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CONVERGENCE OF FIRST-ORDER QUASILINEAR
HYPERBOLIC SYSTEMS TO
HYPERBOLIC-PARABOLIC SYSTEMS

Yue-Jun Peng, Shuimiao Du

► To cite this version:

Yue-Jun Peng, Shuimiao Du. CONVERGENCE OF FIRST-ORDER QUASILINEAR HYPERBOLIC SYSTEMS TO HYPERBOLIC-PARABOLIC SYSTEMS. 2025. hal-04889272

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https://uca.hal.science/hal-04889272v1

Preprint submitted on 15 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
CONVERGENCE OF FIRST-ORDER QUASILINEAR HYPERBOLIC
SYSTEMS TO HYPERBOLIC-PARABOLIC SYSTEMS

YUE-JUN PENG¹,\* AND SHUIMIAO DU²

¹Université Clermont Auvergne / CNRS, Laboratoire de Mathématiques Blaise Pascal
F-63000 Clermont-Ferrand, France

²Sino-European School of Technology of Shanghai University
200444 Shanghai, China

**Abstract.** We provide a framework to study the zero relaxation time limit of Cauchy problem for first-order quasilinear hyperbolic systems with relaxation in several space dimensions. For this purpose, we construct an approximate solution by a formal asymptotic expansion with initial layer corrections. The system of the leading term in the expansion is governed by a hyperbolic-parabolic system. Under appropriate structural and partial dissipation conditions, we justify rigorously the validity of the asymptotic expansion on a time interval independent of the relaxation time, provided that the system of the leading term admits a local-in-time smooth solution. The main theorem of the present paper includes the results obtained in previous works and applies to additional examples of models arising in fluid mechanics.

**Keywords:** first-order quasilinear hyperbolic system, structural and stability conditions, hyperbolic-parabolic limit, asymptotic analysis

**AMS Subject Classification (2000):** 35C20, 35L60, 76M45

# 1. INTRODUCTION

This paper concerns the singular limit to Cauchy problem for first-order quasilinear hyperbolic systems with relaxation of the form

$$ (1.1) \qquad \partial_t U + \frac{1}{\epsilon} \sum_{j=1}^{d} A_j(\epsilon, U) \partial_{x_j} U = \frac{S(\epsilon, U)}{\epsilon^2}, $$

with initial condition

$$ (1.2) \qquad U(0, x) = \bar{U}^{\epsilon}(x), $$

for time $t \in \mathbb{R}^+$ and position $x = (x_1, \dots, x_d) \in \mathbb{R}^d$. Here $U : \mathbb{R}^+ \times \mathbb{R}^d \to G \subset \mathbb{R}^N$ is the unknown, $G$ is the state set of $U$ and $\epsilon \in (0, 1]$ is a small parameter. In physical models, $\epsilon$ often stands for a relaxation time. We suppose that the given functions $S : [0, 1] \times G \to \mathbb{R}^N$ and $A_j : [0, 1] \times G \to \mathcal{M}_N(\mathbb{R})$ are sufficiently smooth, where $\mathcal{M}_N(\mathbb{R})$ denotes the set of square matrices of order $N$.

E-mail address: yue-jun.peng@uca.fr

<footnote>\* The corresponding author.</footnote>
In (1.1), the variable $t$ should be understood as a slow time linked with the usual time $t'$ by $t = \varepsilon t'$. Therefore, (1.1) is equivalent to

$$ (1.3) \qquad \partial_{t'} \tilde{U} + \sum_{j=1}^{d} A_j(\varepsilon, \tilde{U}) \partial_{x_j} U = \frac{S(\varepsilon, \tilde{U})}{\varepsilon}, \quad \text{with } \tilde{U}(t', x) = U(t, x). $$

System (1.3) presents a limit behavior different from that of (1.1). Its formal limit as $\varepsilon \to 0$ is governed by first-order quasilinear hyperbolic systems. The mathematical analysis and physical models to (1.3) can be found in [42, 23, 15, 30, 43, 38, 44] and references therein.

In what follows, we suppose that (1.1) is uniformly symmetrizable hyperbolic (see [9]): i.e., there exist a constant $a_0 > 0$ and a symmetric positive definite matrix $A_0(\varepsilon, U)$, called symmetrizer, such that for all $(\varepsilon, U) \in [0, 1] \times G$,

(i) $A_0(\varepsilon, U)\xi \cdot \xi \ge a_0|\xi|^2$, for all $\xi \in \mathbb{R}^N$,

(ii) $\tilde{A}_j(\varepsilon, U) \stackrel{\text{def}}{=} A_0(\varepsilon, U)A_j(\varepsilon, U)$ is symmetric for all $1 \le j \le d$, where “·” and $| \cdot |$ are the inner product and the Euclidean norm of $\mathbb{R}^N$, respectively.

Let $m \ge 1$ be an integer. For the sake of simplicity, we denote $H^m = H^m(\mathbb{R}^d)$. Assume, for each $\varepsilon \in (0, 1]$, $\bar{U}^\varepsilon \in H^m$ with $m > \frac{d}{2} + 1$. According to Lax [22] and Kato [17], there exists a maximal time $T_\varepsilon > 0$ and a unique smooth solution $U^\varepsilon$ to (1.1)-(1.2) satisfying

$$ U^\varepsilon \in C([0, T_\varepsilon], H^m) \cap C^1([0, T_\varepsilon], H^{m-1}). $$

We want to prove that $U^\varepsilon$ converges to solutions of second-order nonlinear hyperbolic-parabolic systems as $\varepsilon \to 0$, and $\inf_{\varepsilon \in (0,1]} T_\varepsilon > 0$.

Let us denote

$$ U = \begin{pmatrix} u \\ v \end{pmatrix}, \quad u \in \mathbb{R}^{N-n}, \quad v \in \mathbb{R}^n, \quad \text{with } 1 \le n \le N, $$

in which the variables $u$ and $v$ play different roles. For a vector $V \in \mathbb{R}^N$ and a matrix $M \in \mathcal{M}_N(\mathbb{R})$, we denote

$$ V = \begin{pmatrix} V^I \\ V^{II} \end{pmatrix}, \quad M = \begin{pmatrix} M^{11} & M^{12} \\ M^{21} & M^{22} \end{pmatrix}, $$

with $V^I \in \mathbb{R}^{N-n}$, $M^{11} \in \mathcal{M}_{N-n}(\mathbb{R})$ and $M^{22} \in \mathcal{M}_n(\mathbb{R})$. Then

$$ S = \begin{pmatrix} S^I \\ S^{II} \end{pmatrix}. $$

Set

$$ s(U) = S^{II}(0, U). $$

In previous works, the limit problem for (1.1)-(1.2) was considered for special models in [10, 26, 25, 2]. It was investigated using compensated compactness method for a one-dimensional $2 \times 2$ system in [27, 16, 20]. In [21], the authors studied the multi-dimensional problem in the case where

$$ (1.4) \qquad A_j(\varepsilon, U) = A_j(\varepsilon U), \quad S(\varepsilon, U) = S(U). $$

Since

$$ A_j(\varepsilon U) = A_j(0) + O(\varepsilon), $$

the singular part on the left of (1.1) is linear. The results can be applied to a linear wave equation of heat conduction and a generalized discrete two-velocity model. In [33], the authors further investigated the problem in the case where

$$ (1.5) \qquad A_j(\varepsilon, U) = A_j(U), \quad S = S(\varepsilon, U), $$
and apply the results to Euler equations with damping, the M1-model arising in the radiative transfer theory and Euler-Maxwell systems with relaxation. We mention that when $A_j^{11}(0, U) \neq 0$, differential constraints appear in the system of the leading term, which is the case of Euler-Maxwell systems. In the present paper, we provide a more general framework for (1.1)-(1.2) by considering the case where both $A_j$ and $S$ depend on $\varepsilon$. Our results (see Theorem 1.1) include those in [21] and also in [33] if differential constraints are not concerned, and can be applied to additional systems such as the Timoshenko-Cattaneo system and hyperbolic Navier-Stokes equations. These systems cannot be written in the form (1.1) with (1.4) or (1.5).

In the justification of the convergence of $U^\varepsilon$ as $\varepsilon \to 0$, it is important to make minimum structural conditions on $A_j$ and $S$. In this paper, we assume

$$ (1.6) \qquad A_j^{11}(0, U) = 0, \quad \forall U \in G, $$

$$ (1.7) \qquad S^I(0, U) = 0, \quad \partial_\varepsilon S^I(0, U) = 0, \quad \forall U \in G, $$

and

$$ (1.8) \qquad s(u, v) = 0 \iff v = 0, \text{ and } \partial_v s(u, 0) \text{ is invertible for all } (u, 0) \in G, $$

where $\partial_\varepsilon$ and $\partial_v$ stand for the partial derivatives with respect to $\varepsilon$ and $v$, respectively. Conditions (1.6)-(1.7) exclude differential constraints for the approximate solution (see [33]), and (1.7)-(1.8) show that $S(0, U) = 0$ if and only if $v = 0$. More precisely, $S(\varepsilon, U)$ is of the form

$$ (1.9) \qquad S(\varepsilon, U) = \begin{pmatrix} 0 \\ s(u, v) \end{pmatrix} + \varepsilon \begin{pmatrix} 0 \\ \partial_\varepsilon S^{II}(0, U) \end{pmatrix} + \frac{\varepsilon^2}{2} \partial_\varepsilon^2 S(0, U) + O(\varepsilon^3). $$

These conditions are reduced to those imposed in [21] in case (1.4) and in [33] in case (1.5).

As in the previous works, we also need the following partial dissipation condition. There is a constant $c_0 > 0$ such that

$$ (1.10) \qquad A_0(0, u, 0) \partial_u S(0, u, 0) \xi \cdot \xi \le -c_0 |\xi^{II}|^2, \quad \forall (u, 0) \in G, \forall \xi \in \mathbb{R}^N. $$

This condition implies a time-dissipation of variable $v$. It is similar to that proposed in [43] where both $A_j$ and $S$ are independent of $\varepsilon$. From (1.7)-(1.10), we have similarly (see [43] or Lemma 3.1 in [33])

$$ (1.11) \qquad A_0^{12}(0, u, 0) = 0. $$

Hence, (1.10) is equivalent to

$$ (1.12) \qquad A_0^{22}(0, u, 0) \partial_v s(u, 0) \xi^{II} \cdot \xi^{II} \le -c_0 |\xi^{II}|^2, \quad \forall (u, 0) \in G, \forall \xi^{II} \in \mathbb{R}^n. $$

Let $U_0 = \begin{pmatrix} u_0 \\ v_0 \end{pmatrix}$ be the formal limit of the solution $U^\varepsilon$ to system (1.1) as $\varepsilon \to 0$. Under the above assumptions, we see that $v_0 = 0$ and $u_0$ satisfies a second-order system of partial differential equations of the form

$$ (1.13) \qquad \partial_t u_0 + \sum_{i,j=1}^d A_{ij}(u_0) \partial_{x_ix_j}^2 u_0 + \sum_{j=1}^d B_j(u_0) \partial_{x_j} u_0 + \sum_{i,j=1}^d C_{ij}(u_0) \partial_{x_i} u_0 \partial_{x_j} u_0 = \frac{1}{2} \partial_\varepsilon^2 S^I(0, u_0, 0), $$

where

$$ (1.14) \qquad A_{ij}(u_0) = A_i^{12}(0, u_0, 0) \partial_v s(u_0, 0)^{-1} A_j^{21}(0, u_0, 0), $$

$$ (1.15) \qquad \begin{aligned} B_j(u_0) = & \partial_\varepsilon A_j^{11}(0, u_0, 0) - A_j^{12}(0, u_0, 0) \partial_u [\partial_v s(u_0, 0)^{-1} \partial_\varepsilon S^{II}(0, u_0, 0)] \\ & - \partial_v A_j^{11}(0, u_0, 0) \partial_v s(u_0, 0)^{-1} \partial_\varepsilon S^{II}(0, u_0, 0), \end{aligned} $$
$$ (1.16) \qquad \begin{aligned} C_{ij}(u_0) = & A_i^{12}(0, u_0, 0) \partial_u [\partial_v s(u_0, 0)^{-1} A_j^{21}(0, u_0, 0)] \\ & + \partial_v A_j^{11}(0, u_0, 0) \partial_v s(u_0, 0)^{-1} A_i^{21}(0, u_0, 0). \end{aligned} $$

The detailed derivation of these equations is given in Section 2. System (1.13) is in general of hyperbolic-parabolic type and its principal part is governed by the second-order partial differential operator

$$ \partial_t + \sum_{i,j=1}^{d} A_{ij}(u_0) \partial_{x_i x_j}^2. $$

It is proved in [34] that, when

$$ A_j(\varepsilon, U) = A_j(U), \quad S = S(U), $$

in a neighborhood of $U_*$ such that $S(U_*) = 0$, the Shizuta-Kawashima condition [39] implies that (1.13) is strictly parabolic, and $U^\varepsilon$ converges for all time. On the other hand, it is easy to see that, if $A_j^{21} = 0$ for all $j$, then $A_{ij} = 0$ and $C_{ij} = 0$, hence, (1.13) is reduced to a nonlinear first-order system. Here we do not intend to solve system (1.13) in the general case. We refer the readers to [19, 18, 39] for the local-in-time existence of solutions. For the Timoshenko-Cattaneo system and hyperbolic Navier-Stokes equations, (1.13) is hyperbolic-parabolic type and admits a unique local smooth solution.

