ANNALES

DE LA FACULTÉ
DES SCIENCES

DE
TOULOUSE
Mathématiques

GUY MÉTIVIER, BENJAMIN TEXIER, KEVIN ZUMBRUN

*Existence of quasilinear relaxation shock profiles in systems with characteristic velocities*

Tome XXI, n° 1 (2012), p. 1-23.

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# Existence of quasilinear relaxation shock profiles in systems with characteristic velocities

GUY MÉTIVIER$^{(1)}$, BENJAMIN TEXIER$^{(2)}$, KEVIN ZUMBRUN$^{(3)}$

**ABSTRACT.** — We revisit the existence problem for shock profiles in quasi-linear relaxation systems in the case that the velocity is a characteristic mode, implying that the profile ODE is degenerate. Our result states existence, with sharp rates of decay and distance from the Chapman–Enskog approximation, of small-amplitude quasilinear relaxation shocks. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman–Enskog expansion and the macro–micro decomposition of Liu and Yu. In the quasilinear case, however, in order to close the analysis, we find it necessary to apply a parameter-dependent Nash–Moser iteration due to Texier and Zumbrun, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.

**RÉSUMÉ.** — Pour des systèmes de relaxation quasi-linéaires, dans le cas dégénéré où la vitesse est un mode caractéristique, nous donnons un résultat d'existence de profils de relaxation de petite amplitude, avec des taux de décroissance. Comme dans le cas semi-linéaire traité dans un

<footnote>(*) Reçu le 12/03/2010, accepté le 17/04/2011</footnote>

<footnote>(1) IMB, Université de Bordeaux, CNRS, IMB, 33405 Talence Cedex, France
metivier@math.u-bordeaux.fr.

G.M. thanks Indiana University for its hospitality during a visit in which this work was
partly carried out</footnote>

<footnote>(2) Université Paris Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR CNRS 7586
texier@math.jussieu.fr</footnote>

<footnote>Research of B.T. was partially supported under NSF grant number DMS-0505780.</footnote>

<footnote>(3) Indiana University, Bloomington, IN 47405
kzumbrun@indiana.edu

K.Z. thanks the University of Bordeaux I and the Institut Henri Poincaré for their hospitality during the visits in which this work was carried out. Research of K.Z. was partially supported under NSF grants number DMS-0070765 and DMS-0300487.</footnote>

<footnote>Article proposé par Thierry Gallay.</footnote>
travail antérieur de Métivier et Zumbrun, nous construisons un profil ap-
proché par un développement de Chapman-Enskog et nous utilisons la
décomposition “micro-macro” de Liu et Yu. L’ingrédient nouveau dans le
cas quasi-linéaire est le recours à un théorème de Nash-Moser à paramètre,
du à Texier et Zumbrun, par opposition au cas semi-linéaire dans lequel
un simple argument de point fixe permet de conclure la preuve.

# 1. Introduction

We consider the problem of existence of relaxation profiles

$$
U(x,t) = \bar{U}(x - st), \quad \lim_{z \to \pm\infty} \bar{U}(z) = U_{\pm} \tag{1.1}
$$

of a general relaxation system

$$
U_t + A(U)U_x = Q(U), \tag{1.2}
$$

$$
U = \begin{pmatrix} u \\ v \end{pmatrix}, \quad A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}, \quad Q = \begin{pmatrix} 0 \\ q \end{pmatrix}, \quad (1.3)
$$

in one spatial dimension, $u \in \mathbb{R}^n$, $v \in \mathbb{R}^r$, where, for some smooth $v_*$ and $f$,

$$
q(u, v_*(u)) \equiv 0, \quad \Re \sigma(\partial_v q(u, v_*(u))) \le -\theta, \quad \theta > 0, \quad (1.4)
$$

$\sigma(\cdot)$ denoting spectrum, and

$$
(A_{11} \quad A_{12}) = (\partial_u f \quad \partial_v f). \tag{1.5}
$$

Here, we are thinking particularly of the case *n* bounded and *r* ≫ 1 arising
through discretization or moment closure approximation of the Boltzmann
equation or other kinetic models; that is, we seek estimates and proof inde-
pendent of the dimension of *v*.

For fixed $n, r$, the existence problem has been treated in [26, 11] under
the additional assumption

$$
\det(A - sI) \neq 0 \tag{1.6}
$$

corresponding to nondegeneracy of the traveling-wave ODE. However, as
pointed out in [12, 13], this assumption is unrealistic for large models, and
in particular is not satisfied for the Boltzmann equations, for which the
eigenvalues of A are constant particle speeds of all values, hence cannot be
uniformly satisfied for discrete velocity or moment closure approximations.
Our goal here, therefore, is to revisit the existence problem without the
assumption (1.6).
The latter problem was treated in [17] for the semilinear case, which
includes discrete velocity approximations of Boltzmann’s equations, and for
Boltzmann’s equation (semilinear but infinite-dimensional) in [18]. We men-
tion also the proof, by similar methods, of positivity of Boltzmann shock
profiles in [9] and the original proof, by different methods, of existence of
Boltzmann profiles in [2]. The new application here is to moment closure
approximations of Boltzmann’s and other kinetic equations, which are in
general quasilinear.

Our main result is to show existence with sharp rates of decay and dis-
tance from the Chapman–Enskog approximation of small-amplitude quasi-
linear relaxation shocks in the general case that the profile ODE may become
degenerate. See Sections 2 and 3 for model assumptions and description of
the Chapman–Enskog approximation, and Section 4 for a statement of the
main theorem. Our method of analysis, as in [17, 18] is based on Chapman–
Enskog expansion and the macro-micro decomposition of [9]. The main dif-
ference in this analysis from those of the previous works is that, due to
a subtle loss of derivatives, *in the quasilinear case, we find it necessary
to apply Nash-Moser iteration to close the analysis*, whereas in the semi-
linear case a simple contraction-mapping argument sufficed. (See Remark
5.9 for further discussion of this point.) Indeed, we require a nonstandard,
parameter-dependent, Nash–Moser iteration scheme, indexed by amplitude
ε → 0, for which the linear solution operator loses not only derivatives but
powers of ε. In this, we make convenient use of a general scheme developed
in [23] for the treatment of such problems, which also arise in certain hy-
perbolic problems involving oscillatory solutions with large amplitudes or
times of existence (see [23], Section 4).

We note that spectral stability has been shown for general small-ampli-
tude quasilinear relaxation profiles in [13], without the assumption (1.6),
under the assumption that the profile exist and satisfy exponential bounds
like those of the viscous case. The results obtained here verify that assump-
tion, completing the analysis of [13]. Existence results in the absence of
condition (1.6) have been obtained in special cases in [14, 4] by quite dif-
ferent methods. (For example, center-manifold expansion near an assumed
single degenerate point [4]. However, the decay bounds as stated, though
exponential, are not sufficiently sharp with respect to ε for the needs of
[13].)
## 2. Model, assumptions, and the reduced system

Taking without loss of generality $s = 0$, we study the traveling-wave ODE

$$A(U)U' = Q(U), \qquad (2.1)$$

$$U = \begin{pmatrix} u \\ v \end{pmatrix}, \quad A = \begin{pmatrix} \partial_u f(u, v) & \partial_v f(u, v) \\ A_{21}(u, v) & A_{22}(u, v) \end{pmatrix}, \quad Q = \begin{pmatrix} 0 \\ q(u, v) \end{pmatrix} \quad (2.2)$$

governing solutions (1.1), where

$$q(u, v_*(u)) \equiv 0, \quad \Re\sigma(\partial_v q(u, v_*(u))) \le -\theta, \quad \theta > 0. \qquad (2.3)$$

We make the standard assumption of symmetric-dissipativity [25]:

**ASSUMPTION 2.1 (SD).** — There exists a smooth, symmetric and uniformly positive definite matrix $S(U)$ such that

(i) for all $U$, $S(U)A$ is symmetric,

(ii) for all equilibria $U_* = (u, v_*(u))$, $\Re S dQ(U_*)$ is nonpositive with

$$\dim \ker \Re S dQ = \dim \ker dQ \equiv n. \qquad (2.4)$$

In (2.4) and below, $\Re M$ denotes symmetric part of the matrix $M$, i.e.

