z, s+\lambda t)=A_\lambda (y, s)\quad \text { for } (z, t)\in {\mathbb {Z}}^{d+1}. \end{aligned}$$

Let $\chi ^\lambda =\chi ^\lambda (y, s)=(\chi _1^\lambda (y, s), \ldots , \chi _d^\lambda (y, s))$, where $\chi _j^\lambda =\chi _j^\lambda (y, s)$ is the weak solution of the parabolic cell problem

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s \chi _j^\lambda -\text { div} \big ( A_\lambda \nabla \chi _j^\lambda \big ) =\text { div}\big ( A_\lambda \nabla y_j\big ) \quad \text { in } {\mathbb {R}}^{d+1},\\&\chi _j^\lambda \text { is } (1, \lambda ) \text {-periodic in } (y, s),\\&\int _0^\lambda \!\!\!\int _{{\mathbb {T}}^d} \chi _j^\lambda (y, s)\, \mathrm{d}y \mathrm{d}s=0, \end{aligned} \right. \end{aligned}$$

(2.2)

where ${\mathbb {T}}^d=[0, 1)^d\cong {\mathbb {R}}^d /{\mathbb {Z}}^d$. By the energy estimates,

(2.3)

where C depends only on d and $\mu$. Since

$$\begin{aligned} \partial _s \int _{{\mathbb {T}}^d} \chi _j^\lambda (y, s)\, \mathrm{d}y=0, \end{aligned}$$

we obtain, by the integral condition in (2.2),

$$\begin{aligned} \int _{{\mathbb {T}}^d} \chi _j^\lambda (y, s) \, \mathrm{d}y=0. \end{aligned}$$

(2.4)

This, together with (2.3) and Poincaré’s inequality, gives

(2.5)

where C depends only on d and $\mu$. Since $\chi ^\lambda$ and $\nabla \chi ^\lambda$ are $(1, \lambda )$-periodic in (y, s), it follows from (2.3) and (2.5) that if $r\ge 1+\sqrt{\lambda }$,

(2.6)

for any $Q_r =Q_r (x, t)$, where C depends only on d and $\mu$.

Let

(2.7)

Lemma 2.1

There exists $C>0$, depending only on d and $\mu$, such that $|\widehat{A_\lambda }|\le C$. Moreover,

$$\begin{aligned} \mu |\xi |^2 \le \xi \cdot \widehat{A_\lambda } \xi \end{aligned}$$

(2.8)

for any $\xi \in {\mathbb {R}}^d$.

Proof

The inequality $|\widehat{A_\lambda }|\le C$ follows readily from (2.3). To see (2.8), we note that

for any $\xi \in {\mathbb {R}}^d$, where we have used the fact $\int _0^\lambda \!\!\int _{{\mathbb {T}}^d} \nabla \chi ^\lambda \, \mathrm{d}y \mathrm{d}s=0$. $\quad \square$

It is well known that for a fixed $\lambda >0$, the homogenized operator for the parabolic operator

$$\begin{aligned} \partial _t +{\mathcal {L}}_{\varepsilon , \lambda }= \partial _t -\text { div} \big (A_\lambda (x/\varepsilon , t/\varepsilon ^2)\nabla \big ) \end{aligned}$$

(2.9)

is given by $\partial _t -\text { div} \big ( \widehat{A_\lambda }\nabla \big )$ [10]. In particular, if $k=2$, the homogenized operator for the operator in (1.1) is given by $\partial _t -\text { div} \big ( \widehat{A_\lambda }\nabla \big )$ with $\lambda =1$.

To introduce the homogenized operator for $\partial _t +{\mathcal {L}}_\varepsilon$ in (1.1) for $k\ne 2$, we first consider the case $0<k<2$. Let $\chi ^\infty =\chi ^\infty (y, s)= (\chi _1^\infty (y, s), \ldots , \chi _d^\infty (y, s) )$, where $\chi _j^\infty =\chi _j^\infty (y, s)$ denotes the weak solution of the (elliptic) cell problem

$$\begin{aligned} \left\{ \begin{aligned}&-\text { div} \big ( A\nabla \chi _j^\infty ) =\text { div}(A\nabla y_j) \quad \text { in } {\mathbb {R}}^{d+1},\\&\chi _j^\infty \text { is 1-periodic in } (y, s),\\&\int _{{\mathbb {T}}^d} \chi _j^\infty (y, s)\, \mathrm{d}y=0. \end{aligned} \right. \end{aligned}$$

(2.10)

By the energy estimates and Poincaré’s inequality,

$$\begin{aligned} \int _{{\mathbb {T}}^d} \left( |\nabla \chi _j^\infty (y, s)|^2 +|\chi _j^\infty (y, s)|^2\right) \mathrm{d}y \le C, \end{aligned}$$

(2.11)

for almost everywhere $s\in {\mathbb {R}}$, where C depends only on d and $\mu$. Let

$$\begin{aligned} \widehat{A_\infty } =\int _0^1 \!\!\! \int _{{\mathbb {T}}^d} \left( A + A \nabla \chi ^\infty \right) \mathrm{d}y\mathrm{d}s. \end{aligned}$$

(2.12)

It follows from (2.11) that $|\widehat{A_\infty }|\le C$, where C depends only on d and $\mu$. By the same argument as in the proof of Lemma 2.1, one may also show that

$$\begin{aligned} \mu |\xi |^2 \le \xi \cdot \widehat{A_\infty } \xi \end{aligned}$$

(2.13)

for any $\xi \in {\mathbb {R}}^d$. For $0<k<2$, the homogenized operator for the parabolic operator in (1.1) is given by $\partial _t -\text { div} \big (\widehat{A_\infty }\nabla \big )$ (see [10]).

Next, we consider the case $2<k<\infty$. Define

$$\begin{aligned} {\overline{A}}={\overline{A}}(y) =\int _0^1 A(y, s)\, \mathrm{d}s. \end{aligned}$$

(2.14)

Let $\chi ^0 =\chi ^0 (y)=( \chi ^0_1 (y), \ldots , \chi _d^0 (y) )$, where $\chi _j^0=\chi _j^0 (y)$ is the weak solution of the (elliptic) cell problem

$$\begin{aligned} \left\{ \begin{aligned}&-\text { div} \left( {\overline{A}}\nabla \chi _j^0 \right) =\text { div} \left( {\overline{A}}\nabla y_j\right) \quad \text { in } {\mathbb {R}}^d,\\&\chi _j^0 \text { is 1-periodic in } y,\\&\int _{{\mathbb {T}}^d} \chi _j^0\, \mathrm{d}y =0. \end{aligned} \right. \end{aligned}$$

(2.15)

As in the case $0<k<2$, by the energy estimates and Poincaré’s inequality,

$$\begin{aligned} \int _{{\mathbb {T}}^d} \left( |\nabla \chi _j^0(y)|^2 +|\chi _j^0(y)|^2\right) \mathrm{d}y \le C, \end{aligned}$$

(2.16)

where C depends only on d and $\mu$. Let

$$\begin{aligned} \widehat{A_0} =\int _0^1 \!\!\! \int _{{\mathbb {T}}^d} \left( A + A \nabla \chi ^0 \right) \mathrm{d}y\mathrm{d}s = \int _{{\mathbb {T}}^d} \left( {\overline{A}} + {\overline{A}} \nabla \chi ^0 \right) \mathrm{d}y. \end{aligned}$$

(2.17)

It follows from (2.16) that $|\widehat{A_0}|\le C$, where C depends only on d and $\mu$. By the same argument as in the proof of Lemma 2.1, we obtain

$$\begin{aligned} \mu |\xi |^2 \le \xi \cdot \widehat{A_0} \xi \end{aligned}$$

(2.18)

for any $\xi \in {\mathbb {R}}^d$. For $2<k<\infty$, the homogenized operator for $\partial _t +{\mathcal {L}}_\varepsilon$ in (1.1) is given by $\partial _t -\text { div} \big (\widehat{A_0}\nabla \big )$ (see [10]).

In the remaining of this section we study the asymptotic behavior of the matrix $\widehat{A_\lambda }$, as $\lambda \rightarrow \infty$ and as $\lambda \rightarrow 0$. We begin with a lemma on the higher integrability of $\nabla \chi ^\lambda$.

Lemma 2.2

Let $\chi ^\lambda$ be defined by (2.2). Then there exists $q>2$, depending on d and $\mu$, such that

(2.19)

where C depends only on d and $\mu$.

Proof

Let $u(y, s)=y_j +\chi _j^\lambda$. Then $\partial _s u -\text { div} ( A_\lambda \nabla u) =0 \text { in } {\mathbb {R}}^{d+1}.$ By Meyers-type estimates for parabolic systems (see for example [2, Appendix]), there exist $q>2$ and $C>0$, depending only on d and $\mu$, such that

(2.20)

for any $Q_r =Q_r (x, t) =B(x, r) \times (t-r^2, t)$. It follows that

(2.21)

Choose $r>1+\sqrt{\lambda }$ so large that ${\mathbb {T}}^d \times (0, \lambda )\subset Q_r$. Since $\nabla \chi _j^\lambda$ is 1-periodic in y and $\lambda$-periodic in s, we obtain

where we have used (2.6) for the last step. $\quad \square$

Theorem 2.3

Assume $A=A(y, s)$ satisfies conditions (1.3) and (1.4). Then

$$\begin{aligned} \widehat{A_\lambda } \rightarrow \widehat{A_\infty } \quad \text { as }\lambda \rightarrow \infty . \end{aligned}$$

(2.22)

Moreover, if $\Vert \partial _s A\Vert _\infty <\infty$, then

$$\begin{aligned} |\widehat{A_\lambda } -\widehat{A_\infty }|\le C \lambda ^{-1} \Vert \partial _s A\Vert _\infty \end{aligned}$$

(2.23)

for any $\lambda >1$, where C depends only on d and $\mu$.

Proof

We first prove (2.23). Observe that

$$\begin{aligned} \widehat{A_\lambda }-\widehat{A_\infty } =\int _0^1\!\!\! \int _{{\mathbb {T}}^d} A (y, s) \nabla \left\{ \chi ^\lambda (y, \lambda s)-\chi ^\infty (y, s) \right\} \mathrm{d}y\mathrm{d}s. \end{aligned}$$

It follows by the Cauchy inequality that

$$\begin{aligned} | \widehat{A_\lambda }-\widehat{A_\infty }| \le C \left( \int _0^1\!\!\! \int _{{\mathbb {T}}^d} | \nabla \left\{ \chi ^\lambda (y, \lambda s)-\chi ^\infty (y, s) \right\} |^2 \, \mathrm{d}y\mathrm{d}s\right) ^{1/2}. \end{aligned}$$

(2.24)

By the definitions of $\chi ^\lambda$ and $\chi ^\infty$,

$$\begin{aligned} \frac{1}{\lambda } \frac{\partial }{\partial s} \big \{ \chi _j^\lambda (y, \lambda s) \big \} -\text { div} \big \{ A(y, s) \nabla \big ( \chi _j^\lambda (y, \lambda s) -\chi _j^\infty (y, s) \big ) \big \}=0 \quad \text { in } {\mathbb {T}}^{d+1}. \end{aligned}$$

This leads to

$$\begin{aligned} \begin{aligned}&\int _0^1\!\!\! \int _{{\mathbb {T}}^d} A(y, s) \nabla \big \{ \chi _j^\lambda (y, \lambda s) -\chi _j ^\infty (y, s) \big \} \cdot \nabla \big \{ \chi _j^\lambda (y, \lambda s) -\chi _j^\infty (y, s) \big \} \, \mathrm{d}y\mathrm{d}s\\&=-\frac{1}{\lambda } \int _0^1\!\!\! \int _{{\mathbb {T}}^d} \frac{\partial }{\partial s} \big \{ \chi _j^\lambda (y, \lambda s) \big \} \cdot \big \{ \chi _j^\lambda (y, \lambda s) -\chi _j ^\infty (y, s) \big \} \, \mathrm{d}y\mathrm{d}s\\&=-\frac{1}{\lambda } \int _0^1\!\!\! \int _{{\mathbb {T}}^d} \frac{\partial }{\partial s} \big \{ \chi _j^\infty (y, s) \big \} \cdot \big \{ \chi _j^\lambda (y, \lambda s) -\chi ^\infty _j (y, s) \big \} \, \mathrm{d}y\mathrm{d}s, \end{aligned} \end{aligned}$$

where we have used the fact

$$\begin{aligned} \int _0^1\!\!\! \int _{{\mathbb {T}}^d} \frac{\partial }{\partial s} \big \{ \chi _j^\lambda (y, \lambda s) -\chi _j^\infty (y, s) \big \} \cdot \big \{ \chi _j^\lambda (y, \lambda s) -\chi _j^\infty (y, s) \big \} \, \mathrm{d}y\mathrm{d}s =0 \end{aligned}$$

for the last step. Hence, by (1.3) and the Cauchy inequality,

$$\begin{aligned} \begin{aligned}&\mu \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |\nabla \big \{ \chi _j^\lambda (y, \lambda s) -\chi _j^\infty (y, s) \big \} |^2\, \mathrm{d}y\mathrm{d}s\\&\le \frac{1}{\lambda } \left( \int _0^1\!\!\! \int _{{\mathbb {T}}^d} |\chi _j^\lambda (y, \lambda s) -\chi _j^\infty (y, s)|^2\, \mathrm{d}y\mathrm{d}s \right) ^{1/2} \left( \int _0^1\!\!\! \int _{{\mathbb {T}}^d} |\partial _s \chi _j^\infty (y, s)|^2\, \mathrm{d}y\mathrm{d}s\right) ^{1/2}. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} \int _{{\mathbb {T}}^d} \chi _j^\lambda (y, \lambda s)\, \mathrm{d}y =\int _{{\mathbb {T}}^d} \chi _j^\infty (y, s)\, \mathrm{d}y=0, \end{aligned}$$

by Poincaré’s inequality, we obtain

$$\begin{aligned}&\left( \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |\nabla \big \{ \chi _j^\lambda (y, \lambda s) -\chi _j^\infty (y, s) \big \} |^2\, \mathrm{d}y\mathrm{d}s\right) ^{1/2} \nonumber \\&\quad \le \frac{C}{\lambda } \left( \int _0^1\!\!\! \int _{{\mathbb {T}}^d} |\partial _s \chi _j^\infty (y, s)|^2\, \mathrm{d}y\mathrm{d}s\right) ^{1/2}. \end{aligned}$$

In view of (2.24) we have proved that

$$\begin{aligned} |\widehat{A_\lambda } -\widehat{A_\infty }| \le \frac{C}{\lambda } \left( \int _0^1\!\!\! \int _{{\mathbb {T}}^d} |\partial _s \chi ^\infty (y, s)|^2\, \mathrm{d}y\mathrm{d}s\right) ^{1/2}, \end{aligned}$$

(2.25)

where C depends only on d and $\mu$.

