the growth of $s_{\alpha}$ we adapt an argument given in [10, p.139] and we
set

(8.3)
$Y_{\alpha}(t):=\sum_{\beta\in\mathcal{A}(\alpha)}|\vartheta_{\beta}|+\sum_{\beta\in\mathcal{I}}|\vartheta_{\beta}|+|\xi_{k}|,$

where $\mathcal{A}(\alpha)$ is the set of waves approaching $\alpha$ at the
time $t$ and $\mathcal{I}$ is the set of waves of the families $1,\dots,k-1$.
The reason why we need to include the term with $\mathcal{I}$ is to handle
Case 3 below when the hitting wave is not characteristic. Finally, we set

(8.4) $I(t):=K_{1}Q(t)+K_{1}R(t)+K_{2}S(t)+K_{3}Z(t),\qquad
L_{\alpha}(t):=|s_{\alpha}(t)|\exp\big{(}K_{5}[Y_{\alpha}(t)+I(t)]\big{)},$

where $K_{5}$ is a suitable constant satisfying

(8.5) $K_{5}\geq\max\\{1,4C_{5}\\},\qquad K_{5}\delta\leq\frac{1}{2}.$

Assume for a moment that we have shown that the function $L$ is monotone non-
increasing, then

$\begin{split}|s_{\alpha}(t)|&\leq|s_{\alpha}(t)|\exp\big{(}K_{5}[Y_{\alpha}(t)+I(t)]\big{)}\leq|s_{\alpha}(\tau_{\alpha})|\exp\big{(}K_{5}[Y_{\alpha}(\tau_{\alpha})+I(\tau_{\alpha})]\big{)}\\\
&\stackrel{{\scriptstyle\eqref{e:assumption}}}{{\leq}}r_{\varepsilon}\underbrace{\exp\big{(}K_{5}\delta\big{)}}_{:=C_{9}}\quad\text{for
every $t\in]\tau_{\alpha},T_{\alpha}[$}\end{split}$

that is (8.2). To show that the function $L$ is monotone non-increasing we
assume that a given time $\tau$ an interaction or a discontinuity in the
boundary datum occurs and by using an elementary computation based on the
convexity of the exponential function we get

(8.6) $\begin{split}\Delta
L(\tau)&\leq\exp\big{(}K_{5}[Y_{\alpha}(\tau^{+})+I(\tau^{+})]\big{)}\Big{[}\Delta|s_{\alpha}|(\tau)+K_{5}|s_{\alpha}(\tau^{-})|[\Delta
Y_{\alpha}(\tau)+\Delta I(\tau)]\Big{]}\end{split}$

We now prove $\Delta L(\tau)\leq 0$ by separately considering the following
cases.
Case 1: the interaction occurs inside the domain and does not involve
$\alpha$, but rather two wave fronts with sizes $s_{\beta}$ and $s_{\gamma}$.
We have

$\Delta|s_{\alpha}|(\tau)=0.$

Since $\xi_{k}(\tau^{+})=\xi_{k}(\tau^{-})$ then by (7.38) and (7.39) we get
$\Delta Y_{\alpha}(\tau)\leq AC_{5}|s_{\beta}||s_{\gamma}|$ and by following
the analysis in §7.5 and in particular (7.49) we conclude

$\Delta Y_{\alpha}(\tau)+\Delta I(t)\leq 0$

and by combining the above inequalities with (8.6) we conclude that $\Delta
L(\tau)\leq 0$.
Case 2: the interaction occurs inside the domain and involves $\alpha$, which
interacts with a physical front $\beta$ with strength $s_{\beta}$. If $\alpha$
and $\beta$ belong to the same family, $i_{\alpha}=i_{\beta}$, then $\beta$
must be a shock front (because rarefaction wave fronts of the same family do
not cross each other) and then $\Delta|s_{\alpha}|(\tau)\leq 0$ owing to
cancellation. If $i_{\beta}\neq i_{\alpha}$ then owing to (7.38) we have
$\Delta|s_{\alpha}|(\tau)\leq C_{5}|s_{\alpha}s_{\beta}|$. Also,

$\Delta
Y_{\alpha}(\tau)\stackrel{{\scriptstyle\eqref{e:iebressan}}}{{\leq}}-|s_{\beta}|+AC_{5}|s_{\alpha}||s_{\beta}|+\stackrel{{\scriptstyle
AC_{5}\delta\leq 8^{-1}}}{{\leq}}-\frac{1}{4}|s_{\beta}|$