$$\Re M := \frac{1}{2}(M + M^*). \qquad (2.5)$$

By the change of coordinates $v \to v - v_*(u, v)$, we may take without loss of generality

$$v_*(u, v) \equiv 0, \quad dQ = \begin{pmatrix} 0 & 0 \\ 0 & \partial_v q \end{pmatrix} \qquad (2.6)$$

changing neither the assumed structure (2.1) nor (since it is coordinate-independent) the property of symmetrizability. Note that symmetry of $SdQ$, together with (2.4), then implies both block-diagonal structure

$$S = \begin{pmatrix} S_{11} & 0 \\ 0 & S_{22} \end{pmatrix} \qquad (2.7)$$

and definiteness and proper rank of $\Re S_{22} \partial_v q$. Likewise, symmetry of $SA$ together with (2.7) yields symmetry of $S_{11}A_{11}$ and $S_{22}A_{22}$ as well as

$$(S_{11}A_{12})^T = S_{22}A_{21}. \qquad (2.8)$$

We make the simplifying assumption (2.6) throughout the paper. We make also the Kawashima assumption of genuine coupling [8]:
ASSUMPTION 2.2 (GC). — For all equilibria $U_* = (u, v_*(u))$, there ex-
ists no eigenvector of A in the kernel of $dQ(U_*)$. Equivalently [8], given
Assumption 2.1, there exists in a neighborhood $\mathcal{N}$ of the equilibrium mani-
fold a skew symmetric $K = K(U)$ such that

$$
\Re(KA - SdQ)(U) \geq c > 0, \quad \text{for all } U \in \mathcal{N}. \tag{2.9}
$$

Recall [25] that the reduced, Navier–Stokes type equations obtained by
Chapman–Enskog expansions are

$$
f_*(u)' = (b_*(u)u')' \tag{2.10}
$$

where, under the simplifying assumption (2.6),

$f_*(u) := f(u, 0)$

$$
b_*(u)u' := -A_{12}\partial_v q^{-1}A_{21}(u, 0). \tag{2.11}
$$

For the reduced system (2.10), symmetric-dissipativity becomes:

(sd) There exists $s(u)$ symmetric positive definite such that $s df_*$ is
symmetric and $sb_*$ is symmetric positive semidefinite, with $\dim \ker \Re sb_* =$
$\dim \ker b_*$.

We have likewise a notion of genuine coupling [8]:

(gc) There is no eigenvector of $df_*$ in $\ker b_*$.

We note first the following important observation of [25].

PROPOSITION 2.3 [25]. — Let (2.1) as described above be a symmetric–
dissipative system satisfying the genuine coupling condition (GC). Then, the
reduced system (2.10) is a symmetric–dissipative system satisfying genuine
coupling condition (gc).

Proof. — Assuming without loss of generality (2.6), we find that $s = S_{11}$ is a symmetrizer, since $sdf_* = S_{11}A_{11}$ is symmetric as already observed, and $sb_* = -S_{11}A_{12}(S_{22}\partial_vq)^{-1}S_{22}A_{21}$ is definite with proper rank by the corresponding properties of $S_{22}\partial_vq$ together with (2.8). Computing that (gc) is the condition that no eigenvector of $A_{11}$ lie in $\ker A_{21}$, we see that (GC) and (gc) are equivalent. $\square$

Besides the basic properties guaranteed by Lemma 2.3, we assume that
the reduced system satisfies the following important additional conditions.
ASSUMPTION 2.4. — (i) The matrix $b_*(u)$ has constant left kernel, with associated eigenprojector $\pi_*$ onto ker $b_*$, and (ii) The matrix $a_* := \pi_* df_*$ $\pi_*(u)|_{\ker b_*}$ is uniformly invertible.

Assumption 2.4 ensures that the zero-speed profile problem for the reduced system,

$$f_*(u)' = (b_*(u)u)'; \quad \lim_{z \to \pm\infty} u(z) = u_{\pm} \qquad (2.12)$$

or, after integration from $-\infty$ to $x$,

$$b_*(u)u' = f_*(u) - f_*(u_{\pm}), \qquad (2.13)$$

may be expressed as a nondegenerate ODE in $u_2$, coordinatizing $u = (u_1, u_2)$ with $u_1 = \pi_*u$ and $u_2 = (I - \pi_*)u$ [13, 27, 5]. Next, we assume that the classical theory of weak shocks can be applied to (2.12), assuming that the flux $f_*$ has a genuinely nonlinear eigenvalue near 0:

ASSUMPTION 2.5. — In a neighborhood $U_*$ of a given base state $u^0$, $df_*$ has a simple eigenvalue $\alpha$ near zero, with $\alpha(u^0) = 0$, and such that the associated hyperbolic characteristic field is genuinely nonlinear, i.e., after a choice of orientation, $\nabla\alpha \cdot r(u^0) < 0$, where $r$ denotes the eigendirection associated with $\alpha$.

Remark 2.6. — Assumption 2.5 is standard, and is satisfied in particular for the compressible Navier–Stokes equations resulting from Chapman–Enskog approximation of the Boltzmann equation. Assumptions 2.1 and 2.2 are verified in [25] for a wide variety of discrete kinetic models.¹ Assumptions 2.4 and 2.5 on the reduced equations must be checked in individual cases.

### 3. Chapman–Enskog approximation

We construct in this Section an approximate solution $U_{CE} = (u_{CE}, v_{CE})$ to the traveling-wave ODE (2.1) that satisfies $U_{CE} \to U_{\pm} = (u_{\pm}, 0)$ at $\pm\infty$, under a smallness assumption for the amplitude

$$|u_+ - u_-| := \varepsilon. \qquad (3.1)$$

We work in an $O(\varepsilon)$ neighborhood of the base state $u^0$ given in Assumption 2.5, in the sense that, for some $C > 0$,

$$|u_{\pm} - u^{0}| \le C\varepsilon. \qquad (3.2)$$

<footnote>(1) For example, both discrete kinetic models [21] used to approximate the Boltzmann equation [21] and BGK models [7, 19] used to approximate general hyperbolic conservation laws; see pp. 289–294 [25]. Note for each of these examples that the symmetrizer $S$ is not constant, but depends nontrivially on $U$.</footnote>
Integrating the first equation of (2.1), we obtain

$$ \begin{cases} f(u, v) = f_*(u_-), \\ q(u, v) = A_{21}(u, v)u' + A_{22}(u, v)v'. \end{cases} \tag{3.3} $$

Our ansatz is

$$ U(x) = \dot{U}(\varepsilon x) = (\dot{u}(\varepsilon x), \dot{v}(\varepsilon x)), \quad (\dot{u}, \dot{v}) := \sum_{k=0}^{N} \varepsilon^k (u_k, v_k), \tag{3.4} $$

where the profiles $U_k := (u_k, v_k)$ satisfy

$$ \sup_{\varepsilon} \|U_k\|_{W^{k+1, \infty}} < \infty, \tag{3.5} $$

and the boundary conditions

$$ \lim_{\pm\infty} u_0 = u_{\pm}, \quad \lim_{\pm\infty} (u_{k+1}, v_k) = (0, 0), \quad k \ge 0. \tag{3.6} $$

## 3.1. Leading term

By (2.6), we necessarily have $v_0 = 0$. Taylor expanding (3.3) and neglecting $O(\varepsilon^2)$ terms, we then obtain

$$ \left\{ \begin{array}{l} f(u_0, 0) + \varepsilon \partial_u f(u_0, 0) u_1 + \varepsilon \partial_v f(u_0, 0) v_1 = f_*(u_-), \\ \varepsilon \partial_v q(u_0, 0) v_1 = \varepsilon A_{21}(u_0, 0) u'_0, \end{array} \right. \tag{3.7} $$

Equation (3.7) can be solved for $u_0, u_1$ satisfying (3.5) only under the polarization condition

$$ f_*(u_0) - f_*(u_-) = O(\varepsilon), \tag{3.8} $$

uniformly in $x$. If $\varepsilon$ is small enough, then the condition (3.2), together with simplicity (hence regularity) of the eigenvalue $\alpha$ given in Assumption 2.5, implies $\alpha(u_-) = O(\varepsilon)$. Then, under $v_0 = 0$, condition (3.8) is equivalent to

$$ \Pi_-(u_0 - u_-) = O(\sqrt{\varepsilon}), \quad (1 - \Pi_-)(u_0 - u_-) = O(\varepsilon), \tag{3.9} $$

uniformly in $x$, where $\Pi_-$ is the projection onto the eigendirection $r(u_-)$ associated with $\alpha(u_-)$. Under (3.8), the system (3.7) becomes

$$ \left\{ \begin{array}{l} \varepsilon^{-1}(f_*(u_-) - f_*(u_0)) + f'_*(u_0)u_1 = -(A_{12}\partial_v q^{-1}A_{21})(u_0, 0)u'_0, \tag{3.10} \\ v_1 = (\partial_v q^{-1}A_{21})(u_0, 0)u'_0, \end{array} \right. $$
Then, under the uniform polarization condition for $u_1$ :

$$ (1 - \Pi_0)u_1 = O(\epsilon), \qquad (3.11) $$

where $\Pi_0$ is the projection onto $r(u_0)$, we obtain the approximate viscous profile ODE

$$ b_*(u_0)u'_0 = \frac{1}{\epsilon}(f_*(u_0) - f_*(u_-)), \qquad (3.12) $$

with $b_*$ defined in (2.11).

