Convergence rate from systems of balance laws to isotropic parabolic systems,a periodic case

Abstract

It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper.

Keywords

Convergence rate first order balance laws partial dissipation isotropic parabolic system stream function

1. Introduction

We consider the periodic problem in a d-dimensional torus of the following first-order quasilinear hyperbolic system with stiff source terms, which is of the form $\begin{matrix} (1.1) & \partial_{t^{'}} U + \sum_{j = 1}^{d} A_{j} (U) \partial_{x_{j}} U = - \frac{Q (U)}{ε}, \end{matrix}$ with the initial condition $\begin{matrix} (1.2) & U (0, x) = U_{0}^{ε} (x), x \in T^{d} . \end{matrix}$ Here $T^{d} = {(R / 2 π Z)}^{d}$ is a torus in $R^{d}$ , $U : R_{t^{'}}^{+} \times T_{x}^{d} \to S \subset R^{n}$ is the unknown variable, $ε \in (0, 1]$ is a small parameter standing often for the relaxation time in physical models (see Section 5), $t^{'} > 0$ is the usual time and $x = (x_{1}, \dots, x_{d}) \in R^{d}$ is the space variable. The source term $Q : S \to R^{n}$ and the matrices $A_{j} : S \to R^{n}$ ( $1 ⩽ j ⩽ d$ ) are all smooth functions. The set S is called the state space. We assume (1.1) is symmetrizable hyperbolic, i.e., there exists a symmetric positive definite matrix $A_{0} (U)$ , called symmetrizer, such that $A_{0} (U) A_{j} (U)$ is symmetric for all $U \in S$ and $1 ⩽ j ⩽ d$ .

Usually, the source term $Q (U)$ is of the form $\begin{matrix} Q (U) = [\begin{matrix} 0 \\ q (U) \end{matrix}], \end{matrix}$ where $q : S \to R^{r}$ is a smooth function, $1 ⩽ r ⩽ n$ . With the same partition, we denote $\begin{matrix} U = [\begin{matrix} u \\ v \end{matrix}], u \in R^{n - r}, v \in R^{r}, \end{matrix}$ as well as the initial data $\begin{matrix} U_{0}^{ε} (x) = [\begin{matrix} u_{0}^{ε} (x) \\ v_{0}^{ε} (x) \end{matrix}], u_{0}^{ε} (x) \in R^{n - r}, v_{0}^{ε} (x) \in R^{r} . \end{matrix}$ More generally, under the same partition, a vector $V \in R^{n}$ and an $n \times n$ matrix M will be denoted by $V = [\begin{matrix} V^{I} \\ V^{II} \end{matrix}]$ and $M = [\begin{matrix} M^{11} & M^{12} \\ M^{21} & M^{22} \end{matrix}]$ , respectively. Here $M^{11}$ and $M^{22}$ are matrices of order $(n - r) \times (n - r)$ and $r \times r$ respectively and $M^{12}$ and $M^{21}$ are defined accordingly.

A large number of physical models of the form (1.1) can be found in [20,29]. The local existence of smooth solutions to (1.1)-(1.2) is well-known due to Lax [12] and Kato [10], see also [17]. Generally speaking, smooth solutions of the Cauchy problem (1.1)–(1.2) exist only locally in time and singularities may appear in finite time. However, the dissipative structure of the system may prevent the formation of the singularities and lead to global smooth solutions in a neighborhood of an equilibrium state $U_{e} = [\begin{matrix} u_{e} \\ v_{e} \end{matrix}] \in S$ for (1.1), i.e., $q (U_{e}) = 0$ . The global existence for (1.1)-(1.2) was proved by Hanouzet–Natalini [5] in one space dimension and was extended by Yong [31] to the case of several space dimensions. In the proof of these results, they need essentially two main conditions. The first one is a partially dissipative condition and the second one is the Shizuta–Kawashima condition (SK) (see [25]) at an equilibrium state.

Now we consider a partially conserved model of (1.1) in the sense that only u satisfies the equations of conservation laws. A partially conserved model for u means that there exist smooth vectors $f_{j} : S \to R^{n - r}$ , such that (1.1)–(1.2) can be rewritten into the following $\begin{matrix} (1.3) & \{\begin{matrix} \partial_{t^{'}} u + \sum_{j = 1}^{d} \partial_{x_{j}} f_{j} (U) = 0, \\ \partial_{t^{'}} v + \sum_{j = 1}^{d} (A_{j}^{21} (U) \partial_{x_{j}} u + A_{j}^{22} (U) \partial_{x_{j}} v) = - \frac{q (U)}{ε}, \end{matrix} \end{matrix}$ with the initial condition $\begin{matrix} (1.4) & U (0, x) = U_{0}^{ε} (x), x \in T^{d} . \end{matrix}$ System (1.3) can be regarded as a special case included in (1.1) if we denote $\begin{matrix} (1.5) & A_{j} (U) = [\begin{matrix} \partial_{u} f_{j} (U) & \partial_{v} f_{j} (U) \\ A_{j}^{21} (U) & A_{j}^{22} (U) \end{matrix}], 1 ⩽ j ⩽ d . \end{matrix}$

When the slow time $t = ε t^{'}$ is introduced and the following change of variables is made $\begin{matrix} (1.6) & u^{ε} (t, x) = u (t^{'}, x), v^{ε} (t, x) = \frac{1}{ε} v (t^{'}, x), \end{matrix}$ (1.3)–(1.4) becomes $\begin{matrix} (1.7) & \{\begin{matrix} \partial_{t} u^{ε} + \frac{1}{ε} \sum_{j = 1}^{d} \partial_{x_{j}} f_{j} (u^{ε}, ε v^{ε}) = 0, \\ ε^{2} \partial_{t} v^{ε} + \sum_{j = 1}^{d} (A_{j}^{21} (u^{ε}, ε v^{ε}) \partial_{x_{j}} u^{ε} + A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (ε v^{ε})) = - \frac{q (u^{ε}, ε v^{ε})}{ε}, \end{matrix} \end{matrix}$ with initial conditions $\begin{matrix} (1.8) & u^{ε} (0, x) = u_{0}^{ε} (x), v^{ε} (0, x) = \frac{v_{0}^{ε} (x)}{ε} . \end{matrix}$

In physics, the limit $ε \to 0$ is often called the zero relaxation limit. This is because when $ε \to 0$ , the system formally converges to the equilibrium state. For this kind of convergence problems, there are rich literatures. Under sub-characteristic conditions [16], system (1.1) converges formally to a hyperbolic system. We refer the readers to [1,2,8,16,24] and references therein for mathematical results. In particular, under stability conditions, the uniform local smooth solutions with respect to ε and their convergence as $ε \to 0$ were investigated in [30]. Under reasonable conditions on $A_{j}$ and q, system (1.1) converges formally to parabolic-type equations (see [19]). The justification of the local-in-time convergence was proved in [21]. See also [11] for a study close to semilinear case. Moreover, in a neighborhood of an equilibrium state, the uniform existence of global smooth solutions with respect to ε and their global-in-time convergence as $ε \to 0$ were obtained in [22].

Another interesting topic is the convergence rate problem. In the local-in-time convergence result, the convergence rate was clearly shown and it depends on the local existence time [21]. For the global convergence rate problems, few results have been obtained. See Junca–Rascle [9] for the isothermal Euler system with damping, Goudon–Lin [4] for the so-called M1-model in the radiative transfer theory and Li–Peng–Zhao [13] for the general hyperbolic systems of balance laws in the whole space. All of these results are studied in the one-dimensional space. For multi-dimensional cases, to the authors’ best knowledge, no results have been obtained for the general first order quasilinear systems.

The aim of the present paper is to study the global-in-time convergence rate problem of (1.7)–(1.8) as $ε \to 0$ in a multi-dimensional torus. The global convergence rate is hard to obtain mainly due to the difficulty in obtaining the global-in-time $L^{2}$ -estimate. In detail, when constructing the error system, which are established by subtracting the limiting equations from the original system, the structure of the equations changes. To be clear, it leads to the loss of symmetrizable hyperbolicity and the existence of a strictly convex entropy. That is the reason why we can not obtain the highest order estimates nor the $L^{2}$ -estimate. However, a stream function technique can overcome this difficulty and provide a $L^{2}$ error estimate [4,9,13].

The stream function acts as a percise interconnection between the unknowns and the singular source term. Thus it provides an alternative way to estimate the singular terms. However, the previous studies show that this technique is only efficient for the one-dimensional system of conservation laws in the whole space, but not for multi-dimensional cases. In fact, considering the following one-dimensional conservation law, $\begin{matrix} \partial_{t} z + \partial_{x} h (z) = 0, \end{matrix}$ with $h (\cdot)$ sufficiently smooth. We call ϕ a stream function of the above if ϕ satisfies $\begin{matrix} \{\begin{matrix} \partial_{t} ϕ = - h (z), \\ \partial_{x} ϕ = z . \end{matrix} \end{matrix}$ This can be formally achieved by letting $\begin{matrix} ϕ (t, x) = - \int h (z (τ, x)) d τ, \end{matrix}$ and be uniquely determined if we pose an initial condition as $\begin{matrix} ϕ (0, x) = \int_{0}^{x} z (0, y) d y . \end{matrix}$ However, when we consider the periodic problem in a multi-dimensional space, the construction of the stream function becomes more complicated. For example, for the first equation in (1.3), the usual stream function ϕ satisfies the following $\begin{matrix} (1.9) & \{\begin{matrix} \partial_{t} ϕ = - F (U), \\ div ϕ = u, \\ ϕ (0, x) = \int_{0}^{x} u_{0}^{ε} (y) d y, \end{matrix} \end{matrix}$ where $F \in R^{(n - r) \times 3}$ , such that the j-th column vector of F, ${col}_{j} F = f_{j}$ . However, these are not sufficient. Compared to the case of one-dimensional whole space, many mixed partial derivative terms will appear. One direct illustration is that for a certain $1 ⩽ i ⩽ d$ , $‖ \partial_{x_{i}} ϕ ‖_{s}$ can not be controlled by $‖ div ϕ ‖_{s}$ , so that we can not use (1.9) to continue the proof. Here $‖ \cdot ‖_{s}$ denotes the norm of the classical Sobolev space $H^{s} \overset{def}{=} H^{s} (T^{d})$ . Consequently, we need to pose additional conditions on the stream function obtained, which makes it less possible for the existence of a proper ϕ. For example, in order to make $‖ \partial_{x_{i}} ϕ ‖_{s}$ controlled by $‖ div ϕ ‖_{s}$ , in the paper, we require that the stream function satisfies the following conditions $\begin{matrix} (1.10) & M (ϕ) \overset{def}{=} \int_{T^{d}} ϕ (t, x) d x = 0, \nabla \times ϕ = 0 . \end{matrix}$ As a result, by using the Poincaré inequality, we have the following estimate for $\partial_{x_{i}} ϕ$ and ϕ, $\begin{matrix} ‖ \partial_{x_{i}} ϕ ‖_{s} + ‖ ϕ ‖_{s} ⩽ C ‖ \nabla ϕ ‖_{s} ⩽ C ‖ \nabla \times ϕ ‖_{s} + C ‖ div ϕ ‖_{s} ⩽ C ‖ div ϕ ‖_{s}, 1 ⩽ i ⩽ d . \end{matrix}$ Unfortunately, (1.9)–(1.10) are overdetermined. To overcome it, we introduce a new method for constructing the stream function which is valid for our problem. As a sacrifice, additional terms will appear on some of the right hand sides of (1.9). Due to the percise structure of the stream function, these additional terms will bring difficulties in the proof.

Moreover, even in one-dimensional space, additional conditions are needed in the previous studies. In detail, in the one-dimensional whole space, (1.7) becomes $\begin{matrix} \{\begin{matrix} \partial_{t} u^{ε} + \frac{1}{ε} \partial_{x} f (u^{ε}, ε v^{ε}) = 0, \\ ε^{2} \partial_{t} v^{ε} + \partial_{x} g (u^{ε}, ε v^{ε}) = - \frac{q (u^{ε}, ε v^{ε})}{ε}, x \in R . \end{matrix} \end{matrix}$ In order to obtain the global convergence rate, the authors in [13] need a strong connection condition between the flux term and the source term, (H)

There exists a constant matrix $D \in R^{(n - r) \times r}$ such that $\begin{matrix} f (u, v) = D q (u, v), \forall (u, v) \in G_{e}, \end{matrix}$ where $G_{e}$ is a neighbourhood of the equilibrium state.

This condition eliminates all the terms that do not contain derivatives of the stream function, which are hard to handle without the Poincaré inequality. However in a torus, we can avoid using this condition. It is worth mentioning that in order to be able to use the Poincaré inequality, we have to let the average of the stream function ϕ over the torus be zero. That is the main reason why the condition (1.10) is needed.

Another difficulty is the singularity of $v^{ε}$ in $L^{\infty} (R^{+}; H^{s})$ in the slow time scaling, in other words, we have $‖ v^{ε} (t) ‖_{s} = O (ε^{- 1})$ for all $t > 0$ (see (2.2) in the next section). Especially, it is difficult to obtain the estimates for nonlinear terms containing $v^{ε}$ , among which, terms containing $\partial_{t} v^{ε}$ are the most difficult since they can be very singular up to order $O (ε^{- 2})$ . One can observe formally through the second equation in (1.7) together with the estimate (2.2) in the next section. To overcome this difficulty, we introduce a decomposition method (see (3.20) in Section 3), in which according to the degree of singularity, we divide the singular terms into two categories. One contains terms that are very singular but with better regularity when ε is fixed while the other is just opposite. The method requires complicated calculations in which a correct division of the singular terms and a careful choice on the power of ε is needed.

The aim of the present paper is to establish the global-in-time error estimates between the solutions to (1.7) and those to the limiting system, which is parabolic and isotropic (see condition (H1) in Section 2). We study these problems in a three-dimensional torus or in a one-dimensional one. Our results are not applicable for $d = 2$ or $d > 3$ mainly due to the fact that we can not find a suitable rotation operator (see (3.3) in Section 3). In the paper, we introduce a method for constructing the generalized stream function which is valid for multi-dimensional periodic cases and we do not need any additional conditions on the matrix D defined in condition (H), which is necessary in the previous studies. For initial data sufficiently close to the constant equilibrium state in $H^{s}$ , our error estimates are showed in $L^{2} (R^{+}; H^{s}) \cap L^{\infty} (R^{+}; H^{s - 1})$ for u and in $L^{2} (R^{+}; H^{s - 2})$ for v. The estimates are uniform for all $ε \in (0, 1]$ , contrarily to the local convergence case where ε should be small [21].

The paper is organized as follows. In the next section, we give some preliminaries and state the main results of this paper. Sections 3 and 4 are devoted to the proof of the main results in a three-dimensional torus or in a one-dimensional one, respectively. In the last section, we give several examples of systems which include those studied in [4,9,13,21].

