$\|(L_{\mu_{1},N}^{\kappa})^{-1}\|_{\mathcal{H}^{-1}_{{\rm
mic},\kappa,N}\to\mathcal{H}^{1}_{{\rm mic},\kappa,N}}\lesssim 1\,.$ (5.5)

###### Proof of Lemma 5.3.

The mapping property (5.4) follows from the mapping properties of
$\Gamma_{\kappa}[\mu_{0}^{-\nicefrac{{1}}{{2}}}\mu_{1},\cdot]+\Gamma_{\kappa}[\cdot,\mu_{0}^{-\nicefrac{{1}}{{2}}}\mu_{1}]$
on the smooth, well-localized Hermite functions. To complete the proof, we
prove that $L_{\mu_{1},N}^{\kappa}$ is bounded below with uniform estimates
when $\mu_{1}$ is near $\mu_{0}$. That is, when $f\in\mathcal{H}^{1}_{{\rm
mic},\kappa,N}$, we have

$\displaystyle\langle L_{\mu_{1},N}^{\kappa}f,f\rangle$ $\displaystyle=\langle
L_{\mu_{1}}^{\kappa}f,f\rangle$ $\displaystyle=\langle
L_{\mu_{0}}^{\kappa}f,f\rangle-\langle\Gamma_{\kappa}[\mu_{0}^{-\nicefrac{{1}}{{2}}}(\mu_{1}-\mu_{0}),f],f\rangle-\langle
f,\Gamma_{\kappa}[\mu_{0}^{-\nicefrac{{1}}{{2}}}(\mu_{1}-\mu_{0}),f]\rangle$
$\displaystyle\geqslant
C^{-1}\|f\|_{\mathcal{H}^{1}_{\kappa}}^{2}-o_{(\varrho_{1},u_{1},\theta_{1})\to(1,0,1)}(1)\|f\|_{\mathcal{H}^{1}_{\kappa}}^{2}\,,$

with constants independent of $\kappa$. The operators in question are finite
dimensional, so this furnishes invertibility and the estimates. ∎

###### Proof of Proposition 5.2.

_1\. Microscopic estimates_.

We begin with the analogue of the basic microscopic estimate (4.3). The key
points are that (i) the energy estimates do not “see” the projections
$\Pi_{\leqslant N}$, and (ii) the $-\eta\partial_{x}^{2}f$ term provides a new
advantageous contribution, except in (iii) the Kawashima energy estimate, but
its error can be estimated using the new advantageous terms. First, we
multiply the equation by $f$ and integrate by parts, using that
$f=\Pi_{\leqslant N}f$. We have

$\eta\int|\partial_{x}f|^{2}\,\mathrm{d}v\,\mathrm{d}x+\int L_{\rm
NS}^{\kappa}f\cdot f\,\mathrm{d}v\,\mathrm{d}x=\int
zf\,\mathrm{d}v\,\mathrm{d}x\,.$ (5.6)

When we pass one spatial derivative through the equation and multiply by
$\partial_{x}f$, we obtain

$\displaystyle\eta\int$
$\displaystyle|\partial_{x}^{2}f|^{2}\,\mathrm{d}v\,\mathrm{d}x+\int L_{\rm
NS}^{\kappa}\partial_{x}f\cdot\partial_{x}f\,\mathrm{d}v\,\mathrm{d}x-\int\Gamma_{\kappa}[\mu_{0}^{-\nicefrac{{1}}{{2}}}\partial_{x}\mu_{\rm
NS},f]\partial_{x}f\,\mathrm{d}v\,\mathrm{d}x$
$\displaystyle-\int\Gamma_{\kappa}[f,\mu_{0}^{-\nicefrac{{1}}{{2}}}\partial_{x}\mu_{\rm
NS}]\partial_{x}f\,\mathrm{d}v\,\mathrm{d}x=\int\partial_{x}z\partial_{x}f\,\mathrm{d}v\,\mathrm{d}x\,.$
(5.7)

When we apply $K$ to the equation and multiply by $\partial_{x}f$, we obtain

$-\eta\int\partial_{x}^{2}Kf\partial_{x}f\,\mathrm{d}v\,\mathrm{d}x+\int(Kv_{1}\partial_{x}f)\partial_{x}f+\int
KL_{\rm NS}^{\kappa}f\cdot\partial_{x}f\,\mathrm{d}v\,\mathrm{d}x=\int
Kz\partial_{x}f\,\mathrm{d}v\,\mathrm{d}x\,.$ (5.8)

Notice the absence of projections $\Pi_{\leqslant N}$ in the above equalities.
The desired energy estimate is obtained by summing (5.6), (5.7), and a small
multiple of (5.8). The term
$-(2C_{III})^{-1}\eta\int\partial_{x}^{2}Kf\partial_{x}f$ can therefore be
absorbed by the terms $\eta\int|\partial_{x}f|^{2}$ and
$\eta\int|\partial_{x}^{2}f|^{2}$ via Young’s inequality to obtain, exactly as
before,

$\|\partial_{x}f\|_{L^{2}\mathcal{H}^{1}_{\kappa}}+\|P^{\perp}f\|_{L^{2}\mathcal{H}^{1}_{\kappa}}\lesssim\varepsilon\|Pf\|_{L^{2}}+\|z,\partial_{x}z\|_{L^{2}\mathcal{H}^{-1}_{\kappa}}\,,$
(5.9)

Subsequently, the estimate

$\|\partial_{x}^{2}f\|_{L^{2}\mathcal{H}^{1}_{\kappa}}+\|P^{\perp}\partial_{x}f\|_{L^{2}\mathcal{H}^{1}_{\kappa}}\lesssim\varepsilon^{2}\|Pf\|_{L^{2}}+\|\varepsilon
z,\partial_{x}z,\partial_{x}^{2}z\|_{L^{2}\mathcal{H}^{-1}_{\kappa}}$ (5.10)

is obtained by differentiating the Galerkin truncated equation once and
applying (5.9). Crucially, the implicit constants in (5.9)–(5.10) do not
depend on $\kappa$.

_2\. Macroscopic estimates_. We now require the analogue of the linearized
Chapman-Enskog expansion. Write $F=\sqrt{\mu_{0}}f$ and
$\mathcal{E}=\sqrt{\mu_{0}}z$. We write
$\mathsf{L}_{N}^{\kappa}=\sqrt{\mu_{0}}L_{\mu_{0},N}^{\kappa}\sqrt{\mu_{0}}^{-1}$
and $\mathsf{L}_{{\rm NS},N}^{\kappa}=\sqrt{\mu_{0}}L_{{\rm
NS},N}^{\kappa}\sqrt{\mu_{0}}^{-1}$. It will be convenient to write
$\mathsf{L}_{{\rm NS},N}^{\kappa}$ as a block matrix

$\begin{bmatrix}0&0\\\
\mathsf{L}_{21,N}^{\kappa}&\mathsf{L}_{22,N}^{\kappa}\end{bmatrix}\,,$

where $\mathsf{L}_{21,N}^{\kappa}=\mathsf{P}_{\mu_{0}}^{\perp}\mathsf{L}_{{\rm
NS},N}^{\kappa}\mathsf{P}_{\mu_{0}}$ and
$\mathsf{L}_{22,N}^{\kappa}=\mathsf{P}_{\mu_{0}}^{\perp}\mathsf{L}_{{\rm
NS},N}^{\kappa}\mathsf{P}_{\mu_{0}}^{\perp}$. (This notation suppresses the
$x$-dependence of the operators.) Again, we write
$F=U+\mathsf{P}^{\perp}_{\mu_{0}}F$, where $U=\mathsf{P}_{\mu_{0}}F$. The
finite-dimensional analogue of the projection $\mathsf{P}_{\rm NS}$ onto the
tangent space at $\mu_{\rm NS}$ is

$\mathsf{P}_{{\rm
NS},N}^{\kappa}=\mathsf{P}_{\mu_{0}}-(\mathsf{L}_{22,N}^{\kappa})^{-1}\mathsf{L}_{21,N}^{\kappa}\mathsf{P}_{\mu_{0}}\,.$

This is the projection onto the kernel of the operator
$\mathsf{L}^{\kappa}_{{\rm NS},N}$. The complementary projection
$\mathsf{P}^{\perp,\kappa}_{{\rm NS},N}:=I-\mathsf{P}_{{\rm NS},N}^{\kappa}$
is a projection onto the microscopic subspace. The linearized Chapman-Enskog
ansatz is

$F=\mathsf{P}_{{\rm NS},N}^{\kappa}U+V\,,$

which we take as the definition of $V$. Plugging the ansatz into the equation,
we obtain

$-\eta\partial_{x}^{2}\mathsf{P}_{{\rm
NS},N}^{\kappa}U-\eta\partial_{x}^{2}V+\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})\partial_{x}\mathsf{P}_{{\rm
NS},N}^{\kappa}U+\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})\partial_{x}V+\mathsf{L}_{{\rm
NS},N}^{\kappa}V=\mathsf{\Pi}_{\leqslant N}\mathcal{E}\,,$ (5.11)

where we used that $\mathsf{L}_{{\rm NS},N}^{\kappa}\mathsf{P}_{{\rm
NS},N}^{\kappa}=0$. We apply the projection $\mathsf{P}_{{\rm
NS},N}^{\perp,\kappa}$ and invert $\mathsf{L}_{{\rm NS},N}^{\kappa}$ to obtain
an expression for $V$:

$V=-(\mathsf{L}_{{\rm NS},N}^{\kappa})^{-1}\mathsf{P}_{{\rm
NS},N}^{\perp,\kappa}(\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})\partial_{x}\mathsf{P}_{{\rm
NS},N}^{\kappa}U+\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})\partial_{x}V-\mathsf{\Pi}_{\leqslant N}\mathcal{E}+\eta h)\,,$
(5.12)

where

$h=(\partial_{x}^{2}\mathsf{P}_{{\rm
NS},N}^{\kappa})U+2\partial_{x}\mathsf{P}_{{\rm
NS},N}^{\kappa}\partial_{x}U+\partial_{x}^{2}V\,.$ (5.13)

We substitute the expression back into the equations projected by
$\mathsf{P}_{\mu_{0}}$, obtaining

$\mathsf{P}_{\mu_{0}}(v_{1}-s_{0})\partial_{x}\mathsf{P}_{{\rm
NS},N}^{\kappa}U+\mathsf{P}_{\mu_{0}}(v_{1}-s_{0})\partial_{x}V=0\,.$ (5.14)

Let888While we do not prove that $\nabla H_{N}^{\kappa}$ is the gradient of a
flux function $H_{N}^{\kappa}$, we adopt this notation nonetheless.

$\displaystyle\nabla H_{N}^{\kappa}$
$\displaystyle:=\mathsf{P}_{\mu_{0}}v_{1}\mathsf{P}_{{\rm
NS},N}^{\kappa}\,,\qquad
B_{N}^{\kappa}:=\mathsf{P}_{\mu_{0}}v_{1}(\mathsf{L}_{{\rm
NS},N}^{\kappa})^{-1}\mathsf{P}_{{\rm
NS},N}^{\perp,\kappa}\mathsf{\Pi}_{\leqslant N}v_{1}\mathsf{P}_{{\rm
NS},N}^{\kappa}\,,$

and

$g_{N}^{\kappa}=\mathsf{P}_{\mu_{0}}v_{1}(\mathsf{L}_{{\rm
NS},N}^{\kappa})^{-1}\mathsf{P}_{{\rm
NS},N}^{\perp,\kappa}(\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})(\partial_{x}\mathsf{P}_{{\rm
NS},N}^{\kappa})U+\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})\partial_{x}V-\mathsf{\Pi}_{\leqslant N}E+\eta h)\,.$ (5.15)

Then the equation (5.14), integrated once, becomes

$(\nabla H_{N}^{\kappa}-s_{0}{\rm
Id})U=B_{N}^{\kappa}\partial_{x}U+g_{N}^{\kappa}\,.$

For now, we take for granted the following property of the coefficients:

$|\nabla H_{N}^{\kappa}-(\nabla H)(U_{\rm
NS})|+|B_{N}^{\kappa}-B_{\kappa}(U_{\rm NS})|=o_{N\to+\infty}(1)\,,$ (5.16)

uniformly in $x$, with convergence possibly depending on $\kappa$. This will
be justified below at the end of this subsection. Then have the further
comparison $|B_{\kappa}(U_{\rm NS})-B(U_{\rm NS})|\lesssim\kappa$. We consider
$U$ as solving

$[(\nabla H)(U_{\rm NS})-s_{0}{\rm Id}]U=B(U_{\rm NS})\partial_{x}U+(\nabla
H(U_{\rm NS})-\nabla H_{N}^{\kappa})U+(B_{N}^{\kappa}-B(U_{\rm
NS}))\partial_{x}U+g_{N}^{\kappa}\,,$

with the desired phase condition. Then by the macroscopic estimate in
Proposition 2.4, we have

$\|U\|_{L^{2}}\lesssim\varepsilon^{-1}o_{N\to+\infty}(1)\|U\|_{L^{2}}+\varepsilon^{-1}(o_{N\to+\infty}(1)+\kappa)\|\partial_{x}U\|_{L^{2}}+\varepsilon^{-1}\|g_{N}^{\kappa}\|_{L^{2}}\,.$

It remains to ‘close the loop’. Examining the expression (5.15) for
$g_{N}^{\kappa}$, we see that it will be necessary to estimate quantities like
$\|\mu_{0}^{-\nicefrac{{1}}{{2}}}(v_{1}-s)(\partial_{x}\mathsf{P}_{{\rm
NS},N}^{\kappa})U\|_{L^{2}\mathcal{H}^{-1}_{\kappa}}$. More specifically, we
require

$\displaystyle\|(v_{1}-s_{0})P^{\perp}P_{{\rm
NS},N}^{\kappa}\|_{\mathcal{H}^{1}_{\kappa}\to\mathcal{H}^{-1}_{\kappa}}\lesssim_{\kappa}\varepsilon\,,\quad\|(v_{1}-s_{0})\partial_{x}P_{{\rm
NS},N}^{\kappa},\varepsilon^{-1}(v_{1}-s_{0})\partial_{x}^{2}P_{{\rm
NS},N}^{\kappa}\|_{\mathcal{H}^{1}_{\kappa}\to\mathcal{H}^{-1}_{\kappa}}\lesssim_{\kappa}\varepsilon^{2}\,,$
(5.17) $\displaystyle\qquad\qquad\qquad\|\partial^{2}_{x}P_{{\rm
NS},N}^{\kappa}\|_{\mathcal{H}^{1}_{\kappa}\to\mathcal{H}^{-1}_{\kappa}}\lesssim_{\kappa}\varepsilon^{3}\,,$
(5.18)

whose justification we temporarily delay. Note that the implicit constants may
depend on $\kappa$. By Lemma 5.3, we have

$\|g_{N}^{\kappa}\|_{\mathcal{H}^{1}_{\kappa}}\lesssim\varepsilon^{2}\|U\|+\|\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})\partial_{x}V\|_{\mathcal{H}^{-1}_{\kappa}}+\|\mathsf{\Pi}_{\leqslant
N}\mathcal{E}\|_{\mathcal{H}^{-1}_{\kappa}}+\eta\|h\|_{\mathcal{H}^{-1}_{\kappa}}\,,$

where we use the estimate on $\partial_{x}P_{{\rm NS},N}^{\kappa}$ from
(5.17). _We emphasize the decisive use of the following property of the
$\kappa$-regularized function spaces, which follows from Lemma 3.1:_

$\|f,vf\|_{\mathcal{H}^{-1}_{\kappa}}\lesssim_{\kappa}\|f\|_{\mathcal{H}^{1}_{\kappa}}\,.$

We required such an estimate for terms like
$\|(v_{1}-s_{0})\partial_{x}V\|_{\mathcal{H}^{-1}_{\kappa}}$, just as in
(4.13). Like before, we have

$P^{\perp}\partial_{x}F=P^{\perp}\partial_{x}(P_{{\rm
NS},N}^{\kappa}U)+\partial_{x}V\,,$

and, by the estimates on $P^{\perp}_{\mu_{0}}P_{{\rm NS},N}^{\kappa}$ and
$P^{\perp}_{\mu_{0}}(\partial_{x}P_{{\rm NS},N}^{\kappa})$ from (5.17), we
have

$\|(v_{1}-s_{0})\mu_{0}^{-\nicefrac{{1}}{{2}}}\partial_{x}V\|_{L^{2}\mathcal{H}^{-1}_{\kappa}}\lesssim_{\kappa}\varepsilon^{2}\|U\|_{L^{2}}+\varepsilon\|\partial_{x}U\|_{L^{2}}+\|\mu_{0}^{-\nicefrac{{1}}{{2}}}(v_{1}-s_{0})P^{\perp}\partial_{x}F\|_{L^{2}\mathcal{H}^{-1}_{\kappa}}\,.$

Meanwhile, from the expression (5.13) for $h$, we have

$\|h\|_{\mathcal{H}^{-1}_{\kappa}}\lesssim_{\kappa}\varepsilon^{3}\|U\|+\varepsilon^{2}\|\partial_{x}U\|+\|\partial_{x}^{2}V\|_{\mathcal{H}^{-1}_{\kappa}}\,,$

by the estimate (5.18). Of course,

$P^{\perp}\partial_{x}^{2}F=P^{\perp}\partial_{x}^{2}(P_{{\rm
NS},N}^{\kappa}U)+\partial_{x}^{2}V\,,$

so that

$\|(v_{1}-s_{0})\mu_{0}^{-\nicefrac{{1}}{{2}}}\partial_{x}^{2}V\|_{L^{2}\mathcal{H}^{-1}_{\kappa}}\lesssim_{\kappa}\varepsilon^{3}\|U\|_{L^{2}}+\varepsilon^{2}\|\partial_{x}U\|_{L^{2}}+\varepsilon\|\partial_{x}^{2}U\|_{L^{2}}+\|\mu_{0}^{-\nicefrac{{1}}{{2}}}(v_{1}-s_{0})P^{\perp}\partial_{x}^{2}F\|_{L^{2}\mathcal{H}^{-1}_{\kappa}}\,.$

Combining the estimates is enough to close the loop. ∎

We now proceed with the analysis of the coefficients (5.16), together with
(5.17) and (5.18). We necessarily utilize a property of our Galerkin basis;
namely, the decay and regularity of a function is reflected in the decay of
its coefficients in the Galerkin basis. Given $f\in L^{2}$, denote by
$(f_{\alpha})$ the coefficients in the Hermite basis. Then

$f,Tf,\ldots,T^{k}f\in
L^{2}\iff\sum_{\alpha}\langle\lambda_{\alpha}\rangle^{2k}|f_{\alpha}|^{2}<+\infty\,,$

with equivalence of norms:

$\|f,Tf,\ldots,T^{k}f\|_{L^{2}}\approx_{k}\left(\sum_{\alpha}\langle\lambda_{\alpha}\rangle^{2k}|f_{\alpha}|^{2}\right)^{\nicefrac{{1}}{{2}}}\,.$

The key will be to estimate

$((\mathsf{L}_{22,N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant
N}-\mathsf{L}_{22}^{-1})\mathcal{E}$ (5.19)

on well-localized, high regularity, purely microscopic functions
$\mathcal{E}=\mu_{0}^{\nicefrac{{1}}{{2}}}z$.

######  Lemma 5.4.

For $0<\varepsilon\ll 1$, $0\leqslant\kappa\leqslant 1$, and purely
microscopic $z$ satisfying the normalization

$\|\mu_{0}^{-\nicefrac{{1}}{{8}}}\partial_{\beta}z\|_{L^{2}}\leqslant
1\,,\quad\forall|\beta|\leqslant 1000\,,$

we have

$\|\mu_{0}^{-\nicefrac{{1}}{{2}}}((\mathsf{L}_{22,N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant
N}-(\mathsf{L}_{22}^{\kappa})^{-1})\mathcal{E}\|_{H^{2}_{\varepsilon}\mathcal{H}^{1}_{\kappa}}\lesssim_{\kappa}N^{-50}\,.$

A typical example is $z=P^{\perp}L_{\rm NS}^{\kappa}g$ with $g\in\ker L$.

###### Proof of Lemma 5.4.

Let $F_{N}=(\mathsf{L}_{22,N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant
N}\mathcal{E}$ and $F=(\mathsf{L}^{\kappa}_{22})^{-1}\mathcal{E}$, both of
which are purely microscopic. Then

$L_{\rm NS}^{\kappa}f=z\,,\quad\Pi_{\leqslant N}L_{\rm
NS}^{\kappa}f_{N}=\Pi_{\leqslant N}z\,.$ (5.20)

Therefore,

$\Pi_{\leqslant N}L_{\rm NS}^{\kappa}(\Pi_{\leqslant
N}f-f_{N})=-\Pi_{\leqslant N}L_{\rm NS}^{\kappa}\Pi_{>N}f\,.$

The estimates on the non-truncated problem yield
$\|\mu_{0}^{-\nicefrac{{1}}{{16}}}\partial_{\beta}f\|_{L^{2}}\lesssim 1$ for
all $|\beta|\leqslant 1001$. Hence,

$\|\Pi_{\leqslant
N}f-f_{N}\|_{\mathcal{H}^{1}}\lesssim\|\Pi_{>N}f\|_{\mathcal{H}^{1}}\lesssim
N^{-100}\,,$

due to the decay of the Galerkin coefficients of $f$. In particular, we have

$\|\langle v\rangle^{10}(f-f_{N})\|_{\mathcal{H}^{1}}\lesssim\|\langle
v\rangle^{10}(\Pi_{\leqslant N}f-f_{N})\|_{\mathcal{H}^{1}}+\|\langle
v\rangle^{10}\Pi_{>N}f\|_{\mathcal{H}^{1}}\lesssim N^{-50}\,.$ (5.21)

We next demonstrate how to estimate spatial derivatives of (5.19) when
$\mathcal{E}$ is fixed. The desired estimates are on $\langle
v\rangle^{10}\partial_{x}(f-f_{N})$ in $L^{2}$. We differentiate (5.20) once
to obtain

$L\partial_{x}f=2\Gamma_{\kappa}(\mu^{-1/2}\partial_{x}\mu_{\rm
NS},f)\,,\quad\Pi_{\leqslant N}Lf_{N}=2\Pi_{\leqslant
N}\Gamma_{\kappa}(\mu^{-1/2}\partial_{x}\mu_{\rm NS},f_{N})\,.$

The relevant equation is

$\Pi_{\leqslant N}L\partial_{x}(\Pi_{\leqslant
N}f-f_{N})=2\Gamma_{\kappa}(\mu^{-\nicefrac{{1}}{{2}}}\partial_{x}\mu_{\rm
NS},\Pi_{\leqslant
N}f-f_{N})+2\Gamma_{\kappa}(\mu^{-\nicefrac{{1}}{{2}}}\partial_{x}\mu_{\rm
NS},\Pi_{>N}f)-\Pi_{\leqslant N}L\Pi_{>N}\partial_{x}f\,.$

The estimates on the non-truncated problem yield
$\|\mu_{0}^{-\nicefrac{{1}}{{16}}}\partial_{\beta}\partial_{x}f\|_{L^{2}}\lesssim\varepsilon^{2}$
for all $|\beta|\leqslant 1001$. This yields

$\|\partial_{x}(\Pi_{\leqslant
N}f-f_{N})\|_{\mathcal{H}^{1}}\lesssim\varepsilon^{2}\|\Pi_{>N}f,\partial_{x}f\|_{\mathcal{H}^{1}}\lesssim
N^{-100}\,,$ $\|\langle
v\rangle^{10}\partial_{x}(f-f_{N})\|_{\mathcal{H}^{1}}\lesssim\varepsilon^{2}N^{-50}\,,$

by reasoning similar to (5.21). Analogous reasoning for the second derivative
$\partial_{x}^{2}(f-f_{N})$ yields

$\|\langle
v\rangle^{10}\partial_{x}^{2}(f-f_{N})\|_{\mathcal{H}^{1}}\lesssim\varepsilon^{3}N^{-50}\,.$

∎

###### Proof of (5.17) and (5.18).

Recall that $\mathsf{P}_{{\rm
NS},N}^{\kappa}=(I-(\mathsf{L}_{22,N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant
N}\mathsf{L}_{{\rm NS}}^{\kappa})\mathsf{P}_{\mu_{0}}$ and, therefore,
$\mathsf{P}_{\mu_{0}}^{\perp}\mathsf{P}_{{\rm
NS},N}^{\kappa}=-(\mathsf{L}_{22,N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant
N}\mathsf{L}_{{\rm NS}}^{\kappa}\mathsf{P}_{\mu_{0}}$. If
$(\mathsf{L}_{22,N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant N}$ were replaced by
$\mathsf{L}_{22}^{-1}$, then the desired estimates (5.17) and (5.18) would be
evident. Moreover, Lemma 5.4 contains the necessary error estimates between
the two operators. ∎

Finally, we explain the convergence of the coefficients.