## 3.2. First corrector

Further expanding (3.3) and neglecting $O(\epsilon^3)$ terms, we obtain a triangular system in the second corrector $U_2$ :

$$ \left\{ \begin{aligned} \partial_u f(u_0, 0)u_2 + \partial_v f(u_0, 0)v_2 &= -\frac{1}{\epsilon}f'_*(u_0) \cdot u_1 - \frac{1}{2}d^2 f(u_0, 0) \cdot (U_1, U_1), \\ \partial_v q(u_0, 0) \cdot v_2 &= A_{21}(u_0, 0)u'_1 + (\partial_u A_{21}(u_0, 0) \cdot u_1 \\ & \qquad + \partial_v A_{21}(u_0, 0) \cdot v_1)u'_0 \\ & \qquad + A_{22}(u_0, 0)v'_1 - \partial_v^2 q(u_0, 0) \cdot (v_1, v_1). \end{aligned} \right. \qquad (3.13) $$

We impose the uniform polarization condition

$$ (1 - \Pi_0)u_2 = O(\epsilon). \qquad (3.14) $$

By the triangular structure of system (3.13), equation (3.13)(ii) can be solved for $v_2$ as a linear function of $u'_1$, with a source depending on $u_0$ :

$$ v_2 = \partial_v q(u_0, 0)^{-1} (A_{21}u'_1 + (\partial_u A_{21} \cdot U_1)u'_0 + A_{22}v'_1 - \partial_v^2 q \cdot (v_1, v_1)). \qquad (3.15) $$

Then, equation (3.13)(i) can be solved under a compatibility condition that states that the right-hand side belongs to the image of the matrix to the left-hand side; under (3.11) and (3.14), this condition takes the form of a differential equation in $u_1$ with quadratic non-linearity:

$$ b_*(u_0, 0)u'_1 = \tilde{f}_*(u_0)u_1 + \frac{1}{2}f''_*(u_0) \cdot (u_1, u_1) + \mathbf{u}_1, \qquad (3.16) $$

where

$$ \tilde{f}_*(u_0)u_1 := -(\partial_u b^*(u_0, 0) \cdot u_1)u'_0 + \frac{1}{\epsilon}f'_*(u_0)u_1 + \partial_{uv}^2 f(u_0, 0) \cdot (u_1, v_1), $$

and the source $\mathbf{u}_1$ depends on derivatives of the lower-order terms:

$$ \mathbf{u}_1 := \frac{1}{2} \partial_v^2 f(u_0, 0) \cdot (v_1, v_1) \\ -(A_{12} \partial_v q^{-1})(u_0, 0) \left( A_{22}(u_0, 0)v'_1 - \frac{1}{2} \partial_v^2 q(u_0, 0) \cdot (v_1, v_1) \right). $$
### 3.3. Higher-order terms

By induction, we can continue this process of Chapman–Enskog expansion to all orders, and, for $k \ge 2$, under the polarization conditions

$$ (1 - \Pi_0)u_{k'} = O(\epsilon), \quad k' \le k, \qquad (3.17) $$

formally derive linear equations

$$ \left\{ \begin{array}{l} b_*(u_0, 0)u'_k = \tilde{f}_*(u_0)u_k + f''_*(u_0) \cdot (u_k, u_1) + \mathbf{u}_k, \\ v_{k+1} := (\partial_v q^{-1} A_{21})(u_0, 0)u'_k + \mathbf{v}_k, \end{array} \right. \qquad (3.18) $$

where $\mathbf{u}_k$ is linear in $u_k$, and $\mathbf{u}_k$ and $\mathbf{v}_k$ both depend on $\partial^{k''} u_{k'}$, for $0 \le k'' \le k' < k$, with $0 < k''$ if $k' = 0$.

**Remark 3.1.** — Equation (3.18)(i) for the higher-order corrector is the linearization at $(u_1, 0)$ of equation (3.16) for the first-order corrector, whereas in typical Chapman-Enskog expansions [3], the equation for the first corrector is linear, being the linearization of the equation for the leading term.

### 3.4. Existence and decay bounds

Small amplitude shock profiles solutions of (3.12) are constructed using the center manifold analysis of [20] under conditions (i)-(ii) of Assumption 2.4; see discussion in [14].

**PROPOSITION 3.2.** — Under Assumptions 2.4 and 2.5, in a neighborhood of $(u^0, u^0) \in \mathbf{R}^n \times \mathbf{R}^n$, there is a smooth manifold $S$ of dimension $n$ passing through $(u^0, u^0)$, such that for $(u_-, u_+) \in S$ with amplitude $\epsilon := |u_+ - u_-| > 0$ sufficiently small, and direction $(u_+ - u_-)/\epsilon$ sufficiently close to $r(u^0)$, the zero speed shock profile equation (3.12) has a unique (up to translation) solution $u_0$ in the neighborhood $\mathcal{U}_*$ of $u^0$ introduced in Assumption 2.5, with $u_0$ satisfying (3.9), and, for $k \ge 1$, the corrector equations (3.16), (3.18)(i), have unique (up to translation) solutions $u_k$ in $\mathcal{U}_*$ satisfying (3.11) and (3.17).

Moreover, there is $\theta > 0$ and for all $k, k'$, there is $C_{k,k'} > 0$, independent of $(u_-, u_+)$ and $\epsilon$, such that

$$ |\partial_x^{k'} (u_0 - u_\pm)| \le \epsilon C_{0,k'} e^{-\theta|x|}, \quad x \ge 0. \qquad (3.19) $$

and, for $k \ge 1$,

$$ |\partial_x^{k'} u_k| \le \epsilon C_{k,k'} e^{-\theta|x|}, \quad x \ge 0. \qquad (3.20) $$
The shock profile $u_0$ is necessarily of Lax type: i.e., with dimensions of the unstable subspace of $df_*(u_-)$ and the stable subspace of $df_*(u_+)$ summing to one plus the dimension of $u$, that is $n+1$. We denote by $S_+$ the set of $(u_-, u_+) \in S$ with amplitude $\varepsilon := |u_+ - u_-| > 0$ sufficiently small and direction $(u_+ - u_-)/\varepsilon$ sufficiently close to $r(u_0)$ such that the profile $U_{CE}$ exists. Given $(u_-, u_+) \in S_+$, with associated profiles $u_0, \dots, u_N$, given in Proposition 3.2, we define $v_1, \dots, v_N$ by (3.10)(ii), (3.15), (3.18)(ii), and

$$ U_{CE} := (u_{CE}, v_{CE}) := \sum_{k=0}^{N} \varepsilon^k (u_k, v_k)(\varepsilon x). \quad (3.21) $$

It is an approximate solution of (3.3) in the following sense:

**COROLLARY 3.3.** — For fixed $u_-$ and amplitude $\varepsilon := |u_+ - u_-|$ sufficiently small, the remainder $\mathcal{R} := (\mathcal{R}_1, \mathcal{R}_2)$, defined by

$$ \begin{aligned} \mathcal{R}_{1\pm} &:= f(u_{CE}, v_{CE}) - f_*(u_\pm), & x \ge 0, \\ \mathcal{R}_2 &:= A_{21}(u_{CE}, v_{CE})u'_{CE} + A_{22}(u_{CE}, v_{CE})v'_{CE} - q(u_{CE}, v_{CE}). \end{aligned} \quad (3.22) $$

satisfies, for $k \ge 0$,

$$ |\partial_x^k \mathcal{R}_{1\pm}(x)| \le \tilde{C}_{k,N} \varepsilon^{N+k+2} e^{-\theta\varepsilon|x|}, \quad |\partial_x^k \mathcal{R}_{2}(x)| \le \tilde{C}_{k,N} \varepsilon^{N+k+1} e^{-\theta\varepsilon|x|}, \quad (3.23) $$

uniformly in $x$, where the constants $\tilde{C}_{k,N} > 0$ are independent of $(u_-, u_+)$ and $\varepsilon = |u_+ - u_-|$.