2. Preliminaries and main results

In this paper, we study (1.7)–(1.8) in a three dimensional torus or a one-dimensional one based on the results in [13,22]. In the following, let $s > \frac{d}{2} + 1$ be an integer and $C > 0$ be a generic constant independent of ε and any time. We denote by $‖ \cdot ‖$ , $‖ \cdot ‖_{l}$ and $‖ \cdot ‖_{\infty}$ the usual norm of $L^{2} \overset{def}{=} L^{2} (T^{d})$ , $H^{l} \overset{def}{=} H^{l} (T^{d})$ , and $L^{\infty} \overset{def}{=} L^{\infty} (T^{d})$ , respectively with integers $l ⩾ 1$ . In the proof, we use the fact that the embedding from $H^{l} (T^{d})$ to $L^{\infty} (T^{d})$ is continuous for all integer $l > \frac{d}{2}$ . We denote $⟨ \cdot, \cdot ⟩$ the inner product in $L^{2}$ .

We first recall results on the uniform global existence and the global-in-time convergence of the system established in [22]. We assume in the following $G_{e} \subset S$ and $Ω_{e}$ be open sets satisfying $U_{e} \in G_{e}$ and $Ω_{e} \times {0} \subset G_{e}$ . For the periodic problem (1.7)–(1.8), the assumptions needed in [22] are as follows.

(A1)

For all $U = [\begin{matrix} u \\ v \end{matrix}] \in G_{e}$ , $\begin{matrix} q (u, v) = 0 ⟺ v = 0 . \end{matrix}$

(A2)

The matrix $\partial_{v} q (u_{e}, 0)$ is invertible.

(A3)

For all $u \in Ω_{e}$ and $1 ⩽ j ⩽ d$ , $\partial_{u} f_{j} (u, 0) = 0$ .

(A4)

For all $u \in Ω_{e}$ , $\partial_{v} A_{0}^{11} (u, 0) = 0$ .

(A5)

There exists a constant $c_{0} > 0$ such that $\begin{matrix} ξ^{⊤} A_{0} (u, 0) Q^{'} (u, 0) ξ ⩾ c_{0} {| ξ^{II} |}^{2}, \forall ξ = [\begin{matrix} ξ^{I} \\ ξ^{II} \end{matrix}] \in R^{n}, \forall u \in Ω_{e}, \end{matrix}$ where $| \cdot |$ denotes the usual Euclidean norm and ⊤ denotes the transpose of a vector or a matrix.

(SK)

The kernel of the Jacobian $Q^{'} (U_{e})$ , $Ker (Q^{'} (U_{e}))$ , contains no eigenvector of the matrix $A (ω, u_{e}) \overset{def}{=} \sum_{j = 1}^{d} ω_{j} A_{j} (U_{e})$ , for any $ω = (ω_{1}, \dots, ω_{d}) \in S^{d - 1}$ (the unit sphere in $R^{d}$ ), where the matrices $A_{j}$ are defined in (1.5).

Remark 2.1.

(A1) means that the equilibrium state is of the form $(u_{e}, 0)$ , i.e., $v_{e} = 0$ .

(A2)–(A3) are necessary to derive the limiting system as second-order parabolic partial differential equations (see [19,22]).

(A4) is a technical assumption, which is satisfied when the system is semilinear or $n - r = 1$ . The latter is the case for gas dynamics equations.

(SK) is a dissipative condition, which implies a time dissipative estimate of $\nabla U$ .

Remark 2.2.
(A5) is the partially dissipative condition, which implies a time dissipative estimate of v. Under this condition, it is proved in [30] (see also Proposition 2.1 in [22]) that there is a neighbourhood $Ω_{1} \subset Ω_{e}$ of $u_{e}$ , such that $\begin{matrix} A_{0}^{12} (u, 0) = 0, \forall u \in Ω_{1} . \end{matrix}$ It follows that, for all $u \in Ω_{1}$ , $\begin{matrix} A_{0} (u, 0) Q^{'} (u, 0) = [\begin{matrix} A_{0}^{11} (u, 0) & 0 \\ 0 & A_{0}^{22} (u, 0) \end{matrix}] \cdot [\begin{matrix} 0 & 0 \\ 0 & \partial_{v} q (u, 0) \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & A_{0}^{22} (u, 0) \partial_{v} q (u, 0) \end{matrix}] . \end{matrix}$ Thus (A5) shows that $A_{0}^{22} (u, 0) \partial_{v} q (u, 0)$ is a positive definite matrix in $Ω_{1}$ and $\begin{matrix} (2.1) & {(ξ^{II})}^{⊤} A_{0}^{22} (u, 0) \partial_{v} q (u, 0) ξ^{II} ⩾ c_{0} {| ξ^{II} |}^{2}, \forall ξ^{II} \in R^{r}, \forall u \in Ω_{1} . \end{matrix}$

Under these assumptions, in a neighbourhood of the equilibrium state, the uniform global existence of smooth solutions to (1.7)–(1.8) was proved.
Proposition 2.1.
Let $d ⩾ 1$ and integer $s > \frac{d}{2} + 1$ . Let conditions (A1)–(A5) and (SK) hold. Then in a neighbourhood of the constant equilibrium state, there exists a unique global solution $(u^{ε}, v^{ε})$ to ( 1.7 )–( 1.8 ). Moreover, the following estimate holds $\begin{array}{rcl} (2.2) & \begin{matrix} {‖ u^{ε} (t) - u_{e} ‖}_{s}^{2} + ε^{2} {‖ v^{ε} (t) ‖}_{s}^{2} + \int_{0}^{t} ({‖ v^{ε} (τ) ‖}_{s}^{2} + {‖ \nabla u^{ε} (τ) ‖}_{s - 1}^{2}) d τ \\ ⩽ C ({‖ u_{0}^{ε} - u_{e} ‖}_{s}^{2} + {‖ v_{0}^{ε} ‖}_{s}^{2}), \forall t > 0, \end{matrix} \end{array}$ where $C > 0$ is a generic constant independent of ε and any time.
Remark 2.3.
We point out that the above result is also valid for system (1.1)–(1.2).

We now recall the results on the global-in-time convergence as $ε \to 0$ . From (A3), we see that $f_{j} (u, 0)$ is constant. It follows that $\begin{matrix} \partial_{x_{j}} f_{j} (u, v) = \partial_{x_{j}} [f_{j} (u, v) - f_{j} (u, 0)] . \end{matrix}$ Hence, we may suppose $f_{j} (u, 0) = 0$ (otherwise, we replace $f_{j} (u, v)$ by $f_{j} (u, v) - f_{j} (u, 0)$ ). Together with (A1), there are smooth matrix functions $f_{j}^{1} (u, v)$ of order $(n - r) \times r$ and $q^{1} (u, v)$ of order $r \times r$ such that $\begin{matrix} (2.3) & \{\begin{matrix} f_{j} (u, v) = f_{j}^{1} (u, v) v, \\ q (u, v) = q^{1} (u, v) v, \end{matrix} \end{matrix}$ with $\begin{matrix} \{\begin{matrix} f_{j}^{1} (u, 0) = \partial_{v} f_{j} (u, 0), \\ q^{1} (u, 0) = \partial_{v} q (u, 0) . \end{matrix} \end{matrix}$

By (2.3), (1.7)–(1.8) can be written as $\begin{matrix} (2.4) & \{\begin{matrix} \partial_{t} u^{ε} + \sum_{j = 1}^{d} \partial_{x_{j}} (f_{j}^{1} (u^{ε}, ε v^{ε}) v^{ε}) = 0, \\ ε^{2} \partial_{t} v^{ε} + \sum_{j = 1}^{d} (A_{j}^{21} (u^{ε}, ε v^{ε}) \partial_{x_{j}} u^{ε} + A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (ε v^{ε})) = - q^{1} (u^{ε}, ε v^{ε}) v^{ε}, \\ u^{ε} (0, x) = u_{0}^{ε} (x), v^{ε} (0, x) = \frac{v_{0}^{ε} (x)}{ε} . \end{matrix} \end{matrix}$ If we denote the limit of $(u^{ε}, v^{ε})$ as $(\bar{u}, \bar{v})$ , then formally $(\bar{u}, \bar{v})$ satisfies $\begin{matrix} (2.5) & \{\begin{matrix} \partial_{t} \bar{u} + \sum_{j = 1}^{d} \partial_{x_{j}} (f_{j}^{1} (\bar{u}, 0) \bar{v}) = 0, \\ \sum_{j = 1}^{d} (A_{j}^{21} (\bar{u}, 0) \partial_{x_{j}} \bar{u}) = - q^{1} (\bar{u}, 0) \bar{v}, \\ \bar{u} (0, x) = {\bar{u}}_{0} (x) . \end{matrix} \end{matrix}$ By (A2), $q^{1} (\bar{u}, 0)$ is invertible when $\bar{u} - u_{e}$ is sufficiently small. Then we obtain the following proposition.
Proposition 2.2.
Let the conditions in Proposition 2.1 hold. Let ${\bar{u}}_{0} \in H^{s}$ . If $\begin{matrix} u_{0}^{ε} ⇀ {\bar{u}}_{0}, weakly in H^{s}, \end{matrix}$ then $\begin{array}{l} u^{ε} ⇀ \bar{u}, weakly- * in L^{\infty} (R^{+}; H^{s}), \\ v^{ε} ⇀ \bar{v}, weakly in L^{2} (R^{+}; H^{s}), \end{array}$ where $\bar{u} - u_{e} \in L^{\infty} (R^{+}; H^{s})$ is the unique solution to the periodic problem for a system of second-order partial differential equations $\begin{matrix} (2.6) & \{\begin{matrix} \partial_{t} \bar{u} - \sum_{i = 1}^{d} \partial_{x_{i}} {f_{i}^{1} (\bar{u}, 0) {[q^{1} (\bar{u}, 0)]}^{- 1} \sum_{j = 1}^{d} A_{j}^{21} (\bar{u}, 0) \partial_{x_{j}} \bar{u}} = 0, \\ \bar{u} (0, x) = {\bar{u}}_{0} (x), \end{matrix} \end{matrix}$ and $\begin{matrix} (2.7) & \bar{v} = - {[q^{1} (\bar{u}, 0)]}^{- 1} \sum_{j = 1}^{d} A_{j}^{21} (\bar{u}, 0) \partial_{x_{j}} \bar{u} . \end{matrix}$ In addition, the system ( 2.6 ) is parabolic (see [ 22 ] and Lemma 2.1 below). Consequently, the following estimate holds for $\bar{u}$ , $\begin{matrix} (2.8) & {‖ \bar{u} (t) - u_{e} ‖}_{s}^{2} + {‖ \partial_{t} \bar{u} (t) ‖}_{s - 2}^{2} + \int_{0}^{t} {‖ \nabla \bar{u} (τ) ‖}_{s}^{2} d τ ⩽ C ‖ {\bar{u}}_{0} - u_{e} ‖_{s}^{2}, \forall t > 0, \end{matrix}$ provided that $‖ {\bar{u}}_{0} - u_{e} ‖_{s}$ is sufficiently small. This together with ( 2.7 ) yields $\begin{matrix} (2.9) & {‖ \bar{v} (t) ‖}_{s - 1}^{2} + \int_{0}^{t} ({‖ \bar{v} (τ) ‖}_{s}^{2} + {‖ \partial_{t} \bar{v} (τ) ‖}_{s - 2}^{2}) d τ ⩽ C ‖ {\bar{u}}_{0} - u_{e} ‖_{s}^{2}, \forall t > 0 . \end{matrix}$

Estimates (2.2) and (2.8)–(2.9) are useful in Sections 3 and 4 for the proof of the convergence rate. We point out that in general, from (2.7) and the initial condition of $v^{ε}$ in (1.8), there are initial layers on variable v in the limiting process from (1.7)–(1.8) to (2.6)–(2.7).

The following lemma ensures that system (2.6) is parabolic, which is proved in Lemma 2.2 in [22].
Lemma 2.1.
Let (A1)–(A3) and (SK) hold. There is a neighbourhood $Ω_{2} \subset Ω_{1}$ of $u_{e}$ , such that ( 2.6 ) is parabolic in the sense that $\begin{matrix} \sum_{i, j = 1}^{d} K_{i, j} (u, 0) ω_{i} ω_{j} \end{matrix}$ is a positive definite matrix for all $u \in Ω_{2}$ and all $ω = (ω_{1}, \dots, ω_{d}) \in S^{d - 1}$ , in which $\begin{matrix} K_{i, j} (u, 0) = \partial_{v} f_{i} (u, 0) {[\partial_{v} q (u, 0)]}^{- 1} A_{j}^{21} (u, 0) . \end{matrix}$

For simplicity, we denote from now on for all $(u, v) \in G_{e}$ and $1 ⩽ i, j ⩽ d$ , $\begin{matrix} (2.10) & D_{i} (u, v) = f_{i}^{1} (u, v) {[q^{1} (u, v)]}^{- 1}, K_{i, j} (u, v) = D_{i} (u, v) A_{j}^{21} (u, v), \end{matrix}$ and $\begin{matrix} (2.11) & (D_{i}^{ε}, K_{i, j}^{ε}) = (D_{i} (u^{ε}, ε v^{ε}), K_{i, j} (u^{ε}, ε v^{ε})), ({\bar{D}}_{i}, {\bar{K}}_{i, j}) = (D_{i} (\bar{u}, 0), K_{i, j} (\bar{u}, 0)) . \end{matrix}$

In the proof, when $d = 3$ , we need to pose the following condition on $K_{i, j}$ such that the limiting equation is isotropic. This condition automatically holds when $d = 1$ .
(H1)
For all $1 ⩽ i, j ⩽ 3$ , there exists a positive definite matrix $K \in R^{(n - r) \times (n - r)}$ , such that the matrices $K_{i, j} (u, v) \in R^{(n - r) \times (n - r)}$ satisfy $\begin{matrix} K_{i, j} (u, v) = δ_{i, j} K, \forall (u, v) \in G_{e}, \end{matrix}$ where $δ_{i, j}$ is the Kronecker symbol.
Remark 2.4.
Under condition (H1), the limiting system (2.6) becomes the following isotropic diffusive second order parabolic equations $\begin{matrix} \{\begin{matrix} \partial_{t} \bar{u} - \sum_{i = 1}^{d} \partial_{x_{i}} (K (\bar{u}, 0) \partial_{x_{i}} \bar{u}) = 0, \\ \bar{u} (0, x) = {\bar{u}}_{0} (x) . \end{matrix} \end{matrix}$

The main result of this paper is stated as follows. Theorem 2.1.
Let (A1)–(A5), (SK) and (H1) hold. Let $d = 1, 3$ and integer $s > \frac{d}{2} + 1$ . Let $(u^{ε}, v^{ε})$ be the unique solution to ( 1.7 )–( 1.8 ), $\bar{u}$ be the unique solution to ( 2.6 ) with initial data ${\bar{u}}_{0}$ being the weak limit of $u_{0}^{ε}$ in $H^{s}$ and $\bar{v}$ be given by ( 2.7 ). There exist positive constants δ, $C_{1}$ and $C_{2}$ , independent of ε, such that if $\begin{matrix} (2.12) & {‖ u_{0}^{ε} - u_{e} ‖}_{s} + {‖ v_{0}^{ε} ‖}_{s} ⩽ δ, \end{matrix}$ and $\begin{matrix} (2.13) & {‖ u_{0}^{ε} - {\bar{u}}_{0} ‖}_{s} + {‖ v_{0}^{ε} ‖}_{s - 2} ⩽ C_{1} ε^{ζ}, \end{matrix}$ then for all $ε \in (0, 1]$ , we have the following estimate $\begin{array}{l} sup_{t \in R^{+}} ({‖ u^{ε} (t) - \bar{u} (t) ‖}_{s - 1}^{2} + ε^{2} {‖ v^{ε} (t) - \bar{v} (t) ‖}_{s - 2}^{2}) \\ + \int_{0}^{+ \infty} ({‖ u^{ε} (t) - \bar{u} (t) - M (u^{ε} (t) - \bar{u} (t)) ‖}_{s}^{2} d x + {‖ v^{ε} (t) - \bar{v} (t) ‖}_{s - 2}^{2}) d t \\ ⩽ C_{2} ε^{ζ_{1}}, \end{array}$ where $M (\cdot)$ is introduced in ( 1.10 ), $ζ > 0$ is a constant independent of ε and $ζ_{1} = min (1, ζ)$ . In particular, when $d = 1$ , we do not need the condition (H1).