To bound the right-hand side of (2.25), we differentiate in s the elliptic equation for $\chi _j^\infty$ to obtain

$$\begin{aligned} -\text { div} \big (A \nabla \partial _s \chi _j^\infty ) =\text {div} \big (\partial _s A \nabla y_j ) +\text { div} \big (\partial _s A \nabla \chi _j^\infty ). \end{aligned}$$

It follows that

$$\begin{aligned} \int _{{\mathbb {T}}^d} |\nabla \partial _s \chi _j^\infty (y, s)|^2\, \mathrm{d}y \le C \int _{{\mathbb {T}}^d} |\partial _s A(y, s)|^2\, \mathrm{d}y + C \int _{{\mathbb {T}}^d} |\partial _s A(y, s)|^2 |\nabla \chi _j^\infty (y, s)|^2\, \mathrm{d}y. \end{aligned}$$

By Meyer’s estimates, there exists some $q>2$, depending only on d and $\mu$, such that

$$\begin{aligned} \int _{{\mathbb {T}}^d} |\nabla \chi _j^\infty (y, s) |^q \, \mathrm{d}y\le C, \end{aligned}$$

where C depends only on d and $\mu$. Thus, by Hölder’s inequality,

$$\begin{aligned} \left( \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |\nabla \partial _s \chi _j^\infty |^2\, \mathrm{d}y\mathrm{d}s \right) ^{1/2} \le C \left( \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |\partial _s A|^{p_0}\, \mathrm{d}y \mathrm{d}s \right) ^{1/p_0}, \end{aligned}$$

for $p_0 =\frac{2 q}{q-2}$. In view of (2.25) this gives

$$\begin{aligned} |\widehat{A_\lambda } -\widehat{A_\infty }| \le \frac{C}{\lambda } \left( \int _0^1\!\!\! \int _{{\mathbb {T}}^d} |\partial _s A |^{p_0} \, \mathrm{d}y\mathrm{d}s\right) ^{1/p_0}, \end{aligned}$$

(2.26)

by using Poincaré’s inequality. As a consequence, we obtain (2.23).

Finally, to prove (2.22), we let D be a matrix satisfying conditions (1.3) and (1.4). Also assume that D is smooth in (y, s). Let $\widehat{D_\lambda }$ and $\widehat{D_\infty }$ be defined in the same manner as $\widehat{A_\lambda }$ and $\widehat{A_\infty }$, respectively. By using the energy estimates as well as (2.19), it is not hard to show that

$$\begin{aligned} |\widehat{A_\lambda } -\widehat{D_\lambda } | \le C \left( \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |A -D|^{p_0} \, \mathrm{d}y\mathrm{d}s \right) ^{1/p_0}, \end{aligned}$$

where C depends only on d and $\mu$. A similar argument also gives

$$\begin{aligned} |\widehat{A_\infty } -\widehat{D_\infty } | \le C \left( \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |A -D|^{p_0} \, \mathrm{d}y\mathrm{d}s \right) ^{1/p_0}. \end{aligned}$$

Thus, by applying the estimate (2.26) to the matrix D, we obtain

$$\begin{aligned} \begin{aligned} |\widehat{A_\lambda } -\widehat{A_\infty } |&\le |\widehat{A_\lambda } -\widehat{D_\lambda }| +| \widehat{D_\lambda } -\widehat{D_\infty }| +| \widehat{D_\infty } -\widehat{A_\infty }|\\&\le C \left( \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |A -D|^{p_0} \, \mathrm{d}y\mathrm{d}s \right) ^{1/p_0} + \frac{C}{\lambda } \left( \int _0^1\!\!\! \int _{{\mathbb {T}}^d} |\partial _s D |^{p_0} \, \mathrm{d}y\mathrm{d}s\right) ^{1/p_0}. \end{aligned} \end{aligned}$$

It follows that

$$\begin{aligned} \limsup _{\lambda \rightarrow \infty } |\widehat{A_\lambda } -\widehat{A_\infty } | \le C \left( \int _0^1\!\!\!\int _{{\mathbb {T}}^d} |A -D|^{p_0} \, \mathrm{d}y\mathrm{d}s \right) ^{1/p_0}. \end{aligned}$$

Since $p_0 =\frac{2q}{q-2} <\infty$, by using convolution, we may approximate A in $L^{p_0} ({\mathbb {T}}^{d+1})$ by a sequence of smooth matrices satisfying (1.3) and (1.4). As a result, we conclude that $\widehat{A_\lambda } \rightarrow \widehat{A_\infty }$ as $\lambda \rightarrow \infty$. $\quad \square$

Remark 2.4

It follows from the proof of Theorem 2.3 that

By the periodicity this implies that if $r \ge (1+\sqrt{\lambda } ) \varepsilon$, then

(2.27)

The next theorem is concerned with the limit of $\widehat{A_\lambda }$ as $\lambda \rightarrow 0$.

Theorem 2.5

Assume $A=A(y, s)$ satisfies conditions (1.3) and (1.4). Then

$$\begin{aligned} \widehat{A_\lambda } \rightarrow \widehat{A_0} \quad \text { as }\lambda \rightarrow 0. \end{aligned}$$

(2.28)

Moreover, if $\Vert \nabla ^2 A\Vert _\infty <\infty$, then

$$\begin{aligned} |\widehat{A_\lambda } -\widehat{A_0}|\le C \lambda \big \{ \Vert \nabla ^2 A\Vert _\infty + \Vert \nabla A\Vert ^2_\infty \big \}, \end{aligned}$$

(2.29)

where C depends only on d and $\mu$.

Proof

We first prove (2.29). Observe that

$$\begin{aligned} \begin{aligned}\widehat{A_\lambda } -\widehat{A_0}&=\int _0^1\!\!\! \int _{{\mathbb {T}}^d} A (y, s) \nabla \big ( \chi ^\lambda (y, \lambda s) -\chi ^0 (y) \big )\, \mathrm{d}y\mathrm{d}s\\&=\int _0^1\!\!\! \int _{{\mathbb {T}}^d} (A(y, s)-{\overline{A}}(y) ) \nabla \chi ^\lambda (y, \lambda s)\, \mathrm{d}y\mathrm{d}s\\&\qquad + \int _{{\mathbb {T}}^d} {\overline{A}} (y) \nabla \left( \int _0^1 \chi ^\lambda (y, \lambda s)\, \mathrm{d}s -\chi ^0 (y) \right) \mathrm{d}y\\&=I_1 +I_2. \end{aligned} \end{aligned}$$

(2.30)

Write $A(y, s) -{\overline{A}}(y) = \partial _s {\widetilde{A}} (y, s)$, where

$$\begin{aligned} {\widetilde{A}}(y, s)=\int _0^s \big ( A(y, \tau )-{\overline{A}} (y) \big ) \mathrm{d}\tau . \end{aligned}$$

Since ${\widetilde{A}}(y, s)$ is 1-periodic in (y, s), we may use an integration by parts and the Cauchy inequality to obtain

(2.31)

To bound the term $I_2$ in (2.30), we observe that

$$\begin{aligned} -\text { div} \left( \int _0^1 A(y, s)\nabla \chi _j^\lambda (y, \lambda s)\, \mathrm{d}s \right) =\text { div} \left( {\overline{A}}( y)\nabla y_j \right) =-\text { div} \left( {\overline{A}} (y) \nabla \chi _j^0 (y) \right) . \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned}&-\text { div} \left\{ {\overline{A}} (y) \nabla \left( \int _0^1 \chi _j^\lambda (y, \lambda s)\, \mathrm{d}s -\chi _j^0(y) \right) \right\} \\&\quad =\text { div} \left\{ \int _0^1 \left( A(y, s)-{\overline{A}}(y) \right) \nabla \chi _j^\lambda (y, \lambda s) \, \mathrm{d}s \right\} . \end{aligned} \end{aligned}$$

By the energy estimates we obtain

where, for the last step, we have used the integration by parts as in the estimate of $I_1$. As a result, in view of (2.30) and (2.31), we have proved that

(2.32)

To bound the right-hand side of (2.32), we differentiate in y the parabolic equation for $\chi _j^\lambda$ to obtain

$$\begin{aligned} \partial _s \nabla \chi _j^\lambda -\text { div} \big ( A_\lambda \nabla (\nabla \chi _j^\lambda ) \big ) =\text { div}\big ( \nabla A_\lambda \cdot \nabla \chi _j^\lambda \big ) +\text { div} \big ( \nabla A_\lambda \cdot \nabla y_j\big ). \end{aligned}$$

(2.33)

By the energy estimates,

(2.34)

By differentiating (2.33) in y we have

$$\begin{aligned} \begin{aligned}&\partial _s \nabla ^2 \chi _j^\lambda -\text { div} \big (A_\lambda \nabla (\nabla ^2 \chi _j^\lambda )\big )\\&=\text { div} \big ( \nabla A_\lambda \cdot \nabla ^2 \chi _j^\lambda \big ) +\text { div} \big ( \nabla ^2 A_\lambda \cdot \nabla \chi _j^\lambda \big ) +\text { div} \big ( \nabla A_\lambda \cdot \nabla ^2 \chi _j^\lambda )\\&\quad +\text { div} \big ( \nabla ^2 A_\lambda \cdot \nabla y_j). \end{aligned} \end{aligned}$$

Again, by the energy estimates,

It follows by the equation (2.33) that

which, together with (2.32), gives (2.29).

Finally, to see (2.28), we let D be a smooth matrix satisfying (1.3) and (1.4). As in the proof of Theorem 2.3, we have

$$\begin{aligned} \begin{aligned}|\widehat{A_\lambda }-\widehat{A_0}|&\le |\widehat{A_\lambda }-\widehat{D_\lambda }| +|\widehat{D_\lambda }-\widehat{D_0}| +|\widehat{D_0} -\widehat{A_0}|\\&\le C \left( \int _0^1 \!\!\!\int _{{\mathbb {T}}^d} |A-D|^{p_0}\, \mathrm{d}y \mathrm{d}s\right) ^{1/p_0} + C \lambda \Big \{ \Vert \nabla ^2 D\Vert _\infty + \Vert \nabla D\Vert ^2_\infty \Big \}. \end{aligned} \end{aligned}$$

By letting $\lambda \rightarrow 0$ and by approximating A in the $L^{p_0}({\mathbb {T}}^{d+1})$ norm by a sequence of smooth matrices satisfying (1.3) and (1.4), we conclude that $\widehat{A_\lambda } \rightarrow \widehat{A_0}$ as $\lambda \rightarrow 0$. $\quad \square$

Remark 2.6

It follows from the proof of Theorem 2.5 that if $r\ge \varepsilon$,

(2.35)

for $0<\lambda <1$, where C depends only on d and $\mu$.

3 Approximation

Let $A_\lambda$ be the matrix given by (2.1) and ${\mathcal {L}}_{\varepsilon , \lambda } =-\text { div} \big ( A_\lambda (x/\varepsilon , t/\varepsilon ^2)\nabla \big ).$ Let ${\mathcal {L}}_{0, \lambda } =-\text { div} \big ( \widehat{A_\lambda } \nabla )$, where the constant matrix $\widehat{A_\lambda }$ is given by (2.7). The goal of this section is to prove the following theorem:

Theorem 3.1

Suppose A satisfies conditions (1.3) and (1.4). Let $u_{\varepsilon , \lambda }$ be a weak solution of

$$\begin{aligned} (\partial _t +{\mathcal {L}}_{\varepsilon , \lambda } ) u_{\varepsilon , \lambda } =F \quad \text { in } Q_{2r}, \end{aligned}$$

(3.1)

where $r> (1+\sqrt{\lambda }) \varepsilon$ and $F\in L^p(Q_{2r})$ for some $p>d+2$. Then there exists a weak solution of

$$\begin{aligned} (\partial _t +{\mathcal {L}}_{0, \lambda } ) u_{0, \lambda } =F \quad \text { in } Q_r, \end{aligned}$$

(3.2)

such that

(3.3)

and

(3.4)

where $\sigma \in (0, 1)$ and $C>0$ depend only on d, $\mu$ and p.

We begin by introducing the dual correctors $\phi ^\lambda$ for the operator $\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }$. Let

$$\begin{aligned} B_\lambda =A_\lambda +A_\lambda \nabla \chi ^\lambda -\widehat{A_\lambda }, \end{aligned}$$

(3.5)

where the corrector $\chi ^\lambda$ is given by (2.2). Note that $B_\lambda$ is $(1, \lambda )$-periodic in (y, s).

