Homogenization of Parabolic Equations with Non-self-similar Scales

TL;DR

This paper develops quantitative homogenization techniques for second-order parabolic systems with rapidly oscillating periodic coefficients across different scales, providing new estimates and convergence rate results.

Contribution

It introduces higher-order correctors to achieve large-scale Lipschitz and Hölder estimates, advancing the understanding of homogenization in multi-scale parabolic equations.

Findings

Established large-scale interior and boundary Lipschitz estimates

Derived interior $C^{1, eta}$ and $C^{2, eta}$ estimates using higher-order correctors

Analyzed convergence rates for initial-boundary value problems

Abstract

This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different 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.

Equations

$${\partial }_t+{\mathcal{L}}_{\epsilon }$$

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

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

$$A(y+z,s+t)=A(y,s)\text{ for }(z,t)\in {\mathbb{Z}}^{d+1}\text{ and a.e. }(y,s)\in {\mathbb{R}}^{d+1}.$$

$$({\partial }_t+{\mathcal{L}}_{\epsilon })u_{\epsilon }=F\text{ in }{Q}_R=Q_R(x_0,t_0),$$

$${\left({\text{fint}}_{Q_r}|\nabla u_{\epsilon }{|}^2\right)}^{1/2}\le C\left\{{\left({\text{fint}}_{Q_R}|\nabla u_{\epsilon }{|}^2\right)}^{1/2}+R{\left({\text{fint}}_{Q_R}|F{|}^p\right)}^{1/p}\right\},$$

$${\left({\text{fint}}_{D_r}|\nabla u_{\epsilon }{|}^2\right)}^{1/2}\le C\left\{{\left({\text{fint}}_{D_R}|\nabla u_{\epsilon }{|}^2\right)}^{1/2}+R^{-1}\Vert f{\Vert }_{C^{1+\alpha }({\Delta }_R)}+R{\left({\text{fint}}_{D_R}|F{|}^p\right)}^{1/p}\right\},$$

$$({\partial }_t+{\mathcal{L}}_{\epsilon })u_{\epsilon }=F\text{ in }{\Omega }_T\text{ and }{u}_{\epsilon }=f\text{ on }{\partial }_p{\Omega }_T,$$

$$\Vert u_{\epsilon }-u_0{\Vert }_{L^2({\Omega }_T)}$$

$$\displaystyle\leq C\Big{\{}\|u_{0}\|_{L^{2}(0,T;H^{2}(\Omega))}+\|\partial_{t}u_{0}\|_{L^{2}(\Omega_{T})}\Big{\}}\cdot\left\{\begin{aligned} &\varepsilon^{k/2}&\quad&\text{if }\ 0<k\leq 4/3,\\ &\varepsilon^{2-k}&\quad&\text{if }\ 4/3<k<2,\\ &\varepsilon^{k-2}&\quad&\text{if }\ 2<k<3,\\ &\varepsilon&\quad&\text{if }\ k=2\text{ or}\ 3\leq k<\infty,\\ \end{aligned}\right.$$

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

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

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

$$\left\{\begin{aligned} &\partial_{s}\chi_{j}^{\lambda}-\text{\rm div}\big{(}A_{\lambda}\nabla\chi_{j}^{\lambda}\big{)}=\text{\rm div}\big{(}A_{\lambda}\nabla y_{j}\big{)}\quad\text{ in }\mathbb{R}^{d+1},\\ &\chi_{j}^{\lambda}\text{ is }(1,\lambda)\text{-periodic in }(y,s),\\ &\int_{0}^{\lambda}\!\!\!\int_{\mathbb{T}^{d}}\chi_{j}^{\lambda}(y,s)\,dyds=0,\end{aligned}\right.$$

$${\text{fint}}_0^{\lambda }{\int }_{{\mathbb{T}}^d}|\nabla {\chi }_j^{\lambda }{|}^{2}dyds\le C,$$