Remark that if (1.1) can be put in the form of a system of balance laws :

$$ (1.17) \qquad \partial_t U + \frac{1}{\varepsilon} \sum_{j=1}^{d} \partial_{x_j} f_j(\varepsilon, U) = \frac{S(\varepsilon, U)}{\varepsilon^2}, $$

the derivation of (1.13)-(1.16) is easier. We denote

$$ (1.18) \qquad A_j(\varepsilon, U) = \partial_U f_j(\varepsilon, U) = \begin{pmatrix} \partial_u f_j^I(\varepsilon, U) & \partial_v f_j^I(\varepsilon, U) \\ \partial_u f_j^{II}(\varepsilon, U) & \partial_v f_j^{II}(\varepsilon, U) \end{pmatrix}, \quad f_j = \begin{pmatrix} f_j^I \\ f_j^{II} \end{pmatrix}, $$

with $f_j^I: [0, 1] \times G \to \mathbb{R}^{N-n}$ and $f_j^{II}: [0, 1] \times G \to \mathbb{R}^{N-n}$ being sufficiently smooth functions. Since $v_0 = 0$, we have

$$ u = u_0 + O(\varepsilon), \quad v = \varepsilon v_1 + O(\varepsilon^2). $$

We use Taylor formula to obtain

$$ f_j^I(\varepsilon, U) = f_j^I(0, u, 0) + \varepsilon [\partial_\varepsilon f_j^I(0, u_0, 0) + \partial_v f_j^I(0, u_0, 0)v_1] + O(\varepsilon^2). $$

Since $A_j^{11} = \partial_u f_j^I$, condition (1.6) means that $f_j^I(0, u, 0)$ is independent of $u$. It follows from (1.17) and (1.9) that $u_0$ satisfies a system of balance laws

$$ (1.19) \qquad \partial_t u_0 + \sum_{j=1}^{d} \partial_{x_j} [\partial_\varepsilon f_j^I(0, u_0, 0) + \partial_v f_j^I(0, u_0, 0)v_1] = \frac{1}{2} \partial_\varepsilon^2 S^I(0, u_0, 0). $$

Similarly, we have

$$ (1.20) \qquad v_1 = \partial_v s(u_0, 0)^{-1} \left[ \sum_{i=1}^{d} \partial_{x_i} f_i^{II}(0, u_0, 0) \partial_\varepsilon S^{II}(0, u_0, 0) \right]. $$

In this case, the matrix $A_{ij}(u_0)$ is expressed as

$$ A_{ij}(u_0) = \partial_v f_i^I(0, u_0, 0) \partial_v s(u_0, 0)^{-1} \partial_u f_j^{II}(0, u_0, 0). $$
Let $K \ge 2$ be an integer and $T_0 > 0$. Assume that $u_0 \in C([0, T_0], H^m)$ is a smooth solution to (1.13). Similarly to [21, 33], we will construct an approximate solution by an asymptotic expansion of the form

$$ (1.21) \qquad U_{\varepsilon}^{K}(t, x) = \sum_{k=0}^{K} \varepsilon^{k} (U_{k}(t, x) + \chi_{k}(\tau, x)), \quad \text{with} \quad \tau = t/\varepsilon^{2}, $$

where $\chi_k (0 \le k \le K)$ are the initial layer corrections that converge exponentially fast to zero as $\tau$ tends to infinity. This solution is defined on time interval $[0, T_K]$ with $T_K \in (0, T_0]$. Define the error of approximation by

$$ (1.22) \qquad R_{\varepsilon}^{K} = \partial_{t}U_{\varepsilon}^{K} + \frac{1}{\varepsilon}\sum_{j=1}^{d}A_{j}(\varepsilon, U_{\varepsilon}^{K})\partial_{x_{j}}U_{\varepsilon}^{K} - \frac{S(\varepsilon, U_{\varepsilon}^{K})}{\varepsilon^{2}}. $$

We verify in Section 2 that the solution $U_{\varepsilon}^{K}$ satisfies

$$ U_{\varepsilon}^{K} \in C([0, T_{K}], H^{m}) \cap C^{1}([0, T_{K}], H^{m-1}) $$

and

$$ (1.23) \qquad \|U_{\varepsilon}^{K}(0, \cdot) - \bar{U}^{\varepsilon}\|_{m} \le c_{1}\varepsilon^{K}, $$

and for all $t \in [0, T_K]$,

$$ (1.24) \qquad \|U_{\varepsilon}^{K}(t, \cdot) - U_{0}(t, \cdot)\|_{m} \le c_{1}\varepsilon + c_{1}e^{-\frac{\mu t}{\varepsilon^{2}}}, $$

$$ (1.25) \qquad \|\partial_t U_{\varepsilon}^K(t, \cdot)\|_m \le c_1 + c_1 \varepsilon^{-2} e^{-\frac{\mu t}{\varepsilon^2}}, $$

$$ (1.26) \qquad R_{\varepsilon}^{K} = \varepsilon^{K-1} \begin{pmatrix} 0 \\ r_{K} \end{pmatrix} + \varepsilon^{K-1} F_{\varepsilon}^{K}, \quad \|F_{\varepsilon}^{K}(t, \cdot)\|_{m} \le c_{1}\varepsilon + c_{1}e^{-\frac{\mu t}{\varepsilon^{2}}}, $$

where $\| \cdot \|_m$ is the norm of $H^m$, $c_1 > 0$ and $\mu > 0$ are generic constants independent of $\varepsilon$, $r_K \in C([0, T_K], H^m)$ and $F_\varepsilon^K \in C([0, T_K], H^m)$.

The main result of the paper is stated as follows.

**Theorem 1.1.** Let $m > d/2 + 1$ and $K \ge 2$ be integers. Let $U^\varepsilon$ be the exact solution to (1.1)-(1.2) defined on the maximal time interval $[0, T_\varepsilon)$ satisfying $U^\varepsilon \in C([0, T_\varepsilon], H^m) \cap C^1([0, T_\varepsilon], H^{m-1})$, and $U_\varepsilon^K$ be an approximate solution to (1.1)-(1.2) defined on $[0, T_K]$. Assume (1.6)-(1.8), (1.10) and (1.23)-(1.26) hold. Then there exist constants $\varepsilon_0 > 0$ and $c_2 > 0$ such that for all $\varepsilon \in (0, \varepsilon_0]$, we have $T_\varepsilon \ge T_K$ and

$$ (1.27) \qquad \sup_{0 \le t \le T_K} \|U^\varepsilon(t, \cdot) - U_\varepsilon^K(t, \cdot)\|_m \le c_2 \varepsilon^K. $$

Moreover,

$$ (1.28) \qquad \int_{0}^{T_K} \|v^\varepsilon(t, \cdot) - v_\varepsilon^K(t, \cdot)\|_m^2 dt \le c_2 \varepsilon^{2(K+1)}. $$

**Remark 1.1.** Let $U_\varepsilon^K$ be given by (1.21) with

$$ \chi_k = \begin{pmatrix} \chi_k^I \\ \chi_k^{II} \end{pmatrix}, \quad 0 \le k \le K. $$

We will see in the next section that $\chi_0^I = 0$ and $\chi_0^{II}$ converges exponentially fast to zero as $\tau \to +\infty$. Therefore,

$$ \|\chi_0^{II}(\tau, \cdot)\|_m^2 \le c_1 e^{-\mu\tau}, $$
which implies that

$$ \int_{0}^{T_K} \|\chi_0^{II}(t/\epsilon^2, \cdot)\|_m^2 dt \le c_1 \epsilon^2. $$

Thus, we obtain from (1.27)-(1.28) that

$$ \sup_{0 \le t \le T_K} \|u^{\epsilon}(t, \cdot) - u_0(t, \cdot)\|_m \le c_2 \epsilon, $$

and

$$ \int_{0}^{T_K} \|v^{\epsilon}(t, \cdot)\|_{m}^{2} dt \leq c_{2}\epsilon^{2}, $$

provided that $U_k$ and $\chi_k$ are uniformly bounded in $L^\infty([0, T_K], H^m)$. This shows the convergence of $U^\epsilon = \begin{pmatrix} u^\epsilon \\ v^\epsilon \end{pmatrix}$ to $U_0 = \begin{pmatrix} u_0 \\ 0 \end{pmatrix}$.

This paper is organized as follows. In the next section, we construct an approximate solution $U_\epsilon^K$ satisfying (1.23)-(1.26). In particular, we show the detailed derivation of the system for $u_0$. Section 3 is devoted to the proof of Theorem 1.1, which is achieved by a series of lemmas for energy estimates together with the Gronwall Lemma. In the last section, we show examples to which the approximate solutions can be rigorously constructed and thus the theorem can be applied.

## 2. CONSTRUCTION OF AN APPROXIMATE SOLUTION

We first seek for an approximate solution to (1.1)-(1.2) under the form of a power series in $\epsilon$:

$$ \sum_{k=0}^{+\infty} \epsilon^k (U_k(t, x) + \chi_k(\tau, x)), \quad \text{with } U_k = \begin{pmatrix} u_k \\ v_k \end{pmatrix}, \chi_k = \begin{pmatrix} \chi_k^I \\ \chi_k^{II} \end{pmatrix}, \tau = t/\epsilon^2, $$

where $\chi_k (k \in \mathbb{N})$ are the initial layer corrections that converge exponentially fast to zero as $\tau$ tends to infinity. The derivation of the systems satisfied by $U_k$ and $\chi_k$ is formal and is given as follows. Let $f$ be a sufficiently smooth function and

$$ V_{\epsilon} = \sum_{k=0}^{+\infty} \epsilon^k V_k, \quad V = (V_k)_{k \in \mathbb{N}}. $$

We have

$$ f(V_{\epsilon}) = f(V_0) + \sum_{k=1}^{+\infty} \epsilon^k (f'(V_0)V_k + e_k(f, V)), $$

where $e_k(f, V)$ depends only on $f$ and $(V_0, V_1, \dots, V_{k-1})$, with

$$ e_1(f, V) = 0, \quad e_2(f, V) = (f''(V_0)V_1)V_1. $$

Then

$$ \begin{aligned} S(\epsilon, V_\epsilon) = S(0, V_0) &+ \epsilon[\partial_\epsilon S(0, V_0) + \partial_V S(0, V_0)V_1] \\ (2.1) &+ \epsilon^2 \left( \frac{1}{2}[\partial_\epsilon^2 S(0, V_0) + 2\partial_\epsilon \partial_V S(0, V_0)V_1 + (\partial_V^2 S(0, V_0)V_1)V_1] + \partial_V S(0, V_0)V_2 \right) \\ &+ \epsilon^2 \sum_{k=1}^{+\infty} \epsilon^k \left( \frac{1}{(k+2)!} \partial_\epsilon^{k+2} S(0, V_0) + \sum_{l=0}^{k+1} \frac{1}{l!} [\partial_V \partial_\epsilon^l S(0, V_0)V_{k+2-l} + e_{k+2-l}(\partial_\epsilon^l S(0, \cdot), V)] \right), \end{aligned} $$
$$ (2.2) \qquad \begin{aligned} A_j(\varepsilon, V_\varepsilon) \partial_{x_j} V_\varepsilon = & A_j(0, V_0) \partial_{x_j} V_0 + \varepsilon \left[ (\partial_\varepsilon A_j(0, V_0) + \partial_V A_j(0, V_0) V_1) \partial_{x_j} V_0 + A_j(0, V_0) \partial_{x_j} V_1 \right] \\ & + \varepsilon \sum_{k=1}^{+\infty} \varepsilon^k \left[ (\partial_\varepsilon A_j(0, V_0) + \partial_V A_j(0, V_0) V_1) \partial_{x_j} V_k + A_j(0, V_0) \partial_{x_j} V_{k+1} \right] \\ & + \sum_{r=1}^{k} \left( \sum_{l=0}^{r} \frac{1}{l!} \left[ \partial_V \partial_\varepsilon^l A_j(0, V_0) V_{r+1-l} + e_{r+1-l} (\partial_\varepsilon^l A_j(0, \cdot), V) \right] \right. \\ & \qquad \left. + \frac{1}{(r+1)!} \partial_\varepsilon^{r+1} A_j(0, V_0) \right) \partial_{x_j} V_{k-r} \right]. \end{aligned} $$

## 2.1. The equations of $U_k$.

Assume that $U_\varepsilon = \sum_{k=0}^{+\infty} \varepsilon^k U_k$ is a formal solution to (1.1). Then

$$ \partial_t U_\varepsilon + \frac{1}{\varepsilon} \sum_{j=1}^{d} A_j(\varepsilon, U_\varepsilon) \partial_{x_j} U_\varepsilon = \frac{S(\varepsilon, U_\varepsilon)}{\varepsilon^2}. $$