Lemma 3.2

Let $B_\lambda =( b_{ij}^\lambda )$ be given by (3.5). Then there exist $(1, \lambda )$-periodic functions $\phi ^\lambda _{kij}$ and $\phi ^\lambda _{k(d+1)j}$, with $1\le i, j, k\le d$, in $H^1_{loc} ({\mathbb {R}}^{d+1})$ such that

$$\begin{aligned} \left\{ \begin{aligned}b_{ij}^\lambda&= \frac{\partial }{\partial y_k} \phi ^\lambda _{kij} -\partial _s \phi ^\lambda _{i (d+1)j},\\ -\chi _j^\lambda&=\frac{\partial }{\partial y_k} \phi ^\lambda _{k (d+1)j}. \end{aligned} \right. \end{aligned}$$

(3.6)

Moreover, $\phi _{kij}^\lambda =-\phi _{ikj}^\lambda$ and

(3.7)

(3.8)

where C depends only on d and $\mu$.

Proof

The lemma was proved in [14] for the case $\lambda =1$. The case $\lambda \ne 1$ is similar. However, one needs to be careful with the dependence of the constants C on the parameter $\lambda$.

Let $\Delta _{d+1}$ denote the Laplacian operator in $(y, s)\in {\mathbb {R}}^d \times {\mathbb {R}}$. By the definition of $\widehat{A_\lambda }$,

(3.9)

It follows that there exist $(1,\lambda )$-periodic functions $f_{ij}^\lambda \in H^2_{loc}({\mathbb {R}}^{d+1})$ such that $\Delta _{d+1} f_{ij}^\lambda =b_{ij}^\lambda \text { in } {\mathbb {R}}^{d+1}$ for $1\le i, j\le d$. Similarly, there exist $(1, \lambda )$-periodic functions $f^\lambda _{(d+1)j}\in H^2_{loc}({\mathbb {R}}^{d+1})$ such that $\Delta _{d+1} f^\lambda _{(d+1)j } =-\chi _j^\lambda \text { in } {\mathbb {R}}^{d+1}$ for $1\le j \le d$. By the definition of $\chi ^\lambda _j$, we have

$$\begin{aligned} \frac{\partial }{\partial y_i} b_{ij}^\lambda =\partial _s \chi _j^\lambda \quad \text {in } {\mathbb {R}}^{d+1}, \end{aligned}$$

(3.10)

which leads to

$$\begin{aligned} \Delta _{d+1} \left( \frac{\partial f_{ij}^\lambda }{\partial y_i} +\partial _s f_{(d+1)j}^\lambda \right) =0 \quad \text {in } {\mathbb {R}}^{d+1}. \end{aligned}$$

By the periodicity and Liouville Theorem we may conclude that

$$\begin{aligned} \frac{\partial f_{ij}^\lambda }{\partial y_i} +\partial _s f_{(d+1)j}^\lambda \quad \text { is constant in } {\mathbb {R}}^{d+1} \text { for } 1\le j\le d. \end{aligned}$$

(3.11)

This allows us to write

$$\begin{aligned} b_{ij}^\lambda =\frac{\partial }{\partial y_k} \left\{ \frac{\partial f_{ij}^\lambda }{\partial y_k} -\frac{\partial f^\lambda _{kj}}{\partial y_i} \right\} +\partial _s \left\{ \partial _s f_{ij}^\lambda - \frac{\partial f^\lambda _{(d+1) j}}{\partial y_i} \right\} , \end{aligned}$$

and

$$\begin{aligned} -\chi _j^\lambda =\frac{\partial }{\partial y_k} \left\{ \frac{\partial f_{(d+1)j}^\lambda }{\partial y_k} -\partial _s f^\lambda _{kj} \right\} . \end{aligned}$$

We now define $\phi _{kij}^\lambda$ and $\phi _{k (d+1)j}^\lambda$ by

$$\begin{aligned} \left\{ \begin{aligned}\phi _{kij}^\lambda&=\frac{\partial f_{ij}^\lambda }{\partial y_k} -\frac{\partial f_{kj}^\lambda }{\partial y_i},\\ \phi _{k(d+1)j}^\lambda&=\frac{\partial f^\lambda _{(d+1)j}}{\partial y_k} -\partial _s f_{kj}^\lambda \end{aligned} \right. \end{aligned}$$

(3.12)

for $1\le i, j, k\le d$. This gives (3.6). It is easy to see that $\phi _{kij}^\lambda =-\phi _{ikj}^\lambda$.

Finally, to prove estimates (3.7) and (3.8), we use the Fourier series to write

$$\begin{aligned} b_{ij}^\lambda (y, s) =\sum _{\begin{array}{c} n\in {\mathbb {Z}}^d, m\in {\mathbb {Z}}\\ (n, m)\ne (0, 0) \end{array}} a_{n, m} e^{-2\pi i n\cdot y -2\pi i m s\lambda ^{-1}}. \end{aligned}$$

Then

$$\begin{aligned} f_{ij}^\lambda (y, s)=-\frac{1}{4\pi ^2} \sum _{\begin{array}{c} n\in {\mathbb {Z}}^d, m\in {\mathbb {Z}}\\ (n, m)\ne (0, 0) \end{array}} \frac{ a_{n, m}}{ |n|^2 +|m|^2 \lambda ^{-2}} e^{-2\pi i n\cdot y -2\pi i m s\lambda ^{-1}}. \end{aligned}$$

It follows by Parseval’s Theorem that

(3.13)

where C depends only on d and $\mu$. Also note that

(3.14)

where C depends only on d and $\mu$. Similarly, using the estimate (2.5), we obtain

(3.15)

The desired estimates (3.7) and (3.8) follow readily from (3.12)–(3.15). $\quad \square$

We fix $\varphi =\varphi (y, s)=\theta _1 (y) \theta _2 (s)$, where $\theta _1 \in C_0^\infty (B(0, 1))$, $\theta _2\in C^\infty _0 (-1, 0)$, $\theta _1, \theta _2\ge 0$, and $\int _{{\mathbb {R}}^{d}} \theta _1 (y) \, \mathrm{d}y =\int _{{\mathbb {R}}} \theta _2 (s)\, \mathrm{d}s=1$. Define

$$\begin{aligned} S_\delta (f) (x, t) =\int _{{\mathbb {R}}^{d+1}} f(x-y, t-s) \varphi _\delta (y, s)\, \mathrm{d}y\mathrm{d}s, \end{aligned}$$

(3.16)

where $\delta >0$ and $\varphi _\delta (y, s)=\delta ^{-d-2} \varphi (y/\delta , s/\delta ^2)$.

Lemma 3.3

Let $g\in L^2_{loc}({\mathbb {R}}^{d+1})$ and $f\in L^2({\mathbb {R}}^{d+1})$. Then

(3.17)

(3.18)

where C depends only on d.

Proof

By Hölder’s inequality,

$$\begin{aligned} |S_\delta (f) (x, t)|^2 \le \int _{{\mathbb {R}}^{d+1}} | f(y, s)|^2 \varphi _\delta (x-y, t-s)\, \mathrm{d}y\mathrm{d}s. \end{aligned}$$

It follows by Fubini’s Theorem that

where C depends only on d. This gives (3.17). The estimate (3.18) follows in a similar manner. $\quad \square$

Lemma 3.4

Let $S_\delta$ be defined by (3.16). Then

$$\begin{aligned} \Vert g \nabla f - S_\delta (g \nabla f)\Vert _{L^2({\mathbb {R}}^{d+1})}&\le C \delta \Big \{ \Vert \nabla (g \nabla f) \Vert _{L^2({\mathbb {R}}^{d+1})} + \Vert g \partial _t f\Vert _{L^2({\mathbb {R}}^{d+1})}\nonumber \\&\quad + \delta \Vert (\partial _t g) (\nabla f) \Vert _{L^2({\mathbb {R}}^{d+1})} \nonumber \\&\quad + \delta \Vert (\nabla g ) \partial _t f\Vert _{L^2({\mathbb {R}}^{d+1})} \Big \}, \end{aligned}$$

(3.19)

where C depends only on d.

Proof

Write $S_\delta =S_\delta ^1 S_\delta ^2$, where

$$\begin{aligned} \left\{ \begin{aligned}&S_\delta ^1 (f) (x, t)=\int _{{\mathbb {R}}^d} f(x-y, t) \delta ^{-d} \theta _1 (y/\delta )\, \mathrm{d}y,\\&S_\delta ^2 (f) (x, t)=\int _{{\mathbb {R}}} f(x, t-s) \delta ^{-2} \theta _2 (s/\delta ^2)\, \mathrm{d}s. \end{aligned} \right. \end{aligned}$$

(3.20)

By using the Plancherel Theorem, it is easy to see that

$$\begin{aligned} \left\{ \begin{aligned}\Vert f -S_\delta ^1 (f)\Vert _{L^2({\mathbb {R}}^{d+1})}&\le C \delta \Vert \nabla f\Vert _{L^2({\mathbb {R}}^{d+1})},\\ \Vert f -S_\delta ^2 (f)\Vert _{L^2({\mathbb {R}}^{d+1})}&\le C \delta ^2 \Vert \partial _t f\Vert _{L^2({\mathbb {R}}^{d+1})}, \end{aligned} \right. \end{aligned}$$

where C depends only on d. It follows that

$$\begin{aligned} \begin{aligned}\Vert g \nabla f -S_\delta (g \nabla f) \Vert _{L^2({\mathbb {R}}^{d+1})}&\le \Vert g \nabla f -S_\delta ^1 ( g\nabla f) \Vert _{L^2({\mathbb {R}}^{d+1})} + \Vert S_\delta ^1 ( g \nabla f) \\&\quad - S_\delta ( g\nabla f) \Vert _{L^2({\mathbb {R}}^{d+1})}\\&\le C \delta \Vert \nabla (g \nabla f ) \Vert _{L^2({\mathbb {R}}^{d+1})} + C \delta ^2 \Vert \partial _t S_\delta ^1 (g\nabla f) \Vert _{L^2({\mathbb {R}}^{d+1})}. \end{aligned} \end{aligned}$$

To bound the last term in the inequalities above, we note that

$$\begin{aligned} \partial _t (g \nabla f) = (\partial _t g) \nabla f +\nabla (g \partial _t f) - (\nabla g) \partial _t f. \end{aligned}$$

Using the estimates

$$\begin{aligned} \Vert S_\delta ^1 (h)\Vert _{L^2({\mathbb {R}}^{d+1})} \le \Vert h\Vert _{L^2({\mathbb {R}}^{d+1})} \quad \text { and } \quad \Vert \nabla S_\delta ^1 (h)\Vert _{L^2({\mathbb {R}}^{d+1})} \le C \delta ^{-1} \Vert h\Vert _{L^2({\mathbb {R}}^{d+1})}, \end{aligned}$$

we obtain

$$\begin{aligned}&\Vert \partial _t S_\delta ^1 (g \nabla f)\Vert _{L^2({\mathbb {R}}^{d+1})} \le \Vert (\partial _t g ) \nabla f\Vert _{L^2({\mathbb {R}}^{d+1})} + C \delta ^{-1} \Vert g \partial _t f\Vert _{L^2({\mathbb {R}}^{d+1})} \\&\quad +\, \Vert (\nabla g) \partial _t f\Vert _{L^2({\mathbb {R}}^{d+1})}. \end{aligned}$$

This completes the proof. $\quad \square$

Let

$$\begin{aligned} w_\varepsilon =u_{\varepsilon , \lambda } -u_{0, \lambda } - \varepsilon ( \chi _j^\lambda )^\varepsilon K_\varepsilon \left( \frac{\partial u_{0, \lambda } }{\partial x_j}\right) +\varepsilon ^2 \left( \phi _{i (d+1) j}^\lambda \right) ^\varepsilon \frac{\partial }{\partial x_i} K_\varepsilon \left( \frac{\partial u_{0, \lambda } }{\partial x_j} \right) ,\nonumber \\ \end{aligned}$$

(3.21)

where

$$\begin{aligned} (\chi _j^\lambda )^\varepsilon =\chi _j^\lambda (x/\varepsilon , t/\varepsilon ^2),\ \ \ (\phi _{i (d+1)j}^\lambda )^\varepsilon = \phi _{i (d+1) j}^\lambda (x/\varepsilon , t/\varepsilon ^2), \end{aligned}$$

and $K_\varepsilon$ is a linear operator to be specified later.

Lemma 3.5

Suppose that

$$\begin{aligned} (\partial _t +{\mathcal {L}}_{\varepsilon , \lambda })u_{\varepsilon , \lambda } = (\partial _t +{\mathcal {L}}_{0, \lambda } ) u_{0, \lambda } \quad \text { in } \Omega \times (T_0, T_1). \end{aligned}$$

Let $w_\varepsilon$ be defined by (3.21). Then

$$\begin{aligned} \begin{aligned} (\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) w_\varepsilon&= -\text { div} \left( \big ( \widehat{A_\lambda } -A_\lambda (x/\varepsilon , t/\varepsilon ^2)\big ) \big (\nabla u_{0, \lambda } -K_\varepsilon (\nabla u_{0, \lambda }) \big )\right) \\&\quad +\varepsilon \, \text { div} \Big ( A_\lambda (x/\varepsilon , t/\varepsilon ^2)\chi ^\lambda (x/\varepsilon , t/\varepsilon ^2) \nabla K_\varepsilon (\nabla u_{0, \lambda } ) \Big )\\&\quad +\varepsilon \, \frac{\partial }{\partial x_k} \left\{ \phi _{kij}^\lambda (x/\varepsilon , t/\varepsilon ^2) \frac{\partial }{\partial x_i} K_\varepsilon \left( \frac{\partial u_{0, \lambda } }{\partial x_j}\right) \right\} \\&\quad + \varepsilon ^2 \frac{\partial }{\partial x_k} \left\{ \phi ^\lambda _{k (d+1) j} (x/\varepsilon , t/\varepsilon ^2) \partial _t K_\varepsilon \left( \frac{\partial u_{0, \lambda } }{\partial x_j}\right) \right\} \\&\quad -\varepsilon \, \frac{\partial }{\partial x_i} \left\{ a_{ij}^\lambda (x/\varepsilon , t/\varepsilon ^2) \left( \frac{\partial }{\partial x_j} \phi _{\ell (d+1) k}^\lambda \right) (x/\varepsilon , t/\varepsilon ^2) \frac{\partial }{\partial x_\ell } K_\varepsilon \left( \frac{\partial u_{0, \lambda } }{\partial x_k} \right) \right\} \\&\quad -\varepsilon ^2\frac{\partial }{\partial x_i} \left\{ a_{ij}^\lambda (x/\varepsilon , t/\varepsilon ^2) \phi ^\lambda _{\ell (d+1) k }(x/\varepsilon , t/\varepsilon ^2) \frac{\partial ^2}{\partial x_j \partial x_\ell } K_\varepsilon \left( \frac{\partial u_{0, \lambda } }{\partial x_k} \right) \right\} , \end{aligned} \end{aligned}$$

(3.22)

where $A_\lambda = \big ( a_{ij}^\lambda \big )$.