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

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

$${\text{fint}}_0^{\lambda }{\int }_{{\mathbb{T}}^d}|{\chi }_j^{\lambda }{|}^{2}dyds\le C,$$

$${\left({\text{fint}}_{Q_r}\left(|\nabla {\chi }^{\lambda }{|}^2+|{\chi }^{\lambda }{|}^2\right)\right)}^{1/2}\le C$$

$$\widehat{A_{\lambda }}={\text{fint}}_0^{\lambda }{\int }_{{\mathbb{T}}^d}\left(A_{\lambda }+A_{\lambda }\nabla {\chi }^{\lambda }\right)dyds.$$

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

$$\xi \cdot \widehat{A_{\lambda }}\xi$$

$$\ge \mu {\text{fint}}_0^{\lambda }{\int }_{{\mathbb{T}}^d}|\nabla (\xi \cdot y+\xi \cdot {\chi }^{\lambda }){|}^{2}dyds$$

$$=\mu {\text{fint}}_0^{\lambda }{\int }_{{\mathbb{T}}^d}(|\xi {|}^2+|\xi \nabla {\chi }^{\lambda }{|}^2)dyds$$

$$\ge \mu |\xi {|}^2$$

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

$$\left\{\begin{aligned} &-\text{\rm div}\big{(}A\nabla\chi_{j}^{\infty})=\text{\rm div}(A\nabla y_{j})\quad\text{ in }\mathbb{R}^{d+1},\\ &\chi_{j}^{\infty}\text{ is 1-periodic in }(y,s),\\ &\int_{\mathbb{T}^{d}}\chi_{j}^{\infty}(y,s)\,dy=0.\end{aligned}\right.$$

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

$$\widehat{A_{\infty }}={\int }_0^1{\int }_{{\mathbb{T}}^d}\left(A+A\nabla {\chi }^{\infty }\right)dyds.$$

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

$$\overline{A}=\overline{A}(y)={\int }_0^{1}A(y,s)ds.$$

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

Full text

MSC2010:   35B27, 35K40.

Keywords: homogenization; parabolic system; large-scale regularity; convergence rate.

1 Introduction

In this paper we shall be interested in 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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

The qualitative homogenization theory for the operator (1.1) has been known since the 1970s (see e.g. [10]). As $\varepsilon\to 0$, 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 the 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 $2<k<\infty$. For more recent work on multiscale convergence and reiterated homogenization, see [1, 18, 12, 23, 26] and references therein.

In recent years there is 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 [9, 19, 25, 16, 8, 6, 3, 11, 17, 4, 7, 24, 5] and references therein). Some of these work has 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 work in [20, 27, 22, 21]. Most recently, in [15] we study the asymptotic behavior of the fundamental solution and its derivatives and establish sharp estimates for the remainders. We refer the reader to [2] for quantitative stochastic homogenization of parabolic equations.

If $k\neq 2$, the $\varepsilon$ scaling in the coefficient matrix $A(x/\varepsilon,t/\varepsilon^{k})$ is not consistent with the intrinsic scaling of the second-order parabolic equations. To the authors’ best knowledge, very few quantitative results are known in this case, where direct extensions of the techniques developed for elliptic equations fail.

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

[TABLE]

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

[TABLE]

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

The inequality (1.6) is a large-scale interior Lipschitz estimate. We also obtain large-scale $C^{1,\alpha}$ and $C^{2,\alpha}$ excess-decay estimates for solutions of $\partial_{t}+\mathcal{L}_{\varepsilon}$ (see Sections 4 and 5). Regarding the condition $R>r\geq\varepsilon+\varepsilon^{k/2}$, we point out 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 $r^{2}\geq\varepsilon^{k}$ and $2r\geq\sqrt{d}\varepsilon$. This implies that $r\geq(\varepsilon+\varepsilon^{k/2})/4$. As a result, the condition $R>r\geq\varepsilon+\varepsilon^{k/2}$ for (1.6) is more or less necessary.

The next theorem gives the large-scale boundary Lipschitz estimate. Let $\Omega$ be a bounded $C^{1,\alpha}$ domain in $\mathbb{R}^{d}$ for some $\alpha>0$. 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 $\Delta_{R}=D_{R}(x_{0},t_{0})$, where $\varepsilon+\varepsilon^{k/2}<R\leq 1$ and $F\in L^{p}(D_{R})$ for some $p>d+2$. Then for any $\varepsilon+\varepsilon^{k/2}\leq r<R$,

[TABLE]

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

In this paper we also investigate the rate of convergence in $L^{2}(\Omega_{T})$ for the initial-Dirichlet problem,

[TABLE]

where $\partial_{p}\Omega_{T}$ denotes the parabolic boundary of $\Omega_{T}$.