3. Convergence rate in three-dimensional torus

In this section, we let $d = 3$ . In what follows, we assume that the conditions in Theorem 2.1 hold. For convenience, for a given scalar or vector function $g (t, x)$ , we denote its mean value over the torus $T^{3}$ with respect to x as $\begin{matrix} M (g (t)) \overset{def}{=} \int_{T^{3}} g (t, x) d x . \end{matrix}$ Then it follows for integers $l ⩾ 1$ , $\begin{matrix} (3.1) & {‖ M (g (t)) ‖}_{l} = ‖ M (g (t)) ‖ = ‖ \int_{T^{3}} g (t, x) d x ‖ ⩽ C {‖ g (t) ‖}_{s - 1}, \end{matrix}$ in which we have used the continuous embedding $H^{s - 1} ↪ L^{\infty}$ for integers $s ⩾ 3$ .

3.1. Construction of the stream function

We first construct a generalized stream function, which we will use frequently in later proof. For simplicity, for a matrix $E \in R^{m \times n}$ , we denote its i-th row as ${row}_{i} E$ and its k-th column as ${col}_{k} E$ with $1 ⩽ i ⩽ m$ , $1 ⩽ k ⩽ n$ .

Lemma 3.1 (Existence of the stream function).

Let $s ⩾ 3$ be an integer. Let $G^{ε}$ be a matrix of order $(n - r) \times 3$ with its i-th column defined as $\begin{matrix} (3.2) & {col}_{i} G^{ε} = f_{i}^{1} (u^{ε}, ε v^{ε}) v^{ε} + f_{i}^{1} (\bar{u}, 0) {[q^{1} (\bar{u}, 0)]}^{- 1} \sum_{j = 1}^{3} (A_{j}^{21} (\bar{u}, 0) \partial_{x_{j}} \bar{u}), 1 ⩽ i ⩽ 3 . \end{matrix}$ Then there exist stream functions ${ϕ_{k}^{ε}}_{1 ⩽ k ⩽ n - r}$ and functions ${H_{k}^{ε}}_{1 ⩽ k ⩽ n - r}$ , such that $\begin{matrix} (3.3) & \{\begin{matrix} div ϕ_{k}^{ε} = u_{k}^{ε} - {\bar{u}}_{k} - M (u_{k}^{ε} - {\bar{u}}_{k}), \\ \partial_{t} ϕ_{k}^{ε} = - {({row}_{k} G^{ε})}^{⊤} + \nabla \times H_{k}^{ε}, 1 ⩽ k ⩽ n - r, \end{matrix} \end{matrix}$ where $u_{k}^{ε}$ and ${\bar{u}}_{k}$ are the k-th component of $u^{ε}$ and $\bar{u}$ , respectively. Additionally, stream functions ${ϕ_{k}^{ε}}$ satisfy $\begin{matrix} \nabla \times ϕ_{k}^{ε} = 0, M (ϕ_{k}^{ε}) = 0, 1 ⩽ k ⩽ n - r . \end{matrix}$ Moreover, we have the following estimates $\begin{array}{rcl} (3.4) & {‖ ϕ_{k}^{ε} (t) ‖}_{s} ⩽ C {‖ div ϕ_{k}^{ε} (t) ‖}_{s} ⩽ C {‖ u^{ε} (t) - \bar{u} (t) ‖}_{s}^{2}, \forall t > 0, \end{array}$ and $\begin{matrix} (3.5) & \int_{0}^{+ \infty} {‖ \partial_{t} ϕ_{k}^{ε} (τ) ‖}_{s}^{2} d τ ⩽ C δ, \end{matrix}$ where $δ > 0$ is defined in Theorem 2.1 .

Proof.
Subtracting (2.6) from the first equation in (2.4), we have $\begin{matrix} (3.6) & \partial_{t} (u^{ε} - \bar{u}) + \sum_{i = 1}^{3} \partial_{x_{i}} (f_{i}^{1} (u^{ε}, ε v^{ε}) v^{ε} + f_{i}^{1} (\bar{u}, 0) {[q^{1} (\bar{u}, 0)]}^{- 1} \sum_{j = 1}^{3} (A_{j}^{21} (\bar{u}, 0) \partial_{x_{j}} \bar{u})) = 0 . \end{matrix}$ Noticing the definition of $G^{ε}$ in (3.2), we can rewrite (3.6) into $\begin{matrix} (3.7) & \partial_{t} (u^{ε} - \bar{u}) + div G^{ε} = 0 . \end{matrix}$ The proof of the existence of the stream function is constructive. First, let $Φ^{ε}$ be the unique periodic solution to the following Poisson equation $\begin{matrix} (3.8) & \{\begin{matrix} - Δ Φ^{ε} = u^{ε} - \bar{u} - M (u^{ε} - \bar{u}), \\ M (Φ^{ε}) = 0 . \end{matrix} \end{matrix}$ To be clear, we denote $Φ^{ε} = {(Φ_{1}^{ε}, \dots, Φ_{n - r}^{ε})}^{⊤} \in R^{n - r}$ . Thus, let $\begin{matrix} (3.9) & ϕ_{k}^{ε} = - \nabla Φ_{k}^{ε}, 1 ⩽ k ⩽ n - r . \end{matrix}$ Then it is obvious that $ϕ_{k}^{ε} \in R^{3}$ , and $\begin{matrix} M (ϕ_{k}^{ε}) = 0, \nabla \times ϕ_{k}^{ε} = 0, 1 ⩽ k ⩽ n - r . \end{matrix}$ Applying the ‘div’ operator to both sides of (3.9), we have $\begin{matrix} (3.10) & div ϕ_{k}^{ε} = - Δ Φ_{k}^{ε} = u_{k}^{ε} - {\bar{u}}_{k} - M (u_{k}^{ε} - {\bar{u}}_{k}), \end{matrix}$ where $u_{k}^{ε}$ and ${\bar{u}}_{k}$ are the k-th component of $u^{ε}$ and $\bar{u}$ , respectively. Besides, noticing (3.7), we obtain $\begin{array}{rcl} (3.11) & \partial_{t} M (u^{ε} - \bar{u}) = 0 . \end{array}$ Taking the time derivative to both sides of (3.10) and noticing (3.6) yield $\begin{matrix} div \partial_{t} ϕ_{k}^{ε} = \partial_{t} (u_{k}^{ε} - {\bar{u}}_{k}) - \partial_{t} M (u_{k}^{ε} - {\bar{u}}_{k}) = - div {({row}_{k} G^{ε})}^{⊤}, \end{matrix}$ which implies that there are functions $H_{k}^{ε} \in R^{3}$ , satisfying $\begin{matrix} \partial_{t} ϕ_{k}^{ε} + {({row}_{k} G^{ε})}^{⊤} = \nabla \times H_{k}^{ε}, 1 ⩽ k ⩽ n - r . \end{matrix}$

We now establish the estimates for $ϕ_{k}^{ε}$ . Noticing (3.1) and that ${ϕ_{k}^{ε}}_{1 ⩽ k ⩽ n - r}$ are rotation free, by using the Poincaré inequality, we have for $1 ⩽ k ⩽ n - r$ , $\begin{matrix} {‖ ϕ_{k}^{ε} (t) ‖}_{s}^{2} ⩽ C {‖ \nabla ϕ_{k}^{ε} (t) ‖}_{s}^{2} = C {‖ \nabla \times ϕ_{k}^{ε} (t) ‖}_{s}^{2} + C {‖ div ϕ_{k}^{ε} (t) ‖}_{s}^{2} ⩽ C {‖ u^{ε} (t) - \bar{u} (t) ‖}_{s}^{2}, \forall t > 0, \end{matrix}$ which implies (3.4). It remains to prove (3.5). Applying $\partial_{t}$ to both sides of the first equation in (3.8) and using (3.7) and (3.11), we have $\begin{matrix} Δ \partial_{t} Φ^{ε} = - \partial_{t} (u^{ε} - \bar{u}) = div G^{ε} . \end{matrix}$ Energy estimates and the explicit expression of $ϕ_{k}^{ε}$ in (3.9) imply that $\begin{matrix} {‖ \partial_{t} ϕ_{k}^{ε} (t) ‖}_{s} ⩽ C {‖ \nabla \partial_{t} Φ^{ε} (t) ‖}_{s} ⩽ C {‖ G^{ε} (t) ‖}_{s} ⩽ C ({‖ v^{ε} (t) ‖}_{s} + {‖ \nabla \bar{u} (t) ‖}_{s}), \end{matrix}$ in which we have used the Moser-type inequalities. Let $T > 0$ . Noticing the energy estimates (2.2) and (2.8), we have $\begin{matrix} \int_{0}^{T} {‖ \partial_{t} ϕ_{k}^{ε} (τ) ‖}_{s}^{2} d τ ⩽ C \int_{0}^{T} ({‖ v^{ε} (τ) ‖}_{s}^{2} + {‖ \nabla \bar{u} (τ) ‖}_{s}^{2}) d τ ⩽ C δ, \end{matrix}$ which implies (3.5) since $T > 0$ is arbitrary. □
Remark 3.1 (Compatibility conditions).

We now define the initial data of $ϕ_{k}^{ε}$ . From (3.8), the classical elliptic estimates together with the Poincaré inequality implies $\begin{matrix} Φ^{ε} \in L^{\infty} ([0, T]; H^{s}), \partial_{t} Φ^{ε} \in L^{2} ([0, T]; H^{s}), \forall T > 0 . \end{matrix}$ Hence, by the Aubin–Lions lemma, we have $\begin{matrix} Φ^{ε} \in C ([0, T]; H^{s}), \forall T > 0 . \end{matrix}$ Hence, we may define the initial data of $Φ^{ε}$ , i.e., $Φ^{ε} (0)$ , as the unique solution to the following $\begin{matrix} \{\begin{matrix} - Δ Φ^{ε} (0) = u_{0}^{ε} - {\bar{u}}_{0}, \\ M (Φ^{ε} (0)) = 0 . \end{matrix} \end{matrix}$ Similarly, we denote ${(Φ^{ε} (0))}_{k}$ as the k-th component of $Φ^{ε} (0)$ . Now, we can define the initial data of $ϕ_{k}^{ε}$ . By similar methods, we obtain $\begin{matrix} ϕ_{k}^{ε} \in C ([0, T]; H^{s}), \forall T > 0 and 1 ⩽ k ⩽ n - r . \end{matrix}$ As a result, $ϕ_{k}^{ε} (0)$ , as the initial data of $ϕ_{k}^{ε}$ , can be defined by $\begin{matrix} ϕ_{k}^{ε} (0) = - \nabla {(Φ^{ε} (0))}_{k}, 1 ⩽ k ⩽ n - r . \end{matrix}$ Consequently, by (2.13), we have $\begin{matrix} (3.12) & {‖ ϕ_{k}^{ε} (0) ‖}_{s} ⩽ C {‖ \nabla Φ^{ε} (0) ‖}_{s} ⩽ C {‖ u_{0}^{ε} - {\bar{u}}_{0} ‖}_{s} ⩽ C ε^{ζ} . \end{matrix}$

Lemma 3.1 will be frequently used in the later proof. To make the notations easy to follow, for a multi-index $α = (α_{1}, α_{2}, α_{3}) \in N^{3}$ , we denote $\begin{matrix} \partial_{x}^{α} = \frac{\partial^{| α |}}{\partial x_{1}^{α_{1}} \partial x_{2}^{α_{2}} \partial x_{3}^{α_{3}}} with | α | = α_{1} + α_{2} + α_{3}, \end{matrix}$ and for all $1 ⩽ k ⩽ n - r$ , $\begin{matrix} \{\begin{matrix} (u^{α}, v^{α}, ϕ_{k}^{α}) = (\partial_{x}^{α} u^{ε}, \partial_{x}^{α} v^{ε}, \partial_{x}^{α} ϕ^{ε}), \\ ({\bar{u}}^{α}, {\bar{v}}^{α}) = (\partial_{x}^{α} \bar{u}, \partial_{x}^{α} \bar{v}) . \end{matrix} \end{matrix}$

3.2. Convergence rate for $u^{ε}$

The goal of this subsection is to establish the following estimate.

Lemma 3.2 (Convergence rate for $u^{ε}$ ).

Let the conditions in Theorem 2.1 hold. Then $\begin{array}{rcl} (3.13) & {‖ u^{ε} (t) - \bar{u} (t) ‖}_{s - 1}^{2} + \int_{0}^{t} {‖ u^{ε} (τ) - \bar{u} (τ) - M (u^{ε} (τ) - \bar{u} (τ)) ‖}_{s}^{2} d t ⩽ C ε^{2 ζ_{1}}, \forall t > 0, \end{array}$ where $ζ_{1} > 0$ is defined in Theorem 2.1 .