###### Proof of Equation 5.16 (Convergence of coefficients).

First, we address

$\nabla H_{N}^{\kappa}-\nabla H(U_{\rm
NS})=\mathsf{P}_{\mu_{0}}v_{1}(\mathsf{P}_{{\rm
NS},N}^{\kappa}-\mathsf{P}_{\mu_{\rm NS}})\,,$

which converges to zero uniformly in $x$ by Lemma 5.4. Next, we address

$\displaystyle B_{N}^{\kappa}-B=\mathsf{P}_{\mu_{0}}v_{1}[(\mathsf{L}_{{\rm
NS},N}^{\kappa})^{-1}\mathsf{P}_{{\rm
NS},N}^{\perp,\kappa}(\mathsf{\Pi}_{\leqslant N}v_{1}\mathsf{P}_{{\rm
NS},N}^{\kappa})-(\mathsf{L}_{{\rm NS}}^{\kappa})^{-1}\mathsf{P}_{{\rm
NS}}^{\perp,\kappa}(v_{1}\mathsf{P}_{\mu_{\rm NS}})]\,.$

We divide the inside bracket into the terms

$(\mathsf{L}_{{\rm NS},N}^{\kappa})^{-1}\mathsf{P}^{\perp,\kappa}_{{\rm
NS},N}\mathsf{\Pi}_{\leqslant N}v_{1}(\mathsf{P}_{{\rm
NS},N}^{\kappa}-\mathsf{P}_{\mu_{\rm NS}})\,,$ $((\mathsf{L}_{{\rm
NS},N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant N}-L^{-1}_{\rm
NS})\mathsf{P}^{\perp}_{\mu_{\rm NS}}v_{1}\mathsf{P}_{\mu_{\rm NS}}\,,$

and

$(\mathsf{L}_{{\rm NS},N}^{\kappa})^{-1}\mathsf{\Pi}_{\leqslant
N}(\mathsf{P}^{\perp,\kappa}_{{\rm NS},N}-\mathsf{P}^{\perp}_{\mu_{\rm
NS}})v_{1}\mathsf{P}_{\mu_{\rm NS}}\,,$

each of which converges to zero uniformly in $x$ by Lemma 5.4. ∎

###  5.2 Qualitative Galerkin approximation

######  Proposition 5.5.

Let $0<\kappa\ll\varepsilon\ll_{\kappa}$, $N\geqslant 10$, and $\eta\geqslant
0$. There exists a solution $f\in H^{1}\mathcal{H}_{\kappa}^{1}$ to the
approximate system (5.2)-(5.3).

We rewrite the approximate system as a $5+2r$-dimensional first-order ODE in
the image of $\Pi_{\leqslant N}$, where $r=r(N)$ is the dimension of the
finite-dimensional microscopic subspace, namely, the image of the projection
$\Pi^{\perp}\Pi_{\leqslant N}$. The unknown is

$\mathcal{U}=\begin{bmatrix}u\\\ v\\\ w\end{bmatrix}=\begin{bmatrix}PF\\\
P^{\perp}F\\\ \partial_{x}P^{\perp}F\end{bmatrix}\,.$

After $x$-integrating the $u^{\prime\prime}$ equation, we have

$\displaystyle-\eta u^{\prime}+A_{11}u+A_{12}v$ $\displaystyle=0$
$\displaystyle v^{\prime}$ $\displaystyle=w$ $\displaystyle-\eta
w^{\prime}+A_{21}u^{\prime}+A_{22}w+L_{21,N}^{\kappa}u+L_{22,N}^{\kappa}v$
$\displaystyle=z_{N}\,,$

where $A_{11}=P(v_{1}-s_{0})P$, $A_{12}=P(v_{1}-s_{0})P^{\perp}$, etc. We
substitute the $u^{\prime}$ equation into the $w^{\prime}$ equation to obtain
the ODE

$\mathcal{U}^{\prime}=\mathbb{A}(x)\mathcal{U}+\begin{bmatrix}0&0&z_{N}\end{bmatrix}\,,$

where

$\mathbb{A}=\eta^{-1}\begin{bmatrix}A_{11}&A_{12}&0\\\ 0&&\eta I\\\
\eta^{-1}A_{21}A_{11}-L_{21,N}^{\kappa}&\eta^{-1}A_{21}A_{12}-L_{22,N}^{\kappa}&A_{22}\end{bmatrix}\,.$

We write $\mathbb{A}_{L,R}$ to denote the asymptotic operators as
$x\to\mp\infty$. We shall require the following lemma on $\mathbb{A}_{L,R}$.

######  Lemma 5.6.

For $0<\kappa\ll 1$, $0<\varepsilon\ll_{\kappa}1$, $\eta>0$, and
$N\gg_{\kappa,\varepsilon,\eta}1$, the asymptotic matrices $\mathbb{A}_{L,R}$
are hyperbolic, that is, $\sigma(\mathbb{A}_{L,R})\cap i\mathbb{R}=\emptyset$.

###### Proof of Lemma 5.6.

For the sake of contradiction, we suppose that one of the asymptotic matrices
has an imaginary eigenvalue $i\tau$, $\tau\in\mathbb{R}$, with non-trivial
eigenvector $\mathcal{U}$. We drop the $L,R$ notation. Substituting $w=i\tau
v$ and writing $F=\mu_{0}^{1/2}(u+v)$, we have

$\displaystyle\mathsf{P}_{\mu_{0}}(v_{1}-s_{0})F$
$\displaystyle=i\tau\eta\mathsf{P}_{\mu_{0}}F$ (5.22)
$\displaystyle(i\tau\mathsf{\Pi}_{\leqslant N}(v_{1}-s_{0})+\mathsf{L}_{{\rm
NS},N}^{\kappa}+\tau^{2}\eta)F$ $\displaystyle=0\,.$

The first equation is redundant when $\tau\neq 0$ but useful when $\tau=0$.

When $\tau=0$, we have that $\mathsf{L}^{\kappa}_{{\rm NS},N}F=0$, that is,
$F=\mathsf{P}^{\kappa}_{{\rm NS},N}U$ ($U=\mathsf{P}_{\mu_{0}}F$). The first
equation yields

$(\nabla
H_{N}^{\kappa}-s_{0}I)U=\mathsf{P}_{\mu_{0}}(v_{1}-s_{0})\mathsf{P}_{{\rm
NS},N}^{\kappa}U=0\,.$

However, for $N$ sufficiently large, $\nabla H_{N}^{\kappa}-s_{0}I$ has
trivial kernel, which is the desired contradiction.

From now on, assume $\tau\neq 0$. First, we repeat the Kawashima-type energy
estimate from the quantitative Galerkin approximation. We write
$F=\sqrt{\mu_{0}}f$ and obtain

$|\tau|\|f\|_{\mathcal{H}^{1}_{\kappa}}+\eta\tau^{2}\|f\|_{L^{2}}+\|P^{\perp}f\|_{\mathcal{H}^{1}_{\kappa}}\lesssim\varepsilon\|Pf\|\,.$

When $|\tau|+\eta\tau^{2}\geqslant C_{0}\varepsilon$ for $C_{0}\gg 1$, we
obtain the desired contradiction $f=0$.

We now consider $|\tau|+\eta\tau^{2}\leqslant C_{0}\varepsilon$. To simplify
computations, it will be convenient to normalize $\|U\|=1$. We repeat the
Chapman-Enskog argument, which will be relatively simple because derivatives
are replaced by $i\tau$ and we normalize $U$. We write
$F=U+\mathsf{P}^{\perp}F=\mathsf{P}_{{\rm NS},N}^{\kappa}U+V$. Then

$\eta\tau^{2}F+i\tau\mathsf{\Pi}_{\leqslant N}(v_{1}-s_{0})\mathsf{P}_{{\rm
NS},N}^{\kappa}U+i\tau\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})V+\mathsf{L}_{{\rm NS},N}^{\kappa}V=0\,,$

cf. (5.11). Again, we apply $\mathsf{P}_{{\rm NS},N}^{\perp,\kappa}$ and
invert $\mathsf{L}_{{\rm NS},N}^{\kappa}$ to obtain

$V=-(\mathsf{L}_{{\rm NS},N}^{\kappa})^{-1}\mathsf{P}_{{\rm
NS},N}^{\perp,\kappa}(i\tau\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})\mathsf{P}_{{\rm NS},N}^{\kappa}U+i\tau\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})V+\eta\tau^{2}V)\,,$

cf. (5.12). We insert this into the macroscopic equation in (5.22) to obtain

$(\nabla H_{N}^{\kappa}-s_{0}I)U-i\tau B_{N}^{\kappa}U=i\tau\eta
U+g_{N}^{\kappa}$

where

$g_{N}^{\kappa}=P(v_{1}-s_{0})(\mathsf{L}_{{\rm NS},N}^{\kappa})^{-1}[P_{{\rm
NS},N}^{\perp,\kappa}(i\tau\mathsf{\Pi}_{\leqslant
N}(v_{1}-s_{0})V)+\eta\tau^{2}V]=O(|\tau|+\eta\tau^{2})\|V\|_{\mathcal{H}^{1}_{\kappa}}=O(\varepsilon^{2})\,,$

and the coefficients $\nabla H_{N}^{\kappa}$ and $B_{N}^{\kappa}$ are
evaluated at the asymptotic endpoint. We utilize that the microscopic estimate
yields $\|V\|_{\mathcal{H}^{1}_{\kappa}}\lesssim_{\kappa}\varepsilon\|U\|$.
Hence,

$(\nabla H_{N}^{\kappa}-s_{0}I)U-i\tau B_{N}^{\kappa}U-i\tau\eta
U=O(\varepsilon^{2})\,.$

We moreover write

$(\nabla H(U_{\rm NS})-s_{0}I)U-i\tau BU-i\tau\eta U=(\nabla H(U_{\rm
NS})-\nabla H_{N}^{\kappa})U+i\tau(B_{N}^{\kappa}-B)U+O(\varepsilon^{2})\,.$

We wish to invert $\nabla H(U_{\rm NS})-s_{0}I-i\tau B-i\tau\eta I$ with
estimate $\|(\nabla H(U_{\rm NS})-s_{0}I-i\tau B-i\tau\eta
I)^{-1}\|\lesssim\varepsilon^{-1}$. If this is possible, then

$\|U\|\lesssim\varepsilon^{-1}o_{N\to+\infty}(1)\|U\|+\varepsilon^{-1}(o_{N\to+\infty}(1)+\kappa)|\tau|\|U\|+\varepsilon\,,$

which contradicts that $\|U\|=1$.

If $|\tau|+|\tau|\eta\leqslant 2c_{0}$ with $c_{0}\ll 1$, then we consider the
full inverse as a perturbation of $(\nabla H-s_{0}I)^{-1}$ satisfying
$\|(\nabla H-s_{0}I)^{-1}\|\lesssim\varepsilon^{-1}$.

Otherwise, we rely on symmetrizability and the Kawashima theory in Remark B.4.
The compressible Navier-Stokes are symmetrizable at every non-vacuum state,
not merely at $(u_{L},s_{L})$, and the symmetric positive-definite symmetrizer
$A^{0}$ varies continuously. The desired estimate will be obtained as a
consequence of estimates for the inverse of the symmetrized operator
$A^{0}(i(\nabla H-s_{0}I)+\tau(B+\eta I))$, and, in particular, energy
estimates for the equation

$A^{0}(i(\nabla H-s_{0}I)+\tau(B+\eta I))U=G\,,$

for unknown $U$ and right-hand side $G$. If $|\tau|\eta\geqslant
c_{0}\varepsilon$, then a standard energy estimate yields the conclusion,
since $\tau\eta A^{0}$ is coercive. The remaining case is
$c_{0}\varepsilon\leqslant|\tau|\leqslant C_{0}\varepsilon$. $A^{0}(\nabla
H-s_{0}I)$ satisfies the Kawashima condition with respect to $A^{0}B$ because
the condition is invariant under small symmetric perturbations of the
matrices, the constants in the estimates vary in a controlled way, and we
verified the condition at $(u_{L},s_{L})$. The Kawashima-type energy estimates
in Lemma C.2 yield that the matrix is invertible with inverse size
$O(\varepsilon^{-1})$.999Notice that $\tau$ multiplies the damping term
$B+\eta I$, not the oscillation term $i(\nabla H-s_{0}I)$. ∎

######  Remark 5.7.

For $|\tau|\eta\ll\varepsilon$, Lemma 5.6 could also be closed by considering
$-i\tau\eta I$ as a small perturbation of $\nabla H-s_{0}I-i\tau B$ and
applying the techniques in Proposition B.9. However, for the convenient
topological argument in Lemma 5.8, it will be useful to have hyperbolicity of
the endpoint matrices $\mathbb{A}_{L,R}$ for arbitrary $\eta$. We re-invoke
symmetrizability specifically to handle the case when $|\tau|\eta$ is not
small with respect to $\varepsilon$.

######  Lemma 5.8.

For $0<\kappa\ll 1$, $0<\varepsilon\ll_{\kappa}$, $\eta>0$, and
$N\gg_{\kappa,\varepsilon,\eta}1$, the asymptotic matrices $\mathbb{A}_{L,R}$
satisfy ${\rm dim}\,E^{u}(\mathbb{A}_{L,R})+{\rm
dim}\,E^{s}(\mathbb{A}_{R,L})=5+2r+1$, where $E^{s}$ and $E^{u}$ denote the
stable and unstable subspaces, respectively.

###### Proof.

We again drop the $L,R$ notation. The eigenvalues of $\mathbb{A}$ are
continuous in the parameter $\eta$ and, since $\mathbb{A}$ is hyperbolic for
every $\eta$ by Lemma 5.6, they cannot cross the imaginary axis. Therefore, it
is paradoxically enough to determine the dimensions of $E^{u}$ and $E^{s}$
when $\eta\gg 1$. We make the “elliptic” change of variables
$(u,v,w)\mapsto(u,v,\tilde{w})$, where
$\tilde{w}=\eta^{\nicefrac{{1}}{{2}}}w$. The conjugated matrix
$\tilde{\mathbb{A}}$ acting in the new variables is given by

$\eta^{1/2}\tilde{\mathbb{A}}=\begin{bmatrix}0&0&0\\\ 0&0&I\\\
-L_{21}&-L_{22}&0\end{bmatrix}+\eta^{-1/2}\begin{bmatrix}A_{11}&A_{12}&0\\\
0&0&0\\\
\eta^{-\nicefrac{{1}}{{2}}}A_{21}A_{11}&\eta^{-\nicefrac{{1}}{{2}}}A_{21}A_{12}&0\end{bmatrix}=:T_{0}+\delta
T_{1}+\delta^{2}T_{2}\,,$ (5.23)

where $\delta=\eta^{-\nicefrac{{1}}{{2}}}$. Thus we encounter a finite-
dimensional spectral perturbation problem.

First, we analyze the spectrum of $T_{0}$. We will prove that

${\rm dim}\,E^{u}(T_{0})={\rm dim}\,E^{s}(T_{0})=r\text{ and }{\rm dim}\,\ker
T_{0}=5\,.$

We begin with the non-zero eigenvalues. The $(v,\tilde{w})$ subspace is
invariant under $T_{0}$; its restriction to this subspace is
$\tilde{T}_{0}=\begin{bmatrix}0&I\\\ -L_{22,N}^{\kappa}&0\end{bmatrix}$. The
matrix $-L_{22,N}^{\kappa}$ is bounded below:
$\|f\|_{L^{2}}^{2}\lesssim-\langle L_{22,N}^{\kappa}f,f\rangle$ for
microscopic $f$. Hence, its spectrum belongs to $\\{{\rm{Re}}\,\lambda>0\\}$.
We observe that $\tilde{T}_{0}^{2}=\begin{bmatrix}-L_{22,N}^{\kappa}&0\\\
0&-L_{22,N}^{\kappa}\end{bmatrix}$, and therefore $\tilde{T}_{0}$ is
hyperbolic, having no eigenvalues on the imaginary axis. The matrix
$\tilde{T}_{0}$ has the further property that it conjugates to
$-\tilde{T}_{0}$:

$\begin{bmatrix}I&0\\\ 0&-I\end{bmatrix}\begin{bmatrix}0&I\\\
-L_{22,N}^{\kappa}&0\end{bmatrix}\begin{bmatrix}I&0\\\
0&-I\end{bmatrix}=-\begin{bmatrix}0&I\\\ -L_{22,N}^{\kappa}&0\end{bmatrix}\,.$
(5.24)

This is motivated by the observation that $\lambda$ is an eigenvalue of
$\tilde{T}_{0}$ with eigenvector $(v,w)$ if and only if $v=\lambda w$ and
$-L_{22,N}^{\kappa}v=\lambda^{2}v$. In particular, (5.24) implies that
eigenvalues of $\tilde{T}_{0}$ arise only in $\pm$ pairs, and the eigenspaces
satisfy $E_{\lambda}=\begin{bmatrix}I&0\\\ 0&-I\end{bmatrix}E_{-\lambda}$ for
each eigenvalue $\lambda$.

The operator $T_{0}$ has a five-dimensional kernel, namely, $\ker
T_{0}=\\{(u,v,0):u+v=P^{\kappa}_{{\rm NS},N}u\\}$. This accounts for the
remaining eigenvalues of $T_{0}$.

The projection $\begin{bmatrix}P^{\kappa}_{{\rm NS},N}&\\\ &0\end{bmatrix}$
has range $\ker T_{0}$ and kernel $\\{(0,v,\tilde{w})\\}$, which is a
complementary invariant subspace to $\ker T_{0}$. Therefore, $P_{{\rm
NS},N}^{\kappa}$ is the spectral projection onto $\ker T_{0}$, and $I-P_{{\rm
NS},N}^{\kappa}$ is the complementary spectral projection.

We now return to the spectrum of (5.23). Since zero is semisimple, the
analytic perturbation theory yields that the perturbed eigenvalues
$\lambda_{k}$, $1\leqslant k\leqslant 5$ (counted with algebraic
multiplicity), satisfy $\lambda_{k}=\delta\mu_{k}+o(\delta)$, where $\mu_{k}$
is the $k$th eigenvalue of $P_{{\rm NS},N}^{\kappa}T_{1}|_{\ker T_{0}}$. See
[Baumgartel, p. 265-269]. If we parameterize $\ker T_{0}$ by $u\mapsto P_{{\rm
NS},N}u$, where $u\in\ker T_{0}$, then this operator is exactly
$\mu_{0}^{-\nicefrac{{1}}{{2}}}(\nabla
H_{N}-s_{0}I)\mu_{0}^{\nicefrac{{1}}{{2}}}=P(v_{1}-s_{0})P_{{\rm
NS},N}^{\kappa}$, where $\nabla H_{N}$ is evaluated at the asymptotic
endpoint. Recall that $|\nabla H_{N}^{\kappa}-\nabla H(U_{\rm NS})|=o(1)$ as
$N\to+\infty$ and $\kappa\rightarrow 0$ and we recognize $\nabla H(U_{\rm
NS})$ as the Euler flux function in suitable coordinates. This implies that,
at the left endpoint, the null eigenvalues of $T_{0}$ become four stable
eigenvalues and one unstable eigenvalue of $\mathbb{A}_{L}$. At the right
endpoint, the null eigenvalues of $T_{0}$ become five stable eigenvalues and
no unstable eigenvalues of $\mathbb{A}_{R}$. (The shock speed $s_{0}$ is
slower than the fastest characteristic on the left, faster than the fastest
characteristic on the right.) This completes the proof. ∎

###### Proof of Proposition 5.1.

Fix $0<\kappa\ll 1$ and $0<\varepsilon\ll_{\kappa}1$. For $0<\eta\ll 1$ and
$N\gg_{\kappa,\eta}1$, the conditions of Lemma 5.9 (solvability) below are
verified by Lemma 5.6 (hyperbolicity), Lemma 5.8 (dimension counting), and
Proposition 5.2 (_a priori_ estimates and uniqueness). Let $f^{(N,\eta)}$
denote the solution to the approximate (conjugated) system (5.2) with
constraint (5.3). We consider $N\to+\infty$ and $\eta\to 0^{+}$. By the _a
priori_ estimates and weak compactness, there exists a subsequence, which we
do not label, such that $f^{(N,\eta)}\rightharpoonup f$ in
$H^{2}\mathcal{H}^{1}$. Subsequently, we have that
$(v_{1}-s_{0})\partial_{x}f^{(N,\eta)}$ converges to
$(v_{1}-s_{0})\partial_{x}f$, $\Pi_{\leqslant N}E$ converges to $E$, and
$\eta\partial_{x}^{2}f^{(N,\eta)}$ converges to zero, in the sense of
distributions.

To verify the convergence of $L_{{\rm NS},N}^{\kappa}f^{(N,\eta)}$, we
consider $\langle L_{{\rm NS},N}^{\kappa}f^{(N,\eta)},\psi\rangle$, where
$\psi=c(x)\phi(v)$, $c\in C^{\infty}_{0}(\mathbb{R})$ and $\phi$ belongs to
our Galerkin basis. For $N\gg 1$ depending on $\phi$, this expression is
simply $\langle L_{\rm NS}^{\kappa}f^{(N,\eta)},\psi\rangle$. After
integrating by parts in the collision operator to move a $v$ derivative onto
$\phi$, see Section 3.2, it is relatively straightforward to pass to the limit
using the weak convergence.