**Proof.** — A direct consequence of the formal Chapman-Enskog expansion of Sections 3.1 to 3.3 and the existence and decay bounds of Proposition 3.2. $\square$

## 4. Statement of the main theorem

We are now ready to state the main result. Define a base state $U_0 = (u_0, 0)$ and a neighborhood $\mathcal{U} = \mathcal{U}_* \times \mathcal{V}$.

**THEOREM 4.1.** — Let Assumptions 2.1, 2.2 hold in $\mathcal{U}$, with $f, A, Q \in C^\infty$, and let Assumptions 2.4 and 2.5 hold in $\mathcal{U}_*$. Then, there are $\varepsilon_0 > 0$ and $\delta > 0$ such that for $(u_-, u_+) \in S_+$ with amplitude $\varepsilon := |u_+ - u_-| \le \varepsilon_0$, the standing-wave equation (2.1) has a solution $\bar{U} = (\bar{u}, \bar{v})$ in $\mathcal{U}$, with associated Lax-type equilibrium shock $(u_-, u_+)$, satisfying for all $k, N$:

$$ \begin{aligned} |\partial_x^k (\bar{U} - U_{CE})| &\le \varepsilon^{k+N} C_{k,N} e^{-\delta\varepsilon|x|}, \\ |\partial_x^k (\bar{u} - u_\pm)| &\le \varepsilon^{k+1} C_k e^{-\delta\varepsilon|x|}, & x \ge 0, \\ |\partial_x^k (\bar{v} - v_*(\bar{u}))| &\le \varepsilon^{k+2} C_k e^{-\delta\varepsilon|x|}, \end{aligned} \quad (4.1) $$
where $U_{CE}$ is the approximating Chapman–Enskog profile defined in (3.21), and $C_k$, $C_{k,N}$ are independent of $\epsilon$. Moreover, up to translation, this solution is unique within a ball of radius $c\epsilon$ about $U_{CE}$ in norm

$$ \epsilon^{-1/2} \| \cdot \|_{L^2} + \epsilon^{-3/2} \|\partial_x \cdot \|_{L^2} + \dots + \epsilon^{-11/2} \|\partial_x^5 \cdot \|_{L^2}, \quad (4.2) $$

for $c > 0$ sufficiently small.

By (2.6), the equilibrium $v_*$ in (4.1) is $v_* \equiv 0$. Note that $U_{CE} - U_\pm$ is order $O(\epsilon)$ in the norm (4.2), by (4.1)(ii)–(iii).

Theorem 4.1 certainly holds under an assumption of finite, although large, pointwise regularity for $f$, $A$ and $Q$. The uniqueness result in space (4.2) follows from application of Theorem A.5 with $s_0 = 3$, $m = 1$ and $r' = 0$ (see Proposition 5.2).

Bounds (4.1) show that (i) the behavior of profiles is indeed well-described by the Navier–Stokes approximation, and (ii) profiles indeed satisfy the exponential decay rates required for the proof of spectral stability in [13]. From the second observation, we obtain immediately from the results of [13] the following stability result.

**COROLLARY 4.2** [13]. — Under the assumptions of Theorem 4.1, the resulting profiles $\bar{U}$ are spectrally stable for amplitude $\epsilon$ sufficiently small, in the sense that the linearized operator $L := \partial_x A(\bar{U}) - dQ(\bar{U})$ about $\bar{U}$ has no $L^2$ eigenvalues $\lambda$ with $\Re\lambda \ge 0$ and $\lambda \ne 0$.

*Proof.* — In [13], under the same structural conditions assumed here, it was shown that small-amplitude profiles of general quasilinear relaxation systems are spectrally stable, provided that $|\bar{U}'|_{L^\infty} \le C|U_+ - U_-|^2$, $|\bar{U}''(x)| \le C|U_+ - U_-| |\bar{U}'(x)|$, and

$$ \left| \frac{\bar{U}'}{|\bar{U}'|} + \operatorname{sgn}(\eta) R_0 \right| \le C |U_+ - U_-|, \quad R_0 := \begin{pmatrix} r(u_0) \\ dv_*(U_0)r(u_0) \end{pmatrix}, \quad (4.3) $$

where $r(u_0)$ as defined in Theorem 4.1 is the eigenvector of $df_*$ at base point $U_0$ in the principal direction of the shock. These conditions are readily verified using (4.1). □

The remainder of the paper is devoted to the proof of Theorem 4.1.
## 5. Proof

### 5.1. Linear and nonlinear perturbation equations

Defining the perturbation variable $U := \bar{U} - U_{CE}$, where $U_{CE}$ is defined in (3.21), we obtain from (3.3) the nonlinear perturbation equations $\Phi^{\epsilon}(U) = 0$, where

$$ \Phi^{\epsilon}(U) := \left( \begin{array}{c} f(U_{CE} + U) - f_{*}(u_{-}) \\ A_{21}(U_{CE} + U)(u_{CE} + u)' + A_{22}(U_{CE} + U)(v_{CE} + v)' - q(U_{CE} + U) \end{array} \right). \quad (5.1) $$

Formally linearizing $\Phi^{\epsilon}$ about a background profile $\underline{U}$, we obtain

$$ (\Phi^{\epsilon})'(\underline{U})U = \begin{pmatrix} A_{11}u + A_{12}v \\ A_{21}u' + A_{22}v' + b_2\underline{U} - \partial_v q v \end{pmatrix}, \quad (5.2) $$

where

$$ A = A(U_{CE} + \underline{U}), \quad \partial_v q = \partial_v q(U_{CE} + \underline{U}), $$

and

$$ b_2 U = (\partial_u(A_{21} + A_{22})(U_{CE} + \underline{U}) \cdot u + \partial_v(A_{21} + A_{22})(U_{CE} + \underline{U}) \cdot v)(U_{CE} + \underline{U})'. $$

The associated linearized equation for a given forcing term $h = (h_1, h_2)$ is

$$ (\Phi^{\epsilon})'(\underline{U})U = h. \quad (5.3) $$

### 5.2. Functional analytic setting

The coefficients and the error term $\mathcal{R}$ from Corollary 3.3 are smooth functions of $U_{CE}$ and its derivatives, so behave like smooth functions of $\epsilon x$. Thus, it is natural to solve the equations in spaces which reflect this scaling. We observe that

$$ \|f(\epsilon \cdot)\|_{L^2} = \epsilon^{-1/2} \|f\|_{L^2}, \quad \|f(\epsilon \cdot)\|_{H^s} = \epsilon^{-1/2} \sum_{k=0}^{s} \epsilon^k \|\partial_x^k f\|_{L^2}, \quad (5.4) $$

in one space dimension, for $s \in \mathbb{N}$. We do not introduce explicitly the change of variables $\tilde{x} = \epsilon x$, but introduce exponentially weighted norms which correspond to usual weighted $H^s$ norms in the $\tilde{x}$ variable: for $s \in \mathbb{N}$ and $\delta \ge 0$, we let, in accordance with (5.4),

$$ \|f\|_{\epsilon, \delta, s} := \epsilon^{1/2} \sum_{0 \le k \le s} \epsilon^{-k} \|e^{\delta\epsilon(1+|\cdot|^2)^{1/2}} \partial_x^k f\|_{L^2}, \quad (5.5) $$
the exponential weight accounting for the exponential decay of the source
and the solution. For fixed $\delta$, we introduce the spaces $E_s := H^s(\mathbb{R})$, and
$F_s := H^{s+1}(\mathbb{R}) \times H^s(\mathbb{R})$, with norms

$$
|h|_{E_s} := \|h\|_{\varepsilon, \delta, s}, \quad |(h_1, h_2)|_{F_s} := \|h_1\|_{\varepsilon, \delta, s+1} + \|h_2\|_{\varepsilon, \delta, s}.
$$

In particular, the Chapman-Enskog approximate solution of Section 3 satisfies, by (3.19) and (3.20),

$$
|\partial_x^j U_{CE}|_{L^\infty} \le \varepsilon^{j+1} C_j, \quad |\partial_x^{j+1} U_{CE}|_{E_s} \le \varepsilon^{j+2} C_{j,s}, \quad \text{for } j \ge 0, \quad (5.6)
$$

where the constants $C_j > 0$, $C_{j,s} > 0$ do not depend on $\varepsilon$, for all $s \in \mathbb{N}$.