By following again the analysis in §7.5.2 and in particular (7.49) we get
$\Delta I(\tau)\leq 0$. Note that the same estimates hold if the front that
interacts with $\alpha$ is a non-physical one (in this case we actually have
the stronger estimate $\Delta s_{\alpha}(\tau)=0$). Either case, by combining
the above inequalities with (8.5) and (8.6) we conclude that $\Delta
L(\tau)\leq 0.$
Case 3: the interaction occurs at the domain boundary and does not involve
$\alpha$; in this case we have $\Delta|s_{\alpha}|(\tau)=0$ and, owing to the
analysis in §7.2,§7.3 and in particular to (7.17),(7.26),(7.19), we have
$\Delta Y_{\alpha}(\tau)+\Delta I(\tau)\leq 0$ and owing to (8.6) this yields
$\Delta L(\tau)\leq 0$.
Case 4: the interaction occurs at the domain boundary and involves $\alpha$;
we separately consider the following cases.
Case 4A: $\alpha$ belongs to the family $j_{\alpha}=k$. If there is no wave
front of the $k$-th family entering the domain after the interaction, or if
the wave front of the $k$-th family is a a shock, then the rarefaction wave
front ceases to exist at time $\tau$ and there is nothing to prove. If there
is a rarefaction wave front entering the domain after the interaction then we
are in case ii) or iii) in §5.3. Also, we can apply Lemma 6.8 and arrive at

(8.7)
$\Delta|s_{\alpha}|(\tau)\stackrel{{\scriptstyle\eqref{e:rarepiccola}}}{{\leq}}|s_{\alpha}|(\tau^{-})\big{[}-1+C_{8}([\varsigma_{k}]^{-}+|\xi_{k}|)\big{]}\stackrel{{\scriptstyle\eqref{e:sole}}}{{\leq}}-\frac{1}{2}|s_{\alpha}|(\tau^{-}),\quad\quad\Delta
I(\tau)\stackrel{{\scriptstyle\eqref{e:c:dupsilon}}}{{\leq}}0.$

Also, $\Delta
Y_{\alpha}(t)\leq|\xi^{\prime}_{k}|-|\xi_{k}|\leq|\xi^{\prime}_{k}|\leq\delta$
and by using (8.5) and (8.6) this yields $\Delta L(\tau)\leq 0.$ Note that the
previous considerations apply in both the case of the accurate Riemann solver
and the simplified Riemann solver.
Case 4B: $\alpha$ belongs to the family $i_{\alpha}<k$. In this case the wave
front $\alpha$ ceases to exist at time $\tau$ and there is nothing to prove.
Case 5: the boundary datum $\mathbf{u}_{b}^{\varepsilon}$ is discontinuous at
$\tau$. We have $\Delta|s_{\alpha}|(\tau)=0$ and by recalling the analysis in
§7.4 and in particular (7.37) we infer $\Delta Y_{\alpha}(\tau)+\Delta
I(\tau)\leq 0$ and owing to (8.6) this yields $\Delta L(\tau)\leq 0$.∎

We recall that $\omega_{\varepsilon}$ represents the threshold to discriminate
between the accurate and the simplified Riemann and boundary Riemann problem,
see 5.5.

######  Lemma 8.3.

For every $\nu>0$, there is $\chi_{\nu}>0$ such that if
$\omega_{\varepsilon}\leq\chi_{\nu}$ then

(8.8) $\sum_{\alpha\in\mathcal{NP}}|s_{\alpha}|\leq\nu,$

where $\mathcal{NP}$ denotes the set of non-physical fronts.

###### Proof.

The proof is based on the same argument as in [10, pp.139-142], so instead of
providing all the details we only focus on the points where the argument is
different. We proceed according to the following steps.
Step 1: we show that

(8.9) $|s_{\alpha}|\leq\mathcal{O}(1)\omega_{\varepsilon},\quad\text{for every
$\alpha\in\mathcal{NP}$}.$

We recall that, by construction, when a non-physical front is created its
strength is bounded by $\mathcal{O}(1)\omega_{\varepsilon}$. To conclude with
(8.9) we can then use exactly the same argument as in [10, p.140].
Step 2: we assign to each wave-front $\alpha$ a _generation order_
$n_{\alpha}$. To this end, we proceed as follows

  * •

all the waves generated at $t=0$ and at discontinuity times of
$\mathbf{u}^{\varepsilon}_{b}$ have generation order $1$;