Proof

This is proved by a direct computation. See [14, Theorem 2.2] for the case $\lambda =1$. $\quad \square$

Lemma 3.6

Let $Q_r=B(0, r) \times (-r^2, 0)$. Suppose $u_{\varepsilon , \lambda }$ is a weak solution of $( \partial _t +{\mathcal {L}}_{\varepsilon , \lambda } ) u_{\varepsilon , \lambda } =F$ in $Q_2$ for some $F\in L^2(Q_2)$. Then there exists a weak solution of $(\partial _t +{\mathcal {L}}_{0, \lambda } ) u_{0, \lambda } =F$ in $Q_1$ such that

(3.23)

and for $\delta = (1+\sqrt{\lambda } ) \varepsilon$,

(3.24)

where $\sigma \in (0, 1)$ and $C>0$ depend only on d and $\mu$. The operator $K_\varepsilon$ is defined by (3.27).

Proof

We start out by defining $u_{0, \lambda }$ to be the weak solution of the initial-Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t + {\mathcal {L}}_{0, \lambda } ) u_{0, \lambda } =F &{} \text { in } Q_{1},\\ u_{0, \lambda } =u_{\varepsilon , \lambda } &{} \text { on } \partial _p Q_{1}, \end{array} \right. \end{aligned}$$

(3.25)

where $\partial _p Q_{1}$ denotes the parabolic boundary of the cylinder $Q_{1}$. Note that

$$\begin{aligned} (\partial _t +{\mathcal {L}}_{0, \lambda } ) (u_{0, \lambda } -u_{\varepsilon , \lambda } ) =({\mathcal {L}}_{\varepsilon , \lambda } -{\mathcal {L}}_{0, \lambda }) u_{\varepsilon , \lambda } \end{aligned}$$

in $Q_{1}$ and $u_{\varepsilon , \lambda } -u_{0, \lambda } =0$ on $\partial _p Q_{1}$. It follows from the standard regularity estimates for parabolic operators with constant coefficients that

for any $2\le q<\infty$, where C depends only on d, $\mu$ and q. This gives

for any $2<q<\infty$. By the Meyers-type estimates for parabolic systems [2, Appendix], there exist some $q>2$ and $C >0$, depending on d and $\mu$, such that

As a result, we obtain

(3.26)

for some $q>2$ and $C>0$, depending only on d and $\mu$.

To prove (3.24), we let $\delta =(1+\sqrt{\lambda })\varepsilon$. We may assume $\delta \le 1/8$; for otherwise the estimate is trivial. Choose $\eta _\delta \in C_0^\infty ({\mathbb {R}}^{d+1})$ such that $0\le \eta _\delta \le 1$, $\ |\nabla \eta _\delta |\le C/\delta$, $|\partial _t \eta _\delta | +|\nabla ^2 \eta _\delta |\le C /\delta ^2$,

$$\begin{aligned} \eta _\delta =1 \quad \text { in } Q_{1-3\delta } \quad \text { and } \quad \eta _\delta =0 \quad \text{ in } Q_1 {\setminus } Q_{1-2\delta }. \end{aligned}$$

Let $w_\varepsilon$ be defined by (3.21), where the operator $K_\varepsilon$ is given by

$$\begin{aligned} K_\varepsilon (f) = S_\delta ( \eta _\delta f), \end{aligned}$$

(3.27)

with $S_\delta$ defined in (3.16). Note that $w_\varepsilon =0$ in $\partial _p Q_1$. It follows from Lemma 3.5 and energy estimates that

$$\begin{aligned} \begin{aligned}\int _{Q_1}|\nabla w_\varepsilon |^2&\le C \int _{Q_1} |\nabla u_{0, \lambda } -K_\varepsilon (\nabla u_{0, \lambda }) |^2 + C \varepsilon ^2 \int _{Q_1} |( \chi ^\lambda )^\varepsilon \nabla K_\varepsilon (\nabla u_{0, \lambda })|^2\\&\qquad + C \varepsilon ^2 \int _{Q_1} \sum _{k,i,j} |(\phi _{kij}^\lambda )^\varepsilon |^2 |\nabla K_\varepsilon (\nabla u_{0, \lambda })|^2\\&\qquad + C\varepsilon ^4 \int _{Q_1}\sum _{k, j} |(\phi _{k (d+1)j}^\lambda )^\varepsilon |^2 |\partial _t K_\varepsilon (\nabla u_{0, \lambda } )|^2\\&\qquad + C \varepsilon ^2 \int _{Q_1} \sum _{\ell , k} | (\nabla \phi _{\ell (d+1) k} ^\lambda )^\varepsilon |^2 |\nabla K_\varepsilon (\nabla u_{0, \lambda } )|^2\\&\qquad + C\varepsilon ^4 \int _{Q_1} \sum _{\ell , k} |(\phi _{\ell (d+1) k}^\lambda )^\varepsilon |^2 |\nabla ^2 K_\varepsilon (\nabla u_{0, \lambda } )|^2\\&=I_1 +I_2 +I_3 +I_4 +I_5+I_6. \end{aligned} \end{aligned}$$

(3.28)

To bound $I_1$, we use Lemma 3.4. This gives

$$\begin{aligned} \begin{aligned}I_1&\le 2\int _{Q_1} |\nabla u_{0, \lambda } -\eta _\delta (\nabla u_{0, \lambda })|^2 + 2\int _{Q_1} |\eta _\delta (\nabla u_{0, \lambda }) -S_\delta (\eta _\delta (\nabla u_{0, \lambda }))|^2\\&\le C \int _{Q_1 {\setminus } Q_{1-3\delta }} |\nabla u_{0, \lambda }|^2 + C \delta ^2 \int _{Q_{1-2\delta }}\big ( |\nabla ^2 u_{0, \lambda } |^2 +|\partial _t u_{0, \lambda }|^2 \big ). \end{aligned} \end{aligned}$$

By the standard regularity estimates for parabolic systems with constant coefficients [20, 21]

$$\begin{aligned} \int _{Q_{1-2\delta }} \big ( |\nabla ^2 u_{0, \lambda }|^2 + |\partial _t u_{0, \lambda }|^2 \big ) \le C \left\{ \int _{Q_{1-\delta }}\frac{|\nabla u_{0, \lambda }( y, s) |^2\, \mathrm{d}y \mathrm{d}s }{| \text {dist} _p ((y,s), \partial _p Q_1)|^2 } +\int _{Q_1} |F|^2 \right\} , \end{aligned}$$

where $\text { dist}_p ((y, s), \partial _p Q_1)=\inf \big \{ |x-y| +|s-t|^{1/2}: (x, t) \in \partial _p Q_1 \big \}$ denotes the parabolic distance from (y, s) to $\partial _p Q_1$. It follows that

(3.29)

where $q>2$ and we have used Hölder’s inequality for the last step.

To bound $I_2$, $I_3$ and $I_5$, we use Lemma 3.3 and estimates (2.5) and (3.7) as well as the observation $\nabla S_\delta (f)=S_\delta (\nabla f)$. Note that $(\chi ^\lambda )^\varepsilon$, $(\phi _{kij}^\lambda )^\varepsilon$ and $(\nabla \phi _{\ell (d+1) k}^\lambda )^\varepsilon$ are $\varepsilon$-periodic in x and $\varepsilon ^2 \lambda$-periodic in t. Since $\delta =(1+\sqrt{\lambda }) \varepsilon \ge \varepsilon$ and $\delta ^2 \ge \varepsilon ^2 \lambda$, we obtain

for any $(x, t)\in {\mathbb {R}}^{d+1}$. It follows that

(3.30)

To bound $I_6$, we use the inequality (3.18) as well as the estimate (3.8). This leads to

(3.31)

Finally, to handle $I_4$, we use the observation

$$\begin{aligned} \partial _t K_\varepsilon (\nabla u_{0, \lambda })= & {} \partial _t S_\delta (\eta _\delta \nabla u_{0, \lambda }) \nonumber \\= & {} S_\delta ( (\partial _t \eta _\delta ) \nabla u_{0, \lambda }) + S_\delta (\nabla (\eta _\delta \partial _t u_{0, \lambda })) -S_\delta ( (\nabla \eta _\delta ) \partial _t u_{0, \lambda }). \end{aligned}$$

(3.32)

As in the case of $I_6$, we obtain

(3.33)

Let $\sigma =\frac{1}{2}-\frac{1}{q}>0$. In view of (3.29)–(3.32), we have proved that

(3.34)

where we have used (3.26) for the last step. To finish the proof, we let $H_\varepsilon$ be the last term in (3.21). It is easy to see that

$$\begin{aligned} \int _{Q_1} |\nabla H_\varepsilon |^2 \le I_5 + I_6. \end{aligned}$$

This, together with (3.34), gives the estimate (3.24). $\quad \square$

We are now ready to give the proof of Theorem 3.1.

Proof of Theorem 3.1

By translation and dilation we may assume that $r=1$ and $Q_2=B(0, 2) \times (-4, 0)$. We may also assume that $\delta =(1+\sqrt{\lambda }) \varepsilon \le 1/8$. This reduces the problem to the case considered in Lemma 3.6. Observe that $K_\varepsilon (\nabla u_{0, \lambda }) =S_\delta (\nabla u_{0, \lambda })$ on $Q_{1/2}$. Thus, in view of Lemma 3.6, it suffices to show that

(3.35)

is bounded by the right-hand side of (3.24). Furthermore, since $(\partial _t +{\mathcal {L}}_{0, \lambda } )u_{0, \lambda } =F$ in $Q_1$, we have

Also, recall that

$$\begin{aligned} \Vert (\chi ^\lambda )^\varepsilon \Vert _{L^2(Q_1)} +\Vert (\nabla \chi ^\lambda )^\varepsilon \Vert _{L^2(Q_1)}\le C. \end{aligned}$$

(3.36)

As a result, it is enough to show that

(3.37)

is bounded by the right-hand side of (3.24). This, however, follows from (3.36) and the estimate

(3.38)

where $p>d+2$ and $\sigma =1-\frac{d+2}{p}$.

Finally, we point out that (3.38) follows readily from the $C^{1+ \sigma }$ estimates for $\partial _t +{\mathcal {L}}_{0, \lambda }$,

(3.39)

for any $(x, t), (y, s)\in Q_{1/2}$. This completes the proof.

4 Large-Scale Lipschitz and $C^{1, \alpha }$ Estimates

In this section we establish the large-scale Lipschitz and $C^{1, \alpha }$ estimates for $\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }$. As a consequence, we obtain the same estimates for the parabolic operator $\partial _t+ {\mathcal {L}}_\varepsilon$ in (1.1). Let

$$\begin{aligned} \begin{aligned}&P^\lambda _{1, \varepsilon } = \Big \{ P=P(x, t): P(x, t)=\beta _0 + \beta _j (x_j +\varepsilon \chi _j^\lambda (x/\varepsilon , t/\varepsilon ^2) )\\&\quad \text { for some } \beta =(\beta _0, \beta _1, \ldots , \beta _d) \in {\mathbb {R}}^{d+1} \Big \}, \end{aligned} \end{aligned}$$

(4.1)

where the index j is summed from 1 to d. Note that $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) P=0$ in ${\mathbb {R}}^{d+1}$ for any $P\in P_{1, \varepsilon }^\lambda$.

Theorem 4.1

($C^{1, \alpha }$ estimate) Suppose A satisfies conditions (1.3) and (1.4). Let $u_{\varepsilon , \lambda }$ be a weak solution of $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda } =F$ in $Q_R$, where $R> (1+\sqrt{\lambda }) \varepsilon$ and $F\in L^p(Q_R)$ for some $p>d+2$. Then, for any $(1+\sqrt{\lambda })\varepsilon \le r< R$ and $0<\alpha < 1-\frac{d+2}{p}$,

(4.2)

where $C>0$ depends only on d, $\mu$, p and $\alpha$.