Finally, since $Pf^{(N,\eta)}\in H^{2}_{x}$ is uniformly bounded, we can pass
to a further subsequence to ensure that
$\ell_{\varepsilon}(Pf^{(N,\eta)}(0))\to\ell_{\varepsilon}(Pf(0))$, which
verifies the one-dimensional constraint. ∎

Finally, we state the abstract ODE lemma which we are invoking.

######  Lemma 5.9 (Linear ODE solvability).

Consider the ODE

$\dot{u}=A(x)u+f$

where $u:\mathbb{R}\to\mathbb{R}^{n}$, $A\in
L^{\infty}(\mathbb{R};\mathbb{R}^{n\times n})$, and $f\in
L^{\infty}(\mathbb{R};\mathbb{R}^{n})$, supplemented by conditions

$\ell_{1}\cdot u(0)=c_{1},\ldots,\ell_{m}\cdot u(0)=c_{m}\,,$

where $\ell_{1},\ldots,\ell_{m}\in\mathbb{R}^{n}$ are linearly independent,
and $\vec{c}\in\mathbb{R}^{m}$.

Suppose that

  1. 1.

$A$ is continuous and exponentially asymptotic to hyperbolic matrices
$A_{\pm}$ as $x\to\pm\infty$,

  2. 2.

$\dim E^{u}_{-}+\dim E^{s}_{+}=n+m$, where $E^{u}_{-}$ is the unstable
subspace of $A_{-}$ and $E^{s}_{+}$ is the stable subspace of $A_{+}$, and

  3. 3.

$u=0$ is the unique bounded solution to the above problem when $f=0$ and
$\vec{c}=0$.

Then, to each $f\in L^{\infty}(\mathbb{R};\mathbb{R}^{n})$ and
$\vec{c}\in\mathbb{R}^{m}$, there exists a unique bounded solution $u$ to the
problem.

See Appendix B.3, [HenryBook, p. 136-142], or [KapitulaPromislow, Chapter 3]
for related arguments.

###  5.3 Extending the range of $\varepsilon$ and taking $\kappa\to 0^{+}$

We now show how to use the method of continuity to extend the range of
$\varepsilon$:

######  Proposition 5.10.

For all $0<\kappa\ll\varepsilon\ll 1$, purely microscopic $z$ satisfying $z\in
H^{2}\mathcal{H}^{-1}_{-10,\kappa}$, and $d\in\mathbb{R}$, there exists a
unique solution $f\in H^{2}\mathcal{H}^{1}_{-10,\kappa}$ to the
$\kappa$-regularized system satisfying $\ell_{\varepsilon}(Pf(0))=d$.

The key point is the existence for the enlarged range of $\varepsilon$.
Uniqueness is already guaranteed by the _a priori_ estimates.

To prove Proposition 5.10, we require

######  Lemma 5.11 (Method of continuity).

Let $X$ and $Y$ be Banach spaces. Let $A(t):[0,1]\to B(X,Y)$ be a continuous
family of bounded linear operators. Suppose the following _a priori_ estimate:
There exists $C>0$ such that for all $x\in X$ and $t\in[0,1]$, we have

$\|x\|_{X}\leqslant C\|A(t)x\|_{Y}\,.$ (5.25)

Then the following are equivalent:

  1. 1.

There exists $t_{0}$ such that $A(t_{0})$ is surjective.

  2. 2.

For all $t\in[0,1]$, $A(t)$ is surjective.

###### Proof.

($\impliedby$) is obvious, so we prove ($\implies$). The _a priori_ estimate
(5.25) grants that $A(t_{0})$ is injective, so the assumption that it is
surjective implies that it is boundedly invertible and
$\|A(t_{0})^{-1}\|_{Y\to X}\leqslant C$, again by (5.25). Hence, $A(t)$ is
invertible for $|t-t_{0}|\ll_{C,D}1$, where $D=\max_{t\in[0,1]}\|A(t)\|_{X\to
Y}<+\infty$. We repeat this argument $O_{C,D}(1)$ times to demonstrate that
$A(t)$ is invertible for all $t\in[0,1]$. ∎

###### Proof of Proposition 5.10.

Let $X=\\{f:f\in
H^{2}\mathcal{H}^{1}_{-10,\kappa}\,,(v_{1}-s_{0})\partial_{x}f\in
H^{2}\mathcal{H}^{-1}_{-10,\kappa}\,,P(v_{1}-s_{0})\partial_{x}f=0\\}$ and
$Y=\\{z\in H^{2}_{x}\mathcal{H}^{-1}_{\kappa}\cap{\rm
Mic}:H^{2}\mathcal{H}^{-1}_{-10,\kappa}\\}\times\mathbb{R}$ with appropriate
norms. For $0<\varepsilon\ll 1$, we define

$A(\varepsilon)f:=((v_{1}-s_{0})\partial_{x}f+L_{\rm
NS}^{\kappa}f,\ell_{\varepsilon}(Pf))\,.$

Under the assumptions, $A(\varepsilon):X\to Y$ is bounded, continuous in the
parameter $\varepsilon$ and satisfies the _a priori_ estimate
$\|f\|_{X}\leqslant C\|A(\varepsilon)f\|_{Y}$ from Proposition 4.1, which we
proved in subsection 4.2. (The continuity of $\ell_{\varepsilon}$ is contained
in Remark B.10.) Moreover, for $0<\varepsilon\ll_{\kappa}1$, $A(\varepsilon)$
is invertible. Therefore, the method of continuity therefore yields
invertibility in the $\kappa$-independent range where the _a priori_ estimates
hold. ∎

Finally, we complete the proof of Proposition 4.1.

###### Proposition 4.1.

Fix $0<\varepsilon\ll 1$, purely microscopic $z$ satisfying $\langle
v\rangle^{20}z\in H^{10}\mathcal{H}^{1}$ (in particular, it is uniformly
bounded in the $\kappa$-regularized spaces for $0<\kappa\leqslant 1$), and
$d\in\mathbb{R}$. Existence for more general $z$ will be available _a
posteriori_ by approximation. For $0<\kappa\ll\varepsilon\ll 1$, Proposition
5.10 furnishes a solution $f^{\kappa}$ to the $\kappa$-regularized system with
right-hand side $z$ and $\ell_{\varepsilon}(Pf^{\kappa}(0))=d$. The _a priori_
estimates from Proposition 4.1 grant

$\sup_{0<\kappa\ll
1}\|f^{\kappa}\|_{H^{2}\mathcal{H}^{1}_{-10}}+\kappa^{1/2}\|f^{\kappa}\|_{H^{2}\mathcal{H}^{1}_{\rm
R}}<+\infty\,,$

and therefore there exists a sequence, which we do not label, satisfying
$f^{\kappa}\rightharpoonup f$ in $H^{2}\mathcal{H}^{1}_{-10}$ as $\kappa\to
0^{+}$. By compact embedding, $Pf^{\kappa}(0)\to Pf(0)$, so
$\ell_{\varepsilon}(Pf^{\kappa}(0))\to\ell_{\varepsilon}(Pf(0))$, and the
value of the constraint is kept. Evidently
$(v_{1}-s_{0})\partial_{x}f^{\kappa}$ converges to
$(v_{1}-s_{0})\partial_{x}f$ in the sense of distributions, so it remains to
verify the convergence of $L_{\rm NS}^{\kappa}f^{\kappa}$ to $L_{\rm NS}f$.
Let $\phi\in C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{3})$ be a test
function. We have

$\langle L_{\rm NS}^{\kappa}f^{\kappa},\phi\rangle=\langle L_{\rm
NS}f^{\kappa},\phi\rangle+\kappa\langle L^{\rm R}_{\rm
NS}f^{\kappa},\phi\rangle\,.$

Like when we took $N\to+\infty$ in the Galerkin approximation, the term
$\langle L_{\rm NS}f^{\kappa},\phi\rangle$ is handled by integrating by parts
in the collision operator to move a $v$ derivative onto $\phi$, see Section
3.2, after which one may pass to the limit $\langle L_{\rm NS}f,\phi\rangle$
using the weak convergence. Meanwhile, the $\kappa$ term converges to zero:

$\kappa|\langle L^{\rm R}_{\rm
NS}f^{\kappa},\phi\rangle|\lesssim\kappa^{1/2}\|f^{\kappa}\|_{L^{2}\mathcal{H}^{1}_{\rm
R}}\kappa\|\phi\|_{L^{2}\mathcal{H}^{1}_{\rm R}}=O(\kappa^{1/2}\,.$

This completes the proof. ∎

##  6 Fixed point argument

###### Proof of Theorem 1.1.

In sections 4 and 5, we constructed and estimated solutions, $f$, to the
conjugated, linearized equation (2.21) for the residual. In order to solve the
nonlinear problem, we will solve the conjugated version of (2.19); namely,

$\displaystyle(v_{1}-s_{0})\partial_{x}f+L_{\rm NS}f=2\Gamma\left[g_{\rm
NS},f\right]+{\Gamma}[f,f]-\mu^{-\nicefrac{{1}}{{2}}}_{0}\mathcal{E}_{\rm
NS}\,,$

where $g_{\rm NS}=\mu_{0}^{-\nicefrac{{1}}{{2}}}G_{\rm NS}$ is defined using
(2.16). Using Proposition 2.6, we can estimate the purely microscopic right-
hand side

$\displaystyle z[h]:=2\Gamma\left[g_{\rm
NS},h\right]+\Gamma[h,h]-\mu^{-\nicefrac{{1}}{{2}}}_{0}\mathcal{E}_{\rm
NS}\,;$

in particular, the right-hand side satisfies the assmptions of Proposition
4.2. We therefore define the mapping $h\mapsto\mathcal{A}[h]$ by applying
Proposition 4.1, so that $\mathcal{A}[h]$ is the unique (in
$H^{2}\mathcal{H}^{1}_{-10}$) solution to the linear problem

$\displaystyle\left((v_{1}-s_{0})\partial_{x}+L_{\rm
NS}\right)\mathcal{A}[h]=2\Gamma\left[g_{\rm
NS},h\right]+{\Gamma}[h,h]-\mu_{0}^{-\nicefrac{{1}}{{2}}}\mathcal{E}_{\rm
NS}=z[h]\,,$

subject to the constraint

$\displaystyle\ell_{\varepsilon}\left(P\mathcal{A}[h](0)\right)=0\,.$

Recalling the definition of $\mathbb{M}^{N}_{\varepsilon}$ from Proposition
2.2 and applying Proposition 4.2, we have that

$\displaystyle\varepsilon\left\|\mathcal{A}[h]\right\|_{\mathbb{M}^{N}_{\varepsilon}}$
$\displaystyle\lesssim\left\|z[h]\right\|_{\mathbb{Y}^{N}_{\varepsilon,\rm
w}}+\left\|e^{\delta\langle\varepsilon
x\rangle^{\nicefrac{{1}}{{2}}}}z[h]\right\|_{\mathbb{Y}^{N}_{\varepsilon}}\,.$
(6.1)

Define the closed ball

$\displaystyle\mathcal{B}_{\varepsilon}:=\left\\{h\in\mathbb{M}^{N}_{\varepsilon}:\left\|h\right\|_{\mathbb{M}^{N}_{\varepsilon}}\leqslant
M\varepsilon^{2}\right\\}\,,$

where $M$ depends on the (possibly large) implicit constants from (2.48) and
(4.2). By (6.1) and Proposition 2.6, we have that

$\displaystyle\mathcal{A}:\mathcal{B}_{\varepsilon}\mapsto\mathcal{B}_{\varepsilon}$

for $\varepsilon$ sufficiently small depending only on $N$ and universal
constants. To show that it is a contraction, for $h_{1},h_{2}$ we evaluate

$\displaystyle\left((v_{1}-s_{0})\partial_{x}+L_{\rm
NS}\right)\left(\mathcal{A}[h_{1}]-\mathcal{A}[h_{2}]\right)=z[h_{1}]-z[h_{2}],$

and note that

$\displaystyle z[h_{1}]-z[h_{2}]=2\Gamma\left[g_{\rm
NS},h_{1}-h_{2}\right]+\Gamma[h_{1}-h_{2},h_{1}]+\Gamma[h_{2},h_{2}-h_{1}]\,.$

By a variation of Proposition 2.6, we have

$\displaystyle\left\|z[h_{1}]-z[h_{2}]\right\|_{\mathbb{Y}^{N}_{\varepsilon,\rm
w}}+\left\|e^{\delta\langle\varepsilon
x\rangle^{\nicefrac{{1}}{{2}}}}\left(z[h_{1}]-z[h_{2}]\right)\right\|_{\mathbb{Y}^{N}_{\varepsilon}}\lesssim\left(\varepsilon^{2}+\left\|h_{1}\right\|_{\mathbb{M}^{N}_{\varepsilon}}+\left\|h_{2}\right\|_{\mathbb{M}^{N}_{\varepsilon}}\right)\left\|h_{1}-h_{2}\right\|_{\mathbb{M}^{N}_{\varepsilon}}\,.$

Therefore, the contraction mapping theorem implies that there exists a unique
$h_{\ast}\in\mathcal{B}_{\varepsilon}$ with

$\displaystyle\mathcal{A}[h_{\ast}]=h_{\ast}\,.$

This concludes the proof of Theorem 1.1.

∎

##  Appendix A Chapman-Enskog computations

In this appendix, we justify the derivation of the three-dimensional
compressible Navier-Stokes equations (1.4) in traveling wave form with one-
dimensional spatial dependence from the traveling wave Landau equation (1.1).
The computations in this section are essentially classical, and we refer the
reader to [DuanYangYuContactWaves, Appendix A] for a recent treatment of this
derivation, and to the standard references [BardosGolseLevermore1991, BGL2,
CCbook, guo2006boltzmann, UkaiYang]. We however require a slightly non-
standard version of these computations in order to carry out the linearized
Chapman-Enskog expansion for the regularized Landau operator
$Q_{\kappa}=Q+\kappa Q_{\rm R}$ in subsubsection 2.2.1. The main novelty is
that the computation of the viscosity coefficients in the linearized Chapman-
Enskog expansion, cf. 2.35, 2.37, and 2.38, requires inversion of the operator
$L_{\mu_{\rm NS}}^{\kappa}$. Crucially, the viscosity coefficients in the
linearized expansion in subsubsection 2.2.1 are computed according to an
identical procedure as that which produces the full compressible Navier-Stokes
equations in (2.15), save for the substitution of $L^{\kappa}_{\mu_{\rm NS}}$
for $L_{\mu_{\rm NS}}$. Since the case $\kappa=0$ corresponds to the standard
Landau operator, cf. (2.44), we treat in this section general
$\kappa\in[0,1]$. We first derive the compressible Navier-Stokes equations,
thus validating (2.15) and obtaining along the way a formula for the viscosity
coefficients (2.37) for the linearized, regularized equation, before deriving
a stability result comparing $B_{\kappa}$ to $B_{0}$ for $\kappa\ll 1$.

Derivation of (2.15) and viscosity coefficients in (2.37). Recall the
Maxwellians $\mu$ from subsubsection 2.1.1, and the associated macroscopic
space spanned by

$\displaystyle\chi_{0}(v)$
$\displaystyle=\frac{1}{\sqrt{\varrho}}\mu\,,\qquad\chi_{i}(v)=\frac{v_{i}-u_{i}}{\sqrt{\varrho\theta}}\mu\,\,\qquad(i=1,2,3)\,,\qquad\chi_{4}(v)=\frac{1}{\sqrt{6\varrho}}\left(\frac{|v-u|^{2}}{\theta}-3\right)\mu\,,$

where $\langle\cdot,\cdot\rangle$ is the $L^{2}_{v}$ inner product. We recall
the projection operators $\mathsf{P}_{\mu}(h)$ and
$\mathsf{P}_{\mu}^{\perp}(h)=h-\mathsf{P}_{\mu}(h)$ defined in (2.10). Note
crucially that these projection operators and macroscopic spaces play
precisely the same role in the regularized equation

$(v_{1}-s_{0})\partial_{x}F=Q_{\kappa}(F,F)$ (A.1)

and its linearization, given in (2.30), as in the standard Landau equation and
its linearization. Supposing that we have a solution $F$ to the standing wave
equation (A.1) (which includes (1.1) as the case $\kappa=0$), we decompose
$F(t,x,v)=\mu(F)(t,x,v)+f(t,x,v)$, where

$\mathsf{P}_{\mu(F)}(F(t,x,\cdot))(v)=\mu(F)(t,x,v)\,,\qquad
f(t,x,v)=\mathsf{P}^{\perp}_{\mu(F)}(F(t,x,\cdot))(v)\,.$

Substituting the decomposition above into (A.1), we obtain

$\displaystyle(v_{1}-s_{0})\partial_{x}\mu(F)+(v_{1}-s_{0})\partial_{x}f=\mathsf{L}^{\kappa}_{\mu(F)}f+Q_{\kappa}(f,f)\,,$
(A.2)

where $\mathsf{L}^{\kappa}_{\mu(F)}$ is defined in (2.7). We now multiply
(A.2) by $(1,v,\nicefrac{{1}}{{2}}|v|^{2})$ and integrate in $v$ to obtain the
five macroscopic equations

$\displaystyle\begin{cases}&\partial_{x}(\varrho(u_{1}-s_{0}))(t,x)=0\,,\\\
&\partial_{x}(\varrho
u_{1}(u_{1}-s_{0})+p)(t,x)=-\int_{\mathbb{R}^{3}}v_{1}^{2}\partial_{x}f(t,x,v)\,dv\,,\\\
&\partial_{x}(\varrho
u_{i}(u_{1}-s_{0}))(t,x)=-\int_{\mathbb{R}^{3}}v_{1}v_{i}\partial_{x}f(t,x,v)\,dv\qquad\qquad(i=2,3)\,,\\\
&\partial_{x}\left(\varrho\left(e+\frac{|u|^{2}}{2}\right)(u_{1}-s_{0})+pu_{1}\right)(t,x)=-\int_{\mathbb{R}^{3}}\frac{1}{2}v_{1}|v|^{2}\partial_{x}f(t,x,v)\,dv\,,\end{cases}$
(A.3)

where the pressure $p=\varrho\theta=(\gamma-1)\varrho e$. To justify this, we
first note that for all five computations, the integral of the right-hand side
of (A.2) against the collision invariants vanishes. Then the first equation is
immediate by the definitions of $\varrho(t,x)$ and $\varrho(t,x)u_{1}(t,x)$.
For the second equation, we use that

$\displaystyle\int_{\mathbb{R}^{3}}v_{1}(v_{1}-s_{0})\mu(F)$
$\displaystyle=\int_{\mathbb{R}^{3}}v_{1}\theta(t,x)\underbrace{\frac{(v_{1}-u_{1})}{\theta}\mu(F)}_{=-\partial_{v_{1}}\mu(F)}+\int_{\mathbb{R}^{3}}v_{1}(u_{1}-s_{0})\mu(F)$
$\displaystyle=\int_{\mathbb{R}^{3}}\theta\mu(F))+\int_{\mathbb{R}^{3}}v_{1}(u_{1}-s_{0})\mu(F)$
$\displaystyle=\theta\varrho+(u_{1}-s_{0})(u_{1}\varrho)\,.$

For the third and fourth equations, we make very similar computations, except
that no pressure appears since we have now $v_{k}-u_{k}(t,x)$ for $k=2,3$ on
the second line, and then the integral vanishes after integrating by parts.
For the fifth equation, we write that

$\displaystyle\int_{\mathbb{R}^{3}}\frac{|v|^{2}}{2}(v_{1}-s_{0})\mu(F)$
$\displaystyle=(u_{1}-s_{0})\int_{\mathbb{R}^{3}}\frac{|v|^{2}}{2}\mu(F)+\theta\int_{\mathbb{R}^{3}}\frac{|v|^{2}}{2}\frac{(v_{1}-u_{1})}{\theta}\mu(F)$
$\displaystyle=(u_{1}-s_{0})\varrho\left(e+\frac{1}{2}|u|^{2}\right)+\theta\varrho
u_{1}\,,$

where we have integrated by parts again to obtain $\theta\varrho u_{1}$. This
finishes the derivation of (A.3).