  * •

when two waves with generation orders $n_{\alpha}$ and $n_{\beta}$ interact
inside the domain, the generation order of the outgoing waves is assigned as
in [10, pp.140];

  * •

if a wave-front $\alpha$ with generation order $n_{\alpha}$ hits the boundary
and $i_{\alpha}=k$, then wave front of the $k$-th family entering the domain
after the interaction (if any) has generation order $n_{\alpha}$, the other
wave fronts entering the domain have generation order $n_{\alpha}+1$;

  * •

if a wave-front $\alpha$ with generation order $n_{\alpha}$ hits the boundary
and $i_{\alpha}<k$, then all wave fronts entering the domain after the
interaction have generation order $n_{\alpha}+1$.

Note that we are heuristically speaking regarding the boundary as a wave front
with generation order $1$.
Step 3: we fix $n\geq 2$ and we set

$\begin{split}&V^{+}_{n}(t):=\sum_{\begin{subarray}{c}n_{\alpha}\geq n\\\
i_{\alpha}\geq k\end{subarray}}|s_{\alpha}|,\quad
V^{-}_{n}(t):=\sum_{\begin{subarray}{c}n_{\alpha}\geq n\\\
i_{\alpha}<k\end{subarray}}|\vartheta_{\alpha}|,\quad
Q_{n}(t):=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\max\\{n_{\alpha},n_{\beta}\\}\geq
n}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!w_{\alpha\beta}|\vartheta_{\alpha}\vartheta_{\beta}|,\quad
R_{n}(t):=|\xi_{k}|\\!\\!\sum_{\begin{subarray}{c}i_{\alpha}\leq k,\\\
n_{\alpha}\geq n\end{subarray}}|\vartheta_{\alpha}|,\\\
&W_{n}(t):=V^{+}_{n+1}(t)+V^{-}_{n}(t),\end{split}$

where we formally count non-physical fronts as belonging to the family $N+1$.
We also introduce the notation

$S_{n}(t):=\left\\{\begin{array}[]{ll}\displaystyle{\sum_{\begin{subarray}{c}i_{\alpha}=k,\\\
n_{\alpha}\geq
n\end{subarray}}|s_{\alpha}|[\varsigma_{k}(\mathbf{u}_{\alpha},s_{\alpha})]^{-}}&\text{if~{}\eqref{e:gnl}}\\\
\phantom{ciao}\\\ \displaystyle{\sum_{\begin{subarray}{c}i_{\alpha}=k,\\\
n_{\alpha}\geq
n\end{subarray}}|s_{\alpha}|\big{[}\lambda_{k}(\mathbf{u}_{\alpha})\big{]}^{-}}&\text{if~{}\eqref{e:lindeg}}\\\
\end{array}\right.$

and

$I_{n}(t):=K_{1}Q_{n}(t)+K_{1}R_{n}(t)+K_{2}S_{n}(t),\qquad\text{for $n\geq
2$}$

and we set

$V_{1}^{+}(t):=|\xi_{k}|+\sum_{i_{\alpha}\geq k}|s_{\alpha}|,\qquad
V^{-}_{1}(t):=\sum_{i_{\alpha}<k}|\vartheta_{\alpha}|,\qquad
W_{1}(t):=V_{2}^{+}(t)+V_{1}^{-}(t),\qquad I_{1}(t):=I(t),$

where $I$ is as in (8.4). We also term

  * •

$J^{0}_{n}$ the set of times at which two wave fronts $\alpha$ and $\beta$
with $\max\\{n_{\alpha},n_{\beta}\\}=n$ interact;

  * •

$J_{n}^{k}$ the set of times at which a wave-front $\alpha$ with
$i_{\alpha}=k$ and $n_{\alpha}=n$ hits the boundary;

  * •

$J^{-}_{n}$ the set of times where a wave front $\alpha$ with $i_{\alpha}<k$
and $n_{\alpha}=n$ hits the boundary.