Proof

The proof relies on the approximation results in Theorem 3.1 and uses classical regularity estimates for parabolic systems with constant coefficients. By translation and dilation we may assume that $R=2$ and $Q_2 =B(0, 2)\times (-4, 0)$. Let

$$\begin{aligned} (1+\sqrt{\lambda } )\varepsilon< \theta r< r< 1, \end{aligned}$$

where $\theta \in (0, 1/4)$ is to be chosen later. Let $u_{0, \lambda }$ be the weak solution of $(\partial _t +{\mathcal {L}}_{0, \lambda }) u_{0, \lambda } =F$ in $Q_r$, given by Theorem 3.1. By the classical $C^{1+\alpha }$ estimates for parabolic systems with constant coefficients [20, 21],

for any $(x, t)\in Q_{ r/2}$, where $\alpha _p =1-\frac{d+2}{p}$. Let $P(x, t) =\beta _j (x_j +\varepsilon \chi _j^\lambda (x/\varepsilon , t/\varepsilon ^2))$ with $\beta _j =\frac{\partial u_{0, \lambda }}{\partial x_j} (0, 0)$. Then

for any $(x, t)\in Q_{\theta r}$. It follows that

where $C_0$ depends only d, $\mu$ and p. Fix $0<\alpha < \alpha _p$. We choose $\theta \in (0, 1/4)$ so small that $C_0 \theta ^{\alpha _p} \le (1/2)\theta ^{\alpha }$. With $\theta$ chosen, we assume that $r\ge C_\theta (1+\sqrt{\lambda }) \varepsilon$, where $C_\theta >1$ is so large that

$$\begin{aligned} C_0 \theta ^{-\frac{d+2}{2}} C_\theta ^{-\sigma } <(1/2) \theta ^{\alpha }. \end{aligned}$$

This leads to

Since $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) P=0$ in ${\mathbb {R}}^{d+1}$ for any $P\in P_{1, \varepsilon }^ \lambda$, we obtain

(4.3)

for any $C_\theta (1+\sqrt{\lambda })\varepsilon \le r<1$. By an iteration argument it follows that

(4.4)

for any $(1+\sqrt{\lambda }) \varepsilon \le r< 1$. This gives the large-scale $C^{1, \alpha }$ estimate (4.2). $\quad \square$

Theorem 4.2

(Lipschitz estimate) Suppose A satisfies conditions (1.3) and (1.4). Let $u_{\varepsilon , \lambda }$ be a weak solution of $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda }=F$ in $Q_R$, where $R> (1+\sqrt{\lambda }) \varepsilon$ and $F\in L^p(Q_R)$ for some $p>d+2$. Then, for any $(1+\sqrt{\lambda })\varepsilon \le r< R$,

(4.5)

where $C>0$ depends only on d, $\mu$ and p.

Proof

By translation and dilation we may assume that $R=2$ and $Q_2=B(0, 2)\times (-4, 0)$. Define

where $H_r=E_r \cdot (x +\varepsilon \chi ^\lambda (x/\varepsilon , t/\varepsilon ^2) )$, with $E_r \in {\mathbb {R}}^d$, is a function in $P_{1, \varepsilon }^\lambda$ such that

Let $C (1+\sqrt{\lambda })\varepsilon< r<1/2$. Note that

where C depends only on d and $\mu$. It follows that if $r\ge C_1 \varepsilon$ and $C_1>1$ is sufficiently large, then

(4.6)

We remark that the last inequality follows from the fact that $u=H_{2r}-H_r -\beta _0$ is a solution of the second-order parabolic system in divergence form $(\partial _t + {\mathcal {L}}_{\varepsilon , \lambda } )u =0$ in ${\mathbb {R}}^{d+1}$. Such a solution satisfies the Poincaré-type inequality,

$$\begin{aligned} \inf _{\beta _0 \in {\mathbb {R}}} \int _{Q_{r/2}} |u-\beta _0|^2 \le Cr^2 \int _{Q_r} |\nabla u|^2 \end{aligned}$$

(see for example [13, Lemma 2.2]). Hence,

where we have used (4.4) for the last step. By a simple summation this yields

which, together with (4.2), gives the large-scale Lipschitz estimate (4.5). $\quad \square$

Proof of Theorem 1.1

Recall that if $\lambda =\varepsilon ^{k-2}$, then ${\mathcal {L}}_{\varepsilon , \lambda }={\mathcal {L}}_\varepsilon$. Also note that in this case, $(1+\sqrt{\lambda }) \varepsilon =\varepsilon +\varepsilon ^{k/2}$. As a result, Theorem 1.1 follows directly from Theorem 4.2.

Remark 4.3

($C^{1, \alpha }$ estimate) Let $u_\varepsilon$ be a weak solution of $(\partial _t +{\mathcal {L}}_\varepsilon ) u_\varepsilon =F$ in $Q_R$, where $R> \varepsilon +\varepsilon ^{k/2}$ and $F\in L^p(Q_R)$ for some $p>d+2$. It follows from Theorem 4.1 that for $\varepsilon +\varepsilon ^{k/2} \le r< R$ and $0< \alpha < 1-\frac{d+2}{p}$,

(4.7)

where $\lambda =\varepsilon ^{k-2}$ and C depends only on d, $\mu$, p and $\alpha$. Note that $\nabla \chi ^\lambda (x/\varepsilon , t/\varepsilon ^2)$ is $\varepsilon$-periodic in x and $\varepsilon ^k$-periodic in t. One may regard (4.7) as a $C^{1, \alpha }$ excess-decay estimate for the operator $\partial _t +{\mathcal {L}}_\varepsilon$ in (1.1).

Let $E_r\in {\mathbb {R}}^d$ be the constant for which the left-hand side of (4.7) obtains its minimum. It follows from the proof of Theorem 4.2 that

(4.8)

Let $\chi ^\infty$ be defined by (2.10). In view of (2.27) we have

(4.9)

This, together with (4.7) and (4.8), yields

(4.10)

for $0<k<2$. Similarly, for $2<k<\infty$, we obtain

5 Higher-Order Correctors and $C^{2, \alpha }$ Estimates

In this section we introduce the second-order correctors and establish the large-scale $C^{2, \alpha }$ estimates for ${\mathcal {L}}_{\varepsilon , \lambda }$.

Let $A_\lambda =\big ( a_{ij}^\lambda \big )$ and $B_\lambda =\big (b_{k\ell }^\lambda \big )$ be the $(1, \lambda )$-periodic matrices given by (2.1) and (3.5), respectively. For $1\le k, \ell \le d$, the second-order corrector $\chi _{k\ell }^\lambda =\chi _{k\ell }^\lambda (y, s)$ is defined to be the weak solution of the cell problem

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s \chi _{k\ell }^\lambda -\text { div} \big (A_\lambda \nabla \chi _{k\ell }^\lambda \big ) =b_{k\ell }^\lambda + b_{\ell k}^\lambda +\frac{\partial }{\partial y_i} \big ( a_{i \ell }^\lambda \chi _k^\lambda \big ) +\frac{\partial }{\partial y_i} \big ( a^\lambda _{ik} \chi ^\lambda _\ell \big ) \quad \text { in } {\mathbb {R}}^{d+1},\\&\chi _{k\ell }^\lambda \text { is } (1, \lambda )\text {-periodic in } (y, s),\\&\int _0^\lambda \!\!\! \int _{{\mathbb {T}}^d} \chi _{k\ell }^\lambda \, \mathrm{d}y \mathrm{d}s =0, \end{aligned} \right. \end{aligned}$$

(5.1)

where $(\chi _j^\lambda )$ are the first-order correctors defined by (2.2). Since

$$\begin{aligned} \int _0^\lambda \!\!\! \int _{{\mathbb {T}}^d} b^\lambda _{k\ell }\, \mathrm{d}y \mathrm{d}s=0, \end{aligned}$$

the solution to (5.1) exists and is unique. Also, observe that $\chi ^\lambda _{k\ell }=\chi _{\ell k}^\lambda$. Moreover, by the energy estimates,

(5.2)

where C depends only on d and $\mu$.

Lemma 5.1

Let

$$\begin{aligned} u(y, s)=y_k y_\ell + y_k \chi ^\lambda _\ell (y, s) + y_\ell \chi _k^\lambda (y, s) +\chi _{k\ell }^\lambda (y, s). \end{aligned}$$

Then

$$\begin{aligned} \big ( \partial _s -\text { div} (A_\lambda \nabla ) \big ) u = \big ( \partial _s -\text { div} (\widehat{A_\lambda } \nabla ) \big ) (y_k y_\ell ) =- \widehat{a_{\ell k}^\lambda } -\widehat{a_{k\ell }^\lambda } \end{aligned}$$

in ${\mathbb {R}}^{d+1}$, where $\widehat{A_\lambda } = \big ( \widehat{a^\lambda _{k\ell }} \big )$.

Proof

This follows from a direct computation, using the definitions of $\chi _j^\lambda$ and $\chi _{k\ell }^\lambda$. $\quad \square$

Let $P_0 (x, t)=\beta _0 + c_0 t + c_k x_k +c_{k\ell } x_k x_\ell$ and

$$\begin{aligned} \begin{aligned}P_\varepsilon (x, t)&=\beta _0+ c_0 t + c_k \big \{ x_k + \varepsilon \chi _k^\lambda (x/\varepsilon , t/\varepsilon ^2) \big \} \\&\quad +c_{k \ell } \Big \{ x_k x_\ell +\varepsilon x_k \chi _\ell ^\lambda (x/\varepsilon , t/\varepsilon ^2) +\varepsilon x_\ell \chi _k^\lambda (x/\varepsilon , t/\varepsilon ^2) \\&\quad +\varepsilon ^2 \chi _{k \ell } (x/\varepsilon , t/\varepsilon ^2) \Big \}, \end{aligned} \end{aligned}$$

(5.3)

where $\beta _0, c_0, c_k, c_{k\ell } =c_{\ell k} \in {\mathbb {R}}$. It follows from Lemma 5.1 by rescaling that

$$\begin{aligned} (\partial _t +{\mathcal {L}}_{\varepsilon , \lambda } ) P_\varepsilon =(\partial _t +{\mathcal {L}}_{0, \lambda } ) P_0 =c_0-2 c_{k\ell } \widehat{a^\lambda _{k\ell }} \quad \text { in } {\mathbb {R}}^{d+1}. \end{aligned}$$

We shall use $P^\lambda _{2, \varepsilon }$ to denote the set of all functions $P_\varepsilon (x, t)$ in the form of (5.3) such that $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) P_\varepsilon =0$. Let $C^\sigma _p (Q_R)$ denote the space of Hölder continuous functions $u=u(x, t)$ such that

$$\begin{aligned} \Vert u\Vert _{C^\sigma (Q_R)}:= & {} R^\sigma \sup \Bigg \{ \frac{| u(x, t)-u(y, s)|}{ ( |x-y| +|t-s|^{1/2} )^\sigma }: (x, t), (y, s) \\\in & {} Q_R \text { and } (x, t)\ne (y, s) \Bigg \}<\infty , \end{aligned}$$

where $\sigma \in (0, 1)$.

Theorem 5.2

($C^{2, \alpha }$ estimate) Suppose A satisfies conditions (1.3) and (1.4). Let $u_{\varepsilon , \lambda }$ be a weak solution of $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda } =F$ in $Q_R$, where $R> (1+\sqrt{\lambda })\varepsilon$ and $F\in C^{\sigma }(Q_R)$ for some $\sigma \in (0, 1)$. Then, for any $(1+\sqrt{\lambda }) \varepsilon \le r <R$ and $0<\alpha <\sigma$,

(5.4)

where C depends only on d, $\sigma$, $\mu$, and $\alpha$.

Proof

By translation and dilation we may assume that $R=2$ and $Q_2=B(0, 2)\times (-4, 0)$. By subtracting $c_0 t$ from $u_{\varepsilon , \lambda }$, we may also assume that $F(0, 0)=0$, which implies $\Vert F\Vert _{L^\infty (Q_r)} \le C \Vert F\Vert _{C^\sigma (Q_r)}$. Let $(1+\sqrt{\lambda }) \varepsilon< \theta r< r<1$, where $\theta \in (0, 1/4)$ is to be chosen later. Let $u_{0, \lambda }$ be the weak solution of $(\partial _t +{\mathcal {L}}_{0, \lambda }) u_{0, \lambda }=F$ in $Q_r$, given by Theorem 3.1. By the classical $C^{2+\alpha }$ estimates for parabolic systems with constant coefficients [20, 21],

(5.5)

for any $(x, t)\in Q_{\theta r}$, where we have used (3.3) for the last inequality. Let

$$\begin{aligned} P_0 (x, t)=c_0 t +c_ix_i + c_{ij} x_i x_j, \end{aligned}$$

where

$$\begin{aligned} c_0 =\partial _t u_{0, \lambda } (0, 0),\ c_i =\frac{\partial u_{0, \lambda } }{\partial x_i} (0, 0), \text { and } \ c_{ij}=\frac{1}{2} \frac{\partial ^2 u_{0, \lambda }}{\partial x_i \partial x_j} (0, 0). \end{aligned}$$

(5.6)

Note that

$$\begin{aligned} (\partial _t +{\mathcal {L}}_{0, \lambda } ) P_0 = c_0 -2 c_{ij} \widehat{a_{ij}^\lambda } =(\partial _t +{\mathcal {L}}_{0, \lambda }) u_0 (0, 0)=F(0, 0)=0, \end{aligned}$$

(5.7)

and by (5.5),

(5.8)

This, together with the inequality (3.4), gives

(5.9)

Let $P_\varepsilon = P_\varepsilon (x, t)$ be given by (5.3) with the same coefficients as those of $P_0$ in (5.6). Then $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) P_\varepsilon =(\partial _t +{\mathcal {L}}_{0, \lambda }) P_0=0$, and

$$\begin{aligned} |\nabla P_\varepsilon -\nabla P_0 -(\nabla \chi ^\lambda )^\varepsilon (\nabla P_0)| \le \varepsilon |c_{k\ell } \nabla \chi ^\lambda _{k\ell } (x/\varepsilon , t/\varepsilon ^2)|. \end{aligned}$$

(5.10)

In view of (5.9), we obtain

(5.11)

where we have used (5.2) and the assumption that $\theta r \ge (1+\sqrt{\lambda } )\varepsilon$.