We now apply the projection operator $\mathsf{P}^{\perp}_{\mu(F)}$ to (A.2) to
obtain that

$\displaystyle\mathsf{P}^{\perp}_{\mu(F)}\left[(v_{1}-s_{0})\partial_{x}\mu(F)+(v_{1}-s_{0})\partial_{x}f=\mathsf{L}^{\kappa}_{\mu(F)}f+Q_{\kappa}(f,f)\right]\,.$

We expect that $f\ll\mu(F)$, and so to leading order, we may guess that

$f\approx\left(\mathsf{L}^{\kappa}_{\mu(F)}\right)^{-1}\mathsf{P}^{\perp}_{\mu(F)}((v_{1}-s_{0})\partial_{x}\mu(F))\,.$

Now, we first compute

$-\int_{\mathbb{R}^{3}}v_{1}v_{i}\partial_{x}\left(\left(\mathsf{L}^{\kappa}_{\mu(F)}\right)^{-1}\mathsf{P}^{\perp}_{\mu(F)}((v_{1}-s_{0})\partial_{x}\mu(F))\right)\,,\qquad
i=1,2,3\,.$ (A.4)

We first expand the argument of
$(\mathsf{L}^{\kappa}_{\mu(F)})^{-1}\mathsf{P}^{\perp}_{\mu(F)}$, obtaining
that

$\displaystyle(v_{1}-s_{0})\partial_{x}\mu(F)$
$\displaystyle=(v_{1}-s_{0})\left[\frac{\partial_{x}\varrho}{(2\pi\theta)^{\nicefrac{{3}}{{2}}}}-\frac{3}{2}\frac{2\pi\varrho\partial_{x}\theta}{(2\pi\theta)^{\nicefrac{{5}}{{2}}}}\right]\exp\left(-\frac{|v-u|^{2}}{2\theta}\right)$
$\displaystyle\qquad+(v_{1}-s_{0})\mu(F)\left[\frac{(v_{k}-u_{k})\partial_{x}u_{k}}{\theta}+\frac{|v-u|^{2}}{\theta^{2}}\partial_{x}\theta\right]\,.$
(A.5)

Since the first term is a multiple (depending on $(t,x)$) of
$(v_{1}-s_{0})\mu(F)$, it belongs to the kernel of
$\mathsf{P}^{\perp}_{\mu(F)}$, and we may ignore it. In addition, the portion
of the second term with $-s_{0}$ is also in the kernel, and so we may ignore
it. We focus now on expanding the rest of the expression (the part of the
second term with $v_{1}$) in terms of the Burnett functions
[DuanYangYuRarefaction, Appendix]

$\displaystyle\hat{A}^{\kappa}_{j}(v)$
$\displaystyle=\frac{|v|^{2}-5}{2}v_{j}\,,\quad
A^{\kappa}_{j}\left(\frac{v-u}{\sqrt{\theta}}\right)=\left(\mathsf{L}^{\kappa}_{\mu(F)}\right)^{-1}\left[\hat{A}^{\kappa}_{j}\left(\frac{v-u}{\sqrt{\theta}}\right)\mu(F)\right]$
$\displaystyle\hat{B}^{\kappa}_{ij}(v)$
$\displaystyle=v_{i}v_{j}-\frac{1}{3}\delta_{ij}|v|^{2}\,,\qquad
B^{\kappa}_{ij}\left(\frac{v-u}{\sqrt{\theta}}\right)=\left(\mathsf{L}^{\kappa}_{\mu(F)}\right)^{-1}\left[\hat{B}^{\kappa}_{ij}\left(\frac{v-u}{\sqrt{\theta}}\right)\mu(F)\right]\,.$

We first note that the $v_{1}v_{i}$ on the outside of (A.4) can be expanded as

$\displaystyle
v_{1}v_{i}=(v_{1}-u_{1})(v_{i}-u_{i})+\textnormal{kernel}=\theta\hat{B}^{\kappa}_{i1}\left(\frac{v-u}{\sqrt{\theta}}\right)+\textnormal{kernel}\,,$

where by “kernel” we mean terms which, when multiplied by $\mu(F)$, belong to
the kernel of $\mathsf{P}^{\perp}_{\mu(F)}$ and so can be ignored using the
self-adjointness of $(\mathsf{L}^{\kappa}_{\mu(F)})^{-1}$. Now using
[DuanYangYuRarefaction, Lemma 6.1], which asserts that
$\langle\hat{A}^{\kappa}_{i},B^{\kappa}_{jk}\rangle=0$ for any $i,j,k$ and the
self-adjointness of $(\mathsf{L}^{\kappa}_{\mu(F)})^{-1}$ again, we do not
need to consider the portion of $v_{1}\partial_{x}\mu(F)$ (inside the argument
of $\mathsf{L}^{\kappa}_{\mu(F)}\mathsf{P}^{\perp}_{\mu(F)}$) which is
parallel to $\hat{A}_{1}^{\kappa}$. The second part of the second term in
(A.5) can therefore be ignored, since it is a linear combination of
$\hat{A}_{1}^{\kappa}$ and elements of the kernel of
$\mathsf{P}^{\perp}_{\mu(F)}$. Then for the first part, we can subtract
$u_{1}$ for free and expand

$\displaystyle\frac{(v_{k}-u_{k})\partial_{x}u_{k}}{\theta}(v_{1}-u_{1})=\frac{\partial_{x}u_{k}}{\theta}\left(\hat{B}^{\kappa}_{k1}(v-u)+\frac{1}{3}\delta_{k1}|v-u|^{2}\right)=\partial_{x}u_{k}\hat{B}^{\kappa}_{k1}\left(\frac{v-u}{\sqrt{\theta}}\right)+\textnormal{kernel}\,.$

We finally obtain that

$\displaystyle-\int_{R^{3}}v_{1}v_{i}\left(\mathsf{L}^{\kappa}_{\mu(F)}\right)^{-1}(\mathsf{P}^{\perp}_{\mu(F)}((v_{1}-s_{0})\partial_{x}\mu(F)))$
$\displaystyle=-\int_{\mathbb{R}^{3}}\theta\hat{B}^{\kappa}_{i1}\left(\frac{v-u}{\sqrt{\theta}}\right)\partial_{x}u_{k}\left(\mathsf{L}^{\kappa}_{\mu(F)}\right)^{-1}\left[\hat{B}^{\kappa}_{k1}\left(\frac{v-u}{\sqrt{\theta}}\right)\right]$
$\displaystyle=-\partial_{x}u_{k}\theta\int_{\mathbb{R}^{3}}\hat{B}^{\kappa}_{i1}\left(\frac{v-u}{\sqrt{\theta}}\right)B^{\kappa}_{k1}\left(\frac{v-u}{\sqrt{\theta}}\right)\,.$

Using that $-\langle\hat{B}^{\kappa}_{ii},B^{\kappa}_{ii}\rangle$ is positive
and independent of $i$, and
$\langle\hat{B}^{\kappa}_{ij},B^{\kappa}_{kl}\rangle$ is zero unless either
$(i,j)=(k,l)$, $(i,j)=(l,k)$, or $i=j$ and $k=l$ (see again
[DuanYangYuRarefaction, Lemma 6.1]), we have that the only relevant terms are
those with $\langle\hat{B}^{\kappa}_{i1},B^{\kappa}_{i1}\rangle$, i.e. $i=k$.
At this point we set

$\mu^{\kappa}_{i}(\theta)=\theta\int_{\mathbb{R}^{3}}\hat{B}^{\kappa}_{i1}\left(\frac{v-u}{\sqrt{\theta}}\right)B^{\kappa}_{i1}\left(\frac{v-u}{\sqrt{\theta}}\right)$

for $i=1,2,3$, where we are _not_ summing over the repeated index $i$ in the
above formula. Finally, using that $\mu_{2}=\mu_{3}$ and
$\langle\hat{B}^{\kappa}_{11},B^{\kappa}_{11}\rangle=2\langle\hat{B}^{\kappa}_{12},B^{\kappa}_{12}\rangle+\langle\hat{B}^{\kappa}_{11},B^{\kappa}_{22}\rangle$,
where all terms involved are positive, and
$\langle\hat{B}^{\kappa}_{21},B^{\kappa}_{21}\rangle=\langle\hat{B}^{\kappa}_{12},B^{\kappa}_{12}\rangle$,
we define $\mu^{\kappa}(\theta)$ and $\lambda^{\kappa}(\theta)$ by

$\mu^{\kappa}(\theta)=\mu^{\kappa}_{2}(\theta)=\mu^{\kappa}_{3}(\theta)\,,\qquad\lambda^{\kappa}(\theta)=\mu^{\kappa}(\theta)+\theta\int_{\mathbb{R}^{3}}\hat{B}^{\kappa}_{11}\left(\frac{v-u}{\sqrt{\theta}}\right)B^{\kappa}_{22}\left(\frac{v-u}{\sqrt{\theta}}\right)\,dv\,.$

This concludes the derivation of the density and momentum balances for
compressible Navier-Stokes, and thus the derivation of (2.15) from the
$\kappa=0$ case.

Finally, we must still compute the viscosity in the energy balance, namely

$\int_{\mathbb{R}^{3}}\frac{v_{1}}{2}|v|^{2}\partial_{x}\left(\left(\mathsf{L}^{\kappa}_{\mu(F)}\right)^{-1}\mathsf{P}^{\perp}_{\mu(F)}(v_{1}\partial_{x}\mu(F))\right)\,.$
(A.6)

We first expand

$\displaystyle\frac{1}{2}v_{1}|v|^{2}$
$\displaystyle=\frac{1}{2}(v_{1}-u_{1})|v|^{2}+\frac{1}{2}u_{1}|v|^{2}$
$\displaystyle=\frac{1}{2}(v_{1}-u_{1})|v-u|^{2}-\frac{1}{2}(v_{1}-u_{1})\left[|u|^{2}-2\langle
u,v\rangle\right]+\textnormal{kernel}$
$\displaystyle=\frac{1}{2}(v_{1}-u_{1})|v-u|^{2}+v_{1}\langle
u,v\rangle+\textnormal{kernel}\,.$

As usual, by “kernel” we mean terms which are in the kernel of
$\mathsf{P}^{\perp}_{\mu(F)}$. From the term that has $v_{1}\langle
u,v\rangle$, we pick up terms identical to the ones we just calculated above,
just with an extra factor of $u$. So we focus our efforts on the first term,
which is a linear combination of $\hat{A}^{\kappa}_{1}(v-u)$ and elements of
the kernel, which we ignore. Going back to (A.5), we can just consider the
term with $|v-u|^{2}v_{1}$, which will be a linear combination of
$\hat{A}^{\kappa}_{1}$ and elements of the kernel; all other terms involving
$\hat{B}^{\kappa}_{\bullet}$ can be ignored due to the orthogonality condition
$\langle\hat{A}^{\kappa}_{\bullet},B^{\kappa}_{\bullet}\rangle=0$. Counting
factors of $\theta$, we see that we are short a factor of $\theta$ in the
denominator, so we pick up a $\theta$ in the numerator and then set

$\kappa(\theta)=\theta\int_{\mathbb{R}^{3}}\hat{A}^{\kappa}_{1}\left(\frac{v-u}{\sqrt{\theta}}\right)A^{\kappa}_{1}\left(\frac{v-u}{\sqrt{\theta}}\right)\,,$

which is positive; see again [DuanYangYuRarefaction, section 6]. Finally, we
deduce that to leading order,

$\displaystyle\begin{cases}&\partial_{x}(\varrho(u_{1}-s_{0}))(t,x)=0\,,\\\
&\partial_{x}(\varrho
u_{1}(u_{1}-s_{0})+p)(t,x)=\partial_{x}\left(\left(\mu^{\kappa}(\theta)+\lambda^{\kappa}(\theta)\right)\partial_{x}u_{1}\right)\,,\\\
&\partial_{x}(\varrho
u_{i}(u_{1}-s_{0}))=\partial_{x}(\mu^{\kappa}(\theta)\partial_{x}u_{i})\qquad\qquad\qquad(i=2,3)\,,\\\
&\partial_{x}\left(\varrho\left(e+\frac{|u|^{2}}{2}\right)(u_{1}-s_{0})+pu_{1}\right)=\partial_{x}(\kappa^{\kappa}(\theta)\partial_{x}\theta)\\\
&\qquad\qquad\qquad+\partial_{x}\left(\left(\mu^{\kappa}(\theta)+\lambda^{\kappa}(\theta)\right)u_{1}\partial_{x}u_{1}\right)+\partial_{x}\left(\mu^{\kappa}(\theta)u_{2}\partial_{x}u_{2}\right)+\partial_{x}\left(\mu^{\kappa}(\theta)u_{3}\partial_{x}u_{3}\right)\,,\end{cases}$

where the viscosity coefficients $\mu^{\kappa}$, $\lambda^{\kappa}$, and
$\kappa^{\kappa}$ are smooth positive functions of $\theta$ given by the
formulas above.

Stability of viscosity coefficients for $\kappa\ll 1$. From the formulas (A.4)
and (A.6), which crucially are equivalent to (2.37), we have that

$\displaystyle(B_{\kappa}-B_{0})(U_{\rm
NS})=\mathsf{P}_{\mu_{0}}\left[v_{1}\left(\left[\mathsf{L}_{\mu_{\rm
NS}}-\left(\mathsf{L}_{\mu_{\rm
NS}}^{\kappa}\right)^{-1}\right](\mathsf{P}_{\mu_{\rm
NS}}^{\perp}[(v_{1}-s_{0})\mathsf{P}_{\mu_{\rm NS}}])\right)\right]\,.$

We are interested in computing the error in operator norm. Note that
$B_{\kappa}$ and $B_{0}$ are _linear_ operators on the finite-dimensional
vector space $\mathbb{U}$ defined in (2.10), and so there is no need to
specify the norms in any of the following computations. From the standard
expansion of $\mathsf{L}_{\mu_{\rm NS}}^{\kappa}$ in a Neumann series for
$\kappa$ sufficiently small, we have that

$\displaystyle-\mathsf{L}_{\mu_{\rm NS}}^{-1}+\left(\mathsf{L}_{\mu_{\rm
NS}}^{\kappa}\right)^{-1}$ $\displaystyle=-\mathsf{L}_{\mu_{\rm
NS}}^{-1}+\left(\mathsf{L}_{\mu_{\rm
NS}}\left[\operatorname{\mathrm{Id}}+\underbrace{\kappa\mathsf{L}_{\mu_{\rm
NS}}^{-1}\mathsf{L}_{\mu_{\rm NS}}^{\rm
R}}_{:=\mathcal{L}}\right]\right)^{-1}$ $\displaystyle=-\mathsf{L}_{\mu_{\rm
NS}}^{-1}+\left(\sum_{j=0}^{\infty}\mathcal{L}^{j}(-1)^{j}\right)\mathsf{L}_{\mu_{\rm
NS}}^{-1}$
$\displaystyle=\left(\sum_{j=1}^{\infty}\mathcal{L}^{j}(-1)^{j}\right)\mathsf{L}_{\mu_{\rm
NS}}^{-1}\,.$

From this, we see that

$\|B_{\kappa}-B_{0}\|\lesssim\kappa$ (A.7)

for a $\kappa$-independent implicit constant.

##  Appendix B Construction and analysis of the Navier-Stokes shocks

###  B.1 The compressible Navier-Stokes equations

We consider mass, momentum, and energy densities:

$(\varrho,\varrho u,\varrho(\nicefrac{{1}}{{2}}|u|^{2}+e))=(\varrho,m,E)\,.$

These are the “conservative variables.” We require the equation of state

$p=\varrho\theta\,,$

where the absolute temperature $\theta$ is proportional to internal energy
(per unit mass) $e$:

$\theta=(\gamma-1)e\,.$

Here $\gamma>1$ is the adiabatic exponent; $\gamma=\nicefrac{{5}}{{3}}$ for
monatomic gas. We further require the stress tensor

$\tau=2\mu D+\lambda\operatorname{\mathrm{Id}}\operatorname{\mathrm{div}}u\,,$

where

$D=\frac{1}{2}(\nabla u+\nabla u^{T})$

is the deformation tensor. In general, we allow the viscosity coefficients
$\mu,\lambda>0$ and thermal conductivity $\kappa>0$ to be smooth functions of
$\varrho$ and $\theta$. For the compressible Navier-Stokes equations for
monatomic gas derived from the Landau equations with $1$-dimensional spatial
dependence, we saw in the previous appendix that $\lambda$, $\mu$, and
$\kappa$ are smooth positive functions of $\theta$; in particular none of
these coefficients exhibit $\varrho$ dependence.

The compressible Navier-Stokes equations are

$\displaystyle\partial_{t}\varrho+\operatorname{\mathrm{div}}(\varrho u)$
$\displaystyle=0$ $\displaystyle\partial_{t}(\varrho
u)+\operatorname{\mathrm{div}}(\varrho u\otimes u+pI)$
$\displaystyle=\operatorname{\mathrm{div}}\tau$
$\displaystyle\partial_{t}(\varrho(e+\nicefrac{{1}}{{2}}|u|^{2}))+\operatorname{\mathrm{div}}(\varrho
u(e+\nicefrac{{1}}{{2}}|u|^{2})+up)$
$\displaystyle=\operatorname{\mathrm{div}}(\tau\cdot
u)+\operatorname{\mathrm{div}}\left(\kappa\nabla\theta\right)\,,$

and we express them in the variables $(\varrho,m,E)$ as

$\displaystyle\partial_{t}\varrho+\operatorname{\mathrm{div}}m$
$\displaystyle=0$ (B.1)
$\displaystyle\partial_{t}m+\operatorname{\mathrm{div}}\left(\frac{m\otimes
m}{\varrho}+pI\right)$ $\displaystyle=\operatorname{\mathrm{div}}\tau$
$\displaystyle\partial_{t}E+\operatorname{\mathrm{div}}\left(\frac{m}{\varrho}(E+p)\right)$
$\displaystyle=\operatorname{\mathrm{div}}\left(\frac{1}{\varrho}\tau\cdot
m\right)+\operatorname{\mathrm{div}}\left(\kappa\nabla\theta\right)\,,$

where

$\displaystyle\tau(\varrho,m,E)=\mu\left(\nabla\left(\frac{m}{\rho}\right)+\nabla\left(\frac{m}{\rho}\right)^{T}\right)+\lambda\operatorname{\mathrm{Id}}\operatorname{\mathrm{div}}\left(\frac{m}{\varrho}\right)\,$
$\displaystyle,\qquad\theta(\varrho,m,E)=(\gamma-1)\left(\frac{E}{\varrho}-\frac{|m|^{2}}{2\varrho^{2}}\right)\,,$
$\displaystyle p=\varrho\theta=\varrho(\gamma-1)$
$\displaystyle\left(\frac{E}{\varrho}-\frac{|m|^{2}}{2\varrho^{2}}\right)\,.$

We assume that the solutions are translation-invariant in $x_{2}$ and $x_{3}$;
that is, solutions are functions of $x_{1}=x$ only. The equations become

$\displaystyle\partial_{t}\varrho+\partial_{x}m_{1}$ $\displaystyle=0$ (B.2)
$\displaystyle\partial_{t}m+\partial_{x}\left(\frac{mm_{1}}{\varrho}+pe_{1}\right)$
$\displaystyle=\partial_{x}\tau_{\bullet 1}$
$\displaystyle\partial_{t}E+\partial_{x}\left(\frac{m_{1}}{\varrho}(E+p)\right)$
$\displaystyle=\partial_{x}\left(\frac{1}{\varrho}\tau\cdot
m\right)_{1}+\partial_{x}\left(\kappa\partial_{x}(\gamma-1)\left(\frac{E}{\varrho}-\frac{|m|^{2}}{2\varrho^{2}}\right)\right)\,,$

where $\tau_{\bullet 1}$ is the first column of $\tau$, and
$(\nicefrac{{1}}{{\varrho}}\tau\cdot m)_{1}$ is the first component of the
matrix-vector product $\nicefrac{{1}}{{\varrho}}\tau\cdot m$. We may write the
equations (B.2) succinctly in the form

$\partial_{t}U+\partial_{x}F(U)=\partial_{x}(B(U)\partial_{x}U)\,,$

where $U=(\varrho,m,E)$,

$F(U)=\begin{pmatrix}m_{1}\\\ \frac{mm_{1}}{\varrho}+pe_{1}\\\
\frac{m_{1}}{\varrho}(E+p)\end{pmatrix}$

is the flux, and the dissipation matrix $B(U)$ is defined by

$\begin{bmatrix}0&0&0&0&0\\\
\frac{-(2\mu+\lambda)m_{1}}{\varrho^{2}}&\frac{(2\mu+\lambda)}{\varrho}&0&0&0\\\
\frac{-2\mu m_{2}}{\varrho^{2}}&0&\frac{2\mu}{\varrho}&0&0\\\ \frac{-2\mu
m_{3}}{\varrho^{2}}&0&0&\frac{2\mu}{\varrho}&0\\\
\begin{subarray}{c}\frac{|m|^{2}\left(-2\mu+\kappa(\gamma-1)\right)}{\varrho^{3}}\\\
-\frac{\lambda
m_{1}^{2}}{\varrho^{3}}-\frac{\kappa(\gamma-1)}{\varrho^{2}}E\end{subarray}&\frac{m_{1}}{\varrho^{2}}\left((2\mu+\lambda)-{\kappa(\gamma-1)}\right)&\frac{m_{2}}{\varrho^{2}}\left(2\mu-{\kappa(\gamma-1)}\right)&\frac{m_{3}}{\varrho^{2}}\left(2\mu-{\kappa(\gamma-1)}\right)&\frac{\kappa(\gamma-1)}{\varrho}\end{bmatrix}\,.$
(B.3)

Setting the right-hand side to zero in (B.2), one obtains the three-
dimensional Euler equations with one-dimensional spatial dependence. These
equations are in fact hyperbolic, a property which is inherited from the full
three-dimensional model (B.1) [markfelder], since $F$ is simply the flux
function in the first spatial variable. The standard one-dimensional Euler
equations, i.e., (B.2) with $m_{2}=m_{3}=0$, have two genuinely non-linear
eigenvalues, $u\pm c$, and one linearly degenerate eigenvalue, $u$, where
$c=\sqrt{\nicefrac{{\gamma p}}{{\varrho}}}$ is the sound speed. In the fully
one-dimensional setting, the wave speed $u+c$ is associated with shocks, $u-c$
with rarefaction waves, and $u$ with contact discontinuities. In our setting,
the eigenvalue $u$ is a triple eigenvalue, with two additional directions
associated to the $m_{2}$ and $m_{3}$ variables.

The differential of the flux function is

$\nabla F=\left[\begin{array}[]{@{}c|ccc|c@{}}0&1&0&0&0\\\
\hline\cr-\frac{m_{1}^{2}}{\varrho^{2}}+\partial_{\varrho}p&\frac{2m_{1}}{\varrho}+\partial_{m_{1}}p&\partial_{m_{2}}p&\partial_{m_{3}}p&\partial_{E}p\\\
-\frac{m_{1}m_{2}}{\varrho^{2}}&\frac{m_{2}}{\varrho}&\frac{m_{1}}{\varrho}&0&0\\\
-\frac{m_{1}m_{3}}{\varrho^{2}}&\frac{m_{3}}{\varrho}&0&\frac{m_{1}}{\varrho}&0\\\
\hline\cr-\frac{m_{1}}{\varrho^{2}}(E+p)+\frac{m_{1}}{\varrho}\partial_{\varrho}p&\frac{1}{\varrho}(E+p)+\frac{m_{1}}{\varrho}\partial_{m_{1}}p&\frac{m_{1}}{\varrho}\partial_{m_{2}}p&\frac{m_{1}}{\varrho}\partial_{m_{3}}p&\frac{m_{1}}{\varrho}(1+\partial_{E}p)\end{array}\right]\,.$

For our analysis, it will be enough to compute most quantities at
$(\varrho,u,\theta)=(1,0,0,0,1)$, which corresponds to
$(\varrho,m,E)=(1,0,0,0,(\gamma-1)^{-1})=:U_{L}$. The sound speed
$c=\sqrt{\gamma}$, $\partial_{\varrho}p=0$, $p=1$, and
$\partial_{E}p=\gamma-1$. Therefore,

$(\nabla F)(U_{L})=\begin{bmatrix}0&(1,0,0)&0\\\ 0_{3\times 1}&0_{3\times
3}&(\gamma-1,0,0)^{T}\\\
0&(\nicefrac{{\gamma}}{{(\gamma-1)}},0,0)&0\end{bmatrix}$ (B.4)

The right eigenvectors $r_{p}$ corresponding to $-c$, $0$, and $c$ at $U_{L}$
may be chosen to be, respectively,

$(1,-c,0,0,\nicefrac{{\gamma}}{{(\gamma-1)}})\,,\qquad
e_{1}\,,e_{3}\,,e_{4}\,,\qquad
r_{5}:=-(1,c,0,0,\nicefrac{{\gamma}}{{(\gamma-1)}})\,.$ (B.5)

It can be verified directly that $c=\sqrt{\nicefrac{{\gamma p}}{{\varrho}}}$
is genuinely nonlinear in neighborhood of $U_{L}$:

$\displaystyle\Lambda$
$\displaystyle:=\left(\nabla_{\varrho,m,E}\left(\frac{m_{1}}{\varrho}+c\right)\right)(U_{L})\cdot
r_{5}$
$\displaystyle=\left(-\nicefrac{{1}}{{2}}\gamma^{\nicefrac{{1}}{{2}}},0,0,0,\nicefrac{{1}}{{2}}\gamma^{\nicefrac{{1}}{{2}}}(\gamma-1)\right)\cdot\left(-1,-c,0,0,\frac{-\gamma}{\gamma-1}\right)^{T}$
$\displaystyle=\frac{1}{2}\gamma^{\nicefrac{{1}}{{2}}}(1-\gamma)<0\,.$ (B.6)

The left eigenvectors $l_{p}$ are

$\frac{\gamma-1}{2\gamma}(0,-\nicefrac{{c}}{{\gamma-1}},0,0,1)\,,\quad(1,0,0,0-\nicefrac{{(\gamma-1)}}{{\gamma}})\,,e_{3}\,,e_{4}\,,\quad
l_{5}:=-\frac{\gamma-1}{2\gamma}(0,\nicefrac{{c}}{{\gamma-1}},0,0,1)\,.$

Finally, we compute the dissipation matrix $B(U_{L})$ using (B.3), obtaining

$B(U_{L})=\left[\begin{array}[]{@{}c|ccc|c@{}}0&&&&\\\
\hline\cr&2\mu+\lambda&&&\\\ &&\mu&&\\\ &&&\mu&\\\
\hline\cr-\kappa&&&&(\gamma-1)\kappa\end{array}\right]\,.$ (B.7)

Suppose that $U(x,t)$ is a solution of such a system. Then $(T_{s}U)(x-st,t)$
is also a solution. Indeed, we calculate

$\partial_{t}(T_{s}U(x-st,t))=A_{s}(\partial_{t}U)(x-st,t)-sA_{s}(\partial_{x}U)(x-st,t)$
$\partial_{x}F(T_{s}U(x-st,t))=(\nabla
F)(U(x-st,t))A_{s}(\partial_{x}U)(x-st,t)=A_{s}[(\nabla
F)(U(x-st))+sI](\partial_{x}U)(x-st,t)$
$\partial_{x}B(T_{s}U(x-st,t))\partial_{x}T_{s}U(x-st,t)=A_{s}\partial_{x}B(U(x-st,t))(\partial_{x}U)(x-st,t)\,.$

In our setting, the corresponding linearized equation is

$-s_{L}\partial_{x}U+(\nabla F)(U_{\rm
NS}^{(\varrho_{L},u_{L},\theta_{L},s_{L})})\partial_{x}U=\partial_{x}B(U_{\rm
NS}^{(\varrho_{L},u_{L},\theta_{L},s_{L})})\partial_{x}U\,.$

Then $V=T_{-\varepsilon}U$ satisfies the equation

$-(s_{L}-\varepsilon)\partial_{x}V+(\nabla F)(U_{\rm
NS}^{(\varrho_{L},u_{L}-\varepsilon,\theta_{L},s_{L}-\varepsilon)})\partial_{x}V=\partial_{x}B(U_{\rm
NS}^{(\varrho_{L},u_{L}-\varepsilon,\theta_{L},s_{L}-\varepsilon)})\partial_{x}V\,.$

######  Remark B.1 (Galilean invariance).

For any $s\in\mathbb{R}$, we consider the Galilean boost
$T_{s}:\mathbb{R}^{5}\to\mathbb{R}^{5}$ defined by

$T_{s}(\varrho,m,E)=(\varrho,m+\varrho se_{1},E+(m_{1}+\varrho s)s)\,,$

corresponding to $(\varrho,u,\theta)\mapsto(\varrho,u+s,\theta)$ in the non-
conservative variables. Then $(T_{s})_{s\in\mathbb{R}}$ is a group of linear
transformations satisfying

  * •

(flux transformation) $T_{s}(\nabla F)(U)=((\nabla F)(T_{s}U)-sI)T_{s}$, and

  * •

(viscosity transformation) $T_{s}B(U)=B(T_{s}U)T_{s}$.