Using formulas (2.1)-(2.2) and identifying the terms in powers of $\varepsilon$ yield

$$ (2.3) \qquad \varepsilon^{-2} : \quad S(0, U_0) = 0, $$

$$ (2.4) \qquad \varepsilon^{-1} : \quad \sum_{j=1}^{d} A_j(0, U_0) \partial_{x_j} U_0 - \partial_\varepsilon S(0, U_0) - \partial_U S(0, U_0) U_1 = 0, $$

$$ \varepsilon^0 : \quad \partial_t U_0 + \sum_{j=1}^{d} \left( \left[ \partial_\varepsilon A_j(0, U_0) + \partial_U A_j(0, U_0) U_1 \right] \partial_{x_j} U_0 + A_j(0, U_0) \partial_{x_j} U_1 \right) $$

$$ (2.5) \qquad \begin{aligned} & - \frac{1}{2} \left( \partial_\varepsilon^2 S(0, U_0) + 2\partial_\varepsilon \partial_U S(0, U_0) U_1 + \left( \partial_U^2 S(0, U_0) U_1 \right) U_1 \right) \\ & - \partial_U S(0, U_0) U_2 = 0, \end{aligned} $$

$$ \varepsilon^k (k \ge 1) : \quad \partial_t U_k + \sum_{j=1}^{d} \left( \left[ \partial_\varepsilon A_j(0, U_0) + \partial_U A_j(0, U_0) U_1 \right] \partial_{x_j} U_k + A_j(0, U_0) \partial_{x_j} U_{k+1} \right) $$

$$ (2.6) \qquad \begin{aligned} & + \sum_{j=1}^{d} \sum_{r=1}^{k} \left( \sum_{l=0}^{r} \frac{1}{l!} \left[ \partial_U \partial_\varepsilon^l A_j(0, U_0) U_{r+1-l} + e_{r+1-l} (\partial_\varepsilon^l A_j(0, \cdot), U) \right] \right. \\ & \qquad \left. + \frac{1}{(r+1)!} \partial_\varepsilon^{r+1} A_j(0, U_0) \right) \partial_{x_j} U_{k-r} - \frac{1}{(k+2)!} \partial_\varepsilon^{k+2} S(0, U_0) \right. \\ & \qquad \left. - \sum_{l=0}^{k+1} \frac{1}{l!} \left[ \partial_U \partial_\varepsilon^l S(0, U_0) U_{k+2-l} + e_{k+2-l} (\partial_\varepsilon^l S(0, \cdot), U) \right] = 0. \end{aligned} $$

Equation (2.3) yields $v_0 = 0$ thanks to (1.7)-(1.8). It follows from (1.6)-(1.7) that the first block of (2.4) is trivially satisfied. Since $\partial_u s(u, 0) = 0$, (2.4) is equivalent to

$$ \sum_{j=1}^{d} A_j^{21}(0, U_0) \partial_{x_j} u_0 - \partial_\varepsilon S^{II}(0, U_0) - \partial_v s(U_0) v_1 = 0, $$
which implies that

$$ (2.7) \qquad v_1 = \partial_v s(U_0)^{-1} \left[ \sum_{j=1}^{d} A_j^{21}(0, U_0) \partial_{x_j} u_0 - \partial_{\varepsilon} S^{II}(0, U_0) \right], $$

thanks to (1.8). For (2.5), we use (1.6)-(1.7) to obtain

$$ \partial_U A_j^{11}(0, U_0) = 0, \quad \partial_U S^I(0, U_0) = 0, \quad \partial_{\varepsilon} \partial_U S^I(0, U_0) = 0, \quad \partial_U^2 S^I(0, U_0) = 0. $$

Since $v_0 = 0$, (2.5) is equivalent to two block systems:

$$ (2.8) \quad \partial_t u_0 + \sum_{j=1}^{d} \left[ \partial_{\varepsilon} A_j^{11}(0, U_0) \partial_{x_j} u_0 + A_j^{12}(0, U_0) \partial_{x_j} v_1 + \partial_v A_j^{11}(0, U_0) v_1 \partial_{x_j} u_0 \right] = \frac{1}{2} \partial_{\varepsilon}^2 S^I(0, U_0), $$

and

$$ (2.9) \quad = \frac{1}{2} \left[ \partial_{\varepsilon}^2 S^{II}(0, U_0) + 2 \partial_{\varepsilon} \partial_U S^{II}(0, U_0) U_1 + (s''(U_0)U_1)U_1 \right] + \partial_v s(U_0) v_2. $$

Substituting (2.7) into (2.8) yields system (1.13) for $u_0$.

If system (1.1) is written in the form of balance laws (1.17), from (1.18) we obtain

$$ A_j^{21}(0, u_0, 0) = \partial_u f_j^{II}(0, u_0, 0), $$

and

$$ \begin{aligned} & \partial_{\varepsilon} A_j^{11}(0, u_0, 0) \partial_{x_j} u_0 + A_j^{12}(0, u_0, 0) \partial_{x_j} v_1 + \partial_v A_j^{11}(0, u_0, 0) v_1 \partial_{x_j} u_0 \\ & = \partial_{x_j} \left[ \partial_{\varepsilon} f_j^I(0, u_0, 0) + \partial_v f_j^I(0, u_0, 0) v_1 \right], \end{aligned} $$

which imply (1.19) and (1.20).

Similarly, for $k \ge 1$, (2.6) can be written as two block systems of $N-n$ and $n$ equations, respectively. From (1.6)-(1.7), we have

$$ (2.10) \qquad \partial_t u_k + \sum_{j=1}^{d} A_j^{12}(0, U_0) \partial_{x_j} v_{k+1} + g_k((U_i, \nabla U_i)_{0 \le i \le k}) = 0 $$
and

$$
\begin{align}
& \partial_t v_k + \sum_{j=1}^{d} \left( \begin{aligned}[t]
& \left( [\partial_{\epsilon} A_j^{21}(0, U_0) + \partial_U A_j^{21}(0, U_0) U_1] \partial_{x_j} u_k + A_j^{21}(0, U_0) \partial_{x_j} u_{k+1} \right. \\
& \left. + \partial_{\epsilon} A_j^{22}(0, U_0) + \partial_U A_j^{22}(0, U_0) U_1 \right] \partial_{x_j} v_k + A_j^{22}(0, U_0) \partial_{x_j} v_{k+1} \right) \\
& + \sum_{j=1}^{d} \sum_{r=1}^{k} \left( \sum_{l=0}^{r} \frac{1}{l!} \left[ \partial_U \partial_{\epsilon}^l A_j^{21}(0, U_0) U_{r+1-l} + e_{r+1-l} (\partial_{\epsilon}^l A_j^{21}(0, \cdot), U) \right] \right. \\
& \left. + \frac{1}{(r+1)!} \partial_{\epsilon}^{r+1} A_j^{21}(0, U_0) \right) \partial_{x_j} u_{k-r} \right) \nonumber \\
& + \sum_{j=1}^{d} \sum_{r=1}^{k} \left( \sum_{l=0}^{r} \frac{1}{l!} \left[ \partial_U \partial_{\epsilon}^l A_j^{22}(0, U_0) U_{r+1-l} + e_{r+1-l} (\partial_{\epsilon}^l A_j^{22}(0, \cdot), U) \right] \right. \\
& \left. + \frac{1}{(r+1)!} \partial_{\epsilon}^{r+1} A_j^{22}(0, U_0) \right) \partial_{x_j} v_{k-r} - \frac{1}{(k+2)!} \partial_{\epsilon}^{k+2} S^{II}(0, U_0) \right. \\
& \left. - \sum_{l=0}^{k+1} \frac{1}{l!} \left[ \partial_U \partial_{\epsilon}^l S^{II}(0, U_0) U_{k+2-l} + e_{k+2-l} (\partial_{\epsilon}^l S^{II}(0, \cdot), U) \right] = 0, \tag{2.11}
\end{align}
$$

where

$$
\begin{align*}
g_k((U_i, \nabla U_i)_{0 \le i \le k}) &= \sum_{j=1}^{d} \left[ \partial_{\epsilon} A_j^{11}(0, U_0) \partial_{x_j} u_k + (\partial_{\epsilon} A_j^{12}(0, U_0) + \partial_U A_j^{12}(0, U_0) U_1) \partial_{x_j} v_k \right] \\
&\quad + \sum_{j=1}^{d} \sum_{r=1}^{k} \left( \sum_{l=0}^{r} \frac{1}{l!} \left[ \partial_U \partial_{\epsilon}^l A_j^{11}(0, U_0) U_{r+1-l} + e_{r+1-l} (\partial_{\epsilon}^l A_j^{11}(0, \cdot), U) \right] \right. \\
&\qquad \left. + \frac{1}{(r+1)!} \partial_{\epsilon}^{r+1} A_j^{11}(0, U_0) \right) \partial_{x_j} u_{k-r} \\
&\quad + \sum_{j=1}^{d} \sum_{r=1}^{k} \left( \sum_{l=0}^{r} \frac{1}{l!} \left[ \partial_U \partial_{\epsilon}^l A_j^{12}(0, U_0) U_{r+1-l} + e_{r+1-l} (\partial_{\epsilon}^l A_j^{12}(0, \cdot), U) \right] \right. \\
&\qquad \left. + \frac{1}{(r+1)!} \partial_{\epsilon}^{r+1} A_j^{12}(0, U_0) \right) \partial_{x_j} v_{k-r} - \frac{1}{(k+2)!} \partial_{\epsilon}^{k+2} S^I(0, U_0) \\
&\quad - \sum_{l=0}^{k+1} \frac{1}{l!} \left[ \partial_U \partial_{\epsilon}^l S^I(0, U_0) U_{k+2-l} + e_{k+2-l} (\partial_{\epsilon}^l S^I(0, \cdot), U) \right].
\end{align*}
$$

It is easy to see that $g_k$ is affine with respect to $u_k$ and is independent of $u_{k+1}$. From (1.9), we have

$$
\partial_U S^{II}(0, U_0) U_{k+2} = \partial_v s(U_0) v_{k+2}.
$$

This allows to express (2.11) as

$$
v_{k+2} = \partial_v s(U_0)^{-1} \sum_{j=1}^{d} A_j^{21}(0, U_0) \partial_{x_j} u_{k+1} + \beta_{k+1} u_{k+1} + \beta_{k+2}, \quad k \ge 1,
$$

and this together with (2.9) yields

$$
(2.12) \qquad v_{k+1} = \partial_v s(U_0)^{-1} \sum_{j=1}^{d} A_j^{21}(0, U_0) \partial_{x_j} u_k + \beta_k u_k + \beta_{k+1}, \quad k \ge 1,
$$
where $\beta_k (k \ge 1)$ stand for smooth function of $(U_l)_{0 \le l \le k-1}$ and $v_k$ and their first-order derivatives, but are independent of $u_k$.

Substituting (2.12) into (2.10), we obtain a linear system for $u_k$:

$$ (2.13) \qquad \partial_t u_k + \sum_{i,j=1}^{d} A_{ij}(u_0) \partial_{x_i x_j}^2 u_k + \sum_{j=1}^{d} B_j^k \partial_{x_j} u_k + \sum_{j=1}^{d} C_j^k u_k + f_k = 0, $$

where $B_j^k, C_j^k$ and $f_k$ depend only on $U_0, U_1, \dots, U_{k-1}$ and their first-order derivatives. Remark that the second-order operator is the same for (1.13) and (2.13).

Assume that system (1.13) admits a local smooth solution $u_0$, defined on time interval $[0, T_0]$ with $T_0 > 0$ being independent of $\varepsilon$. Then we have constructed $U_0 = \begin{pmatrix} u_0 \\ 0 \end{pmatrix}$ and $v_1$, which is given by (2.7). By induction on $k$, assume that the linear system (2.13) admits a local smooth solution $u_k$ defined on time interval $[0, T_k]$ with $T_k \in (0, T_{k-1}]$ being independent of $\varepsilon$. Thus we have constructed $U_k$ and $v_{k+1}$, which is given by (2.12).

## 2.2. The equations of $\chi_k$ and the initial data of $(u_k, \chi_k)$.

Recall that $\tau = t/\varepsilon^2$ and denote

$$ \hat{U}_{\varepsilon}(\tau, x) = U_{\varepsilon}(t, x). $$

Then (1.1) is equivalent to

$$ (2.14) \qquad \partial_{\tau} \hat{U}_{\varepsilon} + \varepsilon \sum_{j=1}^{d} A_j(\varepsilon, \hat{U}_{\varepsilon}) \partial_{x_j} \hat{U}_{\varepsilon} = S(\varepsilon, \hat{U}_{\varepsilon}). $$