Theorem 1.3**.**

Assume $A=A(y,s)$ satisfies (1.3) and (1.4). Also assume that $\|\partial_{s}A\|_{\infty}<\infty$ for $0<k<2$ and $\|\nabla^{2}A\|_{\infty}<\infty$ for $k>2$. Let $u_{\varepsilon}$ be the weak solution of (1.8) and $u_{0}$ the homogenized solution, where $\Omega$ is a bounded $C^{1,1}$ domain in $\mathbb{R}^{d}$ and $0<T<\infty$. Then

[TABLE]

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

We now describe our general approach to Theorems 1.1, 1.2, and 1.3. The key insight is to introduce a new scale $\lambda\in(0,\infty)$ and consider the operator

[TABLE]

where $A_{\lambda}(y,s)=A(y,s/\lambda)$. Observe that the coefficient matrix $A_{\lambda}$ 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 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 $\lambda\to\infty$, 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 $\lambda\to 0$, then $\widehat{A_{\lambda}}\to\widehat{A_{0}}$, 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 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, 5, and 6 are motivated by recently developed techniques for studying the large-scale regularity in the homogenization theory for elliptic equations and systems [16, 8, 6, 3, 11, 17, 4, 7]. 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 $\fint_{E}u$ to denote the $L^{1}$ average of $u$ over the set $E$; i.e. $\fint_{E}u=\frac{1}{|E|}\int_{E}u$. For notational simplicity we will assume $m=1$ in the rest of the paper. However, no particular fact pertain 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

[TABLE]

The matrix $A_{\lambda}$ is $(1,\lambda)$-periodic in $(y,s)$; i.e.,

[TABLE]

Let $\chi^{\lambda}=\chi^{\lambda}(y,s)=(\chi_{1}^{\lambda}(y,s),\dots,\chi_{d}^{\lambda}(y,s))$, where $\chi_{j}^{\lambda}=\chi_{j}^{\lambda}(y,s)$ is the weak solution of the parabolic cell problem:

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $r\geq 1+\sqrt{\lambda}$,

[TABLE]

for any $Q_{r}=Q_{r}(x,t)$, where $C$ depends only on $d$ and $\mu$.

Let

[TABLE]

Lemma 2.1**.**

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

[TABLE]

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

Proof.

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

[TABLE]

for any $\xi\in\mathbb{R}^{d}$, where we have used the fact $\int_{0}^{\lambda}\!\!\int_{\mathbb{T}^{d}}\nabla\chi^{\lambda}\,dyds=0$. ∎

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

[TABLE]

is given by \partial_{t}-\text{\rm 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{\rm 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 $k\neq 2$, we first consider the case $0<k<2$. Let $\chi^{\infty}=\chi^{\infty}(y,s)=(\chi_{1}^{\infty}(y,s),\dots,\chi_{d}^{\infty}(y,s))$, where $\chi_{j}^{\infty}=\chi_{j}^{\infty}(y,s)$ denotes the weak solution of the (elliptic) cell problem,

[TABLE]

By the energy estimates and Poincaré’s inequality,

[TABLE]

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

[TABLE]

It follows from (2.11) that $|\widehat{A_{\infty}}|\leq C$, 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

[TABLE]

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{\rm div}\big{(}\widehat{A_{\infty}}\nabla\big{)} (see [10]).

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

[TABLE]

Let $\chi^{0}=\chi^{0}(y)=(\chi^{0}_{1}(y),\dots,\chi_{d}^{0}(y))$, where $\chi_{j}^{0}=\chi_{j}^{0}(y)$ is the weak solution of the (elliptic) cell problem,

[TABLE]

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

[TABLE]

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

[TABLE]

It follows from (2.16) that $|\widehat{A_{0}}|\leq C$, where $C$ depends only on $d$ and $\mu$. By the same argument as in the proof of Lemma 2.1, we obtain

[TABLE]

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{\rm 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 $\lambda\to\infty$ and as $\lambda\to 0$. 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

[TABLE]

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

Proof.