Consequently, $U(x,t)$ is a solution of the compressible Euler/Navier-Stokes
equations and only if $(T_{s}U)(x-st,t)$ is a solution. Likewise, the Rankine-
Hugoniot conditions are Galilean invariant.

For any $U_{L}\in\mathbb{R}^{+}\times\mathbb{R}^{3}\times\mathbb{R}^{+}$,
suppose that $U_{\rm NS}[U_{L},s_{L}]$ is a viscous shock with left end-state
$U_{L}$ traveling with speed $s_{L}$. Then $U_{\rm
NS}[T_{s}U_{L},s_{L}+s]:=T_{s}U_{\rm
NS}^{(\varrho_{L},u_{L},\theta_{L},s_{L})}$ is a viscous shock with left end-
state $T_{s}U_{L}$ and speed $s_{L}+s$. If $U$ is a solution to the linearized
equations

$[(\nabla F)(U_{\rm NS}[U_{L},s_{L}])-s_{L}I]U=B(U_{\rm
NS}[U_{L},s_{L}])U_{x}\,,$

then $T_{s}U$ is a solution to the linearized equations

$[(\nabla F)(U_{\rm NS}[T_{s}U_{L},s_{L}+s])-(s_{L}+s)I]U=B(U_{\rm
NS}[T_{s}U_{L},s_{L}+s])U_{x}\,.$

In the analysis below, it is convenient to fix the left end-state as
$(\varrho,u,\theta)=(1,0,1)$ and keep the shock speed
$s=\sqrt{\nicefrac{{5}}{{3}}}-\varepsilon$ in the ODE. In the remainder of the
paper, it is more convenient to keep the shock speed
$s=s_{0}:=\sqrt{\nicefrac{{5}}{{3}}}$ and vary the left end-state as
$(\varrho,u,\theta)=(1,\varepsilon,1)$. The Galilean invariance means that the
ODE results described below will still be applicable.

###  B.2 Existence of small shocks

From now on, we work with general viscous conservation laws satisfying certain
properties, which we must verify for the compressible Navier-Stokes equations
introduced above. Let $u_{L}\in\mathbb{R}^{n}$ and $f=(f^{1},\ldots,f^{n})$ be
a smooth, hyperbolic flux defined in an open neighborhood of $u_{L}$. In a
sufficiently small neighborhood of $u_{L}$, we may enumerate the (real)
eigenvalues $\lambda_{1}(u),\ldots,\lambda_{n}(u)$ of $\nabla f(u_{L})$ and
consider associated left and right eigenvectors $l_{1}(u),\ldots,l_{n}(u)$ and
$r_{1}(u),\ldots,r_{n}(u)$, respectively, satisfying
$l_{i}(u)r_{j}(u)=\delta_{ij}$. The Rankine-Hugoniot conditions are

$s\llbracket u\rrbracket=\llbracket f\rrbracket\,,$ (B.8)

where $s$ is the shock speed, $\llbracket u\rrbracket=u_{L}-u_{R}$ and
$\llbracket f\rrbracket=f(u_{L})-f(u_{R})$. We assume that
$\lambda_{p}(u_{L})$ is a simple eigenvalue. Then in a sufficiently small
neighborhood of fixed $u_{L}$, the unique solutions $(u_{L},u_{R},s)$ to (B.8)
consist of the trivial solutions $(u_{L},u_{L},s)$, $s\in\mathbb{R}$, and the
points $u_{R}$ on the _Hugoniot curve_ , which may be smoothly parameterized
in $\mu$ to satisfy

$u_{R}(\mu)=u_{L}+\mu r_{p}(u_{L})+O(\mu^{2})\,,\quad
s(\mu)=\lambda_{p}(u_{L})+\frac{\mu}{2}(\nabla\lambda_{p})(u_{L})\cdot
r_{p}(u_{L})+O(\mu^{2})\,;$ (B.9)

see [lefloch, Chapter 6, Theorem 1.1] or [dafermos, Section 8.2]. We say that
the $p^{\rm th}$ characteristic field is _genuinely nonlinear_ in a
neighborhood of $u_{L}$ if $(\nabla\lambda_{p})(u)\cdot r_{p}(u)\neq 0$ in
that neighborhood (see (B.6)). Without loss of generality, we assume that the
sign is negative, so that

$\Lambda=(\nabla\lambda_{p})(u_{L})\cdot r_{p}(u_{L})<0\,.$ (B.10)

In this case, we may parameterize the curve $(u_{R},s)$ in
$\varepsilon:=\lambda_{p}(u_{L})-s=s_{L}-s=-\nicefrac{{\mu\Lambda}}{{2}}+O(\varepsilon^{2})$,
where we adopt the notation $s_{L}=\lambda_{p}(u_{L})$.

######  Proposition B.2 (Existence of small shock profiles).

Let $u_{L}\in\mathbb{R}^{n}$ and $f$ be a smooth, hyperbolic flux defined in a
neighborhood of $u_{L}$. Suppose that $s_{L}=\lambda_{p}(u_{L})$ is a simple
eigenvalue of $\nabla f(u_{L})$, and assume that the $p^{\rm th}$
characteristic field is genuinely nonlinear and satisfies (B.10). Define

$A:=(\nabla f)(u_{L})-s_{L}\operatorname{\mathrm{Id}}\,,$

where $\operatorname{\mathrm{Id}}$ is the $n\times n$ identity matrix. Let
$B(u)$ be a smooth $n\times n$ matrix-valued function defined in a
neighborhood of $u_{L}$ with block decomposition

$A=\begin{bmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{bmatrix}\,,\quad
B=\begin{bmatrix}0&0\\\ B_{21}&B_{22}\end{bmatrix}\,,$ (B.11)

corresponding to the decompositions

$u=(u^{I},u^{II})\in\mathbb{R}^{n-m}\times\mathbb{R}^{m}\,,\quad
f=(f^{I},f^{II})\in\mathbb{R}^{n-m}\times\mathbb{R}^{m}$

of $u$ and $f$ into two components. Suppose the following conditions are
satisfied at $u=u_{L}$:

  1. 1.

The square matrix $\begin{bmatrix}A_{11}\,A_{12}\\\
B_{21}\,B_{22}\end{bmatrix}$, evaluated at $u_{L}$, has full rank.

  2. 2.

Consider the $(n-m)\times n$ matrix

$\begin{bmatrix}A_{11}\,A_{12}\end{bmatrix}=\nabla
f^{I}-s_{L}\begin{bmatrix}\operatorname{\mathrm{Id}}_{(n-m)\times(n-m)}\,0_{(n-m)\times
m}\end{bmatrix}=:\nabla f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}}$

obtained by taking the first $n-m$ rows of the $n\times n$ matrix $A$. We
assume that for any $\xi\in\mathbb{R}\setminus\\{0\\}$,

$(-\xi^{2}B(u_{L})+i\xi A)|_{\mathbb{C}\cdot\ker(\nabla
f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})}\text{ is one-to-one}\,,$
(B.12)

where $\mathbb{C}\cdot\ker(\nabla
f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})\cong\mathbb{C}^{m}$ is the
complexification of $\ker(\nabla
f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})\cong\mathbb{R}^{m}$.

  3. 3.

We assume that

$\alpha:=(l_{p}\cdot Br_{p})(u_{L})>0\,.$ (B.13)

Then for $0<\varepsilon\ll 1$, there exists a family of solutions
$u^{\varepsilon}:\mathbb{R}\to\mathbb{R}^{n}$ to the viscous shock ODE

$f(u)_{x}-s(\varepsilon)u_{x}=(B(u)u_{x})_{x}\,,$ (B.14)

satisfying $u^{\varepsilon}\to u_{L}$ as $x\to-\infty$ and $u^{\varepsilon}\to
u_{R}(\varepsilon)$ as $x\to+\infty$, where we adopt the parameterization in
(B.9). Finally, there exists $M\geqslant 1$ such that these solutions satisfy,
for all $k\geqslant 0$, the decay estimates

$\displaystyle|\partial_{x}^{k}(u_{\varepsilon}-u_{L})|\leqslant
C_{k}\varepsilon^{k+1}e^{-\varepsilon|x|/M}\,,\quad x\leqslant 0\,,$ (B.15a)
$\displaystyle|\partial_{x}^{k}(u_{\varepsilon}-u_{R}(\varepsilon))|\leqslant
C_{k}\varepsilon^{k+1}e^{-\varepsilon|x|/M}\,,\quad x\geqslant 0\,.$ (B.15b)

Before proving Proposition B.2, we comment on the various assumptions and
verify them for the compressible Navier-Stokes equations (B.2) in conservative
variables at $u_{L}=(1,0,0,0,(\gamma-1)^{-1})$, $s_{L}=\sqrt{\gamma}$.

######  Remark B.3 (Non-degeneracy of traveling wave ODE).

Assumption 1 is equivalent to

$\nabla f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}}$ is full rank, and
$B|_{\ker(\nabla f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})}$ is one-to-
one. (B.16)

To see the equivalence, split the domain $\mathbb{R}^{n}=(\ker\nabla
f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})\oplus Q$, where $Q$ is any
complementary subspace. Then
$\nabla(f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})$ is one-to-one on $Q$.
Now rewrite the block matrix $\begin{bmatrix}A_{11}\,A_{12}\\\
B_{21}\,B_{22}\end{bmatrix}$ as a block matrix according to the decomposition
$Q\oplus\ker\nabla f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}}$ in the
domain and $\mathbb{R}^{m}\oplus\mathbb{R}^{n-m}$ in the target, where the
copy of $\mathbb{R}^{m}$ corresponds to the image of $Q$ under $B$. It becomes

$\begin{bmatrix}\nabla(f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})|_{Q}&0\\\
B|_{Q}&B|_{\ker\nabla(f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}})}\end{bmatrix}\,,$

which clarifies the equivalence.

The relevance of Assumption 1 is the following: $\nabla
f^{I}-s_{L}\tilde{\operatorname{\mathrm{Id}}}$ full rank yields, by the
implicit function theorem, that the equation
$f^{I}(u)-f^{I}(u_{L})-s(u^{I}-u_{L}^{I})=0$ describes a smooth
$m$-dimensional manifold locally near $(u,s)=(u_{L},s_{L})$. Its tangent space
at $(u_{L},s_{L})$ is $\ker\nabla f^{I}$. Since $B|_{\ker\nabla f^{I}}:{\rm
ker}\,\nabla f^{I}\to{\rm ran}B$ is invertible, the traveling wave ODE (B.14)
can be a realized as a smooth ODE on this manifold, as we will see below. In
the case of the compressible Navier-Stokes equations, Assumption 1 may be
verified using (B.4), (B.7), $s_{L}=\sqrt{\gamma}$, and a block decomposition
in which $m=4$, so that
$\begin{bmatrix}A_{11}\,A_{12}\end{bmatrix}=\begin{bmatrix}-s_{L}\,1\,0\,0\,0\end{bmatrix}$.

######  Remark B.4 (Symmetrizability and Kawashima condition).

We now remark upon the conditions in Assumption 2 by making the following two
claims, which we then prove.

  1. 1.

Suppose that the system is symmetrizable at $(u_{L},s_{L})$ in the sense that
there exists $A^{0}\in\mathbb{R}^{n\times n}$ symmetric and positive-definite
satisfying

$A^{1}:=\nabla f(u_{L})A^{0}\,\text{ is symmetric, and
}B^{11}:=B(u_{L})A^{0}\text{ symmetric and non-negative }\,.$

Suppose furthermore that the Kawashima condition

$\\{\text{eigenvectors of }A^{1}-s_{L}A^{0}\\}\cap\\{\ker B^{11}\\}=\\{0\\}$

is satisfied. Then Assumption 2 is satisfied.

  2. 2.

The compressible Navier-Stokes equations are symmetrizable at $(u_{L},s_{L})$
and satisfy the Kawashima condition.

###### Proof of Remark B.4.

We first prove claim 1. It is immediate that Assumption 2 will follow from the
invertibility of $-\xi^{2}B(u_{L})+i\xi A$, which itself follows from the
invertibility of $(-\xi^{2}B(u_{L})+i\xi A)A^{0}$ due to the assumption that
$A^{0}$ is real, symmetric, and positive-definite (and hence invertible). But
the invertibility of $(-\xi^{2}B(u_{L})+i\xi A)A^{0}$ is precisely a result of
the assumption that $AA^{0}$ satisfies the Kawashima condition in the sense of
Lemma C.1 and an application of Lemma C.2.

To prove claim 2 in general, we recall that the existence of a viscosity-
compatible convex entropy guarantees symmetrizability [ZumbrunNotes,
Proposition 1.19]. For the compressible Navier-Stokes equations, the entropy
is $\eta=-\varrho S$, with corresponding entropy flux $-u\varrho S$, and its
viscosity-compatibility and convexity are verified in [ZumbrunNotes, Remark
1.24, Example 1.25] and [KS88, Section 4]. In this setting, we may choose
$A^{0}=(\nabla^{2}\eta)^{-1}$. From [KS88, Section 4], we have that

$A_{0}=\begin{bmatrix}1&0&0&0&(\gamma-1)^{-1}\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\
(\gamma-1)^{-1}&0&0&0&\frac{\gamma}{(\gamma-1)^{2}}\end{bmatrix}\,,$

which is easily seen to be symmetric and positive definite. Then referring to
(B.4), we can compute $AA_{0}$ directly, obtaining

$AA_{0}=\begin{bmatrix}-s_{L}&1&0&0&\frac{-s_{L}}{\gamma-1}\\\
1&-s_{L}&0&0&\frac{\gamma}{\gamma-1}\\\ 0&0&-s_{L}&0&0\\\ 0&0&0&-s_{L}&0\\\
\frac{-s_{L}}{\gamma-1}&\frac{\gamma}{\gamma-1}&0&0&\frac{-s_{L}\gamma}{(\gamma-1)^{2}}\end{bmatrix}\,.$

Next, referring to (B.7), we can compute $B(u_{L})A_{0}$, obtaining

$B(u_{L})A_{0}=\begin{bmatrix}0&0&0&0&0\\\ 0&2\mu+\lambda&0&0&0\\\
0&0&\mu&0&0\\\ 0&0&0&\mu&0\\\ 0&0&0&0&\kappa\end{bmatrix}\,.$

Then the kernel of $B(u_{L})A_{0}=\textnormal{span}\\{e_{1}\\}$, and computing
$AA_{0}e_{1}$, we obtain $(-s_{L},1,0,0,\nicefrac{{-s_{L}}}{{\gamma-1}})$, so
$e_{1}$ is not an eigenvector of $AA_{0}$. Therefore the Kawashima condition
is satisfied, and thus Assumption 2 is satisfied by 1. ∎

######  Remark B.5.

Assumption 3 ensures that the one-dimensional ODE arising in the center
manifold reduction below flows in the correct direction. In the case of the
compressible Navier-Stokes equations,

$\alpha:=(l_{p}\cdot
Br_{p})(u_{L})=\frac{1}{2}\left[(2\mu+\gamma)+\frac{\kappa(\gamma-1)^{2}}{\gamma}\right]>0\,.$

###### Proof of Proposition B.2.

The $x$-integrated shock profile equation is

$f(u)-f(u_{L})-s(u-u_{L})=B(u)u_{x}\,.$ (B.17)

From (B.11), the equation in the $I$ component is

$f^{I}(u)-f^{I}(u_{L})-s(u^{I}-u_{L}^{I})=0\,,$ (B.18)

which we may consider as a functional equation $F(u,s)=0$ for unknown pairs
$(u,s)$. More precisely, we have that
$F:\mathbb{R}^{n-m}\times\mathbb{R}^{m}\times\mathbb{R}\rightarrow\mathbb{R}^{n-m}$,
and $u=(u_{1},u_{2})$ with $u_{2}\in X_{2}:=\ker(\nabla
f^{I}(u_{L})-s_{L}\tilde{\operatorname{\mathrm{Id}}})\cong\mathbb{R}^{m}$
belonging to the $m$-dimensional kernel and $u_{1}\in
X_{2}^{\perp}\cong\mathbb{R}^{n-m}$ belonging to the perpendicular space. By
assumption 1 and (B.16), $\nabla
f^{I}(u_{L})-s_{L}\tilde{\operatorname{\mathrm{Id}}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n-m}$
has full rank, and is therefore invertible off of its kernel $X_{2}$.
Therefore, the $(n-m)\times(n-m)$ Jacobian matrix $\frac{\partial F}{\partial
u_{1}}$ is non-singular, and so the implicit function theorem yields a unique,
smooth, local parameterization of the solutions $(u,s)=(u_{1},u_{2},s)$ to
(B.18) as

$(u_{1},u_{2},s)=\left(u_{1}(u_{2},s),u_{2},s\right)\,,\qquad
u_{1}(u_{L,2},s_{L})=u_{L,1}\,,$

where $u_{L}=(u_{L,1},u_{L,2})$. After performing a translation by $u_{L,2}$
in the $u_{2}$ component of $u_{1}(u_{2},s)$, we abuse notation slightly by
notating our new parametrization still by $u_{1}$, which now satisfies
$u_{1}(0,s_{L})=u_{L,1}$. We claim now that $\partial_{s}u_{1}(0,s_{L})=0$,
which follows from the identity

$\partial_{s}u_{1}(0,s_{L})=-\left[\frac{\partial F}{\partial
u_{1}}(u_{L,1},u_{L,2},s)\right]^{-1}\cdot\frac{\partial F}{\partial
s}(u_{L,1},u_{L,2},s)\,.$

Indeed the second matrix (actually an $(n-m)\times 1$ vector) above is easily
computed to be $(u^{I}-u^{I}_{L})|_{u_{L}}$, which gives $0$. Finally, we note
that $\frac{\partial u_{1}}{\partial u_{2}}(0,s_{L})=0$ by applying a similar
formula as above, and using that since $u_{2}\in X_{2}$, $\frac{\partial
F}{\partial u_{2}}(u_{L},s)=0$.

Now recall from (B.16) that $B(u_{L})|_{X_{2}}$ is injective, and recall from
(B.11) that $\textnormal{range}\,B\subseteq\mathbb{R}^{m}$, where
$\mathbb{R}^{m}$ corresponds to the $II$ component of $\mathbb{R}^{n}$.
Therefore $\textnormal{range}\,B(u_{L})\cong\mathbb{R}^{m}$, where
$\mathbb{R}^{m}$ is the $II$ component of $\mathbb{R}^{n}$, and by the
smoothness of $B$ in a neighborhood of $u_{L}$, we furthermore have that
$\textnormal{range}\,B(u)\cong\mathbb{R}^{m}$ for $|u-u_{L}|\ll 1$. Applying
these considerations, taking the $II$ component of equation (B.17) by applying
the $m\times n$ projection matrix $P_{II}^{j\ell}:\mathbb{R}^{n}\rightarrow
X_{2}\cong\mathbb{R}^{m}$, and using (B.11) and $\frac{\partial
u_{1}}{\partial u_{2}}(0,s_{L})=0$, we find that

$\displaystyle P_{II}^{j\ell}$
$\displaystyle\left[f^{\ell}(u_{1}(u_{2},s),u_{2})-f^{\ell}(u_{L,1},u_{L,2})-s\left([u_{1}(u_{2},s),u_{2}]^{\ell}-u_{L}^{\ell}\right)\right]$
$\displaystyle=P_{II}^{j\ell}B^{\ell
k}(u_{1}(u_{2},s),u_{2})\partial_{x}u^{k}$
$\displaystyle=B^{jk}(u_{1}(u_{2},s),u_{2})\begin{pmatrix}\frac{\partial
u_{1}^{k}}{\partial u_{2}^{\ell}}(u_{2},s)\partial_{x}u_{2}^{\ell}\\\
\partial_{x}u_{2}^{k}\end{pmatrix}$
$\displaystyle=B^{jk}(u_{1}(u_{2},s),u_{2})\partial_{x}u_{2}^{k}\,,$ (B.19)

which is an ODE for $u_{2}$. Since $B^{jk}$ is a nonsingular $m\times m$
matrix with (range equal to the $II$ subspace) and inverse which we denote by
$(B^{-1})^{ij}$ (with domain $II$), (B.19) is equivalent to

$\displaystyle\dot{u}_{2}^{i}$
$\displaystyle=(B^{-1})^{ij}(u_{1}(u_{2},s),u_{2})P_{II}^{j\ell}\left[f^{\ell}(u_{1}(u_{2},s),u_{2})-f^{\ell}(u_{L})-s\left([u_{1}(u_{2},s),u_{2}]^{\ell}-u_{L}^{\ell}\right)\right]\,,$
(B.20)

where $\dot{u}_{2}$ signifies differentiation in $x$. Steady states of (B.20)
satisfy the Rankine-Hugoniot conditions, and we have already classified steady
states in a neighborhood of $(u_{L},s_{L})$. Obviously, $u=u_{L}$ is a steady
state regardless of $s$; otherwise, the steady states are $u_{R}(\varepsilon)$
as in (B.9).