In the remaining part of the proof we will always assume $n\geq 3$. We have

(8.10) $\begin{array}[]{rl}\Delta W_{n}(t)=0&t\in J^{0}_{1}\cup\dots\cup
J^{0}_{n-2}\cup J^{k}_{1}\cup\dots\cup J^{k}_{n-1}\cup J^{-}_{1}\cup\dots\cup
J^{-}_{n-1};\\\ \Delta W_{n}(t)+\Delta
I_{n-1}(t)\stackrel{{\scriptstyle\S\ref{ss:inside}}}{{\leq}}0&t\in\displaystyle{\bigcup_{m\geq
n-1}J^{0}_{m}};\\\ \Delta W_{n}(t)+\Delta
I_{n-1}(t)\stackrel{{\scriptstyle\S\ref{ss:accuratek}}}{{\leq}}0&t\in\displaystyle{\bigcup_{m\geq
n}J^{k}_{m}};\\\ \Delta W_{n}(t)+\Delta
I_{n-1}(t)\stackrel{{\scriptstyle\S\ref{ss:accuratem}}}{{\leq}}0&t\in\displaystyle{\bigcup_{m\geq
n}J^{-}_{m}.}\end{array}$

If $t\in J^{0}_{1}\cup\dots\cup J^{0}_{n-2}$ then $\Delta R_{n}(t)=\Delta
S_{n}(t)=0$, so $\Delta I_{n}(t)=K_{1}\Delta Q_{n}(t)$, which owing to (7.43),
(7.49) and to the inequality $W_{n}\leq W_{n-1}$ yields

$\Delta I_{n}(t)+K_{1}C_{5}A\Delta\Upsilon W_{n-1}(t^{-})\leq\Delta
I_{n}(t)+K_{1}C_{5}A\Delta\Upsilon W_{n}(t^{-})\leq 0.$

If $t\in J^{k}_{1}\cup\dots\cup J^{k}_{n-1}$ we have $\Delta S_{n}(t)=0$,
whereas $R_{n}$ can change because the strength $\xi_{k}$ can change at $t$.
This yields $\Delta I_{n}(t)=K_{1}\Delta Q_{n}(t)+K_{1}\Delta R_{n}(t)$, which
owing to (7.20) and (7.26) yields

$\Delta I_{n}(t)+K_{1}C_{2}A\Delta\Upsilon W_{n-1}(t^{-})\leq 0.$

If $t\in J^{-}_{1}\cup\dots\cup J^{-}_{n-1}$ we again have $\Delta
I_{n}(t)=K_{1}\Delta Q_{n}(t)+K_{1}\Delta R_{n}(t)$, which owing to (7.11) and
(7.17) yields

$\Delta I_{n}(t)+K_{1}C_{1}A\Delta\Upsilon W_{n-1}(t^{-})\leq 0.$

If $t\in J^{0}_{n-1}$ then owing to (7.38), (7.39) and (7.49) we have

$\displaystyle\Delta Q_{n}(t)+AC_{5}V(t^{-})\Delta I_{n-1}(t)\leq
0,\qquad\Delta R_{n}(t)+AC_{5}|\xi_{k}(t^{-})|\Delta I_{n-1}(t)\leq 0$

We now point out that there are two possible contributions to $\Delta
S_{n}(t)$: i) new $k$-family waves may be created at the interaction; ii)
$k$-family waves with generation order greater or equal than $n$ change their
strength and speed. Owing to (7.38),(7.39),(7.40), (7.42) and (7.49) yields

$\Delta S_{n}(t)+C_{4}C_{5}\delta\Delta I_{n-1}(t)+C_{3}C_{5}\delta\Delta
I_{n-1}(t)\leq 0.$

If $t\in\displaystyle{\bigcup_{m\geq n}J^{0}_{m}}$ then owing to (7.49) we
have $\Delta I_{n}\leq 0$ owing to (7.26). If
$t\in\displaystyle{\bigcup_{m\geq n}J^{k}_{m}}$ then $\Delta I_{n}\leq 0$. If
$t\in\displaystyle{\bigcup_{m\geq n}J^{-}_{m}}$ then $\Delta S_{n}=0$ (because
the new $k$-family waves created at the interaction, if any, have positive
speed) and, owing to (7.11) and (7.16)

$\Delta Q_{n}(t)+\Delta
R_{n}(t)+\frac{C_{1}+C_{12}+C_{12}C_{1}}{A-C_{1}}\Delta W_{n-1}(t)\delta\leq
0.$