To proceed, we let

It follows from (5.11) that

$$\begin{aligned} \Psi (\theta r) \le C_0 \left\{ \theta ^{1+\sigma } + \left( \frac{(1+\sqrt{\lambda })\varepsilon }{r} \right) ^\sigma \right\} \Psi (2r) \end{aligned}$$

for $(1+\sqrt{\lambda }) \varepsilon< \theta r< r<1$, where $C_0$ depends only on d, $\mu$ and $\sigma$. Fix $\alpha \in (0, \sigma )$. Choose $\theta \in (0, 1/4)$ so small that $C_0\theta ^{1+\sigma } \le (1/2) (\theta /2)^{1+\alpha }$. With $\theta$ chosen, we may choose $C_1>1$ so large that $C_0 C_1^{-\sigma } \le (1/2) (\theta /2) ^{1+\alpha }$. As a result, for $C_1 (1+\sqrt{\lambda }) \varepsilon< \theta r< r<1$, we have

$$\begin{aligned} \Psi (\theta r) \le (\theta /2)^{1+\alpha } \Psi (2r). \end{aligned}$$

By a simple iteration argument this gives $\Psi (r)\le C r^{1+\alpha } \Psi (2)$ for any $(1+\sqrt{\lambda }) \varepsilon \le r< 2$. $\quad \square$

Remark 5.3

(Liouville property) By letting $\lambda =\varepsilon ^{k-2}$ in Theorem 5.2 we obtain a $C^{2, \alpha }$ excess-decay estimate for $\partial _t +{\mathcal {L}}_\varepsilon$ in (1.1) for any $0<k<\infty$. The estimate may be used to establish a Liouville property for the operator. Indeed, let $u_\varepsilon$ be a solution of $(\partial _t +{\mathcal {L}}_\varepsilon )u_\varepsilon =0$ in ${\mathbb {R}}^d \times (-\infty , t_0)$ for some $t_0\in {\mathbb {R}}$. Suppose there exist $C_u>0$ and $\alpha \in (0, 1)$ such that

(5.12)

for any $R>1$. By Caccioppoli’s inequality it follows that

for any $R>1$. This, together with (5.4), implies that $u_\varepsilon =P$ in ${\mathbb {R}}^d \times (-\infty , t_0)$ for some $P\in P_{2, \varepsilon }^\lambda$.

6 Boundary Lipschitz Estimates

In this section we establish large-scale boundary Lipschitz estimates for the operator $\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }$, where ${\mathcal {L}}_{\varepsilon , \lambda } =-\text { div} \big ( A_\lambda (x/\varepsilon , t/\varepsilon ^2)\nabla \big )$. As a consequence, we obtain the large-scale boundary Lipschitz estimate for $\partial _t +{\mathcal {L}}_\varepsilon$ in Theorem 1.2.

Throughout this section we will assume that $\Omega$ is a bounded $C^{1, \alpha }$ domain for some $\alpha \in (0, 1)$. Let

$$\begin{aligned} \begin{aligned}D_r (x_0, t_0)&= \big ( B(x_0, r)\cap \Omega \big ) \times (t_0-r^2, t_0),\\ \Delta _r (x_0, t_0)&= \big ( B(x_0, r)\cap \partial \Omega \big ) \times (t_0-r^2, t_0), \end{aligned} \end{aligned}$$

(6.1)

where $x_0\in \partial \Omega$ and $t_0\in {\mathbb {R}}$. For $\alpha \in (0, 1)$ and $\Delta _r=\Delta _r (x_0, t_0)$, we use $C^{1+\alpha } (\Delta _r)$ to denote the parabolic $C^{1+\alpha }$ space of functions on $\Delta _r$ with the scale-invariant norm,

$$\begin{aligned} \Vert f\Vert _{C^{1+\alpha }(\Delta _r)} :=\Vert f\Vert _{L^\infty (\Delta _r)} + r\Vert \nabla _{\tan } f\Vert _{L^\infty (\Delta _r)} + r \Vert \nabla _{\tan } f \Vert _{C^\alpha (\Delta _r)} + \Vert f\Vert _{C_t^{\frac{1+\alpha }{2}} (\Delta _r)}, \end{aligned}$$

where $\Vert g\Vert _{C^\alpha (\Delta _r)}$ is the smallest constant $C_0$ such that

$$\begin{aligned} | g(x, t)-g(y, s)|\le C_0 r^{-\alpha } (|x-y| +|t-s|^{1/2})^\alpha \end{aligned}$$

for any $(x, t), (y, s)\in \Delta _r$, and

$$\begin{aligned} \Vert f\Vert _{C_t^{\frac{1+\alpha }{2}} (\Delta _r)}= & {} \inf \left\{ C: \ |f(y, \tau )-f(y, s)|\right. \\\le & {} \left. C r^{-1-\alpha } |\tau -s|^{\frac{1+\alpha }{2}} \text { for any } (y, \tau ), (y, s)\in \Delta _r \right\} . \end{aligned}$$

Theorem 6.1

Assume $A=A(y,s)$ satisfies (1.3) and (1.4). Suppose that $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda } =F$ in $D_R=D_R (x_0, t_0)$ and $u_{\varepsilon , \lambda } =f$ on $\Delta _R= \Delta _R (x_0, t_0)$, where $x_0\in \partial \Omega$, $(1+\sqrt{\lambda } ) \varepsilon <R\le 1$, and $F\in L^p(D_R)$ for some $p>d+2$. Then, for any $(1+\sqrt{\lambda }) \varepsilon \le r< R$,

(6.2)

where C depends only on d, $\mu$, p, $\alpha$, and $\Omega$.

To prove Theorem 6.1, we localize the boundary of $\Omega$. Let $\psi : {\mathbb {R}}^{d-1} \rightarrow {\mathbb {R}}$ be a $C^{1, \alpha }$ function such that $\psi (0)=0$ and $\Vert \psi \Vert _{C^{1, \alpha } ({\mathbb {R}}^{d-1})} \le M$. Define

$$\begin{aligned} \begin{aligned}T_r&=\big \{ (x^\prime , x_d): |x^\prime |< r \text { and } \psi (x^\prime )< x_d< 100\sqrt{d} (M+1) \big \} \times (-r^2, 0), \\ I_r&=\big \{ (x^\prime , \psi (x^\prime )): |x^\prime |< r \big \} \times (-r^2, 0), \end{aligned} \end{aligned}$$

(6.3)

where $0<r<\infty$

We begin with an approximation lemma.

Lemma 6.2

Assume A satisfies (1.3) and (1.4). Suppose that $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda }=F$ in $T_{2r}$ and $u_{\varepsilon , \lambda }=f$ on $I_{2r}$ for some $0<r\le 1$. Then there exists a function $u_{0, \lambda }$ such that $(\partial _t +{\mathcal {L}}_{0, \lambda }) u_{0, \lambda } =F$ in $T_r$, $u_{0, \lambda }=f$ on $I_r$, and

(6.4)

where $\sigma \in (0, 1)$ and $C>0$ depend only on d, $\mu$, p, and M.

Proof

The proof is similar to that of Theorem 3.1. By dilation we may assume $r=1$. Let $u_{0, \lambda }$ be the weak solution to the initial-Dirichlet problem,

$$\begin{aligned} (\partial _t +{\mathcal {L}}_{0, \lambda }) u_{0, \lambda } =F \quad \text { in } T_1 \quad \text { and } \quad u_{0, \lambda } =u_{\varepsilon , \lambda } \quad \text { on } \partial _p T_1. \end{aligned}$$

It follows by the Meyers-type estimates and Caccioppoli’s inequality for parabolic systems that

(6.5)

where $q>2$ and $C>0$ depend only on d, $\mu$, $\alpha$ and M. To see (6.4), we define $w_{\varepsilon }$ as in (3.21). Using the same argument as in the proof of Theorem 3.1, we may show that

(6.6)

where $\delta =(1+\sqrt{\lambda } ) \varepsilon$ and $\sigma =\frac{1}{2} -\frac{1}{q}>0$. Since $w_\varepsilon =0$ on $\partial _p T_1$, it follows from Poincaré’s inequality and (6.5) that

(6.7)

This yields (6.4), as $\Vert w_\varepsilon -(u_{\varepsilon , \lambda } -u_{0, \lambda }) \Vert _{L^2(T_1)}$ is also bounded by the right-hand side of (6.7). $\quad \square$

For a function u in $T_r$, define

(6.8)

Lemma 6.3

Suppose that $(\partial _t +{\mathcal {L}}_{0, \lambda }) u=F$ in $T_r$, where $0<r\le 1$ and $F\in L^p(T_r)$ for some $p>d+2$. Then there exists $\theta \in (0, 1/4)$, depending only on d, $\mu$, $\alpha$, p, and M, such that

(6.9)

Proof

Choose $\sigma \in (0, 1)$ such that $\sigma <\min (\alpha , 1-\frac{d+2}{p})$. The proof uses the boundary $C^{1+ \sigma }$ estimate for second-order parabolic systems with constant coefficients in $C^{1, \alpha }$ cylinders. Let $E_0=\nabla u (0, 0)$ and $\beta _0 =u (0, 0)$. Then, for any $(x, t)\in T_{r/2}$,

where C depends only on d, $\mu$, $\alpha$, p, and M. It follows that the left-hand side of (6.9) is bounded by

Since $(\partial _t +{\mathcal {L}}_{0, \lambda }) (E\cdot x +\beta _0)=0$ for any $E\in {\mathbb {R}}^d$ and $\beta _0\in {\mathbb {R}}$, we may replace u by $u-E\cdot x -\beta _0$. As a result, we see that the left-hand side of (6.9) is bounded by

To finish the proof, we choose $\theta \in (0, 1/4)$ so small that $C_0\theta ^\sigma \le (1/2)$. $\quad \square$

Lemma 6.4

Suppose that $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda }=F$ in $T_2$ and $u_{\varepsilon , \lambda }=f$ on $I_{2}$, where $(1+\sqrt{\lambda })\varepsilon < 1$ and $F\in L^p(T_{2})$ for some $p>d+2$. Let $\theta \in (0, 1/4)$ be given by Lemma 6.3. Then for any $(1+\sqrt{\lambda }) \varepsilon \le r\le 1$,

(6.10)

where C depends only on d, $\mu$, p, $\alpha$ and M.

Proof

Fix $(1+\sqrt{\lambda }) \varepsilon \le r\le 1$. Let $u_{0, \lambda }$ be the solution of $(\partial _t +{\mathcal {L}}_{0, \lambda }) u_{0, \lambda }=F$ in $T_{r}$ with $u_{0, \lambda }=f$ on $I_r$, given by Lemma 6.2. Observe that

where we have used Lemma 6.3 for the second inequality. This, together with Lemma 6.2, gives (6.10). $\quad \square$

The proof of the next lemma may be found in [27, pp.157-158].

Lemma 6.5

Let H(r) and h(r) be two nonnegative and continuous functions on the interval [0, 1]. Let $0<\delta <(1/4)$. Suppose that there exists a constant $C_0$ such that

$$\begin{aligned} \max _{r\le t\le 2r} H(t) \le C_0 H(2r) \quad \text { and } \quad \max _{r\le t, s\le 2r} |h(t) -h(s)|\le C_0 H(2r) \end{aligned}$$

(6.11)

for any $r\in [\delta , 1/2]$. Suppose further that

$$\begin{aligned} H(\theta r) \le \frac{1}{2} H(r) +C_0 \eta (\delta / r) \Big \{ H(2r) + h(2r) \Big \} \end{aligned}$$

(6.12)

for any $r\in [\delta , 1/2]$, where $\theta \in (0, 1/4)$ and $\eta (t)$ is a nonnegative and nondecreasing function on [0, 1] such that $\eta (0)=0$ and

$$\begin{aligned} \int _0^1 \frac{\eta (t)}{t} \, \mathrm{d}t <\infty . \end{aligned}$$

(6.13)

Then

$$\begin{aligned} \max _{\delta \le r\le 1} \big \{ H(r) + h(r) \big \} \le C \big \{ H(1) + h(1) \big \}, \end{aligned}$$

(6.14)

where C depends only on $C_0$, $\theta$, and the function $\eta (t)$.

We are now ready to give the proof of Theorem 6.1

Proof of Theorem 6.1

By translation and dilation we may assume that $(x_0, t_0)=(0, 0)$ and $R=1$. Moreover, it suffices to show that for $(1+\sqrt{\lambda }) \varepsilon \le r< 2$,

(6.15)

where $(\partial _t + {\mathcal {L}}_{\varepsilon , \lambda }) u_{\varepsilon , \lambda } =F$ in $T_2$ and $u_{\varepsilon , \lambda } =f$ on $I_2$. To this end, we apply Lemma 6.5 with

and $h(t)=|E_r|$, where $E_r$ is a vector in ${\mathbb {R}}^d$ such that

Note that, by (6.10),

$$\begin{aligned} H(\theta r) \le \frac{1}{2} H(r) +C_0 \left( \frac{\delta }{r} \right) ^\sigma \Big \{ H(2r) + h(2r) \Big \} \end{aligned}$$

for $r\in [\delta , 1]$, where $\delta =(1+\sqrt{\lambda }) \varepsilon$. This gives (6.12) with $\eta (t)=t^\sigma$, which satisfies (6.13).

It is easy to see that H(r) satisfies the first inequality in (6.11). To verify the second, we note that, for $r\le t, s \le 2r$,

where C depends only on d, $\alpha$ and M. Thus, by Lemma 6.5, we obtain

By Caccioppoli’s inequality for parabolic systems (see for example [2, Appendix]),

Since $(\partial _t +{\mathcal {L}}_{\varepsilon , \lambda }) (\beta _0 )=0$ for any $\beta _0 \in {\mathbb {R}}$, we may replace $u_{\varepsilon , \lambda }$ in the right-hand side of the inequality above by $u_{\varepsilon , \lambda }-\beta _0$. This, together with Poincaré-type inequality for parabolic systems, yields (6.15). $\quad \square$

Proof of Theorem 1.2

Since ${\mathcal {L}}_\varepsilon ={\mathcal {L}}_{\varepsilon , \lambda }$ for $\lambda =\varepsilon ^{k-2}$, Theorem 1.2 follows readily from Theorem 6.1.