The proof of Lemma 3.2 follows from a series of lemmas given as follows. Since $δ > 0$ is sufficiently small, (2.2) together with (2.12) implies that $(u^{ε}, ε v^{ε}) \in G_{e}$ and $(u^{ε} - u_{e}, ε v^{ε})$ is uniformly small with respect to ε in $L^{\infty} (R^{+} \times T^{3})$ . In addition, the weak convergence of $u_{0}^{ε}$ to ${\bar{u}}_{0}$ in $H^{k}$ implies that $‖ {\bar{u}}_{0} - u_{e} ‖_{s} ⩽ δ$ . By (2.8), we have $\bar{u} \in Ω_{2}$ and $\bar{u} - u_{e}$ is sufficiently small. In the proof, we let i, k be integers with $1 ⩽ i ⩽ 3$ and $1 ⩽ k ⩽ n - r$ .

To start, let $T > 0$ and $α \in N^{3}$ be a multi-index with $| α | ⩽ s$ , multiplying $D_{i}^{ε}$ , which is defined in (2.10)–(2.11), to the second equation in (2.4) and applying $\partial_{x}^{α}$ to the resulting equation, we have for $1 ⩽ i ⩽ 3$ , $\begin{matrix} (3.14) & ε^{2} \partial_{x}^{α} (D_{i}^{ε} \partial_{t} v^{ε}) + \partial_{x}^{α} \sum_{j = 1}^{3} (K_{i, j}^{ε} \partial_{x_{j}} u^{ε} + D_{i}^{ε} A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (ε v^{ε})) = - \partial_{x}^{α} [f_{i}^{1} (u^{ε}, ε v^{ε}) v^{ε}] . \end{matrix}$ For convenience, we introduce the matrices $B_{1}^{α}, B_{2}^{α}, B_{3}^{α}, B_{4}^{α} \in R^{(n - r) \times 3}$ with their i-th columns defined as follows. $\begin{array}{l} {col}_{i} B_{1}^{α} = \partial_{x}^{α} (D_{i}^{ε} \partial_{t} v^{ε}), \\ {col}_{i} B_{2}^{α} = \sum_{j = 1}^{3} \partial_{x}^{α} (K_{i, j}^{ε} \partial_{x_{j}} u^{ε}), \\ {col}_{i} B_{3}^{α} = \sum_{j = 1}^{3} \partial_{x}^{α} (D_{i}^{ε} A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (ε v^{ε})), \\ {col}_{i} B_{4}^{α} = \partial_{x}^{α} [f_{i}^{1} (u^{ε}, ε v^{ε}) v^{ε}] . \end{array}$ Then, (3.14) becomes $\begin{matrix} ε^{2} B_{1}^{α} + B_{2}^{α} + B_{3}^{α} + B_{4}^{α} = 0, \end{matrix}$ which implies $\begin{matrix} ε^{2} {row}_{k} B_{1}^{α} + {row}_{k} B_{2}^{α} + {row}_{k} B_{3}^{α} + {row}_{k} B_{4}^{α} = 0 . \end{matrix}$ Taking the inner product of this equality with $ϕ_{k}^{α}$ in $L^{2}$ and integrating the resulting equation over $[0, T]$ , we get $\begin{array}{rcl} 0 & = & ε^{2} \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{1}^{α} ⟩ d t + \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{2}^{α} ⟩ d t \\ (3.15) & + \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{3}^{α} ⟩ d t + \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{4}^{α} ⟩ d t . \end{array}$ We will treat the right hand side of the above term by term. In the following, we denote for simplicity that the j-th component of a vector W as ${(W)}_{j}$ . First, we have

Lemma 3.3.
For all $| α | ⩽ s$ and a certain $1 ⩽ k ⩽ n - r$ , it holds $\begin{matrix} (3.16) & | \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{1}^{α} ⟩ d t | ⩽ \frac{1}{4} {‖ ϕ_{k}^{α} (T) ‖}^{2} + C ε^{2 ζ_{1}} + C δ ϕ_{T}^{2} + C δ \int_{0}^{T} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ, \end{matrix}$ where $ζ_{1} > 0$ and $δ > 0$ are defined in Theorem 2.1 , and $ϕ_{T}$ is defined by $\begin{matrix} ϕ_{T} = sup_{0 ⩽ t ⩽ T} (\sum_{k = 1}^{n - r} {‖ ϕ_{k}^{ε} (t) ‖}_{s}) . \end{matrix}$
Proof.
Noticing that $\begin{matrix} {row}_{k} B_{1}^{α} = (\partial_{x}^{α} {(D_{1}^{ε} \partial_{t} v^{ε})}_{k}, \partial_{x}^{α} {(D_{2}^{ε} \partial_{t} v^{ε})}_{k}, \partial_{x}^{α} {(D_{3}^{ε} \partial_{t} v^{ε})}_{k}) . \end{matrix}$ Taking the integration by parts to the first term on the right hand side of (3.15), we have $\begin{array}{rcl} ε^{2} \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{1}^{α} ⟩ d t & = & ε^{2} \int_{0}^{T} \frac{d}{d t} ⟨ ϕ_{k}^{α}, {row}_{k} B_{1, 1}^{α} ⟩ d t - ε^{2} \int_{0}^{T} ⟨ \partial_{t} ϕ_{k}^{α}, {row}_{k} B_{1, 1}^{α} ⟩ d t \\ (3.17) & - ε^{2} \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{1, 2}^{α} ⟩ d t \end{array}$ with matrices $B_{1, 1}^{α}$ and $B_{1, 2}^{α}$ defined as $\begin{array}{l} {row}_{k} B_{1, 1}^{α} = (\partial_{x}^{α} {(D_{1}^{ε} v^{ε})}_{k}, \partial_{x}^{α} {(D_{2}^{ε} v^{ε})}_{k}, \partial_{x}^{α} {(D_{3}^{ε} v^{ε})}_{k}), \\ {row}_{k} B_{1, 2}^{α} = (\partial_{x}^{α} {(\partial_{t} D_{1}^{ε} v^{ε})}_{k}, \partial_{x}^{α} {(\partial_{t} D_{2}^{ε} v^{ε})}_{k}, \partial_{x}^{α} {(\partial_{t} D_{3}^{ε} v^{ε})}_{k}) . \end{array}$ For all $ε \in (0, 1]$ , we use (2.2), (2.8), (2.13), (3.3), (3.4) and (3.12) to obtain $\begin{array}{rcl} | ε^{2} \int_{0}^{T} \frac{d}{d t} ⟨ ϕ_{k}^{α}, {row}_{k} B_{1, 1}^{α} ⟩ d t | & ⩽ & ε^{2} | ⟨ ϕ_{k}^{α} (T), {row}_{k} B_{1, 1}^{α} (T) ⟩ | + ε^{2} | ⟨ ϕ_{k}^{α} (0), {row}_{k} B_{1, 1}^{α} (0) ⟩ | \\ ⩽ & \frac{1}{4} {‖ ϕ_{k}^{α} (T) ‖}^{2} + \frac{1}{4} {‖ ϕ_{k}^{α} (0) ‖}^{2} + C ε^{4} ({‖ v^{ε} (T) ‖}_{s}^{2}) + C ε^{2} {‖ v_{0}^{ε} ‖}_{s}^{2} \\ (3.18) & ⩽ & \frac{1}{4} {‖ ϕ_{k}^{α} (T) ‖}^{2} + C ε^{2 ζ_{1}}, \end{array}$ and using (3.5) and the Moser-type inequalities, $\begin{array}{rcl} ε^{2} | \int_{0}^{T} ⟨ \partial_{t} ϕ_{k}^{α}, {row}_{k} B_{1, 1}^{α} ⟩ d t | & ⩽ & C ε^{2} \int_{0}^{T} {‖ \partial_{t} ϕ_{k}^{α} (τ) ‖}^{2} d τ + C ε^{2} \int_{0}^{T} {‖ v^{ε} (τ) ‖}_{s}^{2} d τ \\ (3.19) & ⩽ & C ε^{2} . \end{array}$ For the last term on the right hand side of (3.17), we need a little more calculations. Noticing for $1 ⩽ i ⩽ 3$ , $\begin{matrix} \partial_{x}^{α} {(\partial_{t} D_{i}^{ε} v^{ε})}_{k} = \partial_{x}^{α} (\partial_{t} ({row}_{k} D_{i}^{ε}) v^{ε}) . \end{matrix}$ Besides, by using (2.4), we have $\begin{array}{rcl} \partial_{t} ({row}_{k} D_{i}^{ε}) & = & \partial_{u} ({row}_{k} D_{i}^{ε}) \partial_{t} u^{ε} + \partial_{v} ({row}_{k} D_{i}^{ε}) \partial_{t} (ε v^{ε}) \\ = & - \partial_{u} ({row}_{k} D_{i}^{ε}) \sum_{j = 1}^{3} \partial_{x_{j}} (f_{j}^{1} (u^{ε}, ε v^{ε}) v^{ε}) - \frac{1}{ε} \partial_{v} ({row}_{k} D_{i}^{ε}) q^{1} (u^{ε}, ε v^{ε}) v^{ε} \\ - \frac{1}{ε} \partial_{v} ({row}_{k} D_{i}^{ε}) (\sum_{j = 1}^{3} (A_{j}^{21} (u^{ε}, ε v^{ε}) \partial_{x_{j}} u^{ε} + A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (ε v^{ε}))) \\ = & - \frac{1}{ε} \partial_{v} ({row}_{k} D_{i}^{ε}) q^{1} (u^{ε}, ε v^{ε}) v^{ε} - \frac{1}{ε} \partial_{v} ({row}_{k} D_{i}^{ε}) \sum_{j = 1}^{3} (A_{j}^{21} (u^{ε}, ε v^{ε}) \partial_{x_{j}} u^{ε}) \\ - \partial_{u} ({row}_{k} D_{i}^{ε}) \sum_{j = 1}^{3} \partial_{x_{j}} (f_{j}^{1} (u^{ε}, ε v^{ε}) v^{ε}) - \partial_{v} ({row}_{k} D_{i}^{ε}) \sum_{j = 1}^{3} \partial_{v} A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} v^{ε} \\ (3.20) & \overset{def}{=} & L_{k, i}^{ε} + R_{k, i}^{ε}, \end{array}$ in which $\begin{array}{l} L_{k, i}^{ε} = - \frac{1}{ε} \partial_{v} ({row}_{k} D_{i}^{ε}) q^{1} (u^{ε}, ε v^{ε}) v^{ε}, \\ \begin{matrix} R_{k, i}^{ε} & = - \partial_{u} ({row}_{k} D_{i}^{ε}) \sum_{j = 1}^{3} \partial_{x_{j}} (f_{j}^{1} (u^{ε}, ε v^{ε}) v^{ε}) - \partial_{v} ({row}_{k} D_{i}^{ε}) \sum_{j = 1}^{3} \partial_{v} A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} v^{ε} \\ - \frac{1}{ε} \partial_{v} ({row}_{k} D_{i}^{ε}) \sum_{j = 1}^{3} (A_{j}^{21} (u^{ε}, ε v^{ε}) \partial_{x_{j}} u^{ε}) . \end{matrix} \end{array}$ Using the Moser-type inequalities and the estimate (2.2), we have for $1 ⩽ k ⩽ n - r$ , $\begin{array}{l} \int_{0}^{T} {‖ ε L_{k, i}^{ε} (τ) ‖}_{s}^{2} d τ ⩽ C \int_{0}^{T} {‖ v^{ε} (τ) ‖}_{s}^{2} d τ ⩽ C δ, \\ {‖ ε R_{k, i}^{ε} (t) ‖}_{s - 1}^{2} ⩽ C {‖ ε v^{ε} (t) ‖}_{s}^{2} + {‖ \nabla u^{ε} (t) ‖}_{s - 1}^{2} ⩽ C δ, \forall t > 0 . \end{array}$ Substituting (3.20) into the last term on the right hand side of (3.17), using (2.2), (3.4), the Young’s inequality, the Cauchy–Schwarz inequality, the Poincaré inequality and the Moser-type inequalities, we have when $α = 0$ , $\begin{array}{rcl} | ε^{2} \int_{0}^{T} ⟨ ϕ_{k}^{ε}, {row}_{k} B_{1, 2}^{0} ⟩ d t | & ⩽ & C ε \sum_{i = 1}^{3} \int_{0}^{T} ‖ ϕ_{k}^{ε} ‖ (‖ ε L_{k, i}^{ε} ‖ + ‖ ε R_{k, i}^{ε} ‖) {‖ v^{ε} ‖}_{s} d t \\ ⩽ & C ε ϕ_{T} (\sum_{i = 1}^{3} \int_{0}^{T} {‖ ε L_{k, i}^{ε} (τ) ‖}_{s}^{2} d τ + \int_{0}^{T} {‖ v^{ε} (τ) ‖}_{s}^{2} d τ) \\ + C ε \sum_{i = 1}^{3} sup_{t \in R^{+}} {‖ ε R_{k, i}^{ε} (t) ‖}_{s - 1} \int_{0}^{T} ‖ div ϕ_{k}^{ε} ‖ {‖ v^{ε} ‖}_{s} d t \\ ⩽ & C δ ϕ_{T}^{2} + C ε^{2} + C δ \int_{0}^{T} {‖ div ϕ_{k}^{ε} (τ) ‖}^{2} d τ . \end{array}$ When $1 ⩽ | α | ⩽ s$ , we let a multi-index $β \in N^{3}$ with $| β | = | α | - 1$ . Then, after taking the integration by parts, noticing that $ϕ_{k}^{α}$ is rotation free, we have $\begin{array}{l} | ε^{2} \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{1, 2}^{α} ⟩ d t | \\ ⩽ C ε \sum_{i = 1}^{3} \int_{0}^{T} ‖ ϕ_{k}^{α} ‖ {‖ ε L_{k, i}^{ε} ‖}_{s} {‖ v^{ε} ‖}_{s} d t + C ε \sum_{i = 1}^{3} \int_{0}^{T} ‖ \nabla ϕ_{k}^{α} ‖ ‖ ε \partial_{x}^{β} (R_{k, i}^{ε} v^{ε}) ‖ d t \\ ⩽ C ε ϕ_{T} (\sum_{i = 1}^{3} \int_{0}^{T} {‖ ε L_{k, i}^{ε} (τ) ‖}_{s}^{2} d τ + \int_{0}^{T} {‖ v^{ε} (τ) ‖}_{s}^{2} d τ) \\ + C ε \sum_{i = 1}^{3} sup_{t \in R^{+}} {‖ ε R_{k, i}^{ε} (t) ‖}_{s - 1} \int_{0}^{T} ‖ div ϕ_{k}^{α} ‖ {‖ v^{ε} ‖}_{s} d t \\ ⩽ C δ ϕ_{T}^{2} + C ε^{2} + C δ \int_{0}^{T} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ . \end{array}$ Combining these estimates and (3.18)–(3.19) yields (3.16). □