Summing up, we have

(8.11) $\begin{array}[]{rl}\Delta
I_{n}(t)+\mathcal{O}(1)\Delta\Upsilon(t)W_{n-1}(t^{-})\leq 0&t\in
J^{0}_{1}\cup\dots\cup J^{0}_{n-2}\cup J^{k}_{1}\cup\dots\cup J^{k}_{n-1}\cup
J^{-}_{1}\cup\dots\cup J^{-}_{n-1}\phantom{\displaystyle{\bigcup}}\\\ \Delta
I_{n}(t)+\mathcal{O}(1)\delta\Delta I_{n-1}(t)\leq 0&t\in
J^{0}_{n-1}\phantom{\displaystyle{\bigcup}}\\\ \Delta I_{n}(t)\leq
0&t\in\displaystyle{\bigcup_{m\geq n}J^{k}_{m}}\\\ \Delta
I_{n}(t)+\mathcal{O}(1)\Delta W_{n-1}(t)V(t^{-})\leq
0&t\in\displaystyle{\bigcup_{m\geq n}J^{-}_{m}}\\\ \end{array}$

Note that (8.10) yields

(8.12) $W_{n}(t^{+})\leq\sum_{0\leq\tau\leq t}[\Delta I_{n-1}(\tau)]_{-},$

where in the previous expression the sum is over the times $\tau$ of
interaction, and that owing to (8.11) we also have

(8.13) $\begin{split}I_{n}(t)&\leq\sum_{0\leq\tau\leq t}[\Delta
I_{n}(\tau)]_{+}\leq\mathcal{O}(1)\sup_{\tau}W_{n-1}(\tau)\underbrace{\sum_{0\leq\tau\leq
t}[\Delta\Upsilon(\tau)]_{-}}_{\leq\Upsilon(0)\leq\delta}+\mathcal{O}(1)\delta\sum_{0\leq\tau\leq
t}[\Delta I_{n-1}(\tau)]_{-}\\\ &+\mathcal{O}(1)\delta\sum_{0\leq\tau\leq
t}[\Delta W_{n-1}(\tau)]_{-}\end{split}$

Note that $W_{n-1}(0)=0$ since $n\geq 3$ and this implies

$\sum_{0\leq\tau\leq t}[\Delta W_{n-1}(\tau)]_{-}=\sum_{0\leq\tau\leq
t}[\Delta
W_{n-1}(\tau)]_{+}-W_{n-1}(t)\stackrel{{\scriptstyle\eqref{stimew}}}{{\leq}}\sum_{0\leq\tau\leq
t}[\Delta I_{n-2}(\tau)]_{-}-W_{n-1}(t)\leq\sum_{0\leq\tau\leq t}[\Delta
I_{n-2}(\tau)]_{-}$

and by plugging the above inequality into (8.13) and recalling (8.12) we
arrive at

$\begin{split}\sum_{0\leq\tau\leq t}[\Delta
I_{n}(\tau)]_{+}\leq\mathcal{O}(1)\delta\sum_{0\leq\tau\leq t}[\Delta
I_{n-1}(\tau)]_{-}+\mathcal{O}(1)\delta\sum_{0\leq\tau\leq t}[\Delta
I_{n-2}(\tau)]_{-}\end{split}$

and since

$\sum_{0\leq\tau\leq t}[\Delta I_{n-1}(\tau)]_{-}\stackrel{{\scriptstyle
I_{n-1}(0)=0}}{{=}}\sum_{0\leq\tau\leq t}[\Delta
I_{n-1}(\tau)]_{+}-I_{n-1}(t)\leq\sum_{0\leq\tau\leq t}[\Delta
I_{n-1}(\tau)]_{+}$

by setting

$\tilde{I}_{n}:=\sum_{0\leq\tau\leq t}[\Delta I_{n}(\tau)]_{+}$

we arrive at

(8.14)
$\tilde{I}_{n}\leq\mathcal{O}(1)\delta\tilde{I}_{n-1}+\mathcal{O}(1)\delta\tilde{I}_{n-2}$

From (8.14) a fairly standard induction argument implies that, if $\delta$ is
sufficiently small then $\tilde{I}_{n}\leq(2\mathcal{O}(1)\delta)^{e_{n}}$,
with $e_{n}=[(n-3)/2]+1$ and $[\cdot]$ denoting the integer part. From this we
can then argue as in [10, p. 141-142], control $I_{n}$ and $W_{n}$ and whence
(by arguing as in [10, (7.77)]) the total strength of non-physical waves. This
concludes the proof of Lemma 8.3. ∎

##  9\. Boundary condition

The goal of this section is to conclude the proof of Theorem 1.2 (and hence of
Theorem 1.1) by establishing the following result.