**5.3. Nash Moser iteration scheme**

LEMMA 5.1. — The application $\Phi^\varepsilon$, defined in (5.1), maps smoothly $E_s$
to $F_{s-1}$, for any $s$. It satisfies Assumption A.1 with $s_0 = 1$, $\gamma_0 = 1$, $\bar{s} = +\infty$,
and Assumption A.3, with $k = N + 1$.

*Proof*. — The bounds of Assumption A.1, describing the action of $\Phi^\varepsilon$
and its first two derivatives, follow directly from Moser’s inequality and the
definition of the weighted Sobolev norms. The bound on $\Phi^\varepsilon(0)$ is immediate
from (3.23) and (5.5). $\square$

PROPOSITION 5.2. — *Under the assumptions of Theorem 4.1, for ε and δ small enough, the map Φε satisfies Assumption A.2 with s₀ = 3, γ = 1, r = 1, r' = 0.*

The proof of this proposition is carried out in Sections 5.4–5.6. Once it
is established, existence and uniqueness follow by Theorems A.4 and A.5
from [23]:

*Proof of Theorem 4.1 (Existence).* — The profiles $U_{CE}$ exist if $\varepsilon$ is small enough, by Proposition 3.2. By Lemma 5.1 and Proposition 5.2, we can apply Theorem A.4, and thus obtain existence of a solution $U^\varepsilon$ of (5.1) with $|U^\varepsilon|_{E_{s+1}} \le C\varepsilon^N$. Defining $\bar{U}^\varepsilon := U_{CE} + U^\varepsilon$, and noting by Sobolev embedding that $|h|_{E_{s+1}}$ controls $|e^{\delta\varepsilon(1+|\cdot|)^{1/2}}h|_{L^\infty}$, we obtain the result. $\square$

*Proof of Theorem 4.1 (Uniqueness).* — Applying Theorem A.5 for $s_0 = 3$, $\gamma_0 = 0$, $\gamma = 1$, $k = 3$, $r = 1$, $r' = 0$, we obtain uniqueness in a ball of radius $c_0\varepsilon$ in $\|\cdot\|_{\varepsilon,0,4}$, $c_0 > 0$ sufficiently small, under the additional phase condition (A.29). We obtain unconditional uniqueness from this weaker version by the observation that phase condition (A.29) may be achieved for any solution $\bar{U} = U_{CE} + U$ with

$$
\|U'\|_{L^{\infty}} \le C\varepsilon^2 \ll U'_{CE}(0) \sim \varepsilon^2
$$
by translation in $x$, yielding $\bar{U}_a(x) := \bar{U}(x+a) = U_{CE}(x) + U_a(x)$ with

$$U_a(x) := U_{CE}(x+a) - U_{CE}(x) + U(x+a)$$

so that, defining $\phi := \bar{U}'/|\bar{U}'|$, we have $\partial_a \langle \phi, U_a \rangle \sim \langle \phi, U'_{CE} + U' \rangle = \langle \phi, (1+o(1))\bar{U}' + U' \rangle = (1+o(1))|\bar{U}'| \sim \varepsilon^2$ and so (by the Implicit Function Theorem applied to $h(a) := \varepsilon^{-2}\langle \phi, U_a \rangle$, together with the fact that $\langle \phi, U_0 \rangle = o(\varepsilon)$ and that $\langle \phi, U'_{NS} \rangle \sim |\bar{U}'_{NS}| \sim \varepsilon^2$) the inner product $\langle \phi, U_a \rangle$, hence also $\Pi U_a$ may be set to zero by appropriate choice of $a = o(\varepsilon^{-1})$ leaving $U_a$ in the same $o(\varepsilon)$ neighborhood, by the computation $U_a - U_0 \sim \partial_a U \cdot a \sim o(\varepsilon^{-1})\varepsilon^2$. $\square$

It remains to prove existence of the linearized solution operator and the linearized bounds of Assumption A.2, which tasks will be the work of the rest of the paper. We concentrate first on estimates, Sections 5.4 and 5.5, and mention next, in Section 5.6, how to prove existence using a viscosity method.

## 5.4. Internal and high frequency estimates

We begin by establishing a priori estimates on solutions of the equation (5.3). This will be done in two stages. In the first stage, carried out in this section, we establish energy estimates showing that “microscopic”, or “internal”, variables consisting of $v$ and derivatives of $(u, v)$ are controlled by and small with respect to the “macroscopic”, or “fluid” variable, $u$. In the second stage, carried out in Section 5.5, we estimate the macroscopic variable $u$ by Chapman–Enskog approximation combined with finite-dimensional ODE techniques such as have been used in the study of fluid-dynamical shocks [16, 15, 22, 27].

### 5.4.1. The basic $H^1$ estimate

Let $s \in \mathbb{N}$, and some background profile $\underline{U} \in H^s$. We consider equation (5.3), and its differentiated form:

$$(AU' - dQ + b)U = (h'_1, h_2), \qquad (5.7)$$

in which $bU := (b_1U, b_2U)$, where $b_2$ is defined in Section 5.1, and $b_1$ is defined similarly, by differentiating the coefficients $A_{11}, A_{12}$ in the first line of (5.3). The coefficients $A, b$, and $dQ$, defined in (2.6), are smooth functions of $U_{CE} + \underline{U}$. The bound for $U_{CE}$, (5.6), and the assumed bound for $\underline{U}$ imply the coefficient bounds

$$\left\{ \begin{array}{ll} |\partial_x^{j+1} C|_{L^\infty} + |\partial_x^j b|_{L^\infty} \le c_j \varepsilon^{2+j}, & 0 \le j \le s-1, \\ \|\partial_x^{k+1} C\|_{L^2} + \|\partial_x^k b\|_{L^2} \le C_k \varepsilon^{1/2+k} (\varepsilon + |\underline{U}|_{\varepsilon,0,s+1}), & 0 \le k \le s, \end{array} \right. \qquad (5.8)$$
where $C = A, Q, K$, the matrix $K$ being the Kawashima multiplier (a smooth function of $A$). In (5.8), the constants $c_j$ depend on $|\partial_x^{j'}(U_{CE} + U)|_{L^\infty}$, for $0 \le j' \le j$, while, by the classical Moser's inequality, the constants $C_k$ depend on $|U_{CE} + U|_{L^\infty}$.

We give in the following Proposition an estimate for the internal variables $U' = (u', v')$ and $v$.

**PROPOSITION 5.3.** — Under the assumptions of Theorem 4.1, for some $C > 0$, for $\varepsilon$ and $\delta$ small enough, given $(h_1, h_2) \in F_1$, if $U$ solves (5.3) with $|\underline{U}|_{E_2} \le \varepsilon$, there holds

$$|U'|_{E_0} + |v|_{E_0} \le C(|H|_{E_0} + \varepsilon|u|_{E_0}), \quad (5.9)$$

where $H = (h_1, h_1', h_2'', h_2, h_2')$.

Proposition 5.3 follows from an $L^2$ estimate given in Lemma 5.4 for the symmetrized equations, defined as follows.