We keep the term containing $B_{2}^{α}$ for the moment. For the term containing $B_{3}^{α}$ , we have the following estimate.
Lemma 3.4.
For all $| α | ⩽ s$ and a certain $1 ⩽ k ⩽ n - r$ , it holds $\begin{matrix} (3.21) & | \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{3}^{α} ⟩ d t | ⩽ C δ \int_{0}^{T} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ + C ε^{2}, \end{matrix}$ where $δ > 0$ is defined in Theorem 2.1 .
Proof.
Recall that $\begin{matrix} {({row}_{k} B_{3}^{α})}_{i} = \sum_{j = 1}^{3} \partial_{x}^{α} ({row}_{k} (D_{i}^{ε} A_{j}^{22} (u^{ε}, ε v^{ε})) \partial_{x_{j}} (ε v^{ε})) . \end{matrix}$ When $α = 0$ , using (2.2), (3.4), the Cauchy–Schwarz inequality and the Young’s inequality, we have $\begin{array}{rcl} | \int_{0}^{T} ⟨ ϕ_{k}^{ε}, {row}_{k} B_{3}^{0} ⟩ d t | & ⩽ & C ε \int_{0}^{T} ‖ div ϕ_{k}^{ε} ‖ ‖ \nabla v^{ε} ‖ d t \\ ⩽ & C δ \int_{0}^{T} {‖ div ϕ_{k}^{ε} (τ) ‖}^{2} d τ + C ε^{2} . \end{array}$ When $1 ⩽ | α | ⩽ s$ , we let a multi-index $β \in N^{3}$ with $| β | = | α | - 1$ , such that after taking the integration by parts, $\begin{array}{rcl} | \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{3}^{α} ⟩ d t | & ⩽ & C \int_{0}^{T} ‖ \nabla ϕ_{k}^{α} ‖ ‖ {row}_{k} B_{3}^{β} ‖ d t \\ ⩽ & C ε \int_{0}^{T} ‖ div ϕ_{k}^{α} ‖ {‖ \nabla v^{ε} ‖}_{s - 1} d t \\ ⩽ & C δ \int_{0}^{T} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ + C ε^{2} . \end{array}$ The above two estimates imply (3.21). □

We now treat the term containing $B_{4}^{α}$ . In view of the expression for $B_{4}^{α}$ and noticing (3.2) and (3.3), we have $\begin{array}{rcl} {({row}_{k} B_{4}^{α})}_{i} & = & \partial_{x}^{α} ({row}_{k} (f_{i}^{1} (u^{ε}, ε v^{ε})) v^{ε}) \\ = & \partial_{x}^{α} {({({row}_{k} G^{ε})}^{⊤})}_{i} - \sum_{j = 1}^{3} \partial_{x}^{α} ({row}_{k} ({\bar{K}}_{i, j}) \partial_{x_{j}} \bar{u}) \\ (3.22) & = & - \partial_{t} {(ϕ_{k}^{α})}_{i} + \partial_{x}^{α} {(\nabla \times H_{k}^{ε})}_{i} - \sum_{j = 1}^{3} \partial_{x}^{α} ({row}_{k} ({\bar{K}}_{i, j}) \partial_{x_{j}} \bar{u}) . \end{array}$ For convenience, we denote the matrix ${\bar{B}}_{2}^{α}$ in the way that $\begin{matrix} {col}_{i} {\bar{B}}_{2}^{α} = \sum_{j = 1}^{3} \partial_{x}^{α} ({\bar{K}}_{i, j} \partial_{x_{j}} \bar{u}) . \end{matrix}$ Then (3.22) implies $\begin{matrix} {row}_{k} B_{4}^{α} = - \partial_{t} ϕ_{k}^{α} + \partial_{x}^{α} (\nabla \times H_{k}^{ε}) - {row}_{k} {\bar{B}}_{2}^{α} . \end{matrix}$ Hence, noticing that $ϕ_{k}^{α}$ is rotation free, we have, $\begin{array}{rcl} ⟨ ϕ_{k}^{α}, {row}_{k} B_{4}^{α} ⟩ & = & ⟨ ϕ_{k}^{α}, - \partial_{t} ϕ_{k}^{α} + \partial_{x}^{α} (\nabla \times H_{k}^{ε}) - {row}_{k} {\bar{B}}_{2}^{α} ⟩ \\ = & - \frac{1}{2} \frac{d}{d t} {‖ ϕ_{k}^{α} ‖}^{2} - ⟨ ϕ_{k}^{α}, {row}_{k} {\bar{B}}_{2}^{α} ⟩ . \end{array}$ Integrating the above over $[0, T]$ implies $\begin{matrix} (3.23) & \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{4}^{α} ⟩ d t = - \frac{1}{2} {‖ ϕ_{k}^{α} (T) ‖}^{2} + \frac{1}{2} {‖ ϕ_{k}^{α} (0) ‖}^{2} - \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} {\bar{B}}_{2}^{α} ⟩ d t . \end{matrix}$ Combining the above, Lemmas 3.3–3.4 and (3.15), we have for all $1 ⩽ k ⩽ n - r$ , $\begin{array}{l} \frac{1}{4} {‖ ϕ_{k}^{α} (T) ‖}_{s}^{2} - \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} (B_{2}^{α} - {\bar{B}}_{2}^{α}) ⟩ d t \\ (3.24) & ⩽ C ε^{2 ζ_{1}} + C δ ϕ_{T}^{2} + C δ \int_{0}^{T} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ . \end{array}$

We now estimate the second term on the left hand side of the above inequality. In view of the expression for $B_{2}^{α}$ and ${\bar{B}}_{2}^{α}$ , we have $\begin{array}{rcl} {({row}_{k} (B_{2}^{α} - {\bar{B}}_{2}^{α}))}_{i} & = & \sum_{j = 1}^{3} \partial_{x}^{α} ({row}_{k} (K_{i, j}^{ε}) \partial_{x_{j}} u^{ε} - {row}_{k} ({\bar{K}}_{i, j}) \partial_{x_{j}} \bar{u}) \\ = & \sum_{j = 1}^{3} \partial_{x}^{α} ({row}_{k} (K_{i, j}^{ε} - {\bar{K}}_{i, j}) \partial_{x_{j}} u^{ε}) + \sum_{j = 1}^{3} \partial_{x}^{α} ({row}_{k} ({\bar{K}}_{i, j}) \partial_{x_{j}} (u^{ε} - \bar{u})) . \end{array}$ For simplicity, we denote $B_{2, 1}^{α}$ and $B_{2, 2}^{α}$ the matrices of order $(n - r) \times 3$ , with $\begin{array}{l} {col}_{i} B_{2, 1}^{α} = \sum_{j = 1}^{3} \partial_{x}^{α} (K_{i, j}^{ε} - {\bar{K}}_{i, j}) \partial_{x_{j}} u^{ε}), \\ {col}_{i} B_{2, 2}^{α} = \sum_{j = 1}^{3} \partial_{x}^{α} ({\bar{K}}_{i, j} \partial_{x_{j}} (u^{ε} - \bar{u})) . \end{array}$ Then $\begin{matrix} B_{2}^{α} - {\bar{B}}_{2}^{α} = B_{2, 1}^{α} + B_{2, 2}^{α} . \end{matrix}$
Lemma 3.5.
For all $| α | ⩽ s$ and a certain $1 ⩽ k ⩽ n - r$ , it holds $\begin{matrix} (3.25) & | \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{2, 1}^{α} ⟩ d t | ⩽ C δ \int_{0}^{T} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s}^{2}) d τ + C ε^{2 ζ_{1}}, \end{matrix}$ where $ζ_{1} > 0$ and $δ > 0$ are defined in Theorem 2.1 .
Proof.
Recall that $\begin{matrix} {row}_{k} B_{2, 1}^{α} = \sum_{j = 1}^{3} \partial_{x}^{α} ({row}_{k} (K_{i, j}^{ε} - {\bar{K}}_{i, j}) \partial_{x_{j}} u^{ε}) . \end{matrix}$ Using the Taylor’s formula, we have $\begin{array}{rcl} {row}_{k} (K_{i, j}^{ε} - {\bar{K}}_{i, j}) & = & \int_{0}^{1} [\partial_{u} {row}_{k} K_{i, j} ({\tilde{U}}^{ε} (θ)) (u^{ε} - \bar{u}) + \partial_{v} {row}_{k} K_{i, j} ({\tilde{U}}^{ε} (θ)) (ε v^{ε})] d θ, \end{array}$ in which $\begin{matrix} {\tilde{U}}^{ε} (θ) = (\bar{u} + θ (u^{ε} - \bar{u}), θ ε v^{ε}) . \end{matrix}$ From (2.2) and (2.8), it is clear that ${\tilde{U}}^{ε} (θ) - U_{e}$ is sufficiently small in $L^{\infty} (R^{+} \times T^{3})$ , uniformly with respect to $ε \in (0, 1]$ and $θ \in [0, 1]$ . Besides, noticing (3.11), we obtain that $M (u^{ε} - \bar{u})$ is independent of t. As a result, using (2.13), $\begin{matrix} {‖ M (u^{ε} - \bar{u}) ‖}_{s} = ‖ M (u^{ε} - \bar{u}) ‖ = ‖ M (u_{0}^{ε} - {\bar{u}}_{0}) ‖ ⩽ C {‖ u_{0}^{ε} - {\bar{u}}_{0} ‖}_{s - 1} ⩽ C ε^{ζ} . \end{matrix}$ Hence, $\begin{array}{rcl} {‖ {row}_{k} (K_{i, j}^{ε} - {\bar{K}}_{i, j}) ‖}_{s} & ⩽ & C {‖ u^{ε} (t) - \bar{u} (t) ‖}_{s} + C ε {‖ v^{ε} (t) ‖}_{s} \\ ⩽ & C \sum_{k = 1}^{n - r} {‖ {(u^{ε})}_{k} (t) - {(\bar{u})}_{k} (t) ‖}_{s} + C ε {‖ v^{ε} (t) ‖}_{s} \\ ⩽ & C \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (t) ‖}_{s} + C {‖ M (u^{ε} - \bar{u}) ‖}_{s} + C ε {‖ v^{ε} (t) ‖}_{s} \\ (3.26) & ⩽ & C \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (t) ‖}_{s} + C ε^{ζ} + C ε {‖ v^{ε} (t) ‖}_{s} . \end{array}$ Then, using (2.2), (3.4), the Poincaré inequality and the Young’s inequality, we have when $α = 0$ , $\begin{array}{rcl} | \int_{0}^{T} ⟨ ϕ_{k}^{ε}, {row}_{k} B_{2, 1}^{0} ⟩ d t | & ⩽ & C \int_{0}^{T} {‖ {row}_{k} (K_{i, j}^{ε} - {\bar{K}}_{i, j}) ‖}_{s} ‖ ϕ_{k}^{ε} ‖ ‖ \nabla u^{ε} ‖ d t \\ ⩽ & C δ \int_{0}^{T} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s}^{2}) d τ + C ε^{2 ζ_{1}} . \end{array}$ When $1 ⩽ | α | ⩽ s$ , by taking the integration by parts, we have $\begin{array}{rcl} | \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{2, 1}^{α} ⟩ d t | & ⩽ & C \int_{0}^{T} {‖ {row}_{k} (K_{i, j}^{ε} - {\bar{K}}_{i, j}) ‖}_{s - 1} ‖ \nabla ϕ_{k}^{α} ‖ {‖ \nabla u^{ε} ‖}_{s - 1} d t \\ ⩽ & C δ \int_{0}^{T} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s}^{2}) d τ + C ε^{2 ζ_{1}}, \end{array}$ which ends the proof. □

Substituting (3.25) into (3.24) and summing up for all $1 ⩽ k ⩽ n - r$ , we have $\begin{array}{l} \frac{1}{4} \sum_{k = 1}^{n - r} {‖ ϕ_{k}^{α} (T) ‖}^{2} - \sum_{k = 1}^{n - r} \int_{0}^{T} ⟨ ϕ_{k}^{α}, {row}_{k} B_{2, 2}^{α} ⟩ d t \\ (3.27) & ⩽ C δ \int_{0}^{T} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s}^{2}) d τ + C ε^{2 ζ_{1}} + C δ ϕ_{T}^{2} . \end{array}$

It remains to estimate the second term on the left hand side of the above inequality. We have Lemma 3.6.
There exists a constant $c_{1} > 0$ , such that for all $| α | ⩽ s$ , it holds $\begin{matrix} (3.28) & \int_{0}^{T} \sum_{k = 1}^{n - r} ⟨ ϕ_{k}^{α}, {row}_{k} B_{2, 2}^{α} ⟩ d t ⩾ (c_{1} - C δ) \int_{0}^{T} \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ - C δ ϕ_{T}^{2}, \end{matrix}$ where $δ > 0$ is defined in Theorem 2.1 .
Proof.
We first take the inner product of two matrices in a different way. If we denote a matrix $ϕ^{ε} \in R^{(n - r) \times 3}$ such that $\begin{matrix} {row}_{k} ϕ^{ε} = ϕ_{k}^{ε}, 1 ⩽ k ⩽ n - r . \end{matrix}$ Consequently, it is easy to see that $\begin{matrix} \sum_{k = 1}^{n - r} ⟨ ϕ_{k}^{α}, {row}_{k} B_{2, 2}^{α} ⟩ = {(\partial_{x}^{α} ({col}_{1} ϕ^{ε}, {col}_{2} ϕ^{ε}, {col}_{3} ϕ^{ε}))}^{⊤} \cdot ({col}_{1} B_{2, 2}^{α}, {col}_{2} B_{2, 2}^{α}, {col}_{3} B_{2, 2}^{α}) . \end{matrix}$ Further noticing the expression of $B_{2, 2}^{α}$ , we can rewrite the above into a much simplier form, which is as follows $\begin{array}{l} \sum_{k = 1}^{n - r} ⟨ ϕ_{k}^{α}, {row}_{k} B_{2, 2}^{α} ⟩ \\ = {(\partial_{x}^{α} ({col}_{1} ϕ^{ε}, {col}_{2} ϕ^{ε}, {col}_{3} ϕ^{ε}))}^{⊤} \partial_{x}^{α} (\bar{K} (\partial_{x_{1}} (u^{ε} - \bar{u}), \partial_{x_{2}} (u^{ε} - \bar{u}), \partial_{x_{3}} (u^{ε} - \bar{u}))) \\ (3.29) & = \sum_{i, j = 1}^{3} {(\partial_{x}^{α} {col}_{i} ϕ^{ε})}^{⊤} \partial_{x}^{α} ({\bar{K}}_{i, j} \partial_{x_{j}} (u^{ε} - \bar{u})), \end{array}$ where the partitioned matrix $\bar{K}$ is defined as $\begin{matrix} \bar{K} \overset{def}{=} [\begin{matrix} {\bar{K}}_{1, 1} & {\bar{K}}_{1, 2} & {\bar{K}}_{1, 3} \\ {\bar{K}}_{2, 1} & {\bar{K}}_{2, 2} & {\bar{K}}_{2, 3} \\ {\bar{K}}_{3, 1} & {\bar{K}}_{3, 2} & {\bar{K}}_{3, 3} \end{matrix}] . \end{matrix}$ Under the condition (H1), there exists a matrix K such that $\bar{K} = K I_{3}$ , where $I_{3}$ is the $3 \times 3$ unit matrix. By taking the integration by parts, we have $\begin{array}{l} \sum_{i, j = 1}^{3} {(\partial_{x}^{α} ({col}_{i} ϕ^{ε}))}^{⊤} \partial_{x}^{α} ({\bar{K}}_{i, j} \partial_{x_{j}} (u^{ε} - \bar{u})) \\ = \sum_{i = 1}^{3} {(\partial_{x}^{α} ({col}_{i} ϕ^{ε}))}^{⊤} \partial_{x}^{α} (K (\bar{u}, 0) \partial_{x_{i}} (u^{ε} - \bar{u})) \\ = - (\sum_{i = 1}^{3} {(\partial_{x}^{α} \partial_{x_{i}} ({col}_{i} ϕ^{ε}))}^{⊤}) \partial_{x}^{α} (K (\bar{u}, 0) (u^{ε} - \bar{u} - M (u^{ε} - \bar{u}))) \\ - \sum_{i = 1}^{3} {(\partial_{x}^{α} ({col}_{i} ϕ^{ε}))}^{⊤} \partial_{x}^{α} (\partial_{x_{i}} K (\bar{u}, 0) (u^{ε} - \bar{u} - M (u^{ε} - \bar{u}))) \\ = - {(\partial_{x}^{α} (div ϕ^{ε}))}^{⊤} K (\bar{u}, 0) \partial_{x}^{α} (div ϕ^{ε}) \\ - {(\partial_{x}^{α} (div ϕ^{ε}))}^{⊤} (\partial_{x}^{α} (K (\bar{u}, 0) div ϕ^{ε}) - K (\bar{u}, 0) \partial_{x}^{α} (div ϕ^{ε})) \\ (3.30) & - \sum_{i = 1}^{3} {(\partial_{x}^{α} ({col}_{i} ϕ^{ε}))}^{⊤} \partial_{x}^{α} (\partial_{x_{i}} K (\bar{u}, 0) div ϕ^{ε}) . \end{array}$