Formally, we have

$$ (2.15) \qquad \sum_{k=0}^{+\infty} \varepsilon^k U_k(t, x) = \sum_{k=0}^{+\infty} \varepsilon^k U_k(\varepsilon^2 \tau, x) = \sum_{k=0}^{+\infty} \varepsilon^k W_k(\tau, x), $$

hence,

$$ (2.16) \qquad \sum_{k=0}^{+\infty} \varepsilon^k (U_k(t, x) + \chi_k(\tau, x)) = \sum_{k=0}^{+\infty} \varepsilon^k (W_k(\tau, x) + \chi_k(\tau, x)), $$

where

$$ W_k(\tau, x) = \sum_{h=0}^{\lfloor k/2 \rfloor} \frac{\tau^l}{l!} \frac{\partial^l U_{k-2l}}{\partial t^l}(0, x), \quad k \in \mathbb{N}, $$

with $\lfloor r \rfloor$ standing for the greatest integer less than or equal to $r$. Since the series on the left-hand side of (2.15) is assumed to be a solution to (1.1), and for $t > 0$, $\chi_k$ converges exponentially fast to zero as $\varepsilon \to 0$, we may also assume that the series in (2.16) is a solution to (1.1)-(1.2). Substituting the series on the right-hand side of (2.15) and (2.16) into (2.14), we have

$$ \left\{ \begin{array}{l} \partial_{\tau} W_0 = S(0, W_0), \\ \partial_{\tau} W_k = \partial_{U} S(0, W_0) W_k + \mathcal{F}_k(W), \quad \forall k \ge 1, \end{array} \right. $$

and

$$ \left\{ \begin{array}{l} \partial_{\tau}(W_0 + \chi_0) = S(0, W_0 + \chi_0), \\ \partial_{\tau}(W_k + \chi_k) = \partial_{U} S(0, W_0 + \chi_0)(W_k + \chi_k) + \mathcal{F}_k(W + \chi), \quad \forall k \ge 1, \end{array} \right. $$
where

$$
\mathcal{F}_k(V) =
\begin{cases}
\displaystyle \partial_{\varepsilon} S(0, V_0) - \sum_{j=1}^{d} A_j(0, V_0) \partial_{x_j} V_0, & k=1, \\
\displaystyle \frac{1}{2} [\partial_{\varepsilon}^2 S(0, V_0) + 2\partial_{\varepsilon} \partial_V S(0, V_0) V_1 + (\partial_V^2 S(0, V_0) V_1) V_1] \\
\quad - [(\partial_{\varepsilon} A_j(0, V_0) + \partial_V A_j(0, V_0) V_1) \partial_{x_j} V_0 + A_j(0, V_0) \partial_{x_j} V_1], & k=2, \\
\displaystyle \frac{1}{k!} \partial_{\varepsilon}^k S(0, V_0) + \sum_{l=1}^{k-1} \frac{1}{l!} [\partial_V \partial_{\varepsilon}^l S(0, V_0) V_{k-l} + e_{k-l} (\partial_{\varepsilon}^l S(0, \cdot), V)] \\
\quad - \sum_{j=1}^{d} [(\partial_{\varepsilon} A_j(0, V_0) + \partial_V A_j(0, V_0) V_1) \partial_{x_j} V_{k-2} + A_j(0, V_0) \partial_{x_j} V_{k-1} \\
\quad + \sum_{r=1}^{k-2} \left( \sum_{l=0}^{r} \frac{1}{l!} [\partial_V \partial_{\varepsilon}^l A_j(0, V_0) V_{r+1-l} + e_{r+1-l} (\partial_{\varepsilon}^l A_j(0, \cdot), V)] \right) \\
\quad + \frac{1}{(r+1)!} \partial_{\varepsilon}^{r+1} A_j(0, V_0) \partial_{x_j} V_{k-2-r}], & k \ge 3.
\end{cases}
$$

Remark that $\mathcal{F}_k$ depends only on $(V_0, V_1, \dots, V_{k-1})$.

It follows from $W_0(\tau, x) = U_0(0, x)$ and $S(0, W_0) = 0$ that $\chi_0$ satisfies a nonlinear system of ordinary differential equations

$$
\partial_{\tau} \chi_{0} = S(0, W_{0} + \chi_{0})
$$

and each $\chi_k$ satisfies a linear system of ordinary differential equations of the form

$$
\partial_{\tau} \chi_k = \partial_U S(0, W_0 + \chi_0) \chi_k + [\partial_U S(0, W_0 + \chi_0) - \partial_U S(0, W_0)] W_k + \mathcal{G}_k, \quad \forall k \ge 1,
$$

where

$$
\mathcal{G}_k(\tau, x) = \mathcal{F}_k(W + \chi) - \mathcal{F}_k(W), \quad k \ge 1.
$$

Since $S^I(0, U) = 0$ for all $U \in G$, we have

$$
\chi_0^I = 0,
$$

and

$$
\chi_k^I(\tau, x) = \mathcal{G}_k^I(\tau, x). \quad \forall k \ge 1.
$$

This together with $\lim_{\tau \to +\infty} \chi_k(\tau, x) = 0$ yields

$$
(2.17) \qquad \chi_0^I = 0, \quad \chi_k^I(\tau, x) = - \int_{\tau}^{+\infty} \mathcal{G}_k^I(\tau', x) d\tau', \quad \forall k \ge 1.
$$

If the initial datum $\bar{U}^\epsilon$ admits a formal asymptotic expansion

$$
\bar{U}^{\epsilon}(x) = \sum_{k=0}^{\infty} \epsilon^k \bar{U}_k(x), \quad \bar{U}_k = \begin{pmatrix} \bar{u}_k \\ \bar{v}_k \end{pmatrix},
$$

then the formal solution $\sum_{k=0}^{+\infty} \epsilon^k (U_k(t, x) + \chi_k(\tau, x))$ to (1.1)-(1.2) should satisfy

$$
U_k(0, x) + \chi_k(0, x) = \bar{U}_k(x),
$$

or equivalently

$$
\begin{cases}
u_k(0, x) + \chi_k^I(0, x) = \bar{u}_k(x), \\
v_k(0, x) + \chi_k^{II}(0, x) = \bar{v}_k(x), \quad \forall k \ge 0.
\end{cases}
$$
Since $v_0 = 0$ and $\chi_0^I = 0$, we obtain

$$ (2.18) \qquad u_0(0, x) = \bar{u}_0(x), \quad \chi_0^{II}(0, x) = \bar{v}_0(x), $$

which are the initial conditions for $u_0$ and $\chi_0^{II}$. Hence, the system of $\chi_k^{II}$ becomes

$$ (2.19) \qquad \left\{ \begin{array}{l} \partial_{\tau} \chi_{0}^{II} = s(\bar{u}_{0}(x), \chi_{0}^{II}), \\ \partial_{\tau} \chi_{k}^{II} = \partial_{v}s(\bar{u}_{0}(x), \chi_{0}^{II})\chi_{k}^{II} + [\partial_{v}s(\bar{u}_{0}(x), \chi_{0}^{II}) - \partial_{v}s(\bar{u}_{0}(x), 0)]W_{k}^{II}(\tau, x) \\ \qquad \qquad + \mathcal{G}_{k}^{II}(\tau, x), \quad \forall k \ge 1. \end{array} \right. $$

Finally, the initial conditions of $(u_k, \chi_k^{II})$ are given by

$$ (2.20) \qquad \left\{ \begin{array}{l} u_k(0, x) = \bar{u}_k(x) - \chi_k^I(0, x), \\ \chi_k^{II}(0, x) = \bar{v}_k(x) - v_k(0, x), \quad \forall k \ge 1, \end{array} \right. $$

where $\chi_k^I(\tau, x)$ and $v_k(t, x)$ are solved in previous steps. Thus, equations (2.17)-(2.20) determine $\chi_k$ and $u_k(0, x)$, and hence, $U_k$ and $\chi_k$ are constructed for all $k \in \mathbb{N}$.

### 2.3. The approximate solution.

Let $K \ge 2$ be an integer. The approximate solution is defined by

$$ U_{\varepsilon}^{K}(t, x) = \sum_{k=0}^{K} \varepsilon^{k} (U_{k}(t, x) + \chi_{k}(\tau, x)). $$

Since $T_k \in (0, T_{k-1}]$ for all $k \in \mathbb{N}$, $U_{\varepsilon}^K$ is defined on time interval $(0, T_K]$, where $T_K > 0$ is independent of $\varepsilon$.

**Proposition 2.1.** Let $m > d/2 + 1$ and $K \ge 2$ be integers. Assume (1.6)-(1.8) and (1.10) hold. Moreover, assume $\bar{U}_k (0 \le k \le K)$ are sufficiently smooth and there is a constant $c_3 > 0$ such that

$$ (2.21) \qquad \left\| \bar{U}^{\varepsilon} - \sum_{k=0}^{K} \varepsilon^{k} \bar{U}_{k} \right\|_{m} \leq c_{3} \varepsilon^{K}, $$

then $U_{\varepsilon}^K$ satisfies (1.23)-(1.26).

**Proof.** By the definition of $U_{\varepsilon}^K$ above, (1.23) is just (2.21).

Since $A_0(\varepsilon, U)$ is symmetric positive definite, so is $A_0^{22}(0, u, 0)$ thanks to (1.11). Then (1.12) implies that each eigenvalue of $\partial_v s(u, 0)$ is negative uniformly with respect to $u$. It follows that, for sufficiently small data $\bar{v}_0$, the first equation in (2.19) admits a unique global solution $\chi_0^{II}(\tau, x)$ which decays exponentially fast to zero as $\tau \to +\infty$ (see [1]). This together with $\chi_0^I = 0$ implies that there exist constants $c_3 > 0$ and $\eta > 0$ such that

$$ \|\chi_0(\tau, \cdot)\|_m \leq c_3 e^{-\eta\tau}. $$

By induction, assume, for $k \ge 1$,

$$ \|\chi_l(\tau, \cdot)\|_m \le c_3 e^{-\eta\tau}, \quad 0 \le l \le k-1. $$

From (2.17), we have

$$ \|\chi_k^I(\tau, \cdot)\|_m \le \int_{\tau}^{+\infty} \|\mathcal{G}_k^I(\tau', x)\|_m d\tau' \le c_3 \sum_{l=0}^{k-1} \|\chi_k(\tau', \cdot)\|_m d\tau' \le c_3 e^{-\eta\tau}. $$

The second equation in (2.19) is linear and can be written as

$$ \partial_{\tau} \chi_{k}^{II} = \partial_{v}s(\bar{u}_{0}, \chi_{0}^{II}) \chi_{k}^{II} + O(\chi_{0}, \chi_{1}, \dots, \chi_{k-1}). $$
By induction on $k$, we obtain

$$ \|\chi_k^{II}(\tau, \cdot)\|_m \leq c_3 e^{-\eta\tau}, $$

hence,

$$ \|\chi_k(\tau, \cdot)\|_m \leq c_3 e^{-\eta\tau}, \quad \forall 0 \leq k \leq K. $$

Thus, we have obviously (1.24). Using (2.17), (2.19) and

$$ \partial_t \chi_k \left( \frac{t}{\epsilon^2}, x \right) = \epsilon^{-2} \partial_\tau \chi_k(\tau, x), $$

we also have (1.25). Finally, (1.26) follows by a similar way as in [21]. □

### 3. PROOF OF THEOREM 1.1

Let $m > d/2 + 1$ and $K \geq 2$ be integers. Let $0 < T < \min(T_\epsilon, T_K)$. Then both the exact solution $U^\epsilon$ and the approximate solution $U_\epsilon^K$ are defined on time interval $[0, T]$, on which we define

$$ W^{\epsilon} = U^{\epsilon} - U_{\epsilon}^{K}. $$

Let $t \in [0, T]$. We assume

$$ (3.1) \qquad \|W^{\epsilon}(t)\|_{m} \leq c\epsilon, $$

where $c > 0$ is a generic constant independent of $\epsilon$. This together with (1.24) yields

$$ (3.2) \qquad \|U^{\epsilon}(t) - U_0(t)\|_{m} \leq c\epsilon + c\nu_{\epsilon}(t), \quad \|U^{\epsilon}(t)\|_{m} \leq c, $$

where

$$ \nu_{\epsilon}(t) = e^{-\frac{\mu t}{\epsilon^2}}. $$

We want to prove that

$$ (3.3) \qquad \|W^{\epsilon}(t)\|_{m}^{2} + \frac{1}{\epsilon^{2}} \int_{0}^{t} \|W^{\epsilon,II}(t')\|_{m}^{2} dt' \leq c\epsilon^{2K}. $$

Since $K \geq 2$ and $c > 0$ is a fixed constant, there exists $\epsilon_0 > 0$ such that $c\epsilon^K < \epsilon$ for all $\epsilon \in (0, \epsilon_0]$. By the same argument developed in [43], we obtain $T_\epsilon \geq T_K$, and hence estimates (1.27)-(1.28) follow from (3.3).

Thus, it suffices to prove (3.3). For this purpose, let $\| \cdot \|$ and $\| \cdot \|_\infty$ be the usual norms of $L^2 \stackrel{\text{def}}{=} L^2(\mathbb{R}^d)$ and $L^\infty \stackrel{\text{def}}{=} L^\infty(\mathbb{R}^d)$, respectively. We denote by $\langle \cdot, \cdot \rangle$ the inner product in $L^2$. For a multi-index $\alpha = (\alpha_1, \dots, \alpha_d) \in \mathbb{N}^d$ with $|\alpha| \leq m$, we denote

$$ \partial^{\alpha} = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1} \dots \partial x_d^{\alpha_d}} \quad \text{with} \quad |\alpha| = \alpha_1 + \dots + \alpha_d. $$

The following Sobolev and Moser-type inequalities can be found in [24, 43]. They will be repeatedly used in the proof of Theorem 1.1.