We now consider the system of ODEs for $(u_{2},s)$ which consists of (B.20)
together with the equation $\dot{s}=0$. Computing the linearization of this
system of ODEs at the equilibrium $(u_{L},s_{L})$ gives

$\displaystyle\dot{u}_{2}^{i}$
$\displaystyle=\left(B^{-1}\right)^{ij}(u_{L})P_{II}^{jl}\left[\partial_{u_{2}^{q}}f^{\ell}(u_{L})u_{2}^{q}-s_{L}u_{2}^{\ell}\right]\,,$
(B.21) $\displaystyle\dot{s}$ $\displaystyle=0\,;$

we note that in the above computation, we have used that
$\partial_{u_{2}}u_{1}(0,s_{L})=0$, and $\partial_{s}u_{1}(0,s_{L})=0$.
Finally, recall that the vector $r_{p}$ is in the kernel of $A$, since it is
the eigenvector corresponding to the $p^{\rm th}$ eigenvalue $s_{L}$ of
$\nabla f(u_{L})$; thus $r_{p}$ also belongs to $X_{2}$. Therefore,
$(u_{2},s)=(r_{p},s)$ is in the kernel of the above linearized operator.

We wish to consider the center manifold associated to the system of ODEs for
the augmented vector field $(\dot{u}_{2},\dot{s})$ at $(u,s)=(u_{L},s_{L})$.
We first prove that its linearization has no non-zero eigenvalues on the
imaginary axis.

######  Lemma B.6.

The spectrum of the linearization satisfies

$\sigma\left[\left(B^{-1}\right)^{kj}(u_{L})P_{II}^{jl}\left[\partial_{u_{2}^{q}}f^{\ell}(u_{L})-s_{L}\delta^{\ell
q}\right]\right]\cap i\mathbb{R}=\\{0\\}\,,$ (B.22)

and zero is an algebraically simple eigenvalue.

###### Proof.

The conclusion (B.22) fails to hold for matrices of the form $B^{-1}C$
precisely when there exists $v\in\mathbb{C}^{m}=\mathbb{C}\cdot\mathbb{R}^{m}$
and $\xi\in\mathbb{R}\setminus\\{0\\}$ such that $B^{-1}Cv=i\xi v$, which is
equivalent to $Cv=Bi\xi v$, or to a non-trivial kernel for the matrix
$-\xi^{2}B+i\xi C$. Thus the assertion (B.22) is equivalent to the injectivity
of

$-\xi^{2}B(u_{L})|_{\mathbb{C}\cdot X_{2}}+i\xi A|_{\mathbb{C}\cdot X_{2}}$

on $\mathbb{C}\cdot X_{2}$, which is the content of assumption 2. In order to
prove that zero is a simple eigenvalue, we first note that its geometric
multiplicity is one by the assumption that $s_{L}$ is a simple eigenvalue of
$\nabla f$. In order to prove that its algebraic multiplicity is one, suppose
towards a contradiction that $v_{2}\in X_{2}$ is a generalized eigenvector
satisfying

$\left[B^{-1}P_{II}(\partial_{u_{2}}f-s_{L}\operatorname{\mathrm{Id}}_{m\times
m})\right]v_{2}=r_{p}\,.$

Then

$P_{II}Av_{2}=Br_{p}\qquad\implies\qquad Av_{2}=Br_{p}\,,$

where the implication is a consequence of the fact that $P_{I}A$ has full
rank, and so $X_{2}=\ker P_{I}A$ must be mapped to $II=I^{\perp}$. Now pairing
with the left eigenvector $l_{p}$, we obtain the contradiction

$l_{p}Av_{2}=0=l_{p}\cdot Br_{p}\overset{\eqref{eq:assumption}}{\neq}0\,.$

∎

Returning to the proof of Proposition B.2, we have that the center subspace
associated to the zero eigenvalue is two-dimensional. Therefore, we invoke the
center manifold theorem to obtain a two-dimensional center manifold for
(B.20), augmented with the equation $\dot{s}=0$. This center manifold is
foliated by one-dimensional invariant submanifolds $\\{s=\text{constant}\\}$.
We parameterize these invariant submanifolds via

$u_{2}=u_{2}(\eta,s)=\eta r_{p}(u_{L})+h(\eta,s)\,,\quad
h(\eta,s):\mathbb{R}\times B_{\delta}(s_{L})\subseteq\mathbb{R}^{2}\rightarrow
X_{2}\,,\quad h(\eta,s)=O(\eta^{2}+|s-s_{L}|^{2})\,,$

where $\delta$ is sufficiently small, and $h(0,s)=0$ for all $s$, so that
$(0,s)$ corresponds to the steady solution $(u_{2},s)=(u_{L,2},s)$ of (B.20).
From now on, $r_{p}$ and $l_{p}$ will always be evaluated at $u_{L}$, and so
we suppress the dependency. Tangency to the center subspace means that
$h(0,s_{L})=0$ and $\nabla_{\eta,s}h(0,s_{L})=0$. Applying the chain rule to
$\dot{u}_{2}$ and using that $\dot{s}=0$, the ODE (B.20) for $u_{2}(x)$
becomes an ODE for $\eta(x)$:

$\displaystyle[r_{p}^{i}+\partial_{\eta}h^{i}(\eta,s)]\dot{\eta}$
$\displaystyle=(B^{-1})^{ij}(u_{1}(\eta r_{p}+h(\eta,s),s),\eta
r_{p}+h(\eta,s))P_{II}^{j\ell}$
$\displaystyle\quad\cdot\left[f^{\ell}(u_{1}(\eta r_{p}+h(\eta,s),s),\eta
r_{p}+h(\eta,s))-f^{\ell}(u_{L})-s\left([u_{1}(\eta r_{p}+h(\eta,s),s),\eta
r_{p}+h(\eta,s)]^{\ell}-u_{L}^{\ell}\right)\right]\,.$

c It will be crucial that $\eta$ is _increasing_ along the flow between the
two steady states $u_{L}$ and $u_{R}$. We know that $\eta=0$ corresponds to
$u_{2}=0$, and thus to $u_{L}$. We set
$\eta=\bar{\eta}(s):=-2\varepsilon/\Lambda+O(\varepsilon^{2})$ to be the value
of $\eta$ corresponding to $u_{R}$, where we have obtained this formula by
recalling the parametrization for the curve $(u_{R},s)$ in terms of
$\varepsilon=s_{L}-s$ from underneath (B.10). Notably, (B.10) implies that
$\bar{\eta}(s)>0$ when $0<\varepsilon\ll 1$, where positive $\varepsilon$
corresponds to shock speeds satisfying the Lax shock inequalities.

To determine the direction, we rewrite

$B^{mi}(\eta,s)[r_{p}^{i}+\partial_{\eta}h^{i}(\eta,s)]\dot{\eta}=P^{m\ell}_{II}\left[f^{\ell}(\eta,s)-f^{\ell}(u_{L})-s\left(u^{\ell}(\eta,s)-u_{L}^{\ell}\right)\right]\,,$
(B.23)

and left-multiply by $l_{p}$. Since $B(0,s_{L})=B(u_{L})$ and
$\partial_{\eta}h(0,s_{L})=0$, we obtain

$[l_{p}^{m}B^{mi}(u_{L})r_{p}^{i}+d(\eta,s)]\dot{\eta}=\underbrace{l_{p}^{m}P^{m\ell}_{II}\left[f^{\ell}(\eta,s)-f^{\ell}(u_{L})-s\left(u^{\ell}(\eta,s)-u_{L}^{\ell}\right)\right]}_{G(\eta,s)}\,,$

where $d=O(|s-s_{L}|+|\eta|)$ and $\alpha:=l_{p}\cdot B(u_{L})r_{p}>0$ by
(B.13). Hence,

$\dot{\eta}=F(\eta,s)\,,\qquad
F(\eta,s)=\frac{G(\eta,s)}{\alpha+d(\eta,s)}\,.$

To simply show the existence of heteroclinic orbits, without quantitative
estimates, it is enough to show that $F>0$ when $0<\eta<\bar{\eta}(s)$. For
example, it would be enough to show that $\partial_{\eta}F(0,s)>0$ for
$s<s_{L}$. We will obtain more precise information by expanding to _second
order_. We know that $\partial_{s}\bar{\eta}<0$ by recalling that
$\bar{\eta}(s)=\frac{-2\varepsilon}{\Lambda}+O(\varepsilon^{2})=\frac{-2(s_{L}-s)}{\Lambda}+O((s_{L}-s)^{2})$,
differentiating in $s$, and using (B.10). Then we claim that

$G(0,s)=0=G(\bar{\eta}(s),s)\,,\quad\nabla_{\eta,s}G(0,s_{L})=0\,.$

In fact the first equality is a consequence of our choice of parametrization
$h$, which satisfies $h(0,s)=0$ for all $s\in B_{\delta}(s_{L})$, the second
equality for $G$ is a consequence of the Rankine-Hugoniot conditions, and the
vanishing of $\nabla_{\eta,s}G(0,s_{L})$ is a consequence of

$\displaystyle\partial_{\eta}G$
$\displaystyle=l_{p}^{m}P_{II}^{m\ell}\left[\nabla_{k}f^{\ell}(\eta,s)\frac{\partial
u^{k}}{\partial\eta}-s\frac{\partial u^{\ell}}{\partial\eta}\right]\,,$
(B.24a) $\displaystyle\partial_{s}G$
$\displaystyle=l_{p}^{m}P_{II}^{m\ell}\left[\nabla_{k}f^{\ell}(\eta,s)\frac{\partial
u^{k}}{\partial s}-s\frac{\partial u^{\ell}}{\partial
s}-(u^{\ell}-u^{\ell}_{L})\right]$ (B.24b) $\displaystyle\frac{\partial
u_{2}}{\partial\eta}\bigg{|}_{(\eta,s)=(0,s_{L})}$
$\displaystyle=r_{p}\,,\quad\frac{\partial u_{1}}{\partial
u_{2}}\bigg{|}_{(u_{2},s)=(0,s_{L})}=0\,,\quad\frac{\partial u_{2}}{\partial
s}\bigg{|}_{(\eta,s)=(0,s_{L})}=0\,.$ (B.24c)

Therefore, $F(0,s_{L})$ and $\nabla_{\eta,s}F(0,s_{L})$ also vanish.
Consequently,
$\nabla_{\eta,s}^{2}F(0,s_{L})=\alpha^{-1}\nabla_{\eta,s}^{2}G(0,s_{L})$, so
we compute the second derivatives of $G$. Evidently,
$(\partial_{s}^{k}G)(0,s_{L})=0$ for all $k$, since differentiating $G$ any
number of times along the families of steady states yields zero; equivalently,
$G(0,s)$ depends on $s$ through $u_{2}(0,s)$, which however vanishes for all
$s$ by our choice of $h$. Next, we compute $\partial_{s}\partial_{\eta}G$ by
using (B.24), obtaining

$\displaystyle\partial_{s}\partial_{\eta}G=l_{p}^{m}P_{II}^{m\ell}\left[\nabla_{kj}f^{\ell}\frac{\partial
u^{j}}{\partial s}\frac{\partial
u^{k}}{\partial\eta}+\nabla_{k}f^{\ell}\frac{\partial^{2}u^{k}}{\partial
s\partial\eta}-\frac{\partial
u^{\ell}}{\partial\eta}-s\frac{\partial^{2}u^{\ell}}{\partial
s\partial\eta}\right]\,.$

At $(\eta,s)=(0,s_{L})$, we use (B.24) to simplify this to

$l_{p}^{m}P_{II}^{m\ell}\left(\left[\nabla_{k}f^{\ell}(u_{L})-s_{L}\delta_{k}^{\ell}\right]\frac{\partial^{2}u^{k}}{\partial
s\partial\eta}(0,s_{L})-r_{p}^{\ell}\right)\,.$

However, at $(\eta,s)=(0,s_{L})$, we have that

$\displaystyle\frac{\partial^{2}u^{k}}{\partial
s\partial\eta}\bigg{|}_{(0,s_{L})}=\frac{\partial}{\partial
s}\left[\frac{\partial u^{k}_{1}}{\partial u_{2}^{j}}\frac{\partial
u_{2}^{j}}{\partial\eta}+\frac{\partial
u_{2}^{k}}{\partial\eta}\right]\bigg{|}_{(0,s_{L})}$
$\displaystyle=\left[\frac{\partial^{2}u^{k}_{1}}{\partial u_{2}^{j}\partial
u_{2}^{m}}\frac{\partial u_{2}^{j}}{\partial\eta}\frac{\partial
u_{2}^{m}}{\partial s}+\frac{\partial u^{k}_{1}}{\partial
u_{2}^{j}}\frac{\partial^{2}u_{2}^{j}}{\partial\eta\partial
s}+\frac{\partial^{2}u_{2}^{k}}{\partial\eta\partial
s}\right]\bigg{|}_{(0,s_{L})}$
$\displaystyle=\frac{\partial^{2}u_{2}^{k}}{\partial\eta\partial
s}\bigg{|}_{(0,s_{L})}\in X_{2}\,.$

Then since $X_{2}=\ker(\nabla f-s_{L}\operatorname{\mathrm{Id}})$, we deduce
that
$\partial_{s}\partial_{\eta}G(0,s_{L})=-l_{p}^{m}P_{II}^{m\ell}r_{p}^{\ell}$.
Note, however, that the left kernel of $A$ is equal to the span of $l_{p}$ by
the assumption that $s_{L}$ is a simple eigenvalue and the fact that row rank
is equal to column rank. Therefore, $l_{p}$ is perpendicular to all the
columns of $A$, and therefore perpendicular to the range of $A$, which
contains $I$ by (B.16). Finally, we deduce then that
$\partial_{s}\partial_{\eta}G(0,s_{L})=-l_{p}^{m}P_{II}^{m\ell}r_{p}^{\ell}=-l_{p}^{\ell}r_{p}^{\ell}=-1$.
Finally,

$\displaystyle\partial_{\eta}^{2}G=l_{p}^{m}P_{II}^{m\ell}\left[\nabla_{kj}f^{\ell}\frac{\partial
u^{k}}{\partial\eta}\frac{\partial
u^{j}}{\partial\eta}+\nabla_{k}f^{\ell}\frac{\partial^{2}u^{k}}{\partial\eta^{2}}-s\delta_{k}^{\ell}\frac{\partial^{2}u^{k}}{\partial\eta^{2}}\right]\,.$

Notice that since $l_{p}^{m}P_{II}^{m\ell}=l_{p}^{\ell}$, the second and third
terms above vanish. Focusing then on the first term evaluated at
$(\eta,s)=(0,s_{L})$, we have

$\displaystyle
l_{p}^{\ell}\nabla_{kj}f^{\ell}(u_{L})r_{p}^{k}r_{p}^{j}=l_{p}^{\ell}\partial_{j}\lambda_{p}(u_{L})r_{p}^{\ell}r_{p}^{j}=\Lambda<0$

by genuine nonlinearity. Finally, we deduce that
$\partial_{\eta}^{2}G(0,s_{L})=\Lambda$.

We now summarize the asymptotics by Taylor expanding $F$ to second order
around $(\eta,s)=(0,s_{L})$, obtaining

$\displaystyle F(\eta,s)$
$\displaystyle=\frac{\alpha^{-1}}{2}\begin{bmatrix}0&-1\\\
-1&\Lambda\end{bmatrix}:\begin{bmatrix}(s-s_{L})^{2}&(s-s_{L})\eta\\\
(s-s_{L})\eta&\eta^{2}\end{bmatrix}+O(|\eta|^{3}+\varepsilon^{3})$
$\displaystyle={\alpha^{-1}}\left(\varepsilon\eta+\frac{\Lambda}{2}\eta^{2}\right)+O(|\eta|^{3}+\varepsilon^{3})\,.$

Differentiating, we obtain

$\partial_{\eta}F(\eta,s)={\alpha^{-1}}\left(\varepsilon+\Lambda\eta\right)+O(|\eta|^{2}+\varepsilon^{2})\,.$

Hence, crucially,

$\partial_{\eta}F(0,s)={\alpha^{-1}}\varepsilon+O(\varepsilon^{2})=:m^{u}\,,$
(B.25)

and

$\partial_{\eta}F(\bar{\eta},s)=-\alpha^{-1}\varepsilon+O(\varepsilon^{2})=:m^{s}\,.$
(B.26)

At this point we have constructed a trajectory
$\eta(x):\mathbb{R}\rightarrow[0,-\nicefrac{{2\varepsilon}}{{\Lambda}}+O(\varepsilon^{2})]$
which corresponds to a solution $u_{2}$ of (B.20), and thus a solution to the
integrated shock profile equation (B.17). We note that $\eta$ is unique up to
translations. Then, fixing a translate, for example, by choosing
$\eta(0)=\bar{\eta}(s)/2$, we obtain quantitative estimates for $\eta(x)$ by
considering the unstable and stable manifolds at the equilibria $F(0,s)$ and
$F(\bar{\eta}(s),s)$, respectively, and using that $\eta$ converges to these
equilibria exponentially; more precisely, we have that

$|\eta(x)|\lesssim\varepsilon e^{c\varepsilon x}\,,\quad x<0\,,$
$|\eta(x)-\bar{\eta}|\lesssim\varepsilon e^{-c\varepsilon x}\,,\quad x>0\,.$

These estimates in turn imply estimates for $u_{2}=u_{2}(\eta,s)=\eta
r_{p}+h(\eta,s)$, which in turn provides the estimates in (B.15) for $k=0$.
The derivative estimates may be obtained by bootstrapping, and we omit further
details. ∎

######  Corollary B.7 (Spectral information).

For $0<\varepsilon\ll 1$, the spectrum of

$\left(B^{-1}\right)^{ij}(u)P_{II}^{j\ell}(\partial_{k}f^{\ell}(u)-s(\varepsilon)\delta_{k}^{\ell}):X_{2}\to
X_{2}$ (B.27)

evaluated at $u=u_{L}$ and $u=u_{R}$ consists of two parts:

  1. 1.

$o_{\varepsilon\to 0^{+}}(1)$ perturbations of the $m-1$ non-central
eigenvalues of

$\left(B^{-1}\right)^{ij}(u_{L})P_{II}^{j\ell}A^{\ell k}$

  2. 2.

An algebraically simple eigenvalue satisfying
$\mu_{L}(\varepsilon)=\alpha^{-1}\varepsilon+O(\varepsilon^{2})$ at $u_{L}$
and $\mu_{R}(\varepsilon)=-\alpha^{-1}\varepsilon+O(\varepsilon^{2})$ at
$u_{R}$.

In particular, (B.27) evaluated at $u_{L}$ and $u_{R}(\varepsilon)$ is
hyperbolic and the dimensions of its unstable subspace evaluated at $u_{L}$
and its the stable subspace evaluated at $u_{R}$ sum to $n+1$.

###### Proof.

(1) The continuity in $\varepsilon$ of the eigenvalues follows from standard
finite-dimensional perturbation theory; we do not prove that the eigenvalues
are semisimple, so we do not prove smoothness in $\varepsilon$. (2) The
central eigenvalue is algebraically simple, so it varies smoothly with respect
to smooth perturbations. The perturbed central eigenvalue is associated to the
direction along the one-dimensional invariant manifold. We rewrote the ODE on
the center manifold in the $\eta$-parameterization, that is,
$u_{2}=u_{2}(\eta,s)$. Since the spectrum of the linearization _at an
equilibrium_ is independent of the parameterization of the ODE, the central
eigenvalues are those from (B.25) and (B.26). Equivalently, one may compute
the central eigenvalue directly and observe the claimed asymptotics; this
computation is identical to the analysis of the ODE for $\eta$. ∎

######  Remark B.8.

The ODE (B.23) on the center manifold is approximately

$[\alpha+O(|s-s_{L}|+|\eta|)]\dot{\eta}=-\eta(s-s_{L})+\frac{\Lambda}{2}\eta^{2}+O(|\eta|^{3}+|s-s_{L}|^{3})\,.$
(B.28)

We recognize this as a perturbation of the profile equation for traveling wave
solutions to viscous Burgers.

###  B.3 Analysis of the linearized macroscopic equations

Fix $0<\varepsilon\ll 1$ and a small shock profile
$\bar{u}(x)=u^{\varepsilon}(x)$ with speed $s=s(\varepsilon)$, constructed
using the assumptions and conclusions of Proposition B.2. We have crucially
that (B.29) is equivalent to solving the macroscopic portion (2.38) of the
linearized Landau equation.

######  Proposition B.9.

For all $g\in L^{2}\cap C(\mathbb{R})$ and $d\in\mathbb{R}$, the linearized
macroscopic equation

$B(\bar{u})u_{x}=(\nabla f(\bar{u})-s\operatorname{\mathrm{Id}})u+g$ (B.29)

has a unique solution $u\in L^{2}\cap C(\mathbb{R})$ satisfying a one-
dimensional constraint

$l_{\varepsilon}(u(0))=d\,.$ (B.30)

The solution obeys the estimate

$\|u\|_{L^{2}}\lesssim\varepsilon^{-1}\|g\|_{L^{2}}+|d|\,.$

######  Remark B.10.

The assumption that $g\in C(\mathbb{R})$ is for convenience. For $g\in L^{2}$,
the solution $u$ decomposes into a part in $L^{2}$ and a continuous part, and
the one-dimensional constraint is imposed on the continuous part. The
constraint (B.30) is that $z^{c}(0)=d\eta$, where $z^{c}$ is defined in (B.35)
and $\eta$ is defined above (B.36). We may choose the shock profiles to be
continuous-in-$\varepsilon$ in the $L^{\infty}$ topology (for example, the
choice of translate in the proof of Proposition B.2 will do), so that
$l_{\varepsilon}$ is continuous-in-$\varepsilon$. The functional
$\ell_{\varepsilon}$ in in Proposition 2.4 is more precisely

$\ell_{\varepsilon}=l_{\varepsilon}\circ T_{-\varepsilon}\circ I\,,$

where $T_{-\varepsilon}$ is the Galilean transformation from Remark B.1 and
$I$, defined in (2.3), extracts hydrodynamic moments.