We will estimate the right hand side of the above equality term by term. First, since $K (\bar{u}, 0)$ is positive definite, which means there exists a constant $c_{1} > 0$ such that $\begin{array}{l} \int_{0}^{T} {(\partial_{x}^{α} (div ϕ^{ε}))}^{⊤} K (\bar{u}, 0) \partial_{x}^{α} (div ϕ^{ε}) d t & ⩾ 2 c_{1} \int_{0}^{T} {‖ \partial_{x}^{α} div ϕ^{ε} (τ) ‖}^{2} d τ \\ = 2 c_{1} \int_{0}^{T} \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ . \end{array}$ For the rest of the terms, we have the following estimates. First, by the Moser-type inequalities, $\begin{array}{l} | \int_{0}^{T} {(\partial_{x}^{α} (div ϕ^{ε}))}^{⊤} (\partial_{x}^{α} (K (\bar{u}, 0) div ϕ^{ε}) - K (\bar{u}, 0) \partial_{x}^{α} (div ϕ^{ε})) d t | \\ ⩽ C \int_{0}^{T} ‖ \partial_{x}^{α} (div ϕ^{ε}) ‖ (\sum_{i = 1}^{3} {‖ \partial_{x_{i}} (K (\bar{u}, 0)) ‖}_{s - 1}) {‖ div ϕ^{ε} ‖}_{s - 1} d t \\ ⩽ C δ \int_{0}^{T} \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ, \end{array}$ and $\begin{array}{l} | \int_{0}^{T} \sum_{i = 1}^{3} {(\partial_{x}^{α} ({col}_{i} ϕ^{ε}))}^{⊤} \partial_{x}^{α} (\partial_{x_{i}} K (\bar{u}, 0) div ϕ^{ε}) d t | \\ ⩽ C \int_{0}^{T} ‖ \partial_{x}^{α} ϕ^{ε} ‖ (\sum_{i = 1}^{3} {‖ \partial_{x_{i}} K (\bar{u}, 0) ‖}_{s}) {‖ div ϕ^{ε} ‖}_{s} d t \\ ⩽ C ϕ_{T} \int_{0}^{T} ‖ \nabla \bar{u} ‖_{s} (\sum_{k = 1}^{n - r} ‖ div ϕ_{k}^{α} ‖) d t \\ ⩽ C δ ϕ_{T}^{2} + c_{1} \int_{0}^{T} \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{α} (τ) ‖}^{2} d τ . \end{array}$ Substituting these estimates into (3.30) and combining (3.29) yield (3.28). □
Proof of Lemma 3.2.
Substituting (3.28) into (3.27), we have for all $| α | ⩽ s$ , $\begin{array}{l} \frac{1}{4} \sum_{k = 1}^{n - r} {‖ ϕ_{k}^{α} (T) ‖}^{2} + c_{1} \int_{0}^{T} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{α} (τ) ‖}^{2}) d τ \\ (3.31) & ⩽ C δ \int_{0}^{T} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s}^{2}) d τ + C ε^{2 ζ_{1}} + C δ ϕ_{T}^{2} . \end{array}$ Summing up the above for all $| α | ⩽ s$ , we have $\begin{array}{l} \sum_{k = 1}^{n - r} {‖ ϕ_{k}^{ε} (T) ‖}_{s}^{2} + \int_{0}^{T} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s}^{2}) d τ ⩽ C ε^{2 ζ_{1}} + C δ ϕ_{T}^{2}, \end{array}$ provided that δ is sufficiently small. Taking the superior limit with respect to T yields $\begin{matrix} \sum_{k = 1}^{n - r} {‖ ϕ_{k}^{ε} (t) ‖}_{s}^{2} + \int_{0}^{t} (\sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s}^{2}) d τ ⩽ C ε^{2 ζ_{1}}, \forall t > 0, \end{matrix}$ provided that δ is sufficiently small. Finally, using (3.1), (3.3) and the Moser-type inequalities, we obtain $\begin{array}{rcl} \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} ‖}_{s}^{2} & = & {‖ u^{ε} - \bar{u} - M (u^{ε} - \bar{u}) ‖}_{s}^{2}, \end{array}$ and as a result, for $\forall t > 0$ , $\begin{array}{rcl} {‖ u^{ε} (t) - \bar{u} (t) ‖}_{s - 1}^{2} & ⩽ & C \sum_{k = 1}^{n - r} {‖ div ϕ_{k}^{ε} (τ) ‖}_{s - 1}^{2} + C {‖ M (u^{ε} - \bar{u}) ‖}_{s}^{2} \\ ⩽ & \sum_{k = 1}^{n - r} {‖ ϕ_{k}^{ε} (t) ‖}_{s}^{2} + C ε^{2 ζ_{1}} \\ ⩽ & C ε^{2 ζ_{1}}, \end{array}$ which imply (3.13). □

3.3. Convergence rate for $v^{ε}$

In this section, we obtain the convergence rate of $v^{ε}$ . For convenience, we introduce $\begin{matrix} w^{ε} = v^{ε} - \bar{v} . \end{matrix}$ From (2.4) and (2.5), $w^{ε}$ satisfies $\begin{matrix} (3.32) & ε^{2} \partial_{t} w^{ε} + q^{1} (\bar{u}, 0) w^{ε} = R^{ε}, \end{matrix}$ where, by noting $v^{ε} = w^{ε} + \bar{v}$ , $\begin{matrix} R^{ε} = R_{1}^{ε} - [q^{1} (u^{ε}, ε v^{ε}) - q^{1} (\bar{u}, 0)] w^{ε}, \end{matrix}$ with $\begin{array}{rcl} - R_{1}^{ε} & = & \sum_{j = 1}^{3} (A_{j}^{21} (u^{ε}, ε v^{ε}) \partial_{x_{j}} u^{ε} - A_{j}^{21} (\bar{u}, 0) \partial_{x_{j}} \bar{u} + A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (ε v^{ε})) \\ + [q^{1} (u^{ε}, ε v^{ε}) - q^{1} (\bar{u}, 0)] \bar{v} + ε^{2} \partial_{t} \bar{v} \\ = & \sum_{j = 1}^{3} ((A_{j}^{21} (u^{ε}, ε v^{ε}) - A_{j}^{21} (\bar{u}, 0)) \partial_{x_{j}} \bar{u}) + [q^{1} (u^{ε}, ε v^{ε}) - q^{1} (\bar{u}, 0)] \bar{v} + ε^{2} \partial_{t} \bar{v} \\ (3.33) & + \sum_{j = 1}^{3} (A_{j}^{21} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (u^{ε} - \bar{u}) + A_{j}^{22} (u^{ε}, ε v^{ε}) \partial_{x_{j}} (ε v^{ε})) . \end{array}$

Recall that $A_{0} (\bar{u}, 0)$ is a symmetric positive definite matrix and $A_{0}^{12} (\bar{u}, 0) = 0$ . Hence, $A_{0}^{22} (\bar{u}, 0)$ is also a symmetric positive definite matrix. It follows that there is a constant $c_{2} > 0$ such that $\begin{matrix} (3.34) & c_{2} ‖ w ‖^{2} ⩽ ⟨ A_{0}^{22} (\bar{u}, 0) w, w ⟩ ⩽ C ‖ w ‖^{2}, \forall w \in R^{r} . \end{matrix}$ Similarly, (2.1) implies that $\begin{matrix} (3.35) & ⟨ A_{0}^{22} (\bar{u}, 0) q^{1} (\bar{u}, 0) w, w ⟩ ⩾ c_{0} ‖ w ‖^{2}, \forall w \in R^{r} . \end{matrix}$