7 Convergence Rates

In this section we investigate the problem of convergence rates for the initial-Dirichlet problem,

$$\begin{aligned} \left\{ \begin{aligned} (\partial _t +{\mathcal {L}}_{\varepsilon , \lambda } ) u_{\varepsilon , \lambda }&=F&\quad&\text { in } \Omega _T,\\ u_{\varepsilon , \lambda }&=f&\quad&\text { on } \partial _p \Omega _T, \end{aligned} \right. \end{aligned}$$

(7.1)

where $\Omega$ is a bounded domain in ${\mathbb {R}}^d$ and $\Omega _T=\Omega \times (0, T)$. As a consequence, we obtain rates of convergence for the operator $\partial _t +{\mathcal {L}}_\varepsilon$ in (1.1).

Let $u_{0, \lambda }$ be the solution of the homogenized problem for (7.1),

$$\begin{aligned} \left\{ \begin{aligned}(\partial _t +{\mathcal {L}}_{0, \lambda } ) u_{0, \lambda }&=F&\quad&\text { in } \Omega _T,\\ u_{0, \lambda }&=f&\quad&\text { on } \partial _p \Omega _T. \end{aligned} \right. \end{aligned}$$

(7.2)

Let $w_{\varepsilon }$ be the two-scale expansion given by (3.21). As before, the operator $K_\varepsilon$ is defined by $K_\varepsilon (f) =S_\delta (\eta _\delta f)$ with $\delta =(1+\sqrt{\lambda } )\varepsilon$. The cut-off function $\eta _\delta =\eta _\delta ^1 (x) \eta _\delta ^2 (t)$ is chosen so that $0\le \eta _\delta \le 1$, $|\nabla \eta _\delta |\le C /\delta$, $|\partial _t \eta _\delta | +|\nabla ^2 \eta _\delta |\le C /\delta ^2$, and

$$\begin{aligned} \eta _\delta =1 \quad \text { in } \Omega _T {\setminus } \Omega _{T, 3\delta } \quad \text { and } \quad \eta _\delta =0 \quad \text { in } \Omega _{T, 2\delta }, \end{aligned}$$

where $\Omega _{T, \rho }$ denotes the (parabolic) boundary layer

$$\begin{aligned} \Omega _{T, \rho } = \Big ( \big \{ x\in \Omega : \, \text { dist} (x, \partial \Omega )<\rho \big \} \times (0, T)\Big ) \cup \Big ( \Omega \times (0, \rho ^2)\Big ) \end{aligned}$$

(7.3)

for $0< \rho \le c$.

Lemma 7.1

Let $\Omega$ be a bounded Lipschitz domain in ${\mathbb {R}}^d$. Let $\Omega _{T, \rho }$ be defined by (7.3). Then

$$\begin{aligned} \Vert \nabla g \Vert _{L^2(\Omega _{T, \rho })} \le C \sqrt{ \rho } \, \Big \{ \Vert \nabla g\Vert _{L^2(\Omega _T)} +\Vert \nabla ^2 g\Vert _{L^2(\Omega _T)} +\Vert \partial _t g \Vert _{L^2(\Omega _T)} \Big \}, \end{aligned}$$

(7.4)

where C depends only on d, $\Omega$ and T.

Proof

Let $\Omega _\rho =\big \{ x\in \Omega : \text {dist} (x, \partial \Omega )< \rho \big \}$. Then

$$\begin{aligned} \Vert \nabla g(\cdot , t) \Vert _{L^2(\Omega _\rho )} \le C \sqrt{\rho } \, \Vert \nabla g(\cdot , t) \Vert _{H^1(\Omega )}. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert \nabla g\Vert _{L^2(\Omega _\rho \times (0, T))} \le C \sqrt{\rho } \, \Big \{ \Vert \nabla g\Vert _{L^2 (\Omega _T)} +\Vert \nabla ^2 g\Vert _{L^2(\Omega _T)}\Big \}. \end{aligned}$$

To estimate $\Vert \nabla g\Vert _{L^2( (\Omega {\setminus } \Omega _\rho ) \times (0, \rho ^2))}$, we choose a cut-off function $\theta \in C_0^\infty (\Omega )$ such that $0\le \theta \le 1$, $\theta =1$ on $\Omega {\setminus } \Omega _\rho$, and $|\nabla \theta |\le C/\rho$. By Fubini’s Theorem we may also choose $t_0\in (T/2, T)$ such that

$$\begin{aligned} \int _\Omega |\nabla g (x, t_0)|^2\, \mathrm{d}x \le \frac{2}{T} \int _{\Omega _T} |\nabla g|^2 \, \mathrm{d}x \mathrm{d}t. \end{aligned}$$

Note that for any $t\in (0, \rho ^2)$,

$$\begin{aligned} \begin{aligned}\int _\Omega |\nabla g(x, t)|^2 \theta (x)\, \mathrm{d}x&\le \int _\Omega |\nabla g(x, t_0)|^2 \theta (x)\, \mathrm{d}x + \Big | \int _t^{t_0} \!\!\! \int _\Omega \partial _s (|\nabla g (x, s)|^2 \theta (x) ) \, \mathrm{d}x \mathrm{d}s \Big |\\&\le \frac{2}{T} \int _{\Omega _T} |\nabla g|^2 + \int _{\Omega _T} |\nabla ^2 g| |\partial _t g| + 2 \int _{\Omega _T} |\nabla g| |\partial _t g| |\nabla \theta |, \end{aligned} \end{aligned}$$

where we have used an integration by parts in x for the last step. By integrating the inequality above in the variable t over the interval $(0, \rho ^2)$, we obtain

$$\begin{aligned} \int _0^{\rho ^2} \!\!\! \int _\Omega |\nabla g |^2 \theta \, \mathrm{d}x \mathrm{d}t \le C \rho \int _{\Omega _T} \Big \{ |\nabla g |^2 + |\nabla ^2 g|^2 + |\partial _t g|^2 \Big \}, \end{aligned}$$

where we also used the Cauchy inequality. This completes the proof. $\quad \square$

Lemma 7.2

Let $\Omega$ be a bounded Lipschitz domain in ${\mathbb {R}}^d$ and $0<T<\infty$. Let $u_{\varepsilon , \lambda }$ be a weak solution of (7.1) and $u_{0, \lambda }$ the homogenized problem (7.2). Let $w_\varepsilon$ be defined by (3.21). Then, for any $\psi \in L^2(0, T; H_0^1(\Omega ))$,

$$\begin{aligned} \begin{aligned}&\Big |\int _0^T \langle \partial _t w_\varepsilon , \psi \rangle _{H^{-1}(\Omega ) \times H^1_0(\Omega )} +\int _{\Omega _T} A_\lambda (x/\varepsilon , t/\varepsilon ^2)\nabla w_\varepsilon \cdot \nabla \psi \Big |\\&\le C \Big \{ \Vert u_{0, \lambda } \Vert _{L^2(0, T; H^2(\Omega ))} + \Vert \partial _t u_{0, \lambda }\Vert _{L^2(\Omega _T)} \Big \} \Big \{ \delta \Vert \nabla \psi \Vert _{L^2(\Omega _T)} + \delta ^{1/2} \Vert \nabla \psi \Vert _{L^2(\Omega _{T, 3\delta })} \Big \}, \end{aligned} \end{aligned}$$

(7.5)

where $\delta =(1+\sqrt{\lambda })\varepsilon$ and C depends only on d, $\mu$, $\Omega$ and T.

Proof

In view of Lemma 7.1, the case $\lambda =1$ follows from [14, Lemma 3.5]. The case $\lambda \ne 1$ is proved in a similar manner. Indeed, by (3.22), the left-hand side of (7.5) is bounded by

$$\begin{aligned} \begin{aligned}&C\int _{\Omega _T} |\nabla u_{0, \lambda } -K_\varepsilon (\nabla u_{0, \lambda })| |\nabla \psi | +C \varepsilon \int _{\Omega _T} |(\chi ^\lambda )^\varepsilon | |\nabla K_\varepsilon (\nabla u_{0, \lambda })| |\nabla \psi |\\&\quad + C\varepsilon \int _{\Omega _T} \sum _{k, i, j} | (\phi _{kij}^\lambda )^\varepsilon | |\nabla K_\varepsilon (\nabla u_{0, \lambda })| |\nabla \psi |\\&\quad + C \varepsilon ^2 \int _{\Omega _T} \sum _{k, j} |(\phi _{k (d+1) j}^\lambda )^\varepsilon | |\partial _t K_\varepsilon (\nabla u_{0, \lambda })| |\nabla \psi |\\&\quad + C \varepsilon \int _{\Omega _T} \sum _{k, j} |(\nabla \phi _{ k (d+1) j}^\lambda )^\varepsilon | |\nabla K_\varepsilon (\nabla u_{0, \lambda })| |\nabla \psi |\\&\quad + C \varepsilon ^2 \int _{\Omega _T} \sum _{k, j} |(\phi _{k (d+1) j} ^\lambda )^\varepsilon | |\nabla ^2 K_\varepsilon (\nabla u_{0, \lambda })| |\nabla \psi |\\&=I_1 +I_2+I_3+I_4+I_5+I_6. \end{aligned} \end{aligned}$$

The estimates of $I_j$ for $j=1, \ldots , 6$ are exactly the same as in the proof of Lemma 3.5 in [14]. Also see the proof of Lemma 3.6 in Section 3. We point out that in the cases of $I_4$ and $I_6$, the estimate

is used. We omit the details. $\quad \square$

The next theorem gives an error estimate for the two-scale expansion

$$\begin{aligned} {\widetilde{w}}_\varepsilon (x, t)=u_{\varepsilon , \lambda } -u_{0, \lambda } -\varepsilon \chi ^\lambda (x/\varepsilon , t/\varepsilon ^2) K_\varepsilon (\nabla u_{0, \lambda } ) \end{aligned}$$

(7.6)

in $L^2(0, T; H^1(\Omega ))$.

Theorem 7.3

Let ${\widetilde{w}}_\varepsilon$ be defined by (7.6). Under the same conditions as in Lemma 7.2, we have

$$\begin{aligned} \Vert \nabla {\widetilde{w}}_\varepsilon \Vert _{L^2(\Omega _T)} \le C \sqrt{\delta } \, \Big \{ \Vert u_{0, \lambda } \Vert _{L^2(0, T; H^2(\Omega ))} + \Vert \partial _t u_{0, \lambda } \Vert _{L^2(\Omega _T)} \Big \}, \end{aligned}$$

(7.7)

where $\delta =(1+\sqrt{\lambda })\varepsilon \le 1$ and C depends only on d, $\mu$, $\Omega$ and T.

Proof

Let $\psi =w_\varepsilon$ in (7.5), where $w_\varepsilon$ is given by (3.21). Since $w_\varepsilon =0$ on $\partial _p \Omega _T$, we see that $\int _0^T \langle \partial _t w_\varepsilon , w_\varepsilon \rangle \ge 0$. It follows that $\Vert \nabla w_\varepsilon \Vert _{L^2(\Omega _T)}$ is bounded by the right-hand side of (7.7). It is not hard to show that $\Vert \nabla (w_\varepsilon -{\widetilde{w}}_\varepsilon )\Vert _{L^2(\Omega _T)}$ is also bounded by the right-hand side of (7.7). This gives the inequality (7.7). $\quad \square$

We now move on to the convergence rate of $u_{\varepsilon , \lambda } -u_{0, \lambda }$ in $L^2 (\Omega _T)$.

Theorem 7.4

Suppose A satisfies (1.3) and (1.4). Let $\Omega$ be a bounded $C^{1, 1}$ domain in ${\mathbb {R}}^d$. Let $u_{\varepsilon , \lambda }$ be a weak solution of (7.1) and $u_{0, \lambda }$ the solution of the homogenized problem (7.2). Then

$$\begin{aligned} \Vert u_{\varepsilon , \lambda } -u_{0, \lambda }\Vert _{L^2(\Omega _T)} \le C \delta \Big \{ \Vert u_{0, \lambda } \Vert _{L^2(0, T; H^2(\Omega ))} + \Vert \partial _t u_{0, \lambda }\Vert _{L^2(\Omega _T)} \Big \}, \end{aligned}$$

(7.8)

where $\delta =(1+\sqrt{\lambda } ) \varepsilon$ and C depends only on d, $\mu$, $\Omega$ and T.

Proof

In view of Lemma 7.1, this theorem was proved in [14, Theorem 1.1] for the case $\lambda =1$. With Lemma 7.2 at our disposal, the case $\lambda \ne 1$ follows by a similar duality argument. We omit the details. $\quad \square$

Finally, we study the problem of convergence rates for the parabolic operator $\partial _t +{\mathcal {L}}_\varepsilon$, where ${\mathcal {L}}_\varepsilon =-\text { div} \big (A(x/\varepsilon , t/\varepsilon ^k)\nabla \big )$ and $0<k<\infty$. Note that the case $k=2$ is already treated in Theorems 7.3 and 7.4 with $\lambda =1$.

For the case $k\ne 2$, we use the fact that ${\mathcal {L}}_\varepsilon ={\mathcal {L}}_{\varepsilon , \lambda }$ with $\lambda =\varepsilon ^{k-2}$. Recall that the homogenized operator for $\partial _t +{\mathcal {L}}_\varepsilon$ is given by $\partial _t -\text { div} \big (\widehat{A_\infty }\nabla \big )$ for $0<k<2$, and by $\partial _t -\text { div} \big (\widehat{A_0}\nabla \big )$ for $2<k<\infty$, where $\widehat{A_\infty }$ and $\widehat{A_0}$ are defined in (2.12 ) and (2.17), respectively.

Theorem 7.5

Assume A satisfies (1.3) and (1.4). Also assume that $\Vert \partial _s A \Vert _\infty \le M$. Let $0<k<2$. Let $u_\varepsilon$ be the weak solution of the initial-Dirichlet problem,

$$\begin{aligned} \partial _t u_\varepsilon -\text { div} \big ( A(x/\varepsilon , t/\varepsilon ^k)\nabla u_\varepsilon \big ) =F \quad \text { in } \Omega _T \quad \text { and } \quad u_\varepsilon =f \quad \text { on } \partial _p \Omega _T, \end{aligned}$$

(7.9)

where $\Omega$ is a bounded $C^{1, 1}$ domain in ${\mathbb {R}}^d$ and $0<T<\infty$. Let $u_0$ be the solution of the homogenized problem. Then

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{L^2(\Omega _T)} \le C (\varepsilon ^{k/2} +\varepsilon ^{2-k}) \Big \{ \Vert u_0\Vert _{L^2(0, T; H^2(\Omega ))} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \Big \} \end{aligned}$$

(7.10)

for $0<\varepsilon \le 1$, where C depends only on d, $\mu$, $\Omega$, T, and M.