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

[TABLE]

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

[TABLE]

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

[TABLE]

where we have used (2.6) for the last step. ∎

Theorem 2.3**.**

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

[TABLE]

Moreover, if $\|\partial_{s}A\|_{\infty}<\infty$, then

[TABLE]

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

Proof.

We first prove (2.23). Observe that

[TABLE]

It follows by the Cauchy inequality that

[TABLE]

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

[TABLE]

This leads to

[TABLE]

where we have used the fact

[TABLE]

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

[TABLE]

Since

[TABLE]

by Poincaré’s inequality, we obtain

[TABLE]

In view of (2.24) we have proved that

[TABLE]

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

[TABLE]

It follows that

[TABLE]

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

[TABLE]

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

[TABLE]

for $p_{0}=\frac{2q}{q-2}$. In view of (2.25) this gives

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

It follows that

[TABLE]

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 $\widehat{A_{\lambda}}\to\widehat{A_{\infty}}$ as $\lambda\to\infty$. ∎

Remark 2.4**.**

It follows from the proof of Theorem 2.3 that

[TABLE]

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

[TABLE]

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

Theorem 2.5**.**

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

[TABLE]

Moreover, if $\|\nabla^{2}A\|_{\infty}<\infty$, then

[TABLE]

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

Proof.

We first prove (2.29). Observe that

[TABLE]

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

[TABLE]

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

[TABLE]

To bound the term $I_{2}$ in (2.30), we observe that

[TABLE]

It follows that

[TABLE]

By the energy estimates we obtain

[TABLE]

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

[TABLE]

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

[TABLE]

By the energy estimates,

[TABLE]

By differentiating (2.33) in $y$ we have

[TABLE]

Again, by the energy estimates,

[TABLE]

It follows by the equation (2.33) that

[TABLE]

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

[TABLE]

By letting $\lambda\to 0$ 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 $\widehat{A_{\lambda}}\to\widehat{A_{0}}$ as $\lambda\to 0$. ∎

Remark 2.6**.**

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

[TABLE]

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{\rm div}\big{(}A_{\lambda}(x/\varepsilon,t/\varepsilon^{2})\nabla\big{)}. Let \mathcal{L}_{0,\lambda}=-\text{\rm 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

[TABLE]

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

[TABLE]

such that

[TABLE]

and

[TABLE]

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

[TABLE]

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 $1\leq i,j,k\leq d$, in $H^{1}_{loc}(\mathbb{R}^{d+1})$ such that

[TABLE]

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

[TABLE]

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

Proof.

The lemma was proved in [14] for the case $\lambda=1$. The case $\lambda\neq 1$ 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 $\mathbb{R}^{d+1}$. By the definition of $\widehat{A_{\lambda}}$,

[TABLE]

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 $1\leq i,j\leq d$. 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 $1\leq j\leq d$. By the definition of $\chi^{\lambda}_{j}$, we have

[TABLE]

which leads to

[TABLE]

By the periodicity and Liouville Theorem we may conclude that

[TABLE]

This allows us to write

[TABLE]

and

[TABLE]

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

[TABLE]

for $1\leq i,j,k\leq d$. 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

[TABLE]

Then

[TABLE]

It follows by Parseval’s Theorem that

[TABLE]

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

[TABLE]

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

[TABLE]

The desired estimates (3.7) and (3.8) follow readily from (3.12), (3.13), (3.14) and (3.15). ∎

Let $\varphi=\varphi(y,s)=\theta_{1}(y)\theta_{2}(s)$, where $\theta_{1}\in C_{0}^{\infty}(B(0,1))$, $\theta_{2}\in C^{\infty}_{0}(-1,0)$, $\theta_{1},\theta_{2}\geq 0$, and $\int_{\mathbb{R}^{d}}\theta_{1}(y)\,dy=\int_{\mathbb{R}}\theta_{2}(s)\,ds=1$. Define

[TABLE]

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

[TABLE]

where $C$ depends only on $d$.

Proof.