Lemma 3.7.
The solution $w^{ε}$ to ( 3.32 ) satisfies $\begin{array}{rcl} (3.36) & ε^{2} {‖ w^{ε} (t) ‖}_{s - 2}^{2} + \int_{0}^{t} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ ⩽ C ε^{2 ζ_{1}}, \forall t > 0, \end{array}$ where $ζ_{1} > 0$ is defined in Theorem 2.1 .
Proof.
Let $T > 0$ . For a multi-index $γ \in N^{3}$ with $| γ | = s - 2$ , applying $\partial_{x}^{γ}$ to both sides of (3.32), taking the inner product of the resulting equation with $A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}$ in $L^{2}$ and integrating this equality over $[0, T]$ , since $A_{0}^{22} (\bar{u}, 0)$ is symmetric, we have $\begin{array}{rcl} ε^{2} \int_{0}^{T} \frac{d}{d t} ⟨ A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}, \partial_{x}^{γ} w^{ε} ⟩ d t & = & - 2 \int_{0}^{T} ⟨ A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} (q^{1} (\bar{u}, 0) w^{ε}), \partial_{x}^{γ} w^{ε} ⟩ d t \\ + 2 \int_{0}^{T} ⟨ A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} R^{ε}, \partial_{x}^{γ} w^{ε} ⟩ d t \\ + ε^{2} \int_{0}^{T} ⟨ \partial_{t} A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}, \partial_{x}^{γ} w^{ε} ⟩ d t \\ (3.37) & \overset{def}{=} & - V_{1} + V_{2} + V_{3}, \end{array}$ with the natural correspondence of $V_{1}$ , $V_{2}$ and $V_{3}$ , which are treated term by term as follows. First, $\begin{array}{rcl} V_{1} & = & 2 \int_{0}^{T} ⟨ A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} (q^{1} (\bar{u}, 0) w^{ε}), \partial_{x}^{γ} w^{ε} ⟩ d t \\ = & \int_{0}^{T} ⟨ A_{0}^{22} (\bar{u}, 0) q^{1} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}, \partial_{x}^{γ} w^{ε} ⟩ d t \\ + \int_{0}^{T} ⟨ A_{0}^{22} (\bar{u}, 0) (\partial_{x}^{γ} (q^{1} (\bar{u}, 0) w^{ε}) - q^{1} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}), \partial_{x}^{γ} w^{ε} ⟩ d t, \end{array}$ where using (3.35), $\begin{matrix} \int_{0}^{T} ⟨ A_{0}^{22} (\bar{u}, 0) q^{1} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}, \partial_{x}^{γ} w^{ε} ⟩ d t ⩾ c_{0} \int_{0}^{T} {‖ \partial_{x}^{γ} w^{ε} (τ) ‖}^{2} d τ, \end{matrix}$ and using the Moser-type inequalities and the Cauchy–Schwarz inequality, we have $\begin{array}{l} | \int_{0}^{T} ⟨ A_{0}^{22} (\bar{u}, 0) (\partial_{x}^{γ} (q^{1} (\bar{u}, 0) w^{ε}) - q^{1} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}), \partial_{x}^{γ} w^{ε} ⟩ d t | \\ ⩽ C \int_{0}^{T} ‖ \nabla \bar{u} ‖_{s - 1} {‖ w^{ε} ‖}_{s - 2}^{2} d t \\ ⩽ C δ \int_{0}^{T} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ . \end{array}$ Hence, $\begin{matrix} (3.38) & V_{1} ⩾ c_{0} \int_{0}^{T} {‖ \partial_{x}^{γ} w^{ε} (τ) ‖}^{2} d τ - C δ \int_{0}^{T} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ . \end{matrix}$ For $V_{2}$ , we need to establish the estimates for $R^{ε}$ . In view of the expression of $R_{1}^{ε}$ in (3.33), similar to the treatment to $K_{i, j}^{ε} - {\bar{K}}_{i, j}$ in (3.26), we have for $1 ⩽ j ⩽ 3$ , $\begin{array}{l} {‖ A_{j}^{21} (u^{ε}, ε v^{ε}) - A_{j}^{21} (\bar{u}, 0) ‖}_{\infty} + {‖ q^{1} (u^{ε}, ε v^{ε}) - q^{1} (\bar{u}, 0) ‖}_{\infty} \\ ⩽ C {‖ u^{ε} (t) - \bar{u} (t) ‖}_{s - 1} + C ε {‖ v^{ε} (t) ‖}_{s - 1} \\ ⩽ C {‖ u^{ε} (t) - \bar{u} (t) - M (u^{ε} (t) - \bar{u} (t)) ‖}_{s - 1} + C ε {‖ v^{ε} (t) ‖}_{s - 1} + {‖ M (u^{ε} (t) - \bar{u} (t)) ‖}_{s - 1} \\ ⩽ C {‖ u^{ε} (t) - \bar{u} (t) - M (u^{ε} (t) - \bar{u} (t)) ‖}_{s - 1} + C ε {‖ v^{ε} (t) ‖}_{s - 1} + C ε^{ζ}, \forall t > 0 . \end{array}$ As a result, by (2.8) and (2.9), $\begin{array}{l} {‖ R_{1}^{ε} ‖}_{s - 2} \\ ⩽ \sum_{j = 1}^{3} ({‖ A_{j}^{21} (u^{ε}, ε v^{ε}) - A_{j}^{21} (\bar{u}, 0) ‖}_{\infty} ‖ \partial_{x_{j}} \bar{u} ‖_{s - 2} + {‖ A_{j}^{21} (u^{ε}, ε v^{ε}) ‖}_{\infty} {‖ \partial_{x_{j}} (u^{ε} - \bar{u}) ‖}_{s - 2}) \\ + \sum_{j = 1}^{3} (ε {‖ A_{j}^{22} (u^{ε}, ε v^{ε}) ‖}_{\infty} {‖ \partial_{x_{j}} v^{ε} ‖}_{s - 2}) + {‖ q^{1} (u^{ε}, ε v^{ε}) - q^{1} (\bar{u}, 0) ‖}_{\infty} ‖ \bar{v} ‖_{s - 2} + ε^{2} ‖ \partial_{t} \bar{v} ‖_{s - 2} \\ ⩽ C ({‖ u^{ε} (t) - \bar{u} (t) - M (u^{ε} (t) - \bar{u} (t)) ‖}_{s - 1} + ε {‖ v^{ε} (t) ‖}_{s - 1}) \\ + C ε^{ζ_{1}} (‖ \nabla \bar{u} ‖_{s - 2} + ‖ \bar{v} ‖_{s - 2} + ‖ \partial_{t} \bar{v} ‖_{s - 2}) . \end{array}$ Consequently, using (2.2), (2.9) and (3.13) obtained in the last section, we have $\begin{matrix} \int_{0}^{T} {‖ R_{1}^{ε} (τ) ‖}_{s - 2}^{2} d τ ⩽ C ε^{2 ζ_{1}} . \end{matrix}$ Moreover, $\begin{array}{l} \int_{0}^{T} {‖ R^{ε} (τ) ‖}_{s - 2}^{2} d τ \\ ⩽ \int_{0}^{T} {‖ R_{1}^{ε} (τ) ‖}_{s - 1}^{2} d τ + \int_{0}^{T} {‖ q^{1} (u^{ε}, ε v^{ε}) - q^{1} (\bar{u}, 0) ‖}_{\infty}^{2} {‖ w^{ε} ‖}_{s - 2}^{2} d t \\ ⩽ C ε^{2 ζ_{1}} + C \int_{0}^{T} {({‖ u^{ε} (t) - \bar{u} (t) - M (u^{ε} (t) - \bar{u} (t)) ‖}_{s - 1} + C ε {‖ v^{ε} (t) ‖}_{s - 1} + C ε^{ζ})}^{2} {‖ w^{ε} ‖}_{s - 2}^{2} d t \\ ⩽ C ε^{2 ζ_{1}} + C δ \int_{0}^{T} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ . \end{array}$ Hence, by the Young’s inequality, $\begin{array}{rcl} | V_{2} | & ⩽ & \frac{c_{0}}{3} \int_{0}^{T} {‖ \partial_{x}^{γ} w^{ε} (τ) ‖}^{2} d τ + C \int_{0}^{T} {‖ R^{ε} (τ) ‖}_{s - 2}^{2} d τ \\ (3.39) & ⩽ & \frac{c_{0}}{3} \int_{0}^{T} {‖ \partial_{x}^{γ} w^{ε} (τ) ‖}^{2} d τ + C ε^{2 ζ_{1}} + C δ \int_{0}^{T} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ . \end{array}$ For $V_{3}$ , from (2.8), the limiting equation (2.6) and the weak convergence of $u_{0}^{ε}$ to ${\bar{u}}_{0}$ in $H^{k}$ , we have $\begin{matrix} {‖ \partial_{t} A_{0}^{22} (\bar{u}, 0) ‖}_{\infty} ⩽ C ‖ \nabla \bar{u} ‖_{s} . \end{matrix}$ It follows that, for all $ε \in (0, 1]$ , $\begin{array}{rcl} | V_{3} | & ⩽ & C ε \int_{0}^{T} {‖ \partial_{t} A_{0}^{22} (\bar{u}, 0) ‖}_{\infty} {‖ \partial_{x}^{γ} w^{ε} ‖}^{2} d t \\ ⩽ & \frac{c_{0}}{3} \int_{0}^{T} {‖ \partial_{x}^{γ} w^{ε} (τ) ‖}^{2} d τ + C ε^{2} \int_{0}^{T} {‖ \nabla \bar{u} (τ) ‖}_{s}^{2} d τ \\ (3.40) & ⩽ & \frac{c_{0}}{3} \int_{0}^{T} {‖ \partial_{x}^{γ} w^{ε} (τ) ‖}^{2} d τ + C ε^{2} . \end{array}$ Combining (3.37) and the estimates (3.38)–(3.40), we have $\begin{array}{rcl} ε^{2} \int_{0}^{T} \frac{d}{d t} ⟨ A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}, \partial_{x}^{γ} w^{ε} ⟩ d t + \frac{c_{0}}{3} \int_{0}^{T} {‖ \partial_{x}^{γ} w^{ε} (τ) ‖}^{2} d τ ⩽ C δ \int_{0}^{T} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ + C ε^{2 ζ_{1}} . \end{array}$ Summing up for all $| γ | ⩽ s - 2$ , we have $\begin{array}{rcl} ε^{2} \sum_{| γ | ⩽ s - 2} \int_{0}^{T} \frac{d}{d t} ⟨ A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}, \partial_{x}^{γ} w^{ε} ⟩ d t + \int_{0}^{T} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ ⩽ C ε^{2 ζ_{1}}, \end{array}$ provided that δ is sufficiently small. We notice that (3.34) implies $⟨ A_{0}^{22} (\bar{u}, 0) \partial_{x}^{γ} w^{ε}, \partial_{x}^{γ} w^{ε} ⟩$ are equivalent to $‖ \partial_{x}^{γ} w^{ε} ‖^{2}$ . Consequently, noticing (2.12), we have $\begin{matrix} {‖ ε w^{ε} (T) ‖}_{s - 2}^{2} + \int_{0}^{T} {‖ w^{ε} (τ) ‖}_{s - 2}^{2} d τ ⩽ C ε^{2 ζ_{1}}, \end{matrix}$ which implies (3.36) since $T > 0$ is arbitrary. □

4. Convergence rate in one-dimensional torus

In this section, we consider the problem over one-dimensional torus. We let $d = 1$ and integer $s ⩾ 2$ . When $d = 1$ , (2.4) becomes $\begin{matrix} (4.1) & \{\begin{matrix} \partial_{t} u^{ε} + \partial_{x} (f^{1} (u^{ε}, ε v^{ε}) v^{ε}) = 0, \\ ε^{2} \partial_{t} v^{ε} + A^{21} (u^{ε}, ε v^{ε}) \partial_{x} u^{ε} + A^{22} (u^{ε}, ε v^{ε}) \partial_{x} (ε v^{ε}) = - q^{1} (u^{ε}, ε v^{ε}) v^{ε}, \\ u^{ε} (0, x) = u_{0}^{ε} (x), v^{ε} (0, x) = \frac{v_{0}^{ε} (x)}{ε}, \end{matrix} \end{matrix}$ with its limiting system $\begin{matrix} (4.2) & \{\begin{matrix} \partial_{t} \bar{u} - \partial_{x} (f^{1} (\bar{u}, 0) {(q^{1} (\bar{u}, 0))}^{- 1} A^{21} (\bar{u}, 0) \partial_{x} \bar{u}) = 0, \\ \bar{u} (0, x) = {\bar{u}}_{0} (x), \end{matrix} \end{matrix}$ and $\begin{matrix} \bar{v} = - q^{1} {(\bar{u}, 0)}^{- 1} A^{21} (\bar{u}, 0) \partial_{x} \bar{u} . \end{matrix}$

For convenience, in accordance with the notations in previous sections, we denote $\begin{matrix} D (u, v) = f^{1} (\bar{u}, 0) {(q^{1} (\bar{u}, 0))}^{- 1}, K (u, v) = f^{1} (\bar{u}, 0) {(q^{1} (\bar{u}, 0))}^{- 1} A^{21} (\bar{u}, 0) . \end{matrix}$ We first construct a stream function over a one-dimensional torus.

Lemma 4.1 (Existence of stream function over one-dimensional torus).

There exists a stream function $ϕ^{ε}$ such that $\begin{matrix} (4.3) & \{\begin{matrix} \partial_{x} ϕ^{ε} = u^{ε} - \bar{u} - M (u^{ε} - \bar{u}), \\ \partial_{t} ϕ^{ε} = - (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) + \frac{1}{2 π} M (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}), \\ M (ϕ^{ε}) = 0 . \end{matrix} \end{matrix}$

Proof.
Subtacting (4.2) from the first equation in (4.1), we have $\begin{matrix} (4.4) & \partial_{t} (u^{ε} - \bar{u}) + \partial_{x} (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) = 0 . \end{matrix}$ Let $ϕ^{ε}$ be the unique solution to the following $\begin{matrix} (4.5) & \{\begin{matrix} \partial_{x} ϕ^{ε} = u^{ε} - \bar{u} - M (u^{ε} - \bar{u}), \\ M (ϕ^{ε}) = 0 . \end{matrix} \end{matrix}$ By (4.4), it is clear that $\begin{matrix} (4.6) & \partial_{t} M (u^{ε} - \bar{u}) = - \int_{T} \partial_{x} (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) d x = 0 . \end{matrix}$ Besides, taking the time derivative to both sides of the first equation in (4.5), we have $\begin{matrix} \partial_{t} \partial_{x} ϕ^{ε} = \partial_{t} (u^{ε} - \bar{u}) = - \partial_{x} (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}), \end{matrix}$ which implies that there exists a function $h (t)$ , such that $\begin{matrix} \partial_{t} ϕ^{ε} + (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) = \frac{1}{2 π} h (t) . \end{matrix}$ Noticing that $M (\partial_{t} ϕ^{ε}) = 0$ , taking the average of the above equation over the torus $T$ gives $\begin{matrix} h (t) = \int_{T} f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u} d x, \end{matrix}$ which implies (4.3). Remark 4.1.
We can define the initial data of $ϕ^{ε}$ similarly to that in Remark 3.1.