(i) The embedding from $H^{m-1}$ to $L^\infty$ is continuous and for $u \in H^{m-1}$,
$$ \|u\|_{\infty} \leq C_0 \|u\|_{m-1}. $$

(ii) The Sobolev space $H^{m-1}$ is an algebra and for $u, v \in H^{m-1}$,
$$ \|uv\|_{m-1} \leq C_0 \|u\|_{m-1} \|v\|_{m-1}. $$

(iii) For all $\alpha \in \mathbb{N}^d$ with $1 - |\alpha| \leq m$ and $u, v \in H^m$,
$$ \|\partial^\alpha(uv) - u\partial^\alpha v\| \leq C_0 \|\nabla u\|_{m-1} \|v\|_{|\alpha|-1}. $$
(iv) Let $g$ be a smooth function on a compact set $D \subset \mathbb{R}^n$ and $k \ge 1$ be an integer. For $u \in H^k \cap L^\infty$ satisfying $u(x) \in D$,

$$ \|D^k g(u)\| \le C_0 \|g\|_{C^k(D)} \|u\|_{\infty}^{k-1} \|u\|_k, $$

where

$$ \|D^k u\| = \sum_{|\alpha|=k} \|\partial^\alpha u\|. $$

From (1.1) and (1.22), we obtain

$$ (3.4) \qquad \partial_t W^\varepsilon + \frac{1}{\varepsilon} \sum_{j=1}^d A_j(\varepsilon, U^\varepsilon) \partial_{x_j} W^\varepsilon = \frac{a^\varepsilon}{\varepsilon} + \frac{b^\varepsilon}{\varepsilon^2} - R_\varepsilon^K, $$

where

$$ (3.5) \qquad a^\varepsilon = \sum_{j=1}^d [A_j(\varepsilon, U_\varepsilon^K) - A_j(\varepsilon, U^\varepsilon)] \partial_{x_j} U_\varepsilon^K $$

and

$$ (3.6) \qquad b^\varepsilon = S(\varepsilon, U^\varepsilon) - S(\varepsilon, U_\varepsilon^K). $$

Thanks to the symmetry of $A_0(\varepsilon, U^\varepsilon)$ and $\tilde{A}_j(\varepsilon, U^\varepsilon)$, we obtain an energy equality :

$$ \begin{aligned} \frac{d}{dt} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha W^\varepsilon, \partial^\alpha W^\varepsilon \rangle = & \langle \operatorname{div}_\varepsilon A(\varepsilon, U^\varepsilon) \partial^\alpha W^\varepsilon, \partial^\alpha W^\varepsilon \rangle + \frac{2}{\varepsilon} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha a^\varepsilon, \partial^\alpha W^\varepsilon \rangle \\ & + \frac{2}{\varepsilon^2} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha b^\varepsilon, \partial^\alpha W^\varepsilon \rangle - 2 \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha R_\varepsilon^K, \partial^\alpha W^\varepsilon \rangle \\ & + \frac{2}{\varepsilon} \sum_{j=1}^d \langle A_0(\varepsilon, U^\varepsilon) f_{\alpha j}^\varepsilon, \partial^\alpha W^\varepsilon \rangle, \end{aligned} $$

where

$$ (3.7) \qquad \operatorname{div}_\varepsilon A(\varepsilon, V) = \partial_t A_0(\varepsilon, V) + \frac{1}{\varepsilon} \sum_{j=1}^d \partial_{x_j} \tilde{A}_j(\varepsilon, V), $$

and

$$ (3.8) \qquad f_{\alpha j}^{\varepsilon} = A_j(\varepsilon, U^{\varepsilon}) \partial_{x_j} (\partial^{\alpha} W^{\varepsilon}) - \partial^{\alpha} (A_j(\varepsilon, U^{\varepsilon}) \partial_{x_j} W^{\varepsilon}). $$

We write this equality as

$$ (3.9) \qquad \frac{d}{dt} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha W^\varepsilon, \partial^\alpha W^\varepsilon \rangle = I_{1,\varepsilon}^\alpha + I_{2,\varepsilon}^\alpha + I_{3,\varepsilon}^\alpha + I_{4,\varepsilon}^\alpha + I_{5,\varepsilon}^\alpha, $$

with the natural correspondence for $I_{1,\varepsilon}^\alpha, \dots, I_{5,\varepsilon}^\alpha$.

In what follows we deal with $I_{i,\varepsilon}^\alpha$ ($1 \le i \le 5$). For simplicity, we drop $\varepsilon$ in $W^\varepsilon$ and $I_{i,\varepsilon}^\alpha$, and we adopt $\| \cdot \|_{-1} = 0$. We want to prove that

$$ (3.10) \qquad |I_i^\alpha(t)| \le \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + \frac{c}{\varepsilon^2} \|W^{II}(t)\|^2_{|\alpha|-1} \\ \qquad \qquad + c \left(1 + \frac{1}{\varepsilon^2} \nu_\varepsilon(t) + \frac{1}{\varepsilon^2} \|W^{II}(t)\|_m\right) \|W(t)\|_m^2 + c\varepsilon^{2K}, \quad \forall i \ne 3, $$

and

$$ (3.11) \qquad |I_3^\alpha(t)| \le \frac{4\delta - 2c_0}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + \frac{c}{\varepsilon^2} \|W^{II}(t)\|^2_{|\alpha|-1} + c \left(1 + \frac{1}{\varepsilon^2} \nu_\varepsilon(t)\right) \|W(t)\|_m^2, $$

where the positive constant $c_0$ is given in (1.10) and $\delta > 0$ stands for a sufficiently small constant (independent of $\varepsilon$) to be chosen later. The first term in (3.11) is obtained due to the dissipation
assumption (1.10). It will absorb the first term in (3.10). The second term in (3.10) and (3.11) can be treated by an induction argument on $|\alpha|$. Finally, applying the Gronwall inequality to (3.9) yields (3.3).

**Lemma 3.1.** For all $\alpha \in \mathbb{N}^d$ with $|\alpha| \le m$ and $t \in [0, T]$, we have

$$ (3.12) \qquad |I_1^\alpha(t)| \le \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + c \left(1 + \frac{1}{\varepsilon^2} \nu_\varepsilon(t) + \frac{1}{\varepsilon^2} \|W^{II}(t)\|_m\right) \|W(t)\|_m^2. $$

**Proof.** By definition, we have

$$ I_1^\alpha = \langle \mathrm{div}_\varepsilon A(\varepsilon, U^\varepsilon) \partial^\alpha W, \partial^\alpha W \rangle, $$

where $\mathrm{div}_\varepsilon A(\varepsilon, U^\varepsilon)$ is defined in (3.7). We first prove that

$$ (3.13) \qquad \left| \left\langle \partial_t A_0(\varepsilon, U^\varepsilon) \partial^\alpha W, \partial^\alpha W \right\rangle \right| \le c \left( 1 + \frac{1}{\varepsilon^2} \nu_\varepsilon + \frac{1}{\varepsilon^2} \|W^{II}\|_m \right) \|W\|_m^2, $$

by writing

$$ \partial_t W = -\frac{1}{\varepsilon} \sum_{j=1}^{d} A_j(\varepsilon, U^\varepsilon) \partial_{x_j} W + \frac{a^\varepsilon}{\varepsilon} + \frac{b^\varepsilon}{\varepsilon^2} - R_\varepsilon^K. $$

From (3.1) and (3.5), we have obviously

$$ \left\| \sum_{j=1}^{d} A_j(\varepsilon, U^\varepsilon) \partial_{x_j} W \right\|_{\infty} \le c \|W\|_m \le c\varepsilon, $$

and

$$ \|a^\varepsilon\|_{\infty} \le c\|W\|_m \le c\varepsilon. $$

Since $K \ge 2$, from (1.26), we also have

$$ \|R_\varepsilon^K\|_{\infty} \le \varepsilon^{K-1} \|r_K\|_{\infty} + \varepsilon^{K-1} \|F_\varepsilon^K\|_{\infty} \le c. $$

The term $b^\varepsilon$ can be written as

$$ \begin{aligned} b^\varepsilon &= S(\varepsilon, U^\varepsilon) - S(\varepsilon, U_\varepsilon^K) \\ &= S(\varepsilon, U^\varepsilon) - S(\varepsilon, U_\varepsilon^K) - \partial_U S(\varepsilon, U_\varepsilon^K) W \\ &\quad + (\partial_U S(\varepsilon, U_\varepsilon^K) - \partial_U S(\varepsilon, U_0)) W + (\partial_U S(\varepsilon, U_0) - \partial_U S(0, U_0)) W + \partial_U S(0, U_0) W. \end{aligned} $$

From (1.7), we have

$$ (3.14) \qquad \partial_U S(0, u, 0) = \begin{pmatrix} 0 & 0 \\ 0 & \partial_v s(u, 0) \end{pmatrix}, \qquad \partial_U S(0, u, 0) W = \begin{pmatrix} 0 \\ \partial_v s(u, 0) W^{II} \end{pmatrix}, \quad \forall u \in \mathbb{R}^{N-n}. $$

It follows from (1.24), (3.1) and (3.2) that

$$ \|b^\varepsilon\|_{\infty} \le c\varepsilon^2 + c\varepsilon\nu_\varepsilon + c\|W^{II}\|_m. $$

Therefore,

$$ \|\partial_t W\|_{\infty} \le c + \frac{c}{\varepsilon}\nu_{\varepsilon} + \frac{c}{\varepsilon^2}\|W^{II}\|_m. $$

This inequality together with (1.25) and

$$ \partial_t A_0(\varepsilon, U^\varepsilon) = \partial_U A_0(\varepsilon, U^\varepsilon) (\partial_t W + \partial_t U_\varepsilon^K) $$

implies (3.13).

Next, since $\tilde{A}_j(\varepsilon, U^\varepsilon)$ is symmetric, we have

$$ \begin{aligned} \langle \partial_{x_j} \tilde{A}_j(\varepsilon, U^\varepsilon) \partial^\alpha W, \partial^\alpha W \rangle = & \langle \partial_{x_j} \tilde{A}_j^{11}(\varepsilon, U^\varepsilon) \partial^\alpha W^I, \partial^\alpha W^I \rangle + 2 \langle \partial_{x_j} \tilde{A}_j^{12}(\varepsilon, U^\varepsilon) \partial^\alpha W^{II}, \partial^\alpha W^I \rangle \\ & + \langle \partial_{x_j} \tilde{A}_j^{22}(\varepsilon, U^\varepsilon) \partial^\alpha W^{II}, \partial^\alpha W^{II} \rangle. \end{aligned} $$
The last two terms above are bounded by

$$c\|W\|_m \|\partial^\alpha W^{II}\| \le \frac{\delta}{d\epsilon} \|\partial^\alpha W^{II}\|^2 + c\epsilon\|W\|_m^2.$$

For the first term on the right-hand side, we use (1.6) and (1.11) to get

$$\tilde{A}_j^{11}(0, u, 0) = 0, \quad \forall (u, 0) \in G.$$

It follows from (3.2) and $v_0 = 0$ that

$$\|\partial_{x_j} \tilde{A}_j^{11}(\varepsilon, U^\varepsilon)\|_\infty \le c\varepsilon.$$

Therefore,

$$|\langle \partial_{x_j} \tilde{A}_j^{11}(\varepsilon, U^\varepsilon) \partial^\alpha W^I, \partial^\alpha W^I \rangle| \le c\varepsilon \|W\|_m^2,$$

and thus,

$$\frac{1}{\varepsilon} \left| \left\langle \sum_{j=1}^{d} \partial_{x_j} \tilde{A}_j(\varepsilon, U^\varepsilon) \partial^\alpha W, \partial^\alpha W \right\rangle \right| \le \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}\|^2 + c \|W\|_m^2.$$

This together with (3.13) yields (3.12). □

**Lemma 3.2.** For all $\alpha \in \mathbb{N}^d$ with $|\alpha| \le m$ and $t \in [0, T]$, we have

$$(3.15) \qquad |I_2^\alpha(t)| \le \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + \frac{c}{\varepsilon^2} \|W^{II}(t)\|_{|\alpha|-1}^2 + c \left(1 + \frac{1}{\varepsilon}\nu_\varepsilon(t)\right) \|W(t)\|_m^2.$$

**Proof.** The definition of $a^\varepsilon$ in (3.5) gives

$$I_2^\alpha = \frac{2}{\varepsilon} \sum_{j=1}^{d} \langle (A_0(\varepsilon, U^\varepsilon) - A_0(0, U_0)) \partial^\alpha [(A_j(\varepsilon, U_\varepsilon^K) - A_j(\varepsilon, U^\varepsilon))\partial_{x_j} U_\varepsilon^K], \partial^\alpha W \rangle \\ + \frac{2}{\varepsilon} \sum_{j=1}^{d} \langle A_0(0, U_0) \partial^\alpha [(A_j(\varepsilon, U_\varepsilon^K) - A_j(\varepsilon, U^\varepsilon))\partial_{x_j} U_\varepsilon^K], \partial^\alpha W \rangle. \tag{3.16}$$

From (3.2), it is clear that the first term on the right-hand side of (3.16) is bounded by $c(1 + \frac{1}{\varepsilon}\nu_\varepsilon)\|W\|_m^2$.