Multiplying (5.7) by symmetrizer $S$ (block-diagonal, (2.7)), we obtain an ODE

$$\tilde{A}U' - \tilde{Q}U + \tilde{b}U = S(h_1', h_2), \quad (5.10)$$

where

$$\tilde{A} = SA, \quad \tilde{Q} = SdQ = \begin{pmatrix} 0 & 0 \\ 0 & \tilde{Q}_{22} \end{pmatrix}, \quad \tilde{b} = Sb, \quad (5.11)$$

with $\tilde{A}$ symmetric, $\Re\tilde{Q}_{22}$ negative definite, and $\tilde{b} = O(\varepsilon^2)$, by (5.8). The genuine coupling condition, valid by Assumption 2.2 for $A$ and $dQ$, still holds for $\tilde{A}$ and $\tilde{Q}$. By the results of [8], this is equivalent to the Kawashima condition, and there is a smooth $\tilde{K} = \tilde{K}(U_{CE} + \underline{U}) = -\tilde{K}^*$, such that $\Re(\tilde{K}\tilde{A} - \tilde{Q})$ is definite positive: there is $c > 0$ such that for $\varepsilon$ small enough, there holds, uniformly in $x \in \mathbb{R}$,

$$\tilde{Q} \le -c\mathrm{Id}, \quad \Re(\tilde{K}\tilde{A} - \tilde{Q}) \ge c\mathrm{Id}. \quad (5.12)$$

**LEMMA 5.4.** — For some $C > 0$, for $\varepsilon$ sufficiently small, given $(h_1, h_2) \in H^2 \times H^1$, if $U \in H^1$ satisfies (5.10) with $\|\underline{U}\|_{\varepsilon,0,2} \le \varepsilon$, there holds

$$\|U'\|_{L^2} + \|v\|_{L^2} \le C(\|h_1\|_{H^2} + \|h_2\|_{H^1}) + \varepsilon\|u\|_{L^2}. \quad (5.13)$$

*Proof.* — Introduce the symmetrizer

$$\mathcal{S} = \partial_x^2 + \partial_x \circ \tilde{K} - \lambda,$$

where $\lambda \in \mathbb{R}$. We bound the (real) $L^2$ scalar product $(\mathcal{S}h, U)_{L^2}$ from above and from below. If $M$ is a differential operator, we note that $(Mu, u)_{L^2} =$
(ReMu, u)_{L^2}, where ReM is defined as in (2.5), M* denoting here the adjoint operator of M. Using only the symmetry of A, we find

$$
\begin{align*}
\Re \partial_x^2 \circ (\tilde{A} \partial_x - \tilde{Q}) &= \frac{1}{2} \partial_x \circ \tilde{A}' \circ \partial_x - \partial_x \circ \Re \tilde{Q} \circ \partial_x - \Re \partial_x \circ \tilde{Q}' \\
\Re \partial_x \circ \tilde{K} (\tilde{A} \partial_x - \tilde{Q}) &= \partial_x \circ \Re K \tilde{A} \circ \partial_x - \Re \partial_x \circ K \tilde{Q} \\
\Re (\tilde{A} \partial_x - \tilde{Q}) &= \frac{1}{2} \tilde{A}' - \tilde{Q}.
\end{align*}
$$

Thus

$$
\begin{align*}
\Re \mathfrak{S} \circ (\tilde{A} \partial_x - \tilde{Q}) &= \partial_x \circ \Re(\tilde{K} \tilde{A} - \tilde{Q}) \circ \partial_x + \frac{1}{2} (\partial_x \circ \tilde{A}' \circ \partial_x - \lambda \tilde{A}') \\
&\quad + \lambda \tilde{Q} - \Re \partial_x \circ (\tilde{Q}' + K \tilde{Q}).
\end{align*}
$$

Therefore, if $U \in H^2(\mathbb{R})$ solves (5.10), then (5.12) implies that

$$
-(\mathfrak{S}S(h'_1, h_2), U)|_{L^2} \ge c\|U'\|_{L^2}^2 + \lambda c\|v\|_{L^2}^2 - \lambda \left(\frac{1}{2}\|\tilde{A}'\|_{L^\infty} + |\tilde{b}|_{L^\infty}\right) \|U\|_{L^2}^2 \\
- \left(\frac{1}{2}|\tilde{A}'|_{L^\infty}\|U'\|_{L^2} + \|\tilde{Q}'\|_{L^\infty}\|U\|_{L^2} + \|K\|_{L^\infty}\|\tilde{Q}_{22}v\|_{L^2}\right) \|U'\|_{L^2} \\
- \left(|\tilde{b}|_{L^\infty}\|U'\|_{L^2} + \|\tilde{b}'\|_{L^2}|U|_{L^\infty} + |\tilde{K}\tilde{b}|_{L^\infty}\|U\|_{L^2}\right) \|U'\|_{L^2}.
$$

Note that we used an $L^2$ bound, and not an $L^\infty$ bound, for the term $\tilde{b}'$ which contains the largest number of derivatives of the background $U_{CE}+\underline{U}$. In the above lower bound, all the terms with a minus sign have small prefactors, by (5.8), except the term $\|\tilde{Q}_{22}v\|_{L^2}\|U'\|_{L^2}$. We handle this term by Young's product inequality:

$$
\|K\|_{L^\infty} \|\tilde{Q}_{22}v\|_{L^2} \|U'\|_{L^2} \le \frac{1}{2} c \|U'\|_{L^2}^2 + \frac{1}{c} \|K\|_{L^\infty}^2 \|\tilde{Q}\|_{L^\infty}^2 \|v\|_{L^2}^2,
$$

and this implies that for some $\lambda$, depending on $c$, $\|K\|_{L^\infty}$ and $\|\tilde{Q}_{22}\|_{L^\infty}$,
the above upper bound can be absorbed in $c(\|U'\|_{L^2}^2 + \lambda\|v\|_{L^2})$. Using (5.8)
together with the assumed bound on $\underline{U}$, which implies $\|\tilde{b}'\|_{L^2} \le C\varepsilon^{5/2}$, and
using the bound

$$
|U|_{L^{\infty}} \lesssim \varepsilon^{-1/2} \|U\|_{L^2} + \varepsilon^{1/2} \|U'\|_{L^2},
$$

we obtain

$$
\|U'\|_{L^2}^2 + \|v\|_{L^2}^2 \le C |(\mathfrak{G}Sh, U)|_{L^2} + \varepsilon^2 C_2 (\|U\|_{L^2}^2 + \|U'\|_{L^2}^2),
$$

where

$$
\varepsilon^2 C_2 := |U'_{CE} + \underline{U}'|_{L^\infty} + \varepsilon^{-1/2} \|U''_{CE} + \underline{U}''\|_{L^2}.
$$

In the opposite direction,

$$
\begin{align*}
| (\mathfrak{G}S(h'_1, h_2), U)|_{L^2} | &\le C_1 (\|h'_1\|_{H^1} + \|h_2\|_{H^1}) \|U'\|_{L^2} \\
&\quad + \lambda (\|h_1\|_{L^2} \|(S_{11}u)'\|_{L^2} + \|h_2\|_{L^2} \|v\|_{L^2}),
\end{align*}
$$
where $C_1$ depends on the $L^\infty$ norm of $U'_{CE} + \underline{U}'$, and where we integrated by parts the term $(h'_1, S_{11}u)_{L^2}$ in order to convert the "fluid" variable $u$ into a "microscopic" variable $u'$, up to an error that depends only on one derivative of the coefficients. The estimate (5.13) follows provided that $\epsilon$ is small enough. This proves the lemma under the additional assumption that $U \in H^2$. When $U \in H^1$, the estimates follows using Friedrichs mollifiers.
□

*Proof.* — [Proof of Proposition 5.3] We use Lemma 5.4 for $\epsilon^{1/2}e^{\delta\epsilon\langle x\rangle}U$, which solves (5.10) with the source term

$$ \epsilon^{1/2}e^{\delta\epsilon\langle x\rangle}((h'_1, h_2) + \delta\epsilon\langle x\rangle' \tilde{A}U), $$

from which (5.9) follows. □

<figure><img src="image_1.png" /></figure>

### 5.4.2. Higher order estimates

**PROPOSITION 5.5.** — For $k \ge 1$, for some $C > 0$, for $\epsilon$ and $\delta$ small enough, given $h \in F_{k+1}$, if $U \in H^k$ satisfies (5.10) with $|\underline{U}|_{E_2} \le \epsilon$, there holds

$$ \begin{aligned} |\partial_x^k U'|_{E_0} + |\partial_x^k v|_{E_0} \le C \biggl( & |\partial_x^k H|_{E_0} + \epsilon^k (|U'|_{E_{k-1}} + \epsilon|v|_{E_{k-1}} + \epsilon|u|_{E_0}) \\ & + C \epsilon^{k+1} |\underline{U}|_{E_{k+2}} (|v|_{E_1} + \epsilon|U|_{E_2}) \biggr), \end{aligned} \quad (5.14) $$

where $H = (h_1, h'_1, h''_1, h_2, h'_2)$.