###### Proof.

The equation (B.29) in the $I$ component is

$0=(\nabla f^{I}(\bar{u})-s\tilde{\operatorname{\mathrm{Id}}})u+P_{I}g\,,$
(B.31)

where $\tilde{\operatorname{\mathrm{Id}}}$ is as in (B.12). Towards a
parametrization of the solutions $u$ to this equation in terms of $u_{2}\in
X_{2}$, we first recall from the proof of Proposition B.2 that $\nabla
f^{I}(u_{L})-s_{L}\tilde{\operatorname{\mathrm{Id}}}$ is invertible on
$X_{1}$. We claim that in fact $\nabla
f^{I}(u_{L})-s_{L}\tilde{\operatorname{\mathrm{Id}}}$ is invertible on
$\ker(B(u_{L}))$. To see this, we claim that we can choose $Q=\ker(B(u_{L}))$
in Remark B.3. If this were not the case, then the intersection of
$\ker(B(u_{L}))$ and $\ker(\nabla
f^{I}(u_{L})-s_{L}\tilde{\operatorname{\mathrm{Id}}})$ would contain a non-
zero vector, which would then lie in the kernel of the operator in assumption
1 from Propositon B.2, which however was assumed to be full rank (invertible).
Now due to the smallness of $0<\varepsilon\ll 1$, we therefore have that
$\nabla f^{I}(\bar{u})-s\tilde{\operatorname{\mathrm{Id}}}$ is injective on
$\ker(B(\bar{u}))$. We thus define a collection of linear isomorphisms
$q_{1}(\bar{u}):\ker(B(u_{L}))\rightarrow\ker(B(\bar{u}))$ parametrized by
$\bar{u}$. In a similar fashion, we claim that $B(\bar{u})$ is invertible on
$\ker(\nabla f^{I}(\bar{u})-s\tilde{\operatorname{\mathrm{Id}}})$. This
follows again from the smallness of $\varepsilon$ and (B.16). Thus there
exists a family of linear isomorphisms $q_{2}(\bar{u},s):\ker(\nabla
f^{I}(u_{L})-s_{L}\tilde{\operatorname{\mathrm{Id}}})=X_{2}\rightarrow\mathbb{R}^{n}$,
parametrized by $\bar{u}$, which map $X_{2}$ to $\ker(\nabla
f^{I}(\bar{u})-s\tilde{\operatorname{\mathrm{Id}}})$.

We therefore can parametrize all solutions of (B.31) as

${u}=q_{1}(\bar{u},s)u_{1}+q_{2}(\bar{u},s)u_{2}\,,\qquad u_{1}\in
X_{1}\,,\quad u_{2}\in X_{2}\,,$

where we may solve for $u_{1}$ from (B.31), obtaining

$u_{1}=-\left[(\nabla
f^{I}(\bar{u})-s\tilde{\operatorname{\mathrm{Id}}})q_{1}(\bar{u},s)\right]^{-1}P_{I}g\,.$
(B.32)

Next, we note that our choice of parametrization and the estimates (B.15)
ensure that

$B(\bar{u})(q_{1}(\bar{u})u_{1})_{x}=B(\bar{u})\partial_{x}(q(\bar{u}))u_{1}=O(\varepsilon^{2})|P_{I}g|\,.$

In particular, no derivatives of $u_{1}$ will enter the ODE for $u_{2}$. It
remains to solve for $\tilde{u}=u-q_{1}(\bar{u})u_{1}=q_{2}(\bar{u})u_{2}$.
Let

$\tilde{g}=P_{II}g-B(\bar{u})(q_{1}(\bar{u},s)u_{1})_{x}\,.$ (B.33)

We have

$B(\bar{u})\partial_{x}\tilde{u}=P_{II}(\nabla
f(\bar{u})-sI)\tilde{u}+\tilde{g}\,,$

which after writing out

$\partial_{x}\tilde{u}(x)=\frac{\partial
q_{2}}{\partial\bar{u}}(x)\partial_{x}\bar{u}(x)u_{2}(x)+q_{2}(\bar{u}(x))\partial_{x}u_{2}(x)$

and using the injectivity of $B(\bar{u})$ on
$\textnormal{ran}\,q_{2}(\bar{u}(x))$ (which follows again from the
injectivity of $B(u_{L})$ on $X_{2}$ and a small choice of $\varepsilon$)
gives an ODE for $u_{2}$:

$\partial_{x}u_{2}=\left(B(\bar{u})q_{2}(\bar{u})\right)^{-1}P_{II}\left[(\nabla
f(\bar{u})-s\operatorname{\mathrm{Id}})q_{2}(\bar{u})u_{2}+\tilde{g}-B(\bar{u})\frac{\partial
q_{2}}{\partial\bar{u}}\partial_{x}\bar{u}u_{2}\right]\,.$

For brevity, we rewrite

$w=u_{2}\,,\quad h=\left(B(\bar{u})q_{2}(\bar{u})\right)^{-1}\tilde{g}\,,$
$m:=\left(B(\bar{u})q_{2}(\bar{u})\right)^{-1}P_{II}\left(\nabla
f(\bar{u})-s\operatorname{\mathrm{Id}}-B(\bar{u})\frac{\partial
q_{2}}{\partial\bar{u}}\partial_{x}\bar{u}\right)\,.$

Then the ODE is

$w_{x}=m(x)w+h\,.$ (B.34)

We now summarize the spectral information on $m$. Notably, $m(x)$ satisfies
the asymptotics

$|\partial_{x}^{k}(m-m_{\pm})|\lesssim\varepsilon^{k+1}e^{-\varepsilon|x|/M}\,,\quad\forall
k\geqslant 0\,,$

by the estimates (B.15a)–(B.15b) in Proposition B.2. The end states $m_{\pm}$
(hence, $m(x)$ itself) are $O(\varepsilon)$ perturbations of a fixed $m_{0}$,
namely, the linearized operator in (B.21); that is,
$m_{\pm}=m_{0}+O(\varepsilon)$. Furthermore, the endstates $m_{\pm}$ satisfy
the spectral conclusions detailed in Corollary B.7, assuming a sufficiently
small choice of $\varepsilon$ relative to $\alpha^{-1}$. The fixed $m_{0}$ has
$k$ stable eigenvalues (counting algebraic multiplicity), $n-k-1$ unstable
eigenvalues, and a simple zero eigenvalue. Hence, for sufficiently small
$\varepsilon$, there exists $\theta>0$ such that $m$ has spectrum consisting
of $k$ “strongly” stable eigenvalues $\lambda$ with ${\rm Re}\,k>\theta$,
$n-k-1$ “strongly” unstable eigenvalues with ${\rm Re}\,k<-\theta$, and a
simple eigenvalue $\mu(x)=O(\varepsilon)$. We know from Corollary B.7 that
$\mu(m_{\pm})=\mp{\alpha^{-1}}\varepsilon+O(\varepsilon^{2})$, and
$|\mu(x)-\mu(m_{\pm})|$ decays exponentially with the same rates as $m$
itself.

Let us denote the spectral projections $P^{s}(m(x))$ onto the strongly stable
subspace of $m(x)$, $P^{u}(m(x))$ onto the strongly unstable subspace, and
$P^{c}(m(x))$ onto the central eigenspace. By standard perturbation theory,
namely, smooth dependence of the resolvents, these projections onto the group
eigenspaces depend smoothly on the argument $m(x)$ in a neighborhood of
$m_{0}$. Since $m$ is fixed, we abbreviate $P^{s}(x)=P^{s}(m(x))$,
$P^{s}_{0}=P^{s}(m_{0})$, etc.

We block diagonalize $m(x)$ along its strongly stable, strongly unstable, and
center subspaces. Set

$Q(x):=P^{s}(x)P^{s}_{0}+P^{c}(x)P^{c}_{0}+P^{u}(x)P^{u}_{0}\,,$

and set

$\displaystyle p(x)$ $\displaystyle=Q^{-1}(x)m(x)Q(x)$
$\displaystyle=\underbrace{Q^{-1}(x)m(x)P^{s}(x)Q(x)}_{m^{s}(x)}+\underbrace{Q^{-1}(x)m(x)P^{c}(x)Q(x)}_{m^{c}(x)}+\underbrace{Q^{-1}(x)m(x)P^{u}(x)Q(x)}_{m^{u}(x)}\,.$

Notice that $Q(x)$ is uniformly-in-$\varepsilon$ invertible, since it is an
$O(\varepsilon)$ perturbation of the identity, and
$Q^{\prime}(x)=O(\varepsilon^{2})$. The maps $m^{\bullet}(x)$ are adapted to
the fixed spectral subspaces $E^{\bullet}_{0}$ of $m_{0}$ for $\bullet=s,c,m$.
Additionally, $m^{s}(x)$ is an $O(\varepsilon)$ perturbation of $m^{s}_{0}$,
which has negative spectrum, and $m^{u}(x)$ is an $O(\varepsilon)$
perturbation of $m^{u}_{0}$, which has positive spectrum. Finally,
$m^{c}(x)v=\mu(x)v$, where $v$ spans $E^{c}_{0}$, and $\mu(x)$ satisfies the
aforementioned asymptotics. If we make the change of variables

$z=Q^{-1}w\,,$

then the ODE (B.34) becomes

$z^{\prime}(x)=p(x)z(x)+\underbrace{Q^{-1}(x)h(x)-(Q^{-1})^{\prime}(x)Q(x)z(x)}_{:=\tilde{h}}\,.$

By our block diagonalization, bounded solutions $z(x)$ may be decomposed as

$z(x)=\underbrace{P_{0}^{s}z(x)}_{:=z^{s}}+\underbrace{P^{c}_{0}z(x)}_{:=z^{c}}+\underbrace{P^{u}_{0}z(x)}_{:=z^{u}}\,,$
(B.35)

where

$(z^{\bullet})^{\prime}(x)=m^{\bullet}(x)z^{\bullet}(x)+P^{\bullet}_{0}\tilde{h}(x)\,,\qquad\bullet=s,c,u\,.$

Furthermore, there exists propagators $\Phi^{s}$ and $\Phi^{u}$ such that

$z^{s}(x)=\int_{-\infty}^{x}\Phi^{s}(x,y)\tilde{h}(y)\,dy\,,$
$z^{u}(x)=-\int_{x}^{\infty}\Phi^{u}(x,y)\tilde{h}(y)\,dy\,.$

Fix $\eta$ with $|\eta|=1$ such that ${\rm image}\,P^{c}_{0}={\rm
span}\,\eta$. For the central eigenvalue, we integrate from $x=0$ to obtain

$z^{c}(x)=d\eta+\int_{0}^{x}e^{\int_{y}^{x}\mu(y^{\prime})\,dy^{\prime}}\tilde{h}(y)\,dy\,.$
(B.36)

The constraint $l_{\varepsilon}(u(0))=d$ is precisely that $z^{c}(0)=d\eta$.
Due to the fact that $m^{s}(x)$ has strictly negative spectrum, and $m^{u}(x)$
has strictly positive spectrum, there exist $M_{1}$ and $M_{2}$ such that
propagators satisfy

$\|\Phi^{s}(x,y)\|\lesssim e^{-(x-y)/M_{1}}\,,\quad x>y\,,$
$\|\Phi^{u}(x,y)\|\lesssim e^{-(y-x)/M_{1}}\,,\quad y<x\,,$

and

$\left|e^{\int_{y}^{x}\mu(y^{\prime})\,dy^{\prime}}\right|\lesssim
e^{-\varepsilon|x-y|/M_{2}}\,,\quad x,y\in\mathbb{R}\,.$

Notice that $\tilde{h}$ contains an $O(\varepsilon^{2})$ linear term in $z$,
which, due to its size, is considered lower order and can be added on
perturbatively by a contraction argument. In conclusion, Duhamel’s formula
yields

$\|u_{2}\|_{L^{2}}\lesssim\varepsilon^{-1}\|\tilde{g}\|_{L^{2}}+|d|\,,$

and, recalling the definitions (B.32) and (B.33) of $u_{1}$ and $\tilde{g}$,
respectively,

$\|u\|_{L^{2}}\lesssim\varepsilon^{-1}\|g\|_{L^{2}}+|d|\,.$

∎

##  Appendix C Kawashima compensators

###  C.1 Basic construction

Let $\mathbb{C}^{m}=X\oplus Y$ (orthogonal decomposition) and let
$A:\mathbb{C}^{m}\to\mathbb{C}^{m}$ be a Hermitian matrix. Let $1\leqslant
n<m$ and assume that $\dim X=n$. We say that $A$ satisfies the _Kawashima
condition_ if

$\text{for each non-zero }x\in X,\;A^{k}x\not\in X\text{ for some
}k\in\\{1,\ldots,n\\}\,.$ (C.1)

This is equivalent to

$\\{\text{eigenvectors of $A$}\\}\cap X=\\{0\\}\,.$ (C.2)

###### Proof of equivalence.

($\neg\eqref{eq:B}\implies\neg\eqref{eq:A}$) If there exists a non-zero
eigenvector $u$ of $A$ in $X$, then $A^{k}u\in X$ for all $k$.

($\neg\eqref{eq:A}\implies\neg\eqref{eq:B}$) Let $x$ be such that $A^{k}x\in
X$ for all $k\in\\{1,\dots,n\\}$. Since $X$ is $n$-dimensional and we have
assumed that $A^{k}x\in X$ for $k\in\\{1,\dots,n\\}$, there exists a smallest
positive integer $d\in\\{1,\dots,n\\}$ for which the set of $d+1$ vectors
$\\{x,Ax,\dots,A^{d}x\\}$ is a linearly dependent set. Since $d$ is the
smallest such integer, any linear relation with

$c_{0}x+c_{1}Ax+\dots+c_{d}A^{d}x=0\,,\qquad c_{i}\textnormal{ not all zero}$

must be such that $c_{d}\neq 0$ to avoid contradicting the minimality of $d$.
Fix such a linear relation, which without loss of generality may be chosen
such that $c_{d}=1$. Consider the operator

$A|_{\textnormal{span}\\{x,Ax,\dots,A^{d-1}x\\}}\,.$

Then since $A^{d}x$ is expressible as a linear combination of
$x,Ax,\dots,A^{d-1}x$, we have that
$A|_{\textnormal{span}\\{x,Ax,\dots,A^{d-1}x\\}}$ is a linear operator with
domain and range both given by $\textnormal{span}\\{x,Ax,\dots,A^{d-1}x\\}$.
This linear operator must then have a non-zero eigenvector $x_{0}\in X$,
showing also that $A:\mathbb{C}^{m}\rightarrow\mathbb{C}^{m}$ has a non-zero
eigenvector in $X$. ∎

The standard set-up is to suppose that $L:\mathbb{R}^{m}\to\mathbb{R}^{m}$ is
a negative semi-definite Hermitian matrix which induces the subspaces $X=\ker
L$, and $Y=(\ker L)^{\perp}$, and that $A$ satisfies the Kawashima condition
with $X=\ker L$. We now assume throughout that $A,L$ satisfy these properties.

######  Lemma C.1.

If the Kawashima condition is satisfied, then there exists a _Kawashima
compensator_ , that is, a skew-adjoint operator $K$ satisfying

$\|P_{X}u\|^{2}\lesssim{\rm Re}\langle KAu,u\rangle+\|P_{Y}u\|^{2}\,.$ (C.3)

Notice that ${\rm Re}\langle KAu,u\rangle=\frac{1}{2}\langle[K,A]u,u\rangle$
by (skew-)adjointness. The model equations to keep in mind are

$\displaystyle\dot{u}$ $\displaystyle=i\tau Au+Lu$ (C.4) $\displaystyle
u(\cdot,0)$ $\displaystyle=u_{0}$

and

$i\tau Au+Lu+f=0\,,$ (C.5)

where $\tau\in\mathbb{R}$ is a non-zero parameter. Solutions to (C.4) and
(C.5) can be estimated via a combination of two energy estimates:

  1. 1.

the standard estimate in which we pair the equation with $u$, and

  2. 2.

a ‘twisted’ estimate in which we apply $iK$ to the equation, pair with $u$,
and take ${\rm Re}$.

######  Lemma C.2 (Steady case).

Let $(u,f)$ be a solution to (C.5), and assume that the Kawashima condition is
satisfied. Then

$\|u\|\lesssim\max(1,|\tau|^{-2})\|f\|\,.$ (C.6)

In particular, $i\tau A+L$ is invertible.

###### Proof.

The standard energy estimate yields

$\|P_{Y}u\|^{2}\lesssim\|P_{Y}f\|^{2}+\|P_{X}f\|\|P_{X}u\|\,.$ (C.7)

The ‘twisted estimate’ yields the identity

$-\tau{\rm Re}\langle KAu,u\rangle+{\rm Re}\langle iKLu,u\rangle=-{\rm
Re}\langle iKf,u\rangle\,,$ (C.8)

which gives

$|\tau|\|P_{X}u\|^{2}\overset{\eqref{eq:Kawashimaconditionestimate}}{\lesssim}\|P_{Y}u\|\|u\|+\|f\|\|u\|+|\tau|\|P_{Y}u\|^{2}\,.$
(C.9)

If $|\tau|\geqslant 1$, we sum (C.7) and $|\tau|^{-1}\times$ (C.9):

$\|u\|^{2}_{L^{2}}\lesssim\|P_{Y}f\|^{2}+\|P_{X}f\|\|P_{X}u\|+|\tau|^{-1}\|P_{Y}u\|\|P_{X}u\|+|\tau|^{-1}\|f\|\|u\|\,,$
(C.10)

after using $\|P_{Y}u\|\|u\|\lesssim\|P_{Y}u\|^{2}+\|P_{Y}u\|\|P_{X}u\|$ and
the estimate (C.7). We then split products in (C.9) using Young’s inequality,
and absorb small $\|Pu\|^{2}_{L^{2}}$ terms into the left-hand side to close
the estimate:

$\|Pu\|^{2}_{L^{2}}\lesssim\|f\|^{2}$

Therefore, we focus on $0<|\tau|\leqslant 1$. We have

$\|P_{Y}u\|^{2}+\tau^{2}\|P_{X}u\|^{2}\lesssim\|P_{Y}f\|^{2}+\|P_{X}f\|\|P_{X}u\|+|\tau|\|P_{Y}u\|\|P_{X}u\|+|\tau|\|f\|\|u\|\,,$

which, after we split products and absorb small terms, begets

$\|P_{Y}u\|^{2}+\tau^{2}\|P_{X}u\|^{2}\lesssim\|P_{Y}f\|^{2}+\tau^{-2}\|P_{X}f\|^{2}\,.$
(C.11)

∎

######  Remark C.3.

Notice that we actually have the more precise estimate (C.11). In particular,
when $P_{X}f=0$, we have the improvement

$\|u\|\lesssim\max(1,|\tau|^{-1})\|f\|\,,$

which is akin to the situation in the steady Landau estimates in Lemma 4.4
(the right-hand side is purely macroscopic, that is, belongs to the damped
subspace).

We may similarly use the Kawashima compensator to recover coercivity in the
time-dependent problem (C.4). In fact, one may choose
$0<\delta\ll(1+|\tau|^{-1})^{-1}$ such that

$\Phi[u]:=\frac{1}{2}\|u\|^{2}+\frac{\delta}{2}({\rm sgn}\,\tau){\rm
Re}\langle iKu,u\rangle$ (C.12)

is a hypocoercivity functional.

######  Lemma C.4 (Time-dependent case).

Let $u$ be a solution to (C.4), and assume that the Kawashima condition is
satisfied. Then

$\|u(t)\|\lesssim e^{-\mu\min(1,|\tau|^{2})t}\|u_{0}\|\,,$

where $\mu>0$ is independent of $u_{0}$.

###### Proof.

The standard energy estimate yields

$\frac{d}{dt}\frac{1}{2}\|u\|^{2}\leqslant-c_{1}\|P_{Y}u\|^{2}\,.$ (C.13)

Next, we compute

$\frac{d}{dt}\frac{1}{2}\langle iKu,u\rangle=\frac{1}{2}\langle iK(i\tau
A+L)u,u\rangle+\frac{1}{2}\langle iKu,(i\tau A+L)u\rangle\,.$

The right-hand side is

$\displaystyle-\underbrace{\frac{1}{2}\tau\langle(KA-AK)u,u\rangle}_{\tau{\rm
Re}\langle KAu,u\rangle}+\frac{1}{2}\langle iKLu,u\rangle+\frac{1}{2}\langle
iKu,Lu\rangle$

Therefore, we obtain

$\displaystyle\frac{d}{dt}\frac{1}{2}({\rm sgn}\,\tau){\rm Re}\langle
iKu,u\rangle$ $\displaystyle\leqslant-|\tau|{\rm Re}\langle
KAu,u\rangle+C_{1}\|P_{Y}u\|\left(\|P_{X}u\|+\|P_{Y}u\|\right)$ (C.14)
$\displaystyle\leqslant-
c_{2}|\tau|\|P_{X}u\|^{2}+C_{2}(1+|\tau|^{-1})\|P_{Y}u\|^{2}\,.$

Finally, we define $\Phi[u]$ as in (C.12) and compute its time derivative
(equivalently, add (C.13) and
$\delta\times~{}\eqref{eq:twistedtimedepwithtau}$). We have

$\frac{d}{dt}\Phi[u]\leqslant-c_{3}\Phi[u]\,,$

provided that $\delta\leqslant\nicefrac{{1}}{{2}}\|iK\|_{\rm HS}$ (Hilbert-
Schmidt norm) to ensure that $\Phi[u]\approx\nicefrac{{1}}{{2}}\|u\|^{2}$ and
$\delta\leqslant c_{1}(2C_{2})^{-1}(1+|\tau|^{-1})^{-1}$ to ensure that the
last term on the right-hand side of
$\delta\times\eqref{eq:twistedtimedepwithtau}$ can be absorbed into the
$-c_{1}\|P_{Y}u\|^{2}$ term on the right-hand side of (C.13). In particular,
$c_{3}\gtrsim\min(\delta c_{2}|\tau|,c_{1})$. When $0<|\tau|\leqslant 1$,
$c_{3}\gtrsim|\tau|^{2}$. ∎

###### Proof of Lemma C.1.

To motivate the construction, we begin with a special case in which $Ax\not\in
X$ for all nonzero $x\in X$.