Since $A_0^{12}(0, u, 0) = 0$ (see (1.11)), the last term in (3.16) can be written as

$$\begin{align} & \frac{2}{\varepsilon} \sum_{j=1}^{d} \langle A_0(0, U_0) \partial^\alpha [(A_j(\varepsilon, U_\varepsilon^K) - A_j(\varepsilon, U^\varepsilon))\partial_{x_j} U_\varepsilon^K], \partial^\alpha W \rangle \nonumber \\ & = \frac{2}{\varepsilon} \sum_{j=1}^{d} \langle A_0^{11}(0, U_0) \partial^\alpha [(A_j^{11}(\varepsilon, U_\varepsilon^K) - A_j^{11}(\varepsilon, U^\varepsilon))\partial_{x_j} U_\varepsilon^K], \partial^\alpha W^I \rangle \nonumber \\ & \quad + \frac{2}{\varepsilon} \sum_{j=1}^{d} \langle A_0^{11}(0, U_0) \partial^\alpha [(A_j^{12}(\varepsilon, U_\varepsilon^K) - A_j^{12}(\varepsilon, U^\varepsilon))\partial_{x_j} v_\varepsilon^K], \partial^\alpha W^I \rangle \nonumber \\ & \quad + \frac{2}{\varepsilon} \sum_{j=1}^{d} \langle A_0^{22}(0, U_0) \partial^\alpha [(A_j^{21}(\varepsilon, U_\varepsilon^K) - A_j^{21}(\varepsilon, U^\varepsilon))\partial_{x_j} u_\varepsilon^K], \partial^\alpha W^{II} \rangle \nonumber \\ & \quad + \frac{2}{\varepsilon} \sum_{j=1}^{d} \langle A_0^{22}(0, U_0) \partial^\alpha [(A_j^{22}(\varepsilon, U_\varepsilon^K) - A_j^{22}(\varepsilon, U^\varepsilon))\partial_{x_j} v_\varepsilon^K], \partial^\alpha W^{II} \rangle. \tag{3.17} \end{align}$$

Obviously, the last two terms in (3.17) are bounded by

$$\frac{c}{\varepsilon} \|W\|_m \|\partial^\alpha W^{II}\| \le \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}\|^2 + c \|W\|_m^2.$$
Since $v_0 = 0$, (1.24) yields $\|v_\varepsilon^K\|_m \le c\varepsilon + c\nu_\varepsilon$. Therefore,

$$ \frac{2}{\varepsilon} \left| \sum_{j=1}^{d} \left\langle A_0^{11}(0, U_0) \partial^{\alpha} \left[ (A_j^{12}(\varepsilon, U_\varepsilon^K) - A_j^{12}(\varepsilon, U^\varepsilon)) \partial_{x_j} v_\varepsilon^K \right], \partial^\alpha W^I \right\rangle \right| \le c \left( 1 + \frac{1}{\varepsilon} \nu_\varepsilon \right) \|W\|_m^2. $$

For the first term on the right-hand side of (3.17), we have

$$ \begin{aligned} A_j^{11}(\varepsilon, U_\varepsilon^K) - A_j^{11}(\varepsilon, U^\varepsilon) &= [A_j^{11}(\varepsilon, u_\varepsilon^K, v_\varepsilon^K) - A_j^{11}(\varepsilon, u^\varepsilon, v_\varepsilon^K)] + [A_j^{11}(\varepsilon, u^\varepsilon, v_\varepsilon^K) - A_j^{11}(\varepsilon, u^\varepsilon, v^\varepsilon)] \\ &= - \int_0^1 \partial_u A_j^{11}(\varepsilon, u_\varepsilon^K + \theta(u^\varepsilon - u_\varepsilon^K), v_\varepsilon^K) W^I d\theta \\ &\quad - \int_0^1 \partial_v A_j^{11}(\varepsilon, u^\varepsilon, v_\varepsilon^K + \theta(v^\varepsilon - v_\varepsilon^K)) W^{II} d\theta. \end{aligned} $$

The second integral above is easily estimated due to the appearance of $W^{II}$. For the first one, we use

$$ \partial_u A_j^{11}(0, u, 0) = 0, \quad \|v_\varepsilon^K\|_m \le c\varepsilon + c\nu_\varepsilon. $$

It follows that

$$ \begin{aligned} & \frac{2}{\varepsilon} \left| \sum_{j=1}^{d} \left\langle A_0^{11}(0, U_0) \partial^{\alpha} \left[ (A_j^{11}(\varepsilon, U_\varepsilon^K) - A_j^{11}(\varepsilon, U^\varepsilon)) \partial_{x_j} u_\varepsilon^K \right], \partial^\alpha W^I \right\rangle \right| \\ & \le \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + \frac{c}{\varepsilon^2} \|W^{II}(t)\|^2_{|\alpha|-1} + c \left(1 + \frac{1}{\varepsilon}\nu_\varepsilon\right) \|W\|_m^2. \end{aligned} $$

This proves (3.15).

**Lemma 3.3.** For all $\alpha \in \mathbb{N}^d$ with $|\alpha| \le m$ and $t \in [0, T]$, we have

$$ (3.18) \qquad I_3^\alpha(t) \le \frac{4\delta - 2c_0}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + \frac{c}{\varepsilon^2} \|W^{II}(t)\|^2_{|\alpha|-1} + c \left(1 + \frac{1}{\varepsilon^2}\nu_\varepsilon(t)\right) \|W(t)\|_m^2, $$

where the positive constant $c_0$ is given in (1.10).

**Proof.** By definition, we have

$$ I_3^\alpha = \frac{2}{\varepsilon^2} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha b^\varepsilon, \partial^\alpha W \rangle, $$

where $b^\varepsilon$ is defined in (3.6). We first write $b^\varepsilon$ as

$$ \begin{aligned} b^\varepsilon = & \partial_U S(0, U_0)W \\ & + \varepsilon \partial_U \partial_\varepsilon S(0, U_0)W \\ & + [S(0, U^\varepsilon) - S(0, U_\varepsilon^K) - \partial_U S(0, U_0)W] \\ & + \varepsilon [\partial_\varepsilon S(0, U^\varepsilon) - \partial_\varepsilon S(0, U_\varepsilon^K) - \partial_U \partial_\varepsilon S(0, U_0)W] \\ & + [S(\varepsilon, U^\varepsilon) - S(0, U^\varepsilon) - \varepsilon \partial_\varepsilon S(0, U^\varepsilon) - (S(\varepsilon, U_\varepsilon^K) - S(0, U_\varepsilon^K) - \varepsilon \partial_\varepsilon S(0, U_\varepsilon^K))] \end{aligned} $$

which implies that

$$ I_3^\alpha = I_{31}^\alpha + I_{32}^\alpha + I_{33}^\alpha + I_{34}^\alpha + I_{35}^\alpha, $$

with the natural correspondence for $I_{31}^\alpha, \dots, I_{35}^\alpha$. Now we estimate each of these terms.

(i) First,

$$ I_{31}^{\alpha} = \frac{2}{\varepsilon^2} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha [\partial_U S(0, U_0)W], \partial^\alpha W \rangle, $$
where

$$A_0(\varepsilon, U^\varepsilon)\partial^\alpha [\partial_U S(0, U_0)W] = A_0(0, U_0)\partial_U S(0, U_0)\partial^\alpha W \\ + A_0(0, U_0)[\partial^\alpha (\partial_U S(0, U_0)W) - \partial_U S(0, U_0)\partial^\alpha W] \\ + [A_0(\varepsilon, U^\varepsilon) - A_0(0, U_0)]\partial^\alpha [\partial_U S(0, U_0)W].$$

The partial dissipation condition (1.10) implies that

$$\langle A_0(0, U_0) \partial_U S(0, U_0) \partial^\alpha W, \partial^\alpha W \rangle \leq -c_0 \|\partial^\alpha W^{II}\|^2.$$

Applying the Moser-type calculus inequalities and using (3.14) and $A_0^{12}(0, U_0) = 0$ yield

$$\begin{aligned} & \langle A_0(0, U_0)[\partial^\alpha (\partial_U S(0, U_0)W) - \partial_U S(0, U_0)\partial^\alpha W], \partial^\alpha W \rangle \\ &= \langle A_0^{22}(0, U_0)[\partial^\alpha (\partial_v s(U_0)W^{II}) - \partial_v s(U_0)\partial^\alpha W^{II}], \partial^\alpha W^{II} \rangle \\ &\leq c \|W^{II}\|_{|\alpha|-1} \|\partial^\alpha W^{II}\| \\ &\leq \frac{\delta}{4} \|\partial^\alpha W^{II}\|^2 + c \|W^{II}\|^2_{|\alpha|-1}. \end{aligned}$$

For the last term in $I_{31}^\alpha$, noting

$$\|A_0(\varepsilon, U^\varepsilon) - A_0(0, U_0)\|_{\infty} \leq c\varepsilon + c\nu_\varepsilon, \quad \nu_\varepsilon^2 \leq \nu_\varepsilon,$$

a similar calculation yields

$$\begin{aligned} & \langle [A_0(\varepsilon, U^\varepsilon) - A_0(0, U_0)] \partial^\alpha [\partial_U S(0, U_0)W], \partial^\alpha W \rangle \\ &= \langle [A_0^{12}(\varepsilon, U^\varepsilon) - A_0^{12}(0, U_0)] \partial^\alpha [\partial_v s(U_0)W^{II}], \partial^\alpha W^I \rangle \\ &\quad + \langle [A_0^{22}(\varepsilon, U^\varepsilon) - A_0^{22}(0, U_0)] \partial^\alpha [\partial_v s(U_0)W^{II}], \partial^\alpha W^{II} \rangle \\ &\leq c (\varepsilon + \nu_\varepsilon) \|W^{II}\|_{|\alpha|} \|W\|_{|\alpha|} \\ &\leq \frac{\delta}{4} \|\partial^\alpha W^{II}\|^2 + c \|W^{II}\|^2_{|\alpha|-1} + c(\varepsilon^2 + \nu_\varepsilon) \|W\|_m^2. \end{aligned}$$

Combining the estimates above, we obtain

$$I_{31}^{\alpha} \leq \frac{\delta - 2c_0}{\varepsilon^2} \|\partial^\alpha W^{II}\|^2 + \frac{c}{\varepsilon^2} \|W^{II}\|_{|\alpha|-1}^2 + c \left(1 + \frac{1}{\varepsilon^2} \nu_\varepsilon\right) \|W\|_m^2.$$

(ii) Using (1.7), we have

$$\partial_U \partial_\varepsilon S(0, U_0) = \begin{pmatrix} 0 & 0 \\ \partial_u \partial_\varepsilon S^{II}(0, U_0) & \partial_v \partial_\varepsilon S^{II}(0, U_0) \end{pmatrix},$$

hence,

$$\begin{aligned} I_{32}^{\alpha} ={}& \frac{2}{\varepsilon} \langle [A_0^{12}(\varepsilon, U^\varepsilon) - A_0^{12}(0, U_0)] \partial^\alpha (\partial_u \partial_\varepsilon S^{II}(0, U_0)W^I), \partial^\alpha W^I \rangle \\ & + \frac{2}{\varepsilon} \langle A_0^{12}(\varepsilon, U^\varepsilon) \partial^\alpha (\partial_v \partial_\varepsilon S^{II}(0, U_0)W^{II}), \partial^\alpha W^I \rangle \\ & + \frac{2}{\varepsilon} \langle A_0^{22}(\varepsilon, U^\varepsilon) \partial^\alpha [\partial_u \partial_\varepsilon S^{II}(0, U_0)W^I + \partial_v \partial_\varepsilon S^{II}(0, U_0)W^{II}], \partial^\alpha W^{II} \rangle, \end{aligned}$$

which implies that

$$|I_{32}^{\alpha}| \leq \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}\|^2 + \frac{c}{\varepsilon^2} \|W^{II}\|^2_{|\alpha|-1} + c \left(1 + \frac{1}{\varepsilon}\nu_\varepsilon\right) \|W\|_m^2.$$
(iii) For $I_{33}^{\alpha}$, since $S^I(0, U_0) = 0$, we have

$$ S(0, U^{\epsilon}) - S(0, U_{\epsilon}^{K}) - \partial_{U}S(0, U_{0})W \\ = \left( \begin{array}{cc} 0 & 0 \\ s(U^{\epsilon}) - s(U_{\epsilon}^{K}) - s'(U_{\epsilon}^{K})W & (s'(U_{\epsilon}^{K}) - s'(U_{0}))W \end{array} \right), $$

which implies that, with $A_0^{12}(0, U_0) = 0$,

$$ \begin{aligned} I_{33}^{\alpha} &= \frac{2}{\epsilon^2} \langle A_0(\epsilon, U_0) \partial^{\alpha} (S(0, U^{\epsilon}) - S(0, U_{\epsilon}^{K}) - \partial_{U}S(0, U_0)W), \partial^{\alpha}W \rangle \\ &= \frac{2}{\epsilon^2} \langle (A_0(\epsilon, U_0) - A_0(0, U_0)) \partial^{\alpha} (S(0, U^{\epsilon}) - S(0, U_{\epsilon}^{K}) - \partial_{U}S(0, U_0)W, \partial^{\alpha}W) \rangle \\ &\quad + \langle A_0^{22}(0, U_0) \partial^{\alpha} (s(U^{\epsilon}) - s(U_{\epsilon}^{K}) - s'(U_{\epsilon}^{K})W + (s'(U_{\epsilon}^{K}) - s'(U_0))W), \partial^{\alpha}W^{II} \rangle. \end{aligned} $$

Thus, as above,

$$ |I_{33}^{\alpha}| \leq \frac{\delta}{\epsilon^2} \|\partial^{\alpha} W^{II}\|^2 + c \left(1 + \frac{1}{\epsilon^2} \nu_{\epsilon}\right) \|W\|_{m}^{2}. $$

(iv) For $I_{34}^{\alpha}$, using $\partial_{\epsilon}S^I(0, U_0) = 0$, it yields

$$ \begin{aligned} |I_{34}^{\alpha}| &= \frac{2}{\epsilon} |\langle A_0(\epsilon, U^{\epsilon}) \partial^{\alpha} (\partial_{\epsilon}S(0, U^{\epsilon}) - \partial_{\epsilon}S(0, U_{\epsilon}^{K}) - \partial_{U}\partial_{\epsilon}S(0, U_0)W), \partial^{\alpha}W \rangle| \\ &\leq \frac{\delta}{\epsilon^2} \|\partial^{\alpha}W^{II}\|^2 + \frac{c}{\epsilon^2} \|W^{II}\|_{|\alpha|-1}^2 + c \left(1 + \frac{1}{\epsilon}\nu_{\epsilon}\right) \|W\|_{m}^2. \end{aligned} $$