*Proof.* — Differentiating (5.10) $k$ times, we obtain

$$ \tilde{A} \partial_x^{k+1} U - \tilde{Q} \partial_x^k U + \tilde{b} \partial_x^k U = (\partial_x^{k+1} h_1, \partial_x^k h_2) + r_k, \quad (5.15) $$

where

$$ r_k = -\partial_x^{k-1}((\partial_x \tilde{A}) \partial_x U) + \partial_x^{k-1}((\partial_x \tilde{Q}) U) - \partial_x^{k-1}((\partial_x \tilde{b}) U). $$

Note that in the case $k=1$, the source $r_1$ in (5.15) does not have the structure of the source term in (5.10). It is however straightforward to adapt the proof of Proposition 5.3 to (5.15) with $k=1$, by the bound

$$ ((\partial_x \tilde{C}) U, \partial_x U)|_{L^2} \le |\partial_x \tilde{C}|_{L^\infty} \|U\|_{L^2} \|U'\|_{L^2}, \quad (5.16) $$

in which $\partial_x \tilde{C} = \partial_x(\tilde{A}, \tilde{Q}, \tilde{b}) = O(\epsilon^2)$ in $L^\infty$, by (5.8), hence the contribution of (5.16) is absorbed as in the proof of Proposition 5.3. Thus we apply Proposition 5.3 to (5.15), and obtain

$$ |\partial_x^k U'|_{E_0} + |\partial_x^k v|_{E_0} \le C (|\partial_x^k (h_1, h'_1, h''_1, h_2, h'_2)|_{E_0} + \epsilon |\partial_x^k u|_{E_0} + |r'_k|_{E_0} + |r_k|_{E_0}), $$
in which there is no $r_k''$ term by the reason indicated above. Thus we are led to estimate terms

$$ \left\| \partial_x^\alpha \left( (\partial_x^{1+k_1} \tilde{C})(\partial_x^{k-k_1-1+\beta} U) \right) \right\|_{E_0}, \quad 0 \le k_1 \le k-1, \quad 0 \le \alpha \le 1, \qquad (5.17) $$

in which $\tilde{C} = \tilde{A}, \tilde{Q}, \tilde{b}$, and $\beta = 1$ if $\tilde{C} = \tilde{A}, \beta = 0$ otherwise. We handle these terms as in the proof of Lemma 5.4, by bounding the coefficients in $L^\infty$, save for the term with the largest numbers of derivatives of the coefficients, namely $(\partial_x^k \tilde{C})(\partial_x^\beta U)$, $(\partial_x^{k+1} \tilde{C})\partial_x^\beta U$, which we bound by taking the $L^2$ norm of the coefficients and the $L^\infty$ norm of $\partial_x^\beta U$, and obtain (5.14). $\square$

## 5.5. Linearized Chapman-Enskog estimate

It remains only to estimate the weighted $L^2$ norm $|u|_{E_0}$ in order to close the estimates and establish the bound claimed in Proposition (5.2). To this end, we work with the first equation in (5.3) and estimate it by comparison with the Chapman-Enskog approximation of Section 3.

### 5.5.1. The linearized profile equation

From the second equation in (5.3), in which, by (5.8), $b = O(\epsilon^2)$, we find, for small $\epsilon$,

$$ v = (\partial_v q - b_{22})^{-1} (A_{21}u' + A_{22}v' + b_{21}u - h_2), \qquad (5.18) $$

where $b_2 U =: b_{21} u + b_{22} v$. Introducing now (5.18) in the first equation of (5.3), we obtain the linearized profile equation

$$ A_{12}(\partial_v q - b_{22})^{-1} A_{21} u' + (A_{11} + A_{12}(\partial_v q - b_{22})^{-1} b_{21}) u = h^{\sharp}, \qquad (5.19) $$

where $h^{\sharp}$ depends on the source $h$ and on $v'$, but not on $v$ nor on $u$:

$$ h^{\sharp} := -A_{12}(\partial_v q - b_{22})^{-1} A_{22} v' + h_1 + A_{12}(\partial_v q - b_{22})^{-1} h_2. $$

### 5.5.2. $L^2$ estimates and proof of the main estimates

Introduce the notation

$$ b^{\sharp} := (A_{12}(\partial_v q - b_{22})^{-1} A_{21}) (U_{CE} + \cdot), $$

$$ f^{\sharp} := (A_{11} + A_{12}(\partial_v q - b_{22})^{-1} b_{21}) (U_{CE} + \cdot). $$

Then (5.19) takes the form

$$ (b^{\sharp} \partial_x - f^{\sharp})(\underline{U})u = -h^{\sharp}. \qquad (5.20) $$

We estimate the solution of (5.20) by the following:
**PROPOSITION 5.6.** — Given $\underline{U} \in H^4$, with $|\underline{U}|_{E_4} \le \varepsilon$, if $\varepsilon$ is sufficiently small, then the operator $(b^\sharp \partial_x - f^\sharp)(\underline{U})$ has a right inverse $(b^\sharp \partial_x - f^\sharp)(\underline{U})^\dagger$, satisfying the bound

$$ \|(b^\sharp \partial_x - f^\sharp)(\underline{U})^\dagger h\|_{E_0} \le C\varepsilon^{-1}\|h\|_{E_0}, \qquad (5.21) $$

and uniquely specified by the property that the solution $u$ to (5.20) satisfies

$$ \ell_{\varepsilon} \cdot u(0) = 0. \qquad (5.22) $$

for certain unit vector $\ell_{\varepsilon}$.

**Proof.** — Standard asymptotic ODE techniques, using the gap and reduction lemmas of [16, 13, 22], where the assumption $\|\underline{U}\|_{E_4} \le C\varepsilon$ gives the needed control on coefficients; see the proof of Proposition 7.1, [17]. $\square$

**PROPOSITION 5.7.** — For some $C > 0$, for $\varepsilon$ and $\delta$ small enough, given $h \in F_2$, and $\underline{U} \in H^4$ satisfying $|\underline{U}|_{E_4} \le \varepsilon$, if $U = (u, v) \in H^2$ satisfies (5.3), with $u$ satisfying (5.22), there holds

$$ |U|_{E_2} \le C\varepsilon^{-1}|h|_{F_2}. \qquad (5.23) $$

**Proof.** — If $U$ solves (5.3), then $u$ solves (5.19), and if in addition $u$ satisfies (5.22), then by Proposition (5.6), there holds

$$ |u|_{E_0} \le C\varepsilon^{-1}|h^\sharp|_{E_0} \le C\varepsilon^{-1}(|h|_{E_0} + |v'|_{E_0}). \qquad (5.24) $$

If we now use Proposition 5.3 to bound $v'$, we are left with a term in $C|u|_{E_0}$ in the upper bound, which a priori cannot be absorbed by the left-hand side of (5.24). We use instead Proposition 5.5 with $k = 1$, which together with Proposition 5.3 gives a better estimate for $v'$, namely

$$ |v'|_{E_0} \lesssim |H'|_{E_0} + \varepsilon|H|_{E_0} + \varepsilon^2|u|_{E_0} + \varepsilon^2|\underline{U}|_{E_3}(|v|_{E_1} + \varepsilon|U|_{E_2}), $$

and with (5.24) we find, for small $\varepsilon$,

$$ \varepsilon|u|_{E_0} \lesssim |h|_{E_0} + \varepsilon|H|_{E_0} + |H'|_{E_0} + \varepsilon|\underline{U}|_{E_3}|U''|_{E_0}. \qquad (5.25) $$

Plugging this estimate in (5.9), we find

$$ |U'|_{E_0} + |v|_{E_0} + \varepsilon|u|_{E_0} \lesssim |h|_{E_0} + \varepsilon|H|_{E_0} + |H'|_{E_0} + \varepsilon|\underline{U}|_{E_3}|U''|_{E_0}, \qquad (5.26) $$

from which we deduce, using again Proposition 5.5 with $k = 1$,

$$ |U''|_{E_0} + |v'|_{E_0} \lesssim |h|_{E_0} + \varepsilon|H|_{E_0} + |H'|_{E_0}. \qquad (5.27) $$

By definition of the $E_2$ and $F_2$ norms, (5.23) follows from (5.26) and (5.27).

<figure><img src="image_2.png" /></figure>

Knowing a bound for $|u|_{E_0}$, Proposition 5.5 implies by induction the following final result.
**PROPOSITION 5.8.** — For $s \ge 3$, for some $C > 0$, for $\varepsilon$ and $\delta$ small enough, given $h \in F_s$ and $\underline{U} \in H^{s+1}$ with $|\underline{U}|_{E_4} \le \varepsilon$, if $U \in H^s$ satisfies (5.3) and (5.22), then

$$|U|_{E_s} \le \varepsilon^{-1} C (|\underline{U}|_{E_{s+1}} |h|_{F_2} + |h|_{F_s}) \quad (5.28)$$

*Remark 5.9.* — The loss of derivative on $\tilde{U}$ comes from the conservative form of the linearized equations, through the microscopic energy estimates on the solution. A similar loss in derivative may be seen in the resolvent equation for linear hyperbolic equations in conservative form, $\lambda U + (A(\tilde{U})u)' = f$; see [23] for further discussion. We could avoid this by writing the differentiated equations in quasilinear form, but this would prevent us from integrating back to carry out linearized Chapman–Enskog estimates. That is, the loss of derivatives is due to a subtle incompatibility between the integrated form needed for linearized Chapman–Enskog estimates and the nonconservative (quasilinear) form needed for optimal energy estimates with no loss of derivative.