Proof

Let $\lambda =\varepsilon ^{k-2}$ and $u_{0, \lambda }$ be the solution of the initial-Dirichlet problem,

$$\begin{aligned} \partial _t u_{0, \lambda } -\text { div} \big ( \widehat{A_\lambda }\nabla u_{0, \lambda } \big ) =F \quad \text { in } \Omega _T \quad \text { and } \quad u_{0, \lambda } =f \quad \text { on } \partial _p \Omega _T. \end{aligned}$$

(7.11)

Note that $(1+\sqrt{\lambda }) \varepsilon = \varepsilon +\varepsilon ^{k/2} \le 2 \varepsilon ^{k/2}$ for $0<\varepsilon \le 1$. It follows by Theorem 7.4 that

$$\begin{aligned} \Vert u_\varepsilon -u_{0, \lambda } \Vert _{L^2(\Omega _T)} \le C \varepsilon ^{k/2} \Big \{ \Vert u_{0, \lambda } \Vert _{L^2(0, T; H^2(\Omega ))} +\Vert \partial _t u_{0, \lambda } \Vert _{L^2 (\Omega _T)} \Big \}. \end{aligned}$$

(7.12)

Next, we observe that $u_{0, \lambda } -u_0=0$ on $\partial _p \Omega _T$ and

$$\begin{aligned} \partial _t (u_{0, \lambda } -u_0) -\text { div } \big ( \widehat{A_\lambda } \nabla (u_{0, \lambda } -u_0) \big ) =\text { div} \big ( (\widehat{A_\lambda } -\widehat{A_\infty }) \nabla u_0 \big ) \end{aligned}$$

in $\Omega _T$. Since $\Omega$ is $C^{1,1}$, it follows by the standard regularity estimates for parabolic systems with constant coefficients that

$$\begin{aligned} \begin{aligned}&\Vert \partial _t (u_0 -u_{0, \lambda }) \Vert _{L^2(\Omega _T)} + \Vert u_0- u_{0, \lambda }\Vert _{L^2(0, T; H^2(\Omega ))}\\&\qquad \qquad \qquad \qquad \le C |\widehat{A_\lambda } -\widehat{A_\infty } | \Vert \nabla ^2 u_0\Vert _{L^2 (\Omega _T)}\\&\qquad \qquad \qquad \qquad \le C \lambda ^{-1} \Vert \partial _s A \Vert _\infty \Vert \nabla ^2 u_0\Vert _{L^2(\Omega _T)}, \end{aligned} \end{aligned}$$

where we have used (2.23) for the last step. This, together with (7.12), yields the estimate (7.10). $\quad \square$

The next theorem treats the case $2<k<\infty$.

Theorem 7.6

Assume A satisfies (1.3) and (1.4). Also assume that $\Vert \nabla ^2 A \Vert _\infty \le M$. Let $2<k<\infty$. Let $u_\varepsilon$ be the weak solution of the initial-Dirichlet problem (7.9), where $\Omega$ is a bounded $C^{1, 1}$ domain in ${\mathbb {R}}^d$ and $0<T<\infty$. Let $u_0$ be the solution of the homogenized problem. Then

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{L^2(\Omega _T)} \le C (\varepsilon +\varepsilon ^{k-2}) \Big \{ \Vert u_0\Vert _{L^2(0, T; H^2(\Omega ))} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \Big \} \end{aligned}$$

(7.13)

for $0<\varepsilon < 1$, where C depends only on d, $\mu$, $\Omega$, T, and M.

Proof

The proof is similar to that of Theorem 7.5. The only modification is that in the place of (2.24), we use the estimate (2.29) to bound $|\widehat{A_\lambda } -\widehat{A_0|}$. Also, note that $\Vert \nabla A\Vert _\infty$ may be bounded by a constant depending on $\mu$ and M. We omit the details. $\quad \square$

Proof of Theorem 1.3

Let $0<\varepsilon <1$. Note that $\varepsilon ^{2-k} \le \varepsilon ^{k/2}$ if $0< k\le 4/3$, and $\varepsilon ^{k/2}\le \varepsilon ^{2-k}$ if $4/3<k< 2$. Also, $\varepsilon \le \varepsilon ^{k-2}$ if $2<k< 3$, and $\varepsilon ^{k-2}\le \varepsilon$ if $k\ge 3$. Thus, by Theorems 7.5 and 7.6,

$$\begin{aligned} \begin{aligned}&\Vert u_\varepsilon -u_0\Vert _{L^2(\Omega _T)}\\&\le C \Big \{ \Vert u_0\Vert _{L^2(0, T; H^2(\Omega ))} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \Big \} \cdot \left\{ \begin{aligned}&\varepsilon ^{k/2}&\quad&\text { if } 0<k\le 4/3,\\&\varepsilon ^{2-k}&\quad&\text { if } 4/3< k< 2,\\&\varepsilon ^{k-2}&\quad&\text { if } 2<k<3,\\&\varepsilon&\quad&\text { if } k=2 \text { or } 3\le k<\infty .\\ \end{aligned} \right. \end{aligned} \end{aligned}$$

Remark 7.7

The results on convergence rates in Theorems 7.5 and 7.6 also hold for initial-Neumann problems. The proof is almost identical to the case of the initial-Dirichlet problem. See [14] for the case $k=2$.

Using Theorem 7.3 we may obtain an error estimate in $L^2(0, T; H^1(\Omega ))$ for a two-scale expansion for $\partial _t + {\mathcal {L}}_\varepsilon$ in (1.1) in terms of its own correctors. The case $k=2$ is contained in Theorem 7.3 with $\lambda =1$. For $k\ne 2$, we let

$$\begin{aligned} v_\varepsilon = \left\{ \begin{aligned}&u_\varepsilon -u_0 -\varepsilon \chi ^\infty (x/\varepsilon , t/\varepsilon ^k) {\widetilde{K}}_\varepsilon (\nabla u_0)&\quad&\text { if } 0<k<2,\\&u_\varepsilon -u_0 -\varepsilon \chi ^0 (x/ \varepsilon ) {\widetilde{K}}_\varepsilon (\nabla u_0)&\quad&\text { if } 2<k<\infty . \end{aligned} \right. \end{aligned}$$

(7.14)

In (7.14), $\chi ^\infty$ and $\chi ^0$ are the correctors defined by (2.10) and (2.15), respectively, for $\partial _t + {\mathcal {L}}_\varepsilon$. Since they satisfy the estimates (2.11) and (2.16), only smoothing in the space variable is needed for the operator ${\widetilde{K}}_\varepsilon$. More precisely, we let ${\widetilde{K}}_\varepsilon (f)= S^1_\delta (\eta _\delta f)$, where

$$\begin{aligned} S^1_\delta (f) (x, t) = \int _{{\mathbb {R}}^d} f(x-y, t) \delta ^{-d} \theta _1 (y/\delta )\, \mathrm{d}y, \end{aligned}$$

$\delta =\varepsilon + \varepsilon ^{k/2}$, and the cut-off function $\eta _\delta$ is the same as in $K_\varepsilon$.

Theorem 7.8

Suppose that A and $\Omega$ satisfy the same conditions as in Theorem 7.5. Let $u_\varepsilon$ be the weak solution of (7.9) and $u_0$ the homogenized solution. Let $v_\varepsilon$ be given by (7.14). Then

$$\begin{aligned} \begin{aligned}&\Vert \nabla v_\varepsilon \Vert _{L^2(\Omega _T)}\\&\le C \Big \{ \Vert u_0\Vert _{L^2(0, T; H^2(\Omega ))} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \Big \} \cdot \left\{ \begin{aligned}&\varepsilon ^{k/4}&\quad&\text { if }\ 0<k\le 8/5,\\&\varepsilon ^{2-k}&\quad&\text { if }\ 8/5< k< 2,\\&\varepsilon ^{k-2}&\quad&\text { if }\ 2<k<5/2,\\&\varepsilon ^{1/2}&\quad&\text { if } \ 5/2\le k<\infty .\\ \end{aligned} \right. \end{aligned} \end{aligned}$$

(7.15)

Proof

The proof uses Theorem 7.3 and the estimates of $u_{0, \lambda } -u_0$ in the proof of Theorems 7.5 and 7.6, where $u_{0, \lambda }$ is the solution of (7.11) with $\lambda =\varepsilon ^{2-k}$.

Let $\lambda =\varepsilon ^{k-2}$. Suppose $0<k<2$. In view of (7.7) it suffices to bound

$$\begin{aligned} I=\Vert \nabla \Big \{ \varepsilon \chi ^\lambda (x/ \varepsilon , t/\varepsilon ^2) K_\varepsilon (\nabla u_{0, \lambda }) -\varepsilon \chi ^\infty (x/\varepsilon , t/\varepsilon ^k ) {\widetilde{K}}_\varepsilon (\nabla u_0)\Big \} \Vert _{L^2(\Omega _T)}. \end{aligned}$$

Note that

$$\begin{aligned} \begin{aligned}I&\le \Vert \big (\nabla \chi ^\lambda (x/\varepsilon , t/\varepsilon ^2) -\nabla \chi ^\infty (x/\varepsilon , t/\varepsilon ^k)\big ) K_\varepsilon (\nabla u_{0, \lambda })\Vert _{L^2(\Omega _T)}\\&\qquad + \Vert \nabla \chi ^\infty (x/\varepsilon , t/\varepsilon ^k) \big ( K_\varepsilon (\nabla u_{0, \lambda }) -{\widetilde{K}}_\varepsilon (\nabla u_0) \big )\Vert _{L^2(\Omega _T)}\\&\qquad +\varepsilon \Vert \chi ^\lambda (x/\varepsilon , t/\varepsilon ^2) \nabla K_\varepsilon (\nabla u_{0, \lambda }) \Vert _{L^2(\Omega _T)}\\&\qquad +\varepsilon \Vert \chi ^\infty (x/\varepsilon , t/\varepsilon ^k) \nabla {\widetilde{K}}_\varepsilon (\nabla u_0)\Vert _{L^2(\Omega _T)}\\&=I_1 +I_2 +I_3 +I_4. \end{aligned} \end{aligned}$$

To bound $I_1$, we use the inequality (3.17). This gives

(7.16)

where we have used (2.27) for the second inequality. To estimate $I_2$, we assume that the function $\theta _1$ is chosen so that $\theta _1 =\theta _{11} * \theta _{11}$, where $\theta _{11} \in C^\infty _0 (B(0, 1))$, $\theta _{11} \ge 0$ and $\int _{{\mathbb {R}}^d} \theta _{11} =1$. This allows us to write $S_\delta ^1 = S_\delta ^{11} \circ S_\delta ^{11}$, where $S_\delta ^{11} (f)=f * (\theta _{11})_\delta$. As a result, we obtain

$$\begin{aligned} \begin{aligned}I_2&\le C \Vert S_\delta ^2\big [ S_\delta ^{11}(\eta _\delta \nabla u_0)\big ] -S^{11}_\delta (\eta _\delta \nabla u_0)\Vert _{L^2(\Omega _T)} \\&\le C\delta ^2 \Vert \partial _t S_\delta ^{11} (\eta _\delta \nabla u_0)\Vert _{L^2(\Omega _T)}\\&= C \delta ^2 \ \Vert S_\delta ^{11} \big \{ (\partial _t \eta _\delta ) (\nabla u_0) +\nabla (\eta _\delta \partial _t u_0) -(\nabla \eta _\delta ) \partial _t u_0 \big \} \Vert _{L^2(\Omega _T)}\\&\le C \delta ^{1/2} \Big \{ \Vert \nabla u_0\Vert _{L^2(\Omega _T)} +\Vert \nabla ^2 u_0\Vert _{L^2(\Omega _T)} + \Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \Big \}. \end{aligned} \end{aligned}$$

It is not hard to see that

$$\begin{aligned} \begin{aligned}I_3 +I_4&\le C \varepsilon \Big \{ \Vert \nabla (\eta _\delta \nabla u_{0, \lambda }) \Vert _{L^2(\Omega _T)} + \Vert \nabla (\eta _\delta \nabla u_{0}) \Vert _{L^2(\Omega _T)}\Big \}\\&\le C \delta ^{1/2} \Big \{ \Vert \nabla u_0\Vert _{L^2(\Omega _T)} +\Vert \nabla ^2 u_0\Vert _{L^2(\Omega _T)} \Big \}. \end{aligned} \end{aligned}$$

In summary, we have proved that

$$\begin{aligned} \Vert \nabla v_\varepsilon \Vert _{L^2(\Omega _T)} \le C \big \{ \varepsilon ^{k/4} +\varepsilon ^{2-k} \big \} \big \{ \Vert u_0\Vert _{L^2(0, T; H^1(\Omega ))} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \big \}\nonumber \\ \end{aligned}$$

(7.17)

for $0<k<2$. A similar argument gives

$$\begin{aligned} \Vert \nabla v_\varepsilon \Vert _{L^2(\Omega _T)} \le C \big \{ \varepsilon ^{1/2} +\varepsilon ^{k-2} \big \} \big \{ \Vert u_0\Vert _{L^2(0, T; H^1(\Omega ))} +\Vert \partial _t u_0\Vert _{L^2(\Omega _T)} \big \}\nonumber \\ \end{aligned}$$

(7.18)

for $2<k<\infty$. The error estimate (7.15) follows readily from (7.17) and (7.18). $\quad \square$