_Special case_. Consider the operator $P_{Y}A|_{X}$. Our assumption yields
that $P_{Y}A|_{X}$ is injective; indeed if $P_{Y}A(x_{1})=P_{Y}A(x_{2})$ for
two nonzero vectors $x_{1}\neq x_{2}$, then $A(x_{1}-x_{2})\in X$, a
contradiction. Therefore we have that $P_{Y}A|_{X}$ is a bijection between $X$
and its image $Z:={\rm im}(P_{Y}A|_{X})$. We recall the decomposition
$\mathbb{C}^{m}=X\oplus Y$ and rewrite $A$ as a block matrix

$A=\begin{bmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{bmatrix}\,,$

where $A_{11}=P_{X}A|_{X}\equiv 0$ by our assumption, $A_{21}=P_{Y}A|_{X}$,
etc. Notably, by our earlier claim that $A_{21}:X\to Z$ is invertible, and the
fact that $A_{21}=A_{12}^{*}$ since $A$ is Hermitian and the projections are
orthogonal, we have that

$A_{12}A_{21}\geqslant\text{const.}\times{\rm Id}_{X\to X}$ (C.15)

is Hermitian positive definite. Using the same block decomposition as above,
we define the Kawashima compensator

$K=\begin{bmatrix}0&A_{12}\\\ -A_{21}&0\end{bmatrix}\,.$

The idea is that $K$ acts as a rotation between $X$ and $Y$ which interchanges
the undamped region $X$ with the damped region $Y$. Recalling that
$A_{11}\equiv 0$, the commutator $[K,A]$ is explicitly computed as

$KA-AK=\begin{bmatrix}2A_{12}A_{21}&A_{12}A_{22}\\\
-A_{22}A_{21}&-2A_{21}A_{12}\end{bmatrix}\,.$

Then

$\displaystyle{\rm Re}\langle KAu,u\rangle$
$\displaystyle=\frac{1}{2}\langle[K,A]u,u\rangle$
$\displaystyle\geqslant\langle A_{12}A_{21}u,u\rangle-|\langle
A_{22}u,A_{21}u\rangle|-|\langle A_{21}A_{12}u,u\rangle|\,.$

Finally, we split the cross terms using Young’s inequality with $\varepsilon$
and combine the above estimate with (C.15) to obtain (C.3).

_General case_. We define the subspaces

$F_{k}:=\\{x\in\mathbb{C}^{m}:x,Ax,\ldots,A^{k}x\in X\\}\,,\quad k\geqslant
0\,.$

With the convention $F_{-1}=\mathbb{C}^{n}$, they satisfy the nesting property

$\cdots\subseteq F_{n}\subseteq\cdots\subseteq F_{1}\subseteq
F_{0}=X\subseteq\mathbb{C}^{n}=F_{-1}\,.$

We note that for $k\geqslant 0$, we can characterize $F_{k}$ as the preimage
$A|_{X}^{-1}(F_{k-1})$. Furthermore, if $1\leqslant k_{0}\leqslant n$ is the
least integer such that Condition (C.1) is satisfied for all $x\in X$, then
$F_{k}=\\{0\\}$ for all $k\geqslant k_{0}$.

For $k\geqslant 0$, we define $E_{k}$ to be the orthogonal complement of
$F_{k}$ in $F_{k-1}$, so that

$E_{k}\oplus F_{k}=F_{k-1}\,.$ (C.16)

Using the convention that $E_{0}=Y$, we have as a consequence the
decomposition

$E_{n}\oplus\cdots\oplus E_{1}=X\,,\qquad E_{n}\oplus\cdots\oplus E_{1}\oplus
E_{0}=X\oplus Y=\mathbb{C}^{m}\,.$

In addition, $E_{k_{0}}=F_{k_{0}-1}$, and $E_{k}=F_{k}=\\{0\\}$ for $k>k_{0}$.
We now crucially observe that whenever $k\leqslant k_{0}$ and $x\in
E_{k}\subseteq F_{k-1}$, we have $x,Ax,\ldots,A^{k-1}x\in X$ but
$A^{k}x\not\in X$. More precisely, while $Ax\in F_{k-1}$ for each $x\in
F_{k}$, we have that $Ax\notin F_{k-1}$ for each $x\in E_{k}$, although $Ax\in
F_{k-2}$. As a consequence of these observations, we have that

${\rm ker}\,P_{E_{k-1}}A|_{E_{k}}=\\{0\\}\,.$

Indeed if $x_{1},x_{2}\in E_{k}$ were such that
$P_{E_{k-1}}Ax_{1}=P_{E_{k-1}}Ax_{1}$, then $P_{E_{k-1}}A(x_{1}-x_{2})=0\in
F_{k-1}$, contradicting the above claim that $Ax\notin F_{k-1}$ for $x\in
E_{k}$. We have now extracted the required coercivity on the subspace $E_{k}$.
The special case corresponds to $k_{0}=1$ and $X=E_{1}$.

Recalling that $E_{k-1}\subseteq F_{k-1}^{\perp}$, the desired compensator at
level $1\leqslant k\leqslant k_{0}$ is

$K_{k}=P_{F^{\perp}_{k-1}}AP_{E_{k}}-P_{E_{k}}AP_{F_{k-1}^{\perp}}\,.$

For convenience we write $K_{k}$ as a block matrix in the subspaces
$F_{k},E_{k},F_{k-1}^{\perp}$, which from (C.16) form an orthogonal
decomposition of $\mathbb{C}^{m}$. Set $A_{11}=P|_{F_{k}}AP_{F_{k}}$,
$A_{12}=P|_{F_{k}}AP_{E_{k}}$, etc., and recall that
$P_{F_{k-1}^{\perp}}AP_{F_{k}}\equiv 0$. Then $A$ and $K_{k}$ may be written
as

$A=\begin{bmatrix}A_{11}&A_{12}&0\\\ A_{21}&A_{22}&A_{23}\\\
0&A_{32}&A_{33}\end{bmatrix}\,,\qquad K_{k}=\begin{bmatrix}0&0&0\\\
0&0&A_{23}\\\ 0&-A_{32}&0\end{bmatrix}\,.$

Computing the commutator leads to

$K_{k}A-AK_{k}=\begin{bmatrix}0&0&-A_{12}A_{23}\\\
0&2A_{23}A_{32}&A_{23}A_{33}-A_{22}A_{23}\\\
-A_{32}A_{21}&-A_{32}A_{22}+A_{33}A_{32}&-2A_{32}A_{23}\end{bmatrix}\,.$

Note crucially that $A_{23}A_{32}$ is coercive on $E_{k}$, although we will
still need to control the contributions from the rest of the commutator. We
compute $\langle[K_{k},A]u,u\rangle$ and repeatedly use Cauchy-Schwarz with
$\varepsilon$; specifically, we choose $\varepsilon_{k}\ll 1$ and
$C_{k}\geqslant\varepsilon_{k}^{-1}$ and use
$\varepsilon_{k}\|P_{F_{k}}u\|^{2}$ to help bound $\langle
A_{32}A_{21}u,u\rangle$ and $\langle A_{12}A_{23}u,u\rangle$, and
$C_{k}\|P_{F_{k-1}^{\perp}}u\|^{2}$ to help bound
$\langle(A_{23}A_{33}-A_{22}A_{23})u,u\rangle$,
$\langle(A_{32}A_{22}-A_{33}A_{32})u,u\rangle$, and $\langle
A_{32}A_{23}u,u\rangle$. As a consequence, we find that for
$\delta_{k},\varepsilon_{k}$ sufficiently small and $C_{k}$ sufficiently
large,

$\displaystyle\delta_{k}\left\|P_{E_{k}}\right\|^{2}\leqslant{\rm
Re}\left\langle
K_{k}Au,u\right\rangle+C_{k}\left\|P_{F_{k-1}^{\perp}}u\right\|^{2}+\varepsilon_{k}\left\|P_{F_{k}}u\right\|^{2}\,.$
(C.17)

Finally, the full Kawashima compensator is defined by

$K=\sum_{i=1}^{k_{0}}\gamma_{i}K_{i}\,,$

where $\gamma_{k}$, $C_{k}$, $\varepsilon_{k}$, and $\delta_{k}$ are chosen
via backwards induction as follows. Beginning at level $k_{0}$ and recalling
that $F_{k_{0}}=\\{0\\}$ and $F_{k_{0}-1}=E_{k_{0}}$, we set
$\gamma_{k_{0}}=1$ so that

$\displaystyle\delta_{k_{0}}\left\|P_{E_{k_{0}}}u\right\|^{2}$
$\displaystyle\leqslant{\rm Re}\left\langle
K_{k_{0}}Au,u\right\rangle+C_{k_{0}}\left\|P_{E_{k_{0}}^{\perp}}u\right\|^{2}$
$\displaystyle={\rm Re}\left\langle
K_{k_{0}}Au,u\right\rangle+C_{k_{0}}\left\|P_{E_{k_{0}-1}}u\right\|^{2}+C_{k_{0}}\left\|P_{(E_{k_{0}}\oplus
E_{k_{0}-1})^{\perp}}u\right\|^{2}\,.$ (C.18)

Now we may choose $\gamma_{k_{0}-1}$ large enough and $\varepsilon_{k_{0}-1}$
small enough so that we can absorb the term with $P_{E_{k_{0}-1}}u$ above and
the term with $P_{F_{k_{0}}-1}u=P_{E_{k_{0}}}u$ from (C.17) with $k=k_{0}-1$,
obtaining the inequality

$\displaystyle\frac{\delta_{k_{0}}}{2}\left\|P_{E_{k_{0}}}u\right\|^{2}+\delta_{k_{0}-1}\left\|P_{E_{k_{0}-1}}u\right\|^{2}$
$\displaystyle\leqslant{\rm
Re}\left\langle(K_{k_{0}}+\gamma_{k_{0}-1}K_{k_{0}-1})Au,u\right\rangle$
$\displaystyle\qquad+C_{k_{0}}\left\|P_{(E_{k_{0}}\oplus
E_{k_{0}-1})^{\perp}}u\right\|^{2}+C_{k_{0}-1}\left\|P_{F_{k_{0}-2}^{\perp}}u\right\|^{2}\,.$

Now recalling that $F_{k_{0}-2}=E_{k_{0}-1}\oplus
F_{k_{0}-1}=E_{k_{0}-1}\oplus E_{k_{0}}$ and re-labeling
$\tilde{E}_{k_{0}-1}=E_{k_{0}}\oplus E_{k_{0}-1}$,
$\tilde{\delta}_{k_{0}-1}=\min(\delta_{k_{0}}/2,\delta_{k_{0}-1})$,
$\tilde{K}_{k_{0}-1}=K_{k_{0}}+\gamma_{k_{0}-1}K_{k_{0}-1}$, and
$\tilde{C}_{k_{0}-1}=2\max(C_{k_{0}},C_{k_{0}-1})$, we have that

$\displaystyle\tilde{\delta}_{k_{0}-1}\left\|P_{\tilde{E}_{k_{0}-1}}u\right\|^{2}$
$\displaystyle\leqslant{\rm
Re}\left\langle\tilde{K}_{k_{0}-1}Au,u\right\rangle+\tilde{C}_{k_{0}-1}\left\|P_{\tilde{E}_{k_{0}-1}^{\perp}}u\right\|^{2}\,.$

But this inequality is identical in character to (C.18), and so it is clear
that we may continue this process inductively. We then effectively reduce to
the special case and obtain that

$\delta\left\|P_{X}u\right\|^{2}\leqslant{\rm Re}\left\langle
KAu,u\right\rangle+C\left\|P_{Y}u\right\|^{2}\,,$

and thus concluding the proof of Lemma C.1. ∎

######  Remark C.5.

Alternatively, one can define the compensator at level $k$ as follows. Let
$k>1$ and consider the operator

$\tilde{K}_{k}=P_{Y}A^{k}P_{E_{k}}-P_{E_{k}}A^{k}P_{Y}\,.$

If one were allowed to take the commutator of $\tilde{K}_{k}$ with $A^{k}$,
then the construction of $K_{k}$ would be identical to the construction of
$K_{1}$. However, we may define
$K_{k}=\tilde{K}_{k}A^{k-1}+A\tilde{K}_{k}A^{k-2}+\dots+A^{k-2}\tilde{K}_{k}A+A^{k-1}\tilde{K}_{k}$,
so that by direct computation,

$\langle\tilde{K}_{k},A^{k}\rangle=\langle K_{k},A\rangle\,.$

This construction of $K_{k}$ demonstrates the direct exchange of coercivity
between $E_{k}$ and $Y$, rather than the indirect exchange between $E_{k}$ and
$Y$ via $E_{k-1},\dots,E_{1}$.

######  Remark C.6.

In [BZ], Beauchard and Zuazua gave a similar construction and moreover provide
several equivalent characterizations of the Kawashima condition; for example,
it is equivalent to the Kalman rank condition familiar from control theory.

###  C.2 Example: Oscillators with partial damping

Consider the damped harmonic oscillator

$\ddot{x}+\kappa\dot{x}+x=0\,,$

where the friction coefficient $\kappa=1$ for convenience of exposition. We
rewrite the second-order ODE as a first order system

$\begin{aligned} \dot{x}&=y\\\ \dot{y}&=-x-y\end{aligned}\,.$

With $u=(x,y)$, the ODE can be rewritten in the form (C.4) with

$A=\begin{bmatrix}0&-i\\\ i&0\end{bmatrix}\,,\quad L=\begin{bmatrix}0&0\\\
0&-1\end{bmatrix}\,.$

The Kawashima compensator is

$K=\begin{bmatrix}0&-i\\\ -i&0\end{bmatrix}\,,$

and the hypocoercivity functional is

$\Phi=\frac{1}{2}|x|^{2}+\frac{1}{2}|y|^{2}-\delta xy\,,$

where $0<\delta\leqslant 1/2$. One can analogously consider, for $n\geqslant
3$, oscillator chains of the type

$\dot{x}_{1}=x_{2}\,,\qquad\dot{x}_{k}=-x_{k-1}+x_{k}\,,\,2\leqslant
k\leqslant n-1\,,\qquad\dot{x}_{n}=-x_{n-1}-x_{n}\,.$

###  C.3 Example: Linearized Landau equation

For $\kappa\in[0,1]$, consider the linearized (and regularized, when
$\kappa\in(0,1]$) Landau equation

$\partial_{t}f+v\cdot\nabla_{x}f=L^{\kappa}f\,,$ (C.19)

where $L^{\kappa}=\frac{2}{\sqrt{\mu}}Q_{\kappa}(\mu,\sqrt{\mu}f)$,
$Q_{\kappa}$ is the collision operator defined in (2.44), and
$\mu=e^{-\frac{|v|^{2}}{2}}$. It is well known that $L$ and $L_{\rm R}$, and
thus $L^{\kappa}$, with appropriate densely-defined domain $D(L)\subseteq
L^{2}$ are closed, self-adjoint, and positive semi-definite. We furthermore
showed in subsection 3.2 that the kernel is precisely

$X:=\ker L^{\kappa}={\rm span}(\sqrt{\mu},v\sqrt{\mu},|v|^{2}\sqrt{\mu})\,.$

Then taking the union of $X$ with its image under multiplication by
$v_{1},v_{2},v_{3}$, we obtain the subspace consisting of thirteen moments,

$M_{13}={\rm
span}(\sqrt{\mu},v_{i}\sqrt{\mu},v_{i}v_{j}\sqrt{\mu},|v|^{2}v_{i}\sqrt{\mu})\,,$

where $1\leqslant i,j\leqslant 3$. The equation (C.19) does not preserve the
subspace $M_{13}$, so we cannot reduce the problem to the ODE system (C.4).
Nonetheless, the Kawashima approach will be effective. It will be convenient
to consider the Fourier transform in the $x$-direction and, without loss of
generality, the Fourier variable $\vec{k}=k\vec{e}_{1}$. With this reduction,
we will consider only nine moments,

$M_{9}={\rm
span}(\sqrt{\mu},v_{i}\sqrt{\mu},v_{1}v_{i}\sqrt{\mu},|v|^{2}\sqrt{\mu},|v|^{2}v_{1}\sqrt{\mu})\,,$

where $1\leqslant i\leqslant 3$. Let $Y$ be the orthogonal complement of $X$
in $M_{9}$. Our goal is to understand the (projected) multiplication
operator101010Note that the operator $v_{1}P_{M_{9}}$ is well-defined with
domain $M_{9}$ and range equal to the image of $M_{9}$ under multiplication by
$v_{1}$, which can be understood as a subset of a Hilbert space with Gaussian
weights, defined as in (2.26). We are therefore justified in considering
$P_{M_{9}}v_{1}P_{M_{9}}$ as an operator from $M_{9}$ to itself. We also
apologize for the abuse of notation; this operator $A$ is not the same as the
$A$ defined in subsection 3.2.

$A=P_{M_{9}}v_{1}P_{M_{9}}:M_{9}\rightarrow M_{9}\,.$

If $A$ satisfies the Kawashima condition, then we can construct an anti-
symmetric operator $K:M_{9}\rightarrow M_{9}$ such that

$\|P_{X}f\|_{L^{2}}^{2}\lesssim{\rm Re}\langle
KAf,f\rangle+\|P_{Y}f\|_{L^{2}}^{2}\,.$

Translated into the notation of sections 2–5, this implies that

$\|Pg\|_{L^{2}}^{2}\lesssim\langle[KP_{\leqslant 9},v_{1}]P_{\leqslant
9}g,P_{\leqslant 9}g\rangle+\|\langle v\rangle^{-100}P_{\leqslant
9}P^{\perp}g\|_{L^{2}}^{2}\,,$ (C.20)

where the $\langle v\rangle^{-100}$ is allowed due to the fact that all
elements of $M_{9}$ have Gaussian decay and so can afford any polynomial
weight. The computation (actually a more detailed version than is required in
our setting) is already contained in the book of Glassey [Glassey, pg. 75]. We
note that by combining this estimate with the standard $L^{2}$ energy
estimate, one can obtain time-asymptotic linear stability.

######  Lemma C.7.

$A$ satisfies the Kawashima condition.

###### Proof.

Since every element of $M_{9}$ contains a factor of
$\mu^{\nicefrac{{1}}{{2}}}$, we can simply focus on the prefactors, which are
polynomials in $\\{v_{1},v_{2},v_{3}\\}$ of degree at most $2$. We claim that
given any non-zero polynomial $p(v)=a+b_{i}v_{i}+c|v|^{2}\in{\rm
span}\\{1,v,|v|^{2}\\}$, either $v_{1}p(v)\notin{\rm span}\\{1,v,|v|^{2}\\}$,
or $v_{1}^{2}p(v)\notin{\rm span}\\{1,v,|v|^{2}\\}$. If $c\neq 0$, then
$v_{1}p(v)$ is degree 3 and we are done. If $c=0$ and there exists
$b_{i_{0}}\neq 0$, then $v_{1}b_{i}v_{i}$ is a homogeneous quadratic
polynomial which is however not in the span of $|v|^{2}$, and we are done.
Finally, if $c=b_{i}=0$, then $a\neq 0$, and $av_{1}^{2}$ is a homogeneous
quadratic polynomial which is again not in the span of $|v|^{2}$. ∎

######  Remark C.8.

The micro-macro energy method introduced by Guo [guo2006boltzmann] (see also
[duan2009stability, duan2011hypocoercivity, strain2012optimal]) is an explicit
example of a Kawashima compensator for the linearized Landau equations.
Indeed, if we define the higher order moments
$\Theta[g]=(\Theta_{ij}[g])_{d\times d}$ and
$\Lambda[g]=(\Lambda_{1}[g],\dots,\Lambda_{d}[g])$ by

$\displaystyle\Theta_{ij}[g]=\left\langle(v_{i}v_{j}-1)\sqrt{\mu},g\right\rangle_{L^{2}_{v}},$
$\displaystyle\Lambda_{i}[g]=\left\langle(|v|^{2}-(d+2))v_{i}\sqrt{\mu},g\right\rangle_{L^{2}_{v}}\,,$

the micro-macro energy used in [guo2006boltzmann] is written as the following,
for each frequency in $x$:

$\displaystyle\mathcal{M}:=\,$
$\displaystyle\Re(\Lambda[(I-P_{X})\widehat{f}_{k}]\cdot(ik\widehat{E}_{k})+b_{1}\Re\big{(}\big{(}\Theta[(I-P_{X})\widehat{f}_{k}]+(2\widehat{E}_{k}I)\big{)}:(ik\widehat{m}_{k}+(ik\widehat{m}_{k})^{T})\big{)}$
$\displaystyle+b_{2}\Re\big{(}\widehat{m}_{k}\cdot(ik\widehat{\rho}_{k})\big{)},$

where $0<b_{2}\ll b_{1}\ll 1$ are small universal constants. However, notice
that this energy can be written in terms of an anti-symmetric operator
$K:M_{13}\to M_{13}$:

$\displaystyle\mathcal{M}=\left\langle P_{\leqslant 13}f,KP_{\leqslant
13}f\right\rangle.$

######  Remark C.9.

Heuristically, the continuity equation for the density $\varrho$ is
$\partial_{t}\varrho+\operatorname{\mathrm{div}}_{x}(\varrho u)=0$, which is
undamped. However, the density exchanges energy with the momentum, which has
its own damping. We verify the Kawashima condition for the linearized
compressible Navier-Stokes equations in its own section, since it requires the
notion of symmetrizability.

######  Remark C.10.

The Kawashima compensator strategy also gives upper bounds on Taylor
dispersion of a passive scalar in a shear flow. Upon applying the Fourier
transform in the longitudinal variable, $X$ is the subspace corresponding to
the mean in the cross-sectional variable.

##  Appendix D Proof of Lemmas 3.3 and 3.5

###### Proof of Lemma 3.3, estimate (3.7).

We first prove (3.7). We begin by recalling (3.3) and writing that

$\displaystyle-\left\langle\langle v\rangle^{2\theta}L^{\rm
R}g,g\right\rangle$ $\displaystyle=\left\langle\langle
v\rangle^{2\theta}A_{\rm R}g,g\right\rangle+\left\langle\langle
v\rangle^{2\theta}K_{\rm R}g,g\right\rangle$
$\displaystyle=\left\langle\langle v\rangle^{2\theta}\partial_{i}[\sigma_{\rm
R}^{ij}\partial_{j}g],g\right\rangle-\left\langle\langle
v\rangle^{2\theta}\sigma_{\rm
R}^{ij}v_{i}v_{j}g,g\right\rangle+\left\langle\langle
v\rangle^{2\theta}\partial_{i}\sigma_{\rm
R}^{i}g,g\right\rangle+\left\langle\langle v\rangle^{2\theta}K_{\rm
R}g,g\right\rangle$ $\displaystyle=-|g|_{\sigma_{\rm
R},\theta}^{2}-\left\langle\partial_{i}\langle
v\rangle^{2\theta}\sigma^{ij}_{\rm
R}\partial_{j}g,g\right\rangle+\left\langle\langle
v\rangle^{2\theta}\partial_{i}\sigma_{\rm
R}^{i}g,g\right\rangle+\left\langle\langle v\rangle^{2\theta}K_{\rm
R}g,g\right\rangle\,.$

From [G02, Lemma 5], we have that for any $m>1$, there exists $0<C(m)<\infty$
such that

$\displaystyle\left|\left\langle\langle v\rangle^{2\theta}K_{\rm
R}g,g\right\rangle\right|\leqslant\frac{C}{m}|g|_{\sigma_{\rm
R},\theta}^{2}+C(m)\int_{|v|\leqslant C(m)}\left|\langle
v\rangle^{\theta}g\right|^{2}\,\mathrm{d}v\leqslant\frac{C}{m}+C(m,\theta)|g|_{\sigma_{\rm
R},\theta-\nicefrac{{1}}{{2}}}$

Choosing $m$ large enough achieves a bound for $K_{\rm R}$ commensurate with
that of (3.7), and so we focus on the remaining error terms from $A_{\rm R}$.
From (3.1) and [G02, Lemma 3], we have that

$\left|\partial_{i}{\rm w}_{\rm R}^{2\theta}\sigma_{\rm
R}^{ij}\right|+\left|{\rm w}_{\rm