We now turn to the global convergence rate of $u^{ε} - \bar{u}$ . Noticing when $d = 1$ , the condition (H1) automatically holds. Let $0 ⩽ l ⩽ s$ be integers. Multiplying D and then applying $\partial_{x}^{l}$ to the second equation in (4.1), we have $\begin{matrix} ε^{2} \partial_{x}^{l} (D \partial_{t} v^{ε}) + \partial_{x}^{l} (K (u^{ε}, ε v^{ε}) \partial_{x} u^{ε}) + \partial_{x}^{l} (D A^{22} (u^{ε}, ε v^{ε}) \partial_{x} (ε v^{ε})) = - \partial_{x}^{l} (f^{1} (u^{ε}, ε v^{ε}) v^{ε}) . \end{matrix}$ Taking the inner product of this equality with $\partial_{x}^{l} ϕ^{ε}$ in $L^{2} (T)$ and integrating over $[0, T]$ , we get $\begin{array}{l} 0 & = \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, ε^{2} \partial_{x}^{l} (D \partial_{t} v^{ε}) ⟩ d t + \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (K (u^{ε}, ε v^{ε}) \partial_{x} u^{ε}) ⟩ d t \\ + \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (D A^{22} (u^{ε}, ε v^{ε}) \partial_{x} (ε v^{ε})) ⟩ d t + \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (f^{1} (u^{ε}, ε v^{ε}) v^{ε}) ⟩ d t \\ (4.7) & \overset{def}{=} I_{1} + I_{2} + I_{3} + I_{4}, \end{array}$ with the natural correspondence of $I_{1}$ , $I_{2}$ , $I_{3}$ and $I_{4}$ , which are treated term by term in the way similar to that in the previous section. First, taking the integration by parts with respect to time for $I_{1}$ , we have $\begin{array}{rcl} I_{1} & = & ε^{2} \int_{0}^{T} \frac{d}{d t} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (D (u^{ε}, ε v^{ε}) v^{ε}) ⟩ d t - ε^{2} \int_{0}^{T} ⟨ \partial_{x}^{l} \partial_{t} ϕ^{ε}, \partial_{x}^{l} (D (u^{ε}, ε v^{ε}) v^{ε}) ⟩ d t \\ - ε^{2} \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (\partial_{t} D (u^{ε}, ε v^{ε}) v^{ε}) ⟩ d t . \end{array}$ Similar to (3.17), denoting $ϕ_{T} = {sup}_{t \in [0, T]} ‖ ϕ^{ε} (t) ‖_{s}$ , we have $\begin{array}{l} | ε^{2} \int_{0}^{T} \frac{d}{d t} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (D (u^{ε}, ε v^{ε}) v^{ε}) ⟩ d t | ⩽ \frac{1}{4} {‖ \partial_{x}^{l} ϕ^{ε} ‖}^{2} + C ε^{2 ζ_{1}}, \\ | ε^{2} \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (\partial_{t} D (u^{ε}, ε v^{ε}) v^{ε}) ⟩ d t | ⩽ C δ ϕ_{T}^{2} + C ε^{2} + C δ \int_{0}^{T} {‖ \partial_{x}^{l + 1} ϕ^{ε} (τ) ‖}^{2} d τ . \end{array}$ Using (2.2), (2.8) and (3.1), we have $\begin{array}{rcl} {‖ \partial_{t} ϕ^{ε} ‖}_{s} & ⩽ & C {‖ (f_{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) ‖}_{s} + C {‖ M (f_{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) ‖}_{s} \\ ⩽ & C ({‖ v^{ε} ‖}_{s} + ‖ \partial_{x} \bar{u} ‖_{s}), \end{array}$ then by the Moser-type inequalities, $\begin{array}{l} | ε^{2} \int_{0}^{T} ⟨ \partial_{x}^{l} \partial_{t} ϕ^{ε}, \partial_{x}^{l} (D (u^{ε}, ε v^{ε}) v^{ε}) ⟩ d t | \\ ⩽ C ε^{2} \int_{0}^{T} {‖ \partial_{t} ϕ^{ε} ‖}_{s} ‖ \partial_{x}^{l} (D (u^{ε}, ε v^{ε}) v^{ε}) ‖ d t \\ ⩽ C ε^{2} \int_{0}^{T} C ({‖ v^{ε} ‖}_{s} + ‖ \partial_{x} \bar{u} ‖_{s}) ‖ \partial_{x}^{l} (D (u^{ε}, ε v^{ε}) v^{ε}) ‖ d t \\ ⩽ C ε^{2} . \end{array}$ These estimates imply $\begin{matrix} | I_{1} | ⩽ \frac{1}{4} {‖ \partial_{x}^{l} ϕ^{ε} ‖}^{2} + C δ ϕ_{T}^{2} + C ε^{2 ζ_{1}} + C δ \int_{0}^{T} {‖ \partial_{x}^{l + 1} ϕ^{ε} (τ) ‖}^{2} d τ . \end{matrix}$ Similar to (3.21), we have $\begin{array}{rcl} | I_{3} | & ⩽ & C δ \int_{0}^{T} {‖ \partial_{x}^{l + 1} ϕ^{ε} (τ) ‖}^{2} d τ + C ε^{2} . \end{array}$ For $I_{4}$ , using (3.23) and the explicit expression of $\partial_{t} ϕ^{ε}$ in (4.3), we have $\begin{array}{rcl} I_{4} & = & \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, - \partial_{t} \partial_{x}^{l} ϕ^{ε} ⟩ d t - \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (K (\bar{u}, 0) \partial_{x} \bar{u}) ⟩ d t \\ + \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \frac{1}{2 π} M (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) ⟩ d t \\ = & - \frac{1}{2} {‖ \partial_{x}^{l} ϕ^{ε} (T) ‖}^{2} + \frac{1}{2} {‖ \partial_{x}^{l} ϕ^{ε} (0) ‖}^{2} - \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (K (\bar{u}, 0) \partial_{x} \bar{u}) ⟩ d t \\ + \int_{0}^{T} \frac{1}{2 π} M (f^{1} (u^{ε}, ε v^{ε}) v^{ε} + K (\bar{u}, 0) \partial_{x} \bar{u}) M (\partial_{x}^{l} ϕ^{ε}) d t \\ = & - \frac{1}{2} {‖ \partial_{x}^{l} ϕ^{ε} (T) ‖}^{2} + \frac{1}{2} {‖ \partial_{x}^{l} ϕ^{ε} (0) ‖}^{2} - \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (K (\bar{u}, 0) \partial_{x} \bar{u}) ⟩ d t . \end{array}$ Combining (4.7) and these estimates, we have $\begin{matrix} \frac{1}{4} {‖ \partial_{x}^{l} ϕ^{ε} (T) ‖}^{2} - \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (K (u^{ε}, ε v^{ε}) \partial_{x} u^{ε} - K (\bar{u}, 0) \partial_{x} \bar{u}) ⟩ d t \\ ⩽ C δ ϕ_{T}^{2} + C ε^{2 ζ_{1}} + C δ \int_{0}^{T} {‖ \partial_{x}^{l + 1} ϕ^{ε} (τ) ‖}^{2} d τ . \end{matrix}$ Similar to (3.25) and (3.28), we have $\begin{array}{l} - \int_{0}^{T} ⟨ \partial_{x}^{l} ϕ^{ε}, \partial_{x}^{l} (K (u^{ε}, ε v^{ε}) \partial_{x} u^{ε} - K (\bar{u}, 0) \partial_{x} \bar{u}) ⟩ d t \\ ⩾ (c_{1} - C δ) \int_{0}^{T} {‖ \partial_{x}^{l + 1} ϕ^{ε} (τ) ‖}^{2} d τ - C δ \int_{0}^{T} {‖ \partial_{x} ϕ^{ε} (τ) ‖}_{s}^{2} d τ - C ε^{2 ζ_{1}} - C δ ϕ_{T}^{2}, \end{array}$ which implies $\begin{matrix} \frac{1}{4} {‖ \partial_{x}^{l} ϕ^{ε} (T) ‖}^{2} + (c_{1} - C δ) \int_{0}^{T} {‖ \partial_{x}^{l + 1} ϕ^{ε} (τ) ‖}^{2} d τ ⩽ C δ \int_{0}^{T} {‖ \partial_{x} ϕ^{ε} (τ) ‖}_{s}^{2} d τ + C ε^{2 ζ_{1}} + C δ ϕ_{T}^{2} . \end{matrix}$ Following the same procedure as we did in (3.31), we have $\begin{matrix} {‖ u^{ε} (t) - \bar{u} (t) ‖}_{s - 1}^{2} + \int_{0}^{t} {‖ u^{ε} (τ) - \bar{u} (τ) - M (u^{ε} (τ) - \bar{u} (τ)) ‖}_{s}^{2} d t ⩽ C ε^{2 ζ_{1}}, \forall t > 0 . \end{matrix}$ The convergence rate of $v^{ε}$ is similar based on the above estimate. Hence, we have $\begin{array}{rcl} ε^{2} {‖ v^{ε} (t) - \bar{v} (t) ‖}_{s - 2}^{2} + \int_{0}^{t} {‖ v^{ε} (τ) - \bar{v} (τ) ‖}_{s - 2}^{2} d τ ⩽ C ε^{2 ζ_{1}}, \forall t > 0, \end{array}$ which ends the proof. □

5. Examples

In this section, we give several examples of systems which fulfill all conditions (A1)–(A5), (SK) and (H1). Then Theorem 2.1 can be applied to the periodic problems of all the examples in the section. In these examples, $t > 0$ is the slow time and the systems are expressed after scaling (1.6). We denote from now on $d = 1, 3$ and ${e_{j}}_{j = 1}^{3}$ be the canonical basis of $R^{3}$ . Example 5.2 in the isothermal case and Example 5.4 were studied in [9] and [4], respectively.

5.1. Examples in multi-dimensional spaces

Example 5.1 (Wave equation of heat conduction).

This concerns a linear equation which is written as (see [6,21] and references therein) $\begin{matrix} ε^{2} \partial_{t t}^{2} w - Δ w + \partial_{t} w = 0, t > 0, x \in T^{d} . \end{matrix}$ Let us introduce $\begin{matrix} u^{ε} = \nabla w, v^{ε} = - \partial_{t} w, \end{matrix}$ then the equation becomes $\begin{matrix} \{\begin{matrix} \partial_{t} u^{ε} + \nabla v^{ε} = 0, \\ ε^{2} \partial_{t} v^{ε} + div u^{ε} = - v^{ε}, \end{matrix} \end{matrix}$ which is of the form (1.7) with $\begin{matrix} f_{j} (u, v) = v e_{j}, q (u, v) = v, A_{j}^{21} = e_{j}, A_{j}^{22} = 0, K_{i, j} = δ_{i, j} . \end{matrix}$ Its corresponding parabolic limiting equation is $\begin{matrix} \partial_{t} \bar{u} - Δ \bar{u} = 0, \end{matrix}$ which is the classical heat equation. It is easily checked that conditions (A1)–(A5), (SK) and (H1) are satisfied. Then Theorem 2.1 can be applied.

Example 5.2 (Euler equations with nonlinear damping).

The model is written as (see [3,13,14,18,22,26]) $\begin{matrix} (5.1) & \{\begin{matrix} \partial_{t} ρ + div (ρ w) = 0, \\ ε^{2} (\partial_{t} (ρ w) + div (ρ w \otimes w)) + \nabla p (ρ) = - α (ρ, w) ρ w, x \in T^{d}, \end{matrix} \end{matrix}$ where $ε > 0$ is the relaxation time, $ρ > 0$ is the density and w is the velocity. The pressure function $p (ρ)$ is supposed to be smooth and strictly increasing. The damping coefficient $α (ρ, w)$ is a smooth function and there exists positive constants $β_{1} > 0$ and $β_{2} > 0$ , such that $\begin{matrix} (5.2) & β_{1} ⩽ α (ρ, w) ⩽ β_{2}, \forall (ρ, w) \in G_{e} . \end{matrix}$ When $ρ > 0$ , system (5.1) is equivalent to the following $\begin{matrix} (5.3) & \{\begin{matrix} \partial_{t} ρ + div (ρ w) = 0, \\ ε^{2} (\partial_{t} w + (w \cdot \nabla) w) + \nabla h (ρ) = - α (ρ, w) w, x \in T^{d}, \end{matrix} \end{matrix}$ where h is the enthalpy function satisfying $\begin{matrix} h^{'} (ρ) = \frac{p^{'} (ρ)}{ρ}, \forall ρ > 0 . \end{matrix}$ System (5.3) is of the form (1.7) with $\begin{matrix} u = ρ, v = w, f_{j}^{1} (u, v) = u e_{j}, q^{1} (u, v) = α (u, v), \end{matrix}$ and $\begin{matrix} A_{j}^{21} (u, v) = h^{'} (u) e_{j}, A_{j}^{22} (u, v) = v_{j} I_{3}, K_{i, j} = δ_{i, j} α^{- 1} p^{'} (u) \end{matrix}$ and its corresponding limiting equation is $\begin{matrix} (5.4) & \partial_{t} \bar{ρ} - div (α^{- 1} \nabla p (\bar{ρ})) = 0, \end{matrix}$ which is in general a nonlinear parabolic equation under condition (5.2). It is easily checked that conditions (A1)–(A5), (SK) and (H1) are satisfied. Then Theorem 2.1 can be applied. For weak solutions, an error estimate for ρ in $L^{2} (R^{+} \times R)$ was obtained in [9] in the isothermal case with linear damping, i.e., $p (ρ) = ρ$ and $α (u, v) = 1$ , which implies that (5.4) is a linear heat equation.

5.2. Examples in one-dimensional space

Example 5.3 (The Euler equations with linear damping in Lagrangian coordinates).

In this example (see [7,13,22]), let $d y = ρ d x - ρ v d t$ and $α = 1$ . Then the Lagrangian coordinates $(t, y)$ are well defined for $ρ > 0$ and the Euler equations in (5.1) are equivalent to (see [28]) $\begin{matrix} (5.5) & \{\begin{matrix} \partial_{t} τ - \partial_{y} v = 0, \\ ε^{2} \partial_{t} v + \partial_{y} \tilde{p} (τ) = - v, t > 0, y \in T, \end{matrix} \end{matrix}$ where $\begin{matrix} τ = 1 / ρ, \tilde{p} (τ) = p (1 / τ) . \end{matrix}$ System (5.5) is still of the form (4.1) with $\begin{matrix} u = τ, f^{1} (u, v) = - q^{1} (u, v) = - 1, A^{21} (u, v) = {\tilde{p}}^{'} (u), A^{22} (u, v) = 0 . \end{matrix}$ The corresponding limiting equation is $\begin{matrix} (5.6) & \partial_{t} \bar{τ} + \partial_{y y} p (1 / \bar{τ}) = 0, \end{matrix}$ which is a nonlinear parabolic equation. It is easily checked that conditions (A1)–(A5) and (SK) are satisfied. Then Theorem 2.1 can be applied. We remark that (5.6) is a nonlinear equation even if p is a linear function.

Example 5.4 (The M1 model).

This model in one space variable is written as $\begin{matrix} \{\begin{matrix} \partial_{t} ρ + \partial_{x} (ρ w) = 0, \\ ε^{2} \partial_{t} (ρ w) + \partial_{x} (ρ B (ε w)) = - ρ w, x \in T, \end{matrix} \end{matrix}$ where $ρ > 0$ is the density, w is the velocity and $B (w)$ is defined by $\begin{matrix} B (w) = \frac{1}{3} + \frac{2 w^{2}}{2 + Δ (w)}, Δ (w) = \sqrt{4 - 3 w^{2}} . \end{matrix}$ This system is of the form (4.1) with $\begin{matrix} u = ρ, v = ρ w, f^{1} (u, v) = q^{1} (u, v) = 1, \end{matrix}$ and $\begin{matrix} A^{21} (u, v) = B (v / u) - (v / u) B^{'} (v / u), A^{22} (u, v) = B^{'} (v / u) . \end{matrix}$ Noticing $B (ε w) \to \frac{1}{3}$ when $ε \to 0$ , we obtain its corresponding limiting equation is $\begin{matrix} \partial_{t} \bar{ρ} - \frac{1}{3} \partial_{x x} \bar{ρ} = 0, \end{matrix}$ which is a linear heat equation. It is easily checked that conditions (A1)–(A5) and (SK) are satisfied. Then Theorem 2.1 can be applied. Remark that for smooth solutions an error estimate for ρ in $L^{2} (R^{+} \times R)$ was obtained in [4].

Example 5.5 (A generalized discrete two-velocity model).

The model is written as (see for instance [13,15,22,23,27]): $\begin{matrix} \{\begin{matrix} ε \partial_{t} f + \partial_{x} f = ε^{- 1} {(f + g)}^{ν} (g - f), \\ ε \partial_{t} g - \partial_{x} g = ε^{- 1} {(f + g)}^{ν} (f - g), t > 0, x \in T, \end{matrix} \end{matrix}$ where ν is a real number and $f + g > 0$ . If we introduce $\begin{matrix} u^{ε} = f + g, v^{ε} = \frac{1}{ε} (f - g), \end{matrix}$ then the equivalent system is $\begin{matrix} \{\begin{matrix} \partial_{t} u^{ε} + \partial_{x} v^{ε} = 0, \\ ε^{2} \partial_{t} v^{ε} + \partial_{x} u^{ε} = - 2 {(u^{ε})}^{ν} v^{ε}, \end{matrix} \end{matrix}$ which is of the form (4.1) with $\begin{matrix} f^{1} (u, v) = 1, q^{1} (u, v) = 2 u^{ν}, A^{21} (u, v) = 1, A^{22} (u, v) = 0 . \end{matrix}$ Its corresponding limiting equation is $\begin{matrix} \partial_{t} \bar{u} - \frac{1}{2} \partial_{x} ({\bar{u}}^{- ν} \partial_{x} \bar{u}) = 0, \end{matrix}$ which is a nonlinear parabolic equation for $\bar{u} > 0$ . It is easily checked that conditions (A1)–(A5) and (SK) are satisfied. Then Theorem 2.1 can be applied.

Footnotes

Acknowledgements

The authors want to express their sincerely thanks to the referees for their valuable remarks and suggestions, which made this paper more readable. The Second author was supported by the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.

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Convergence rate from systems of balance laws to isotropic parabolic systems,a periodic case

Abstract

Keywords

1. Introduction

2. Preliminaries and main results

3.1. Construction of the stream function

Lemma 3.1 (Existence of the stream function).

3.2. Convergence rate for u ε

Lemma 3.2 (Convergence rate for u ε ).

Lemma 4.1 (Existence of stream function over one-dimensional torus).

5.1. Examples in multi-dimensional spaces

Example 5.1 (Wave equation of heat conduction).

Example 5.2 (Euler equations with nonlinear damping).

5.2. Examples in one-dimensional space

Example 5.3 (The Euler equations with linear damping in Lagrangian coordinates).

Example 5.4 (The M1 model).

Example 5.5 (A generalized discrete two-velocity model).

Footnotes

Acknowledgements

References

3.2. Convergence rate for $u^{ε}$

Lemma 3.2 (Convergence rate for $u^{ε}$ ).