By Hölder’s inequality,

[TABLE]

It follows by Fubini’s Theorem that

[TABLE]

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

Lemma 3.4**.**

Let $S_{\delta}$ be define by (3.16). Then

[TABLE]

where $C$ depends only on $d$.

Proof.

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

[TABLE]

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

[TABLE]

where $C$ depends only on $d$. It follows that

[TABLE]

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

[TABLE]

Using the estimates

[TABLE]

we obtain

[TABLE]

This completes the proof. ∎

Let

[TABLE]

where

[TABLE]

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

Lemma 3.5**.**

Suppose that

[TABLE]

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

[TABLE]

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

Proof.

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

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

[TABLE]

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

[TABLE]

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:

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

As a result, we obtain

[TABLE]

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 $\delta\leq 1/8$; for otherwise the estimate is trivial. Choose $\eta_{\delta}\in C_{0}^{\infty}(\mathbb{R}^{d+1})$ such that $0\leq\eta_{\delta}\leq 1$, $\ |\nabla\eta_{\delta}|\leq C/\delta$, $|\partial_{t}\eta_{\delta}|+|\nabla^{2}\eta_{\delta}|\leq C/\delta^{2}$,

[TABLE]

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

[TABLE]

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

[TABLE]

To bound $I_{1}$, we use Lemma 3.4. This gives

[TABLE]

By the standard regularity estimates for parabolic systems with constant coefficients,

[TABLE]

where $\text{\rm dist}_{p}((y,s),\partial_{p}Q_{1})$ denotes the parabolic distance from $(y,s)$ to $\partial_{p}Q_{1}$. It follows that

[TABLE]

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 as well as estimates (2.5) and (3.7), 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 $\delta\geq\varepsilon$ and $\delta^{2}\geq\varepsilon^{2}\lambda$, we obtain

[TABLE]

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

[TABLE]

To bound $I_{6}$, we use the inequality (3.18) as well as the estimate (3.8). This leads to

[TABLE]

Finally, to handle $I_{4}$, we use the observation

[TABLE]

As in the case of $I_{6}$, we obtain

[TABLE]

Let $\sigma=\frac{1}{2}-\frac{1}{q}>0$. In view of (3.29), (3.30), (3.31) and (3.32), we have proved that

[TABLE]

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

[TABLE]

This, together with (3.34), gives the estimate (3.24). ∎

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 $\delta=(1+\sqrt{\lambda})\varepsilon\leq 1/8$. 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

[TABLE]

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

[TABLE]

Also, recall that

[TABLE]

As a result, it is enough to show that

[TABLE]

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

[TABLE]

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 $\partial_{t}+\mathcal{L}_{0,\lambda}$,

[TABLE]

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

[TABLE]

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 $(1+\sqrt{\lambda})\varepsilon\leq r<R$ and $0<\alpha<1-\frac{d+2}{p}$,

[TABLE]

where $C>0$ depend 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

[TABLE]

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,

[TABLE]

for any $(x,t)\in Q_{r/2}$, where $\alpha_{p}=1-\frac{d+2}{p}$. Let $P(x,t)=e_{j}(x_{j}+\varepsilon\chi_{j}^{\lambda}(x/\varepsilon,t/\varepsilon^{2}))$ with $e_{j}=\frac{\partial u_{0,\lambda}}{\partial x_{j}}(0,0)$. Then

[TABLE]

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

[TABLE]

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

[TABLE]

This leads to

[TABLE]

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

[TABLE]

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

[TABLE]

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

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 $(1+\sqrt{\lambda})\varepsilon\leq r<R$,

[TABLE]

where $C>0$ depend 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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

where we have used the fact that $(\partial_{t}+\mathcal{L}_{\varepsilon,\lambda})(H_{2r}-H_{r}-\beta)=0$ in $\mathbb{R}^{d+1}$ for the last inequality. Hence,

[TABLE]

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

[TABLE]

which, together with (4.2), gives the large-scale Lipschitz estimate (4.5). ∎

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 $\varepsilon+\varepsilon^{k/2}\leq r<R$ and $0<\alpha<1-\frac{d+2}{p}$,

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $1\leq k,\ell\leq d$, 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:

[TABLE]

where $(\chi_{j}^{\lambda})$ are the first-order correctors defined by (2.2). Since

[TABLE]

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,

[TABLE]

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

Lemma 5.1**.**

Let

[TABLE]

Then

[TABLE]

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 $\chi_{k\ell}^{\lambda}$. ∎

Let $P_{0}(x,t)=\beta+e_{0}t+e_{k}x_{k}+e_{k\ell}x_{k}x_{\ell}$ and

[TABLE]

where $\beta,e_{0},e_{k},e_{k\ell}=e_{\ell k}\in\mathbb{R}$. It follows from Lemma 5.1 by rescaling that

[TABLE]

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

[TABLE]

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 $(1+\sqrt{\lambda})\varepsilon\leq r<R$ and $0<\alpha<\sigma$,

[TABLE]

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 $e_{0}t$ from $u_{\varepsilon,\lambda}$, we may also assume that $F(0,0)=0$, which implies $\|F\|_{L^{\infty}(Q_{r})}\leq C\|F\|_{C^{\sigma}(Q_{r})}$. 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,

[TABLE]

for any $(x,t)\in Q_{\theta r}$, where we have used (3.3) for the last inequality. Let $P_{0}(x,t)=e_{0}t+e_{i}x_{i}+e_{ij}x_{i}x_{j}$, where

[TABLE]

Note that

[TABLE]

and by (5.5),

[TABLE]

This, together with the inequality (3.4), gives

[TABLE]

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

[TABLE]

In view of (5.9) we obtain

[TABLE]

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

To proceed, we let

[TABLE]

It follows from (5.11) that

[TABLE]

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 $C_{0}\theta^{1+\sigma}\leq(1/2)(\theta/2)^{1+\alpha}$. With $\theta$ chosen, we may choose $C_{1}>1$ so large that $C_{0}C_{1}^{-\sigma}\leq(1/2)(\theta/2)^{1+\alpha}$. As a result, for $C_{1}(1+\sqrt{\lambda})\varepsilon<\theta r<r<1$, we have

[TABLE]

By a simple iteration argument this gives $\Psi(r)\leq Cr^{1+\alpha}\Psi(2)$ for any $(1+\sqrt{\lambda})\varepsilon\leq r<2$. ∎

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

[TABLE]

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

[TABLE]

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{\rm 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

[TABLE]

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,

[TABLE]

where $\|g\|_{C^{\alpha}(\Delta_{r})}$ is the smallest constant $C_{0}$ such that

[TABLE]

for any $(x,t),(y,s)\in\Delta_{r}$, and

[TABLE]

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 $x_{0}\in\partial\Omega$, $(1+\sqrt{\lambda})\varepsilon<R\leq 1$, and $F\in L^{p}(D_{R})$ for some $p>d+2$. Then, for any $(1+\sqrt{\lambda})\varepsilon\leq r<R$,

[TABLE]

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

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

[TABLE]

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 $0<r\leq 1$. 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

[TABLE]

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,

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

This yields (6.4), as $\|w_{\varepsilon}-(u_{\varepsilon,\lambda}-u_{0,\lambda})\|_{L^{2}(T_{1})}$ is also bounded by the right-hand side of (6.7). ∎

For a function $u$ in $T_{r}$, define

[TABLE]

Lemma 6.3**.**

Suppose that $(\partial_{t}+\mathcal{L}_{0,\lambda})u=F$ in $T_{r}$, where $0<r\leq 1$ 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

[TABLE]

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 $(x,t)\in T_{r/2}$,

[TABLE]

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

[TABLE]

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

[TABLE]

To finish the proof, we choose $\theta\in(0,1/4)$ so small that $C_{0}\theta^{\sigma}\leq(1/2)$. ∎

Lemma 6.4**.**

Suppose that $(\partial_{t}+\mathcal{L}_{\varepsilon,\lambda})u_{\varepsilon,\lambda}=F$ in $T_{2}$ and $u=f$ 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 $(1+\sqrt{\lambda})\varepsilon\leq r\leq 1$,

[TABLE]

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

Proof.