(v) For the last term in $I_3^\alpha$, we use

$$ \begin{aligned} S(\epsilon, U^{\epsilon}) - S(0, U^{\epsilon}) - \epsilon \partial_{\epsilon} S(0, U^{\epsilon}) - (S(\epsilon, U_{\epsilon}^{K}) - S(0, U_{\epsilon}^{K}) - \epsilon \partial_{\epsilon} S(0, U_{\epsilon}^{K})) \\ = \epsilon^2 \int_0^1 \int_0^1 \partial_{U} \partial_{\epsilon\epsilon}^2 S(\theta\epsilon, \tau U^{\epsilon} + (1-\tau)U_{\epsilon}^{K}) W d\tau d\theta \end{aligned} $$

to obtain

$$ |I_{35}^{\alpha}| \leq \frac{2}{\epsilon^2} c \epsilon^2 \|W\|_m \|W\|_m \leq c \|W\|_m^2. $$

From the estimates in (i)-(v), we obtain (3.18). $\square$

**Lemma 3.4.** For all $\alpha \in \mathbb{N}^d$ with $|\alpha| \leq m$ and $t \in [0, T]$, we have

$$ (3.19) \qquad |I_4^\alpha(t)| \leq \frac{\delta}{\epsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + c \left(1 + \frac{1}{\epsilon^2} \nu_\epsilon(t)\right) \|W(t)\|_m^2 + c\epsilon^{2K}. $$

**Proof.** We have

$$ I_4^\alpha = -2 \langle [A_0(\epsilon, U^\epsilon) - A_0(0, U_0)] \partial^\alpha R^\epsilon, \partial^\alpha W \rangle - 2 \langle A_0(0, U_0) \partial^\alpha R^\epsilon, \partial^\alpha W \rangle. $$

Using (1.26) and (3.2) we obtain

$$ \begin{aligned} |\langle [A_0(\epsilon, U^{\epsilon}) - A_0(0, U_0)] \partial^{\alpha} R^{\epsilon}, \partial^{\alpha} W \rangle| &\leq c \epsilon^K \left(1 + \frac{1}{\epsilon} \nu_{\epsilon}\right) (\|r_K\|_m + \|F_{\epsilon}^K\|_m) \|W\|_m \\ &\leq c \left(1 + \frac{1}{\epsilon^2} \nu_{\epsilon}\right) \|W\|_m^2 + c \epsilon^{2K}. \end{aligned} $$

For the last term in $I_4^\alpha$, we use $A_0^{12}(0, U_0) = 0$ to get

$$ \begin{aligned} |\langle A_0(0, U_0) \partial^{\alpha} R^{\epsilon}, \partial^{\alpha} W \rangle| &= \epsilon^{K-1} |\langle A_0^{22}(0, U_0) \partial^{\alpha} r_K, \partial^{\alpha} W^{II} \rangle + \langle A_0(0, U_0) \partial^{\alpha} F_{\epsilon}^K, \partial^{\alpha} W \rangle| \\ &\leq \frac{\delta}{2\epsilon^2} \|\partial^{\alpha} W^{II}\|^2 + c \left(1 + \frac{1}{\epsilon^2} \nu_{\epsilon}\right) \|W\|_m^2 + c\epsilon^{2K}. \end{aligned} $$

This proves (3.19). $\square$
**Lemma 3.5.** For all $\alpha \in \mathbb{N}^d$ with $|\alpha| \le m$ and $t \in [0, T]$, we have

$$ (3.20) \qquad |I_5^\alpha(t)| \le \frac{\delta}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 + \frac{c}{\varepsilon^2} \|W^{II}(t)\|_{|\alpha|-1}^2 + c \left(1 + \frac{1}{\varepsilon}\nu_\varepsilon(t)\right) \|W(t)\|_m^2. $$

**Proof.** By definition, we have

$$ I_5^\alpha = \frac{2}{\varepsilon} \sum_{j=1}^d \langle A_0(\varepsilon, U^\varepsilon) f_{\alpha j}^\varepsilon, \partial^\alpha W \rangle, $$

where $f_{\alpha j}^\varepsilon$ is defined in (3.8). Then

$$ \begin{aligned} A_0(\varepsilon, U^\varepsilon) f_{\alpha j}^\varepsilon = & A_0(\varepsilon, U^\varepsilon) \biggl[ (A_j(\varepsilon, U^\varepsilon) \partial_{x_j} (\partial^\alpha W) - \partial^\alpha (A_j(\varepsilon, U^\varepsilon) \partial_{x_j} W)) \\ & - A_0(0, U_0) \biggl[ (A_j(0, U_0) \partial_{x_j} (\partial^\alpha W) - \partial^\alpha (A_j(0, U_0) \partial_{x_j} W)) \\ & + A_0(0, U_0) \biggl[ (A_j(0, U_0) \partial_{x_j} (\partial^\alpha W) - \partial^\alpha (A_j(0, U_0) \partial_{x_j} W)) \biggr] \biggr]. \end{aligned} $$

By the Moser-type calculus inequalities together with (3.2), the first two terms in $I_5^\alpha$ are bounded by $c(1 + \frac{1}{\varepsilon}\nu_\varepsilon) \|W\|_m^2$.

For the last term in $I_5^\alpha$, we use again $A_0^{12}(0, U_0) = 0$ and $A_j^{11}(0, U_0) = 0$ to obtain

$$ \begin{aligned} & \langle A_0(0, U_0) [A_j(0, U_0) \partial_{x_j} (\partial^\alpha W) - \partial^\alpha (A_j(0, U_0) \partial_{x_j} W)], \partial^\alpha W \rangle \\ = & \langle A_0^{11}(0, U_0) [A_j^{12}(0, U_0) \partial_{x_j} (\partial^\alpha W^{II}) - \partial^\alpha (A_j^{12}(0, U_0) \partial_{x_j} W^{II})], \partial^\alpha W^I \rangle \\ & + \langle A_0^{22}(0, U_0) [A_j^{21}(0, U_0) \partial_{x_j} (\partial^\alpha W^I) - \partial^\alpha (A_j^{21}(0, U_0) \partial_{x_j} W^I)], \partial^\alpha W^{II} \rangle \\ & + \langle A_0^{22}(0, U_0) [A_j^{22}(0, U_0) \partial_{x_j} (\partial^\alpha W^{II}) - \partial^\alpha (A_j^{22}(0, U_0) \partial_{x_j} W^{II})], \partial^\alpha W^{II} \rangle, \end{aligned} $$

in which each term on the right-hand side contains $W^{II}$. Therefore, by the Moser-type calculus inequalities,

$$ |\langle A_0(0, U_0) [A_j(0, U_0) \partial_{x_j} (\partial^\alpha W) - \partial^\alpha (A_j(0, U_0) \partial_{x_j} W)], \partial^\alpha W \rangle| \le c \|W^{II}\|_{|\alpha|} \|W\|_m. $$

This implies (3.20). □

**Lemma 3.6.** For all $t \in [0, T]$, (3.3) holds.

**Proof.** Combining (3.9), the estimates in Lemmas 3.1-3.5, and choosing $\delta = c_0/8$, we obtain

$$ \frac{d}{dt} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha W, \partial^\alpha W \rangle + \frac{c_0}{\varepsilon^2} \|\partial^\alpha W^{II}(t)\|^2 \le \frac{c}{\varepsilon^2} \|W^{II}(t)\|_{|\alpha|-1}^2 + c \left(1 + \frac{1}{\varepsilon^2}\nu_\varepsilon(t)\right) \|W(t)\|_m^2 \\ + \frac{c}{\varepsilon^2} \|W^{II}(t)\|_m \|W(t)\|_m^2 + c\varepsilon^{2K}. $$

Let $0 \le k \le m$ be an integer and

$$ \mathcal{L}_k(t) = \sum_{|\alpha| \le k} \langle A_0(\varepsilon, U^\varepsilon) \partial^\alpha W, \partial^\alpha W \rangle. $$

Then

$$ \mathcal{L}'_k(t) + \frac{c_0}{\varepsilon^2} \|W^{II}(t)\|_k^2 \le \frac{c}{\varepsilon^2} \|W^{II}(t)\|_{k-1}^2 + c \left(1 + \frac{1}{\varepsilon^2}\nu_\varepsilon(t)\right) \|W(t)\|_m^2 \\ + \frac{c}{\varepsilon^2} \|W^{II}(t)\|_m \|W(t)\|_m^2 + c\varepsilon^{2K}, \quad \forall 0 \le k \le m. $$

Recall $\|W^{II}\|_{-1} = 0$. By induction on $k$, we have

$$ \mathcal{L}'_m(t) + \frac{c_1}{\varepsilon^2} \|W^{II}(t)\|_m^2 \le c \left(1 + \frac{1}{\varepsilon^2}\nu_\varepsilon(t)\right) \|W(t)\|_m^2 + \frac{c}{\varepsilon^2} \|W^{II}(t)\|_m \|W(t)\|_m^2 + c\varepsilon^{2K}, $$
for some constant $c_1 > 0$. It follows from (3.1) that

$$ \mathcal{L}'_m(t) + \frac{c_1}{\varepsilon^2} \|W^{II}(t)\|_m^2 \le c \left(1 + \frac{1}{\varepsilon^2} \nu_\varepsilon(t)\right) \|W(t)\|_m^2 + c\varepsilon^{2K}, $$

provided that $\varepsilon > 0$ is sufficiently small. Since $A_0(\varepsilon, U^\varepsilon)$ is uniformly positive definite, $\mathcal{L}_m$ is uniformly equivalent to $\|W\|_m^2$. Recall $T \le T_K$. Applying the Gronwall inequality and using

$$ \int_0^T \left( 1 + \frac{1}{\varepsilon^2} \nu_\varepsilon(t') \right) dt' \le c, $$

we obtain (3.3).

<figure><img src="image_3.png" /></figure>

## 4. EXAMPLES

As mentioned in the introduction, Theorem 1.1 includes the results obtained in [21] and also in [33] where differential constraints are not concerned. Therefore, it can be applied to the examples in these papers, except the Euler-Maxwell system which is combined with differential constraints. For the application of Theorem 1.1, here we give additional examples which cannot be written in the form of (1.1) with (1.4) or (1.5).

**Example 4.1.** The Timoshenko-Cattaneo system.

The Timoshenko-Cattaneo system is a one-dimensional model in thermoelasticity (see [40, 41, 5, 14, 8] and references therein):

$$ (4.1) \qquad \left\{ \begin{array}{l} \partial_{tt}\varphi - \partial_x(\partial_x\varphi - \psi) = 0, \\ \partial_{tt}\psi - \partial_x(\sigma(\partial_x\psi)) - (\partial_x\varphi - \psi) + b\partial_x\theta = 0, \\ \partial_t\theta + \partial_xq + b\partial_{tx}\psi = 0, \\ \varepsilon^2\partial_tq + \kappa\partial_x\theta = -q, \end{array} \right. $$

where $b$ and $\kappa$ are positive constants, $\varepsilon > 0$ is a relaxation time, and $\sigma$ is a smooth function satisfying $\sigma'(z) \ge \sigma_*$ for all $z \in \mathbb{R}$ with $\sigma_* > 0$ being a constant. The last equation in (4.1) stands for the Cattaneo law for heat conduction from which we recover the Fourier law as $\varepsilon \to 0$:

$$ q = -\kappa\partial_x\theta. $$

We introduce

$$ w = \partial_t\varphi, \quad h = \partial_x\varphi - \psi, \quad y = \partial_t\psi \quad z = \partial_x\psi, \quad v = \varepsilon q. $$

Then the Timoshenko-Cattaneo system becomes a system of balance laws:

$$ (4.2) \qquad \left\{ \begin{array}{l} \partial_t w - \partial_x h = 0, \\ \partial_t z - \partial_x y = 0, \\ \partial_t \theta + b \partial_x y + \frac{1}{\varepsilon} \partial_x v = 0, \\ \partial_t h - \partial_x w = -y, \\ \partial_t y - \partial_x \sigma(z) + b \partial_x \theta = h, \\ \partial_t v + \frac{\kappa}{\varepsilon} \partial_x \theta = -\frac{1}{\varepsilon^2} v. \end{array} \right. $$
Let

$$ U = \begin{pmatrix} w \\ z \\ \theta \\ h \\ y \\ v \end{pmatrix}, \quad f(\varepsilon, U) = \begin{pmatrix} -\varepsilon h \\ -\varepsilon y \\ \varepsilon by + v \\ -\varepsilon w \\ -\varepsilon \sigma(z) + \varepsilon b \theta \\ \kappa \theta \end{pmatrix}, \quad S(\varepsilon, U) = - \begin{pmatrix} 0 \\ 0 \\ 0 \\ \varepsilon^2 y \\ -\varepsilon^2 h \\ v \end{pmatrix}. $$

Then system (4.2) is written in the form of (1.17), i.e.

$$ \partial_t U + \frac{1}{\varepsilon} \partial_x f(\varepsilon, U) = \frac{S(\varepsilon, U)}{\varepsilon^2}, $$

and

$$ A(\varepsilon, U) \stackrel{\text{def}}{=} \partial_U f(\varepsilon, U) = \begin{pmatrix} 0 & 0 & 0 & -\varepsilon & 0 & 0 \\ 0 & 0 & 0 & 0 & -\varepsilon & 0 \\ 0 & 0 & 0 & 0 & \varepsilon b & 1 \\ -\varepsilon & 0 & 0 & 0 & 0 & 0 \\ 0 & -\varepsilon \sigma'(z) & \varepsilon b & 0 & 0 & 0 \\ 0 & 0 & \kappa & 0 & 0 & 0 \end{pmatrix}. $$

Since $\sigma'(z) \ge \sigma_* > 0$, the matrix $A_0$ defined by

$$ A_0(U) \stackrel{\text{def}}{=} \operatorname{diag}(1, \sigma'(z), 1, 1, 1, \frac{1}{\kappa}) $$

is symmetric positive definite and

$$ A_0(U)A(\varepsilon, U) = \begin{pmatrix} 0 & 0 & 0 & -\varepsilon & 0 & 0 \\ 0 & 0 & 0 & 0 & -\varepsilon\sigma'(z) & 0 \\ 0 & 0 & 0 & 0 & \varepsilon b & 1 \\ -\varepsilon & 0 & 0 & 0 & 0 & 0 \\ 0 & -\varepsilon\sigma'(z) & \varepsilon b & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix}, $$

which is a symmetric matrix. Therefore, system (4.2) is symmetrizable hyperbolic.