## 5.6. Existence for the linearized problem

To complete the proof of Proposition 5.2, it remains to demonstrate existence for the linearized problem. This can be carried out as in [17] by the vanishing viscosity method, with viscosity coefficient $\eta > 0$, obtaining existence for each positive $\eta$ by standard boundary-value theory, and noting that our previous A Priori bounds (5.28) persist under regularization for sufficiently small viscosity $\eta > 0$, so that we can obtain a weak solution in the limit by extracting a weakly convergent subsequence. We omit these details, referring the reader to Section 8, [17]. The asserted estimates then follow in the limit by continuity.

### A. A Nash–Moser Theorem with losses

We give in this appendix the parameter-dependent Nash–Moser theory developed in [23]. The main novelty of this treatment is to allow losses of powers of the parameter $\varepsilon \to 0$ in the linearized solution operator. For a proof of this result, see [23]; for a more general discussion of Nash–Moser iteration methods, see [6, 1, 24], and references therein.

Consider two families of Banach spaces $\{E_s, |\cdot|_{E_s}\}_{s \in \mathbb{R}}$, $\{F_s, |\cdot|_{F_s}\}_{s \in \mathbb{R}}$, where the norms $|\cdot|_{E_s}$ and $|\cdot|_{F_s}$ may be $\varepsilon$-dependent, as in our application here, and a family of equations $\Phi^\varepsilon(u^\varepsilon) = 0$, $u^\varepsilon \in E_s$, indexed by $\varepsilon \in (0, 1)$, where for all $\varepsilon$, $\Phi^\varepsilon \in C^2(E_s, F_{s-1})$, for all $s \le \bar{s}$, and some $\bar{s} \in \mathbb{R}$.
We assume (i) for $s \le s'$, the embeddings $E_{s'} \hookrightarrow E_s$, $F_{s'} \hookrightarrow F_s$, hold, with $|\cdot|_{E_s} \le |\cdot|_{E_{s'}}$, $|\cdot|_{F_s} \le |\cdot|_{F_{s'}}$, (ii) the interpolation property $|\cdot|_{E_{s+\sigma}} \lesssim |\cdot|_{E_s}^{(\sigma'-\sigma)/\sigma'}|\cdot|_{E_{s+\sigma'}}^{\sigma/\sigma'}$, for $0 < \sigma < \sigma'$, and (iii) the existence of a family of regularizing operators $S_\theta: E_s \to E_s$, for $\theta > 0$, such that for all $s \le s'$, $|S_\theta u - u|_{E_s} \lesssim \theta^{s-s'}|u|_{E_{s'}}$, and $|S_\theta u|_{E_{s'}} \lesssim \theta^{s'-s}|u|_{E_s}$. (In Sobolev spaces, we can take $S_\theta$ to be high-frequency truncations.)

**ASSUMPTION A.1.** — For some $s_0 \in \mathbb{R}$, some $\gamma_0 \ge 0$, for all $s$ such that $s_0 + 1 \le s + 1 \le \bar{s}$, for all $u, v, w \in E_{s+1}$,

$$ \begin{aligned} |\Phi^\varepsilon(u)|_{F_s} &\le C_0(1 + |u|_{E_{s+1}} + |u|_{E_{s_0+1}}|u|_{E_s}), \\ |(\Phi^\varepsilon)'(u) \cdot v|_{F_s} &\le C_0(|v|_{E_{s+1}} + |v|_{E_{s_0+1}}|u|_{E_{s+1}}), \\ |(\Phi^\varepsilon)''(u) \cdot (v, w)|_{F_s} &\le C_0(|v|_{E_{s_0+1}}|w|_{E_{s+1}} + |v|_{E_{s+1}}|w|_{E_{s_0+1}} \\ &\phantom{\le} + |u|_{E_{s+1}}|v|_{E_{s_0+1}}|w|_{E_{s_0+1}}) \end{aligned} $$

where $C_0 = C_0(\varepsilon, |u|_{E_{s_0+1}})$ satisfies $\sup_\varepsilon \sup_{|u|_{E_{s_0+1}} \lesssim \varepsilon^{\gamma_0}} C_0 < +\infty$.

**ASSUMPTION A.2.** — For some $\gamma \ge 0$, $r \ge 0$, $r' \ge 0$, for all $s$ such that $s_0 + 1 + \max(r, r') \le s + \max(r, r') \le \bar{s}$, for all $u \in E_{s+r}$ such that $|u|_{E_{s_0+1}} \lesssim \varepsilon^\gamma$, the map $(\Phi^\varepsilon)'(u) : E_{s+1} \to F_s$ has a right inverse $\Psi^\varepsilon(u)$:

$$ (\Phi^\varepsilon)'(u)\Psi^\varepsilon(u) = \mathrm{Id} : F_s \to F_s, $$

satisfying, for all $\phi \in F_{s+r'}$,

$$ |\Psi^\varepsilon(u)\phi|_{E_s} \le \varepsilon^{-1}C(|\phi|_{F_{s_0+1+r'}}|u|_{E_{s+r}} + |\phi|_{F_{s+r'}}), $$

where $C$ is a non-decreasing function of its arguments $s$ and $|u|_{s_0+1+r}$.

**ASSUMPTION A.3.** — There holds the bound

$$ \|\Phi^\varepsilon(0)\|_s \lesssim \varepsilon^k, $$

for some $k$ and $s$ satisfying $\max(2, 1 + \gamma_0, 1 + \gamma) < k$, $C(k) \le \bar{s} - s_0 - 1$, where $C(k)$ is a certain positive function (see [23]) and $s \in [s_0+1, \bar{s} - C(k)]$.

**THEOREM A.4 (Existence).** — Under Assumptions A.1, A.2 and A.3, for $\varepsilon$ small enough, there exists a real sequence $\theta_j^\varepsilon$, satisfying $\theta_j^\varepsilon \to +\infty$ as $j \to +\infty$ and $\varepsilon$ is held fixed, such that the sequence $u_j^\varepsilon := 0$, $u_{j+1}^\varepsilon := u_j^\varepsilon + S_{\theta_j^\varepsilon} v_j^\varepsilon$, $v_j^\varepsilon := -\Psi^\varepsilon(u_j^\varepsilon)\Phi^\varepsilon(u_j^\varepsilon)$, is well defined and converges, as $j \to +\infty$ and $\varepsilon$ is held fixed, to a solution $u^\varepsilon$ of $\Phi^\varepsilon(u^\varepsilon) = 0$, in $s+1$ norm, which satisfies the bound $|u^\varepsilon|_s \lesssim \varepsilon^{k-1}$.
**THEOREM A.5 (Uniqueness).** — *Under Assumptions A.1, A.2 and A.3, for ε small enough, if (Φε)' is invertible, i.e., Ψε is also a left inverse, then the solution described in Thm A.4 is unique in a ball of radius o(εmax(1,γ₀,γ)) in s₀+2+r' norm. More generally, if ûε is a second solution within this ball, then (ûε - uε) is approximately tangent to Ker(Φε)'(uε), in the sense that its distance in s₀ norm from Ker(Φε)'(uε) is o(|ûε - uε|s₀). In particular, if Ker(Φε)'(uε) is finite-dimensional, then u is the unique solution in the ball satisfying the additional “phase condition”*

$$ \Pi_{(\Phi^{\epsilon})'(u^{\epsilon})}(\hat{u}^{\epsilon} - u^{\epsilon}) = 0, \qquad (A.29) $$

where $\Pi_{(\Phi^{\epsilon})'(u^{\epsilon})}$ is any uniformly bounded projection onto $Ker(\Phi^{\epsilon})'(u^{\epsilon})$ (in a Hilbert space, any orthogonal projection onto $Ker(\Phi^{\epsilon})'(u^{\epsilon})$).

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