Fix $(1+\sqrt{\lambda})\varepsilon\leq r\leq 1$. 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

[TABLE]

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

The proof of the following lemma may be found in [24].

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

[TABLE]

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

[TABLE]

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

[TABLE]

Then

[TABLE]

where $C$ deepnds 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 $(1+\sqrt{\lambda})\varepsilon\leq r<2$,

[TABLE]

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

[TABLE]

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

[TABLE]

Note that by (6.10),

[TABLE]

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 $r\leq t,s\leq 2r$,

[TABLE]

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

[TABLE]

By Cacciopoli’s inequality for parabolic systems (see e.g. [2, Appendix]),

[TABLE]

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

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,

[TABLE]

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),

[TABLE]

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 $0\leq\eta_{\delta}\leq 1$, $|\nabla\eta_{\delta}|\leq C/\delta$, $|\partial_{t}\eta_{\delta}|+|\nabla^{2}\eta_{\delta}|\leq C/\delta^{2}$, and

[TABLE]

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

[TABLE]

for $0<\rho\leq c$.

Lemma 7.1**.**

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

[TABLE]

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

[TABLE]

It follows that

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

where we also used the Cauchy inequality. This completes the proof. ∎

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 $\psi\in L^{2}(0,T;H_{0}^{1}(\Omega))$,

[TABLE]

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 $\lambda\neq 1$ is proved in a similar manner. Indeed, by (3.22), the right-hand side of (7.5) is bounded by

[TABLE]

The estimates of $I_{j}$ for $j=1,\dots,6$ 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

[TABLE]

is used. We omit the details. ∎

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

[TABLE]

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

[TABLE]

where $\delta=(1+\sqrt{\lambda})\varepsilon\leq 1$ 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 $\int_{0}^{T}\langle\partial_{t}w_{\varepsilon},w_{\varepsilon}\rangle\geq 0$. It follows that $\|\nabla w_{\varepsilon}\|_{L^{2}(\Omega_{T})}$ is bounded by the right-hand side of (7.7). It is not hard to show that $\|\nabla(w_{\varepsilon}-\widetilde{w}_{\varepsilon})\|_{L^{2}(\Omega_{T})}$ is also bounded by the right-hand side of (7.7). This gives the inequality (7.7). ∎

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

[TABLE]

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 $\lambda\neq 1$ follows by a similar duality argument. We omit the details. ∎

Finally, we study the problem of convergence rates for the parabolic operator $\partial_{t}+\mathcal{L}_{\varepsilon}$, where \mathcal{L}_{\varepsilon}=-\text{\rm 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 $k\neq 2$, 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{\rm div}\big{(}\widehat{A_{\infty}}\nabla\big{)} for $0<k<2$, and by \partial_{t}-\text{\rm 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 $\|\partial_{s}A\|_{\infty}\leq M$. Let $0<k<2$. Let $u_{\varepsilon}$ be the weak solution of the initial-Dirichlet problem,

[TABLE]

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

[TABLE]

for $0<\varepsilon\leq 1$, 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,

[TABLE]

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

[TABLE]

Next, we observe that $u_{0,\lambda}-u_{0}=0$ on $\partial_{p}\Omega_{T}$ and

[TABLE]

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

[TABLE]

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

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

Theorem 7.6**.**

Assume $A$ satisfies (1.3) and (1.4). Also assume that $\|\nabla^{2}A\|_{\infty}\leq M$. 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

[TABLE]

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 $\|\nabla A\|_{\infty}$ may be bounded by a constant depending on $\mu$ and $M$. We omit the details. ∎

Proof of Theorem 1.3.

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

[TABLE]

∎

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 $k\neq 2$, we let

[TABLE]

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

[TABLE]

$\delta=\varepsilon+\varepsilon^{k/2}$, and the cut-off functions $\eta_{\delta}$ 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

[TABLE]

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

[TABLE]

Note that

[TABLE]

To bound $I_{1}$, we use the inequality (3.17). This gives

[TABLE]

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 $\theta_{11}\in C^{\infty}_{0}(B(0,1))$, $\theta_{11}\geq 0$ 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

[TABLE]

It is not hard to see that

[TABLE]

In summary we have proved that

[TABLE]

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

[TABLE]

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