Here $N = 6, n = 1,$

$$ u = (w, z, \theta, h, y)^{\top}, \quad s(u, v) = -v, $$

and

$$ A^{11}(\varepsilon, U) = \varepsilon \begin{pmatrix} 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & b \\ -1 & 0 & 0 & 0 & 0 \\ 0 & -\sigma'(z) & b & 0 & 0 \end{pmatrix}, \quad S^I(\varepsilon, U) = \varepsilon^2 \begin{pmatrix} 0 \\ 0 \\ 0 \\ -y \\ h \end{pmatrix}, $$

where the symbol $\top$ stands for the transpose. Moreover,

$$ A_0(U) \partial_U S(0, U) = \operatorname{diag}(0, 0, 0, 0, 0, -\frac{1}{\kappa}). $$

It is easy to see that assumptions (1.6)-(1.8) and (1.10) are satisfied, and as $\varepsilon \to 0$, system (4.2) converges formally to the Timoshenko-Fourier system

$$ \left\{ \begin{aligned} \partial_t w - \partial_x h &= 0, \\ \partial_t z - \partial_x y &= 0, \\ \partial_t \theta + b \partial_x &= \kappa \partial_{xx} \theta, \\ \partial_t h - \partial_x w &= -y, \\ \partial_t y - \partial_x \sigma(z) + b \partial_x \theta &= h, \end{aligned} \right. $$
and

$$q = -\kappa \partial_x \theta.$$

In the above system, the equation for $\theta$ is parabolic and those for $(w, z, h, y)$ are hyperbolic. Therefore, the system is of hyperbolic-parabolic type. The local-in-time existence of smooth solutions can be found in [14, 8, 4]. Thus, Theorem 1.1 can be applied to (4.2).

**Example 4.2.** Hyperbolic isentropic Navier-Stokes equations.

Let $\lambda > 0$ be the viscosity coefficient. The compressible isentropic Navier-Stokes equations are written as

$$ (4.3) \qquad \begin{cases} \partial_t \rho + \operatorname{div}(\rho w) = 0, \\ \partial_t (\rho w) + \operatorname{div}(\rho w \otimes w) + \nabla p(\rho) = \lambda \Delta w, \quad t > 0, x \in \mathbb{R}^d. \end{cases} $$

Here $\rho > 0$, $w = (w_1, \dots, w_d)^\top$ and $p$ are the density, the velocity and the pressure function, respectively. The symbol $\otimes$ stands for the tensor product. We assume $p'(\rho) > 0$ for all $\rho > 0$. In physical situations, $d = 1, 2, 3$.

Equations (4.3) can be approximated by first-order hyperbolic systems called hyperbolic Navier-Stokes equations. These systems are obtained by the decomposition of $\Delta$ into two first-order operators in space, see [29, 5, 36, 6, 3, 45, 13, 7]. In [32], the first author of the present paper constructed approximate systems with vector variables instead of tensor variables by using Hurwitz-Radon matrices. More precisely, let $\varepsilon > 0$ be a relaxation time and $M_j (1 \le j \le d)$ be constant matrices of order $d \times r$ ($d \le r$) which satisfy

$$ (4.4) \qquad M_i M_i^\top = I_d, \quad \forall 1 \le i \le d \text{ and } M_i M_j^\top + M_j M_i^\top = 0, \quad \forall 1 \le i, j \le d, \ j \ne i. $$

The existence of such matrices was established in [32] for all $d \ge 1$. Then we have a decomposition

$$ \Delta w = \sum_{i=1}^{d} M_i \partial_{x_i} \left( \sum_{j=1}^{d} M_j^\top \partial_{x_j} w \right). $$

Introduce an auxiliary variable $v \in \mathbb{R}^r$ and its first-order evolution equation

$$ \partial_t v + \frac{\sqrt{\lambda}}{\varepsilon} \sum_{j=1}^{d} M_j^\top \partial_{x_j} w = -\frac{v}{\varepsilon^2}. $$

Then the hyperbolic Navier-Stokes equations for (4.3) are (see [32])

$$ (4.5) \qquad \begin{cases} \partial_t \rho + \operatorname{div}(\rho w) = 0, \\ \partial_t (\rho w) + \operatorname{div}(\rho w \otimes w) + \nabla p(\rho) + \frac{\sqrt{\lambda}}{\varepsilon} \sum_{j=1}^{d} M_j \partial_{x_j} v = 0, \\ \partial_t v + \frac{\sqrt{\lambda}}{\varepsilon} \sum_{j=1}^{d} M_j^\top \partial_{x_j} w = -\frac{v}{\varepsilon^2}, \quad t > 0, x \in \mathbb{R}^d. \end{cases} $$

It is a system of balance laws.

Since $\rho > 0$, the second equation in (4.5) is equivalent to

$$ \partial_t w + (w \cdot \nabla)w + \nabla h(\rho) + \frac{\sqrt{\lambda}}{\varepsilon \rho} \sum_{j=1}^{d} M_j \partial_{x_j} v = 0, $$

where $h$ is the enthalpy function defined by $h'(\rho) = p'(\rho)/\rho$ for $\rho > 0$. Let

$$ u = (\rho, w)^\top, \quad U = (\rho, w, v)^\top, $$
$$A_j(\varepsilon, U) = \begin{pmatrix} \varepsilon w_j & \varepsilon \rho e_j^\top & 0 \\ \varepsilon h'(\rho) e_j & \varepsilon w_j I_d & \frac{\sqrt{\lambda}}{\rho} M_j \\ 0 & \sqrt{\lambda} M_j^\top & 0 \end{pmatrix}, \quad S(U) = - \begin{pmatrix} 0 \\ v \end{pmatrix},$$

where $I_d$ is the unit matrix of order $d$ and $(e_1, \cdots, e_d)$ is the standard basis of $\mathbb{R}^d$. Then system (4.5) is of the form (1.1) with $N = 1 + d + r$ and $n = r$. We denote

$$A_0(U) = \begin{pmatrix} h'(\rho) & 0 & 0 \\ 0 & \rho I_d & 0 \\ 0 & 0 & I_r \end{pmatrix},$$

which is symmetric positive definite for $\rho > 0$ and

$$A_0(U)A_j(\varepsilon, U) = \begin{pmatrix} \varepsilon h'(\rho)w_j & \varepsilon p'(\rho)e_j^\top & 0 \\ \varepsilon p'(\rho)e_j & \varepsilon \rho w_j I_d & \sqrt{\lambda} M_j \\ 0 & \sqrt{\lambda} M_j^\top & 0 \end{pmatrix},$$

which is a symmetric matrix. This shows that system (4.3) is symmetrizable hyperbolic.

Since

$$A_j^{11}(\varepsilon, U) = \varepsilon \begin{pmatrix} w_j & \rho e_j^\top \\ h'(\rho)e_j & w_j I_d \end{pmatrix}, \quad S^I = 0, \quad S^{II}(U) = s(u, v) = -v,$$

it is easy to check that assumptions (1.6)-(1.8) and (1.10) are satisfied. Let $v_1$ be the formal limit of $v/\varepsilon$ as $\varepsilon \to 0$. Then the formal limit system of (4.5) as $\varepsilon \to 0$ is given by

$$\left\{ \begin{array}{l} \partial_t \rho + \operatorname{div}(\rho w) = 0, \\ \\ \partial_t (\rho w) + \operatorname{div}(\rho w \otimes w) + \nabla p(\rho) + \sqrt{\lambda} \sum_{j=1}^d M_j \partial_{x_j} v_1 = 0, \\ \\ v_1 = -\sqrt{\lambda} \sum_{j=1}^d M_j^\top \partial_{x_j} w, \quad t > 0, \ x \in \mathbb{R}^d. \end{array} \right.$$

By the expression of $v_1$ above together with (4.4), we have

$$\sum_{j=1}^{d} M_j \partial_{x_j} v_1 = - \sum_{i,j=1}^{d} M_i M_j^\top \partial_{x_i x_j}^2 w = -\Delta w.$$

This shows that the compressible Navier-Stokes equations (4.3) are the formal limit of system (4.5) as $\varepsilon \to 0$. It is known that the Cauchy problem for compressible Navier-Stokes equations (4.3) admits a local-in-time smooth solution [28]. Thus, Theorem 1.1 can be applied to (4.5).

**Example 4.3.** Hyperbolic Navier-Stokes equations with Oldroyd-B constitutive law.

Using the notation in Example 4.2, in one space dimension (4.3) reads

$$\left\{ \begin{array}{l} \partial_t \rho + \partial_x(\rho w) = 0, \\ \\ \partial_t (\rho w) + \partial_x(\rho w^2 + p(\rho)) = \lambda \partial_{xx} w, \quad t > 0, \ x \in \mathbb{R}. \end{array} \right.$$

Let $\varepsilon > 0$ be a relaxation time and $\tau_1$ be an auxiliary variable. The Oldroyd-B constitutive law is (see for instance [31, 37, 35] and the references therein)

$$\varepsilon^2 (\partial_t \tau_1 + w \partial_x \tau_1) - \lambda \partial_x w = -\tau_1.$$

Let us further define

$$\tau = \varepsilon \tau_1.$$
Then

$$ \partial_t \tau + w \partial_x \tau - \frac{\lambda}{\varepsilon} \partial_x w = - \frac{\tau}{\varepsilon^2}. $$

This leads to the following approximate system [11, 12]

$$ (4.6) \qquad \left\{ \begin{array}{l} \partial_t \rho + \partial_x (\rho w) = 0, \\ \partial_t (\rho w) + \partial_x (\rho w^2 + p) - \frac{1}{\varepsilon} \partial_x \tau = 0, \\ \partial_t \tau + w \partial_x \tau - \frac{\lambda}{\varepsilon} \partial_x w = - \frac{\tau}{\varepsilon^2}, \quad t > 0, x \in \mathbb{R}. \end{array} \right. $$

Note that (4.6) is a non conservative hyperbolic system.

By developing the second equation, (4.6) can be written as

$$ (4.7) \qquad \left\{ \begin{array}{l} \partial_t \rho + w \partial_x \rho + \rho \partial_x w = 0, \\ \partial_t w + w \partial_x w + h'(\rho) \partial_x \rho - \frac{1}{\varepsilon \rho} \partial_x \tau = 0, \\ \partial_t \tau + w \partial_x \tau - \frac{\lambda}{\varepsilon} \partial_x w = - \frac{\tau}{\varepsilon^2}, \quad t > 0, x \in \mathbb{R}. \end{array} \right. $$

Let

$$ \begin{gathered} u = (\rho, w)^{\top}, \quad v = \tau, \quad U = (\rho, w, \tau)^{\top}, \\ A(\varepsilon, U) = \begin{pmatrix} \varepsilon w & \varepsilon \rho & 0 \\ \varepsilon h'(\rho) & \varepsilon w & -\frac{1}{\rho} \\ 0 & -\lambda & \varepsilon w \end{pmatrix}, \quad S(v) = - \begin{pmatrix} 0 \\ 0 \\ \tau \end{pmatrix}. \end{gathered} $$

Then (4.7) is of the form (1.1) with $N=3$ and $n=1$. We take

$$ A_0(U) = \begin{pmatrix} h'(\rho) & 0 & 0 \\ 0 & \rho & 0 \\ 0 & 0 & \frac{1}{\lambda} \end{pmatrix}, $$

which is symmetric positive definite such that

$$ A_0(U) A(\varepsilon, U) = \begin{pmatrix} \varepsilon h'(\rho)w & \varepsilon p'(\rho) & 0 \\ \varepsilon p'(\rho) & \varepsilon \rho w & -1 \\ 0 & -1 & \frac{\varepsilon w}{\lambda} \end{pmatrix}, $$

which is symmetric. Therefore, system (4.7) is symmetrizable hyperbolic.

Since

$$ A^{11}(\varepsilon, U) = \varepsilon \begin{pmatrix} w & \rho \\ h'(\rho) & w \end{pmatrix}, $$

we have $A^{11}(0, U) = 0$. It is easy to check that assumptions (1.6)-(1.8) and (1.10) are satisfied and thus, Theorem 1.1 can be applied to (4.7).

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