######  Theorem 9.1.

Under the same assumption as in the statement of Theorem 1.1, let
$\mathbf{v}:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$ be the same as in
(1.8). Then for a.e. $t\in\mathbb{R}_{+}$ the trace
$\mathbf{\bar{v}}(t):=\lim_{x\to+0^{+}}\mathbf{v}(t,x)$ satisfies
$\mathbf{\bar{v}}(t)\sim_{\mathbf{D}}\mathbf{v}_{b}(t)$ in the sense of
Definition 1.1.

The proof of Theorem 9.1 is rather technical and involved, and represents one
of the most innovative parts of the present paper. Before giving some
technical details, a word of warning: in the previous sections, we have always
worked with the variable $\mathbf{u}$ only because in the boundary layers
analysis we have used the fact that the viscous system (1.6) is in the
_normal_ form, in the Kawashima Shizuta sense [27]. In this section, instead,
we investigate the boundary condition and hence we have to use the variable
$\mathbf{v}$ as well, because Definition 1.1 involves the flux function
$\mathbf{f}$. For the reader’s convenience we now provide a sketchy overview
of the proof of Theorem 9.1.

###  9.1. Roadmap of the proof of Theorem 9.1

For the most part, in what follows we assume that the $k$-th characteristic
field is genuinely nonlinear, namely we assume (2.8), and only in §9.6 we
discuss how to adapt the proof to the case of a linearly degenerate (2.9)
$k$-th characteristic field.

In §9.2 we show that

(9.1) $\lim_{m\to+\infty}\mathbf{f}\left(\lim_{x\to
0^{+}}\mathbf{v}_{\varepsilon_{m}}(t,x)\right)=\lim_{x\to
0^{+}}\mathbf{f}\circ\mathbf{v}(t,x),\qquad\text{for a.e.
$t\in\mathbb{R}_{+}$,}$

where $\mathbf{v}_{\varepsilon_{m}}=\mathbf{v}(\mathbf{u}_{\varepsilon_{m}})$
and $\mathbf{u}_{\varepsilon_{m}}$ is the same as in (8.1)191919Note that in
(8.1) $\varepsilon_{m}$ is a specific subsequence and, in principle, the limit
$\mathbf{u}$ depends on the particular choice of the subsequence (as we
mentioned in the introduction, we are confident that one could establish
uniqueness results, but this goes beyond the scopes of the present paper).
However, in the remaining of this section we will always work with the same
sequence $\varepsilon_{m}$ and this uniquely identifies the limit
$\mathbf{u}$.. To this end, the main issue is establishing Lemma 9.2, which
provides a uniform in $\varepsilon$ total variation bound on
$\mathbf{f}\left(\lim_{x\to
0^{+}}\mathbf{v}_{\varepsilon_{m}}(\cdot,x)\right)$. In the proof of Lemma 9.2
we introduce the functional $\Lambda$, defined as in (9.5), and prove that it
is monotone non-increasing by carefully tracking how it varies any time either
a wave front hits the boundary, or the approximate boundary datum has a jump
discontinuity.

In §9.3 we construct the set $\widetilde{E}\subseteq\mathbb{R}_{+}$, which
plays a main role in our analysis and is defined by setting

(9.2) $\widetilde{E}:=F\cup G\cup H\cup D.$

In the previous expression, the sets $F$ and $H$ are rigorously defined in
(9.24) and (9.26) below, but, very loosely speaking, a time $\tau$ belongs to
$F$ if there are a lot of wave fronts of the wave front-tracking approximation
$\mathbf{u}_{\varepsilon_{m}}$ hitting the domain at times that accumulate
around $\tau$ as $m\to+\infty$. Conversely, an instant $\tau$ belongs to $H$
if there are a lot of waves of the wave front-tracking approximation
$\mathbf{u}_{\varepsilon_{m}}$ that interact in proximity of the domain
boundary at times that accumulate around $\tau$ as $m\to+\infty$. By relying
on the rigorous definitions (9.24) and (9.26) it is fairly easy to show that
$F$ and $H$ are both $\mathcal{L}^{1}$-negligible. The set $D$ is the set
where the boundary datum $\mathbf{v}_{b}$ is discontinuous, and it is also
$\mathcal{L}^{1}$-negligible owing to the assumption
$\mathrm{TotVar}\,\mathbf{v}_{b}<+\infty$. Finally, the (not necessarily
negligible) set $G$ in formula (9.2) is rigorously defined by (9.25) and
heuristically speaking represents the set of times where there is a sequence
of Lax admissible shocks for $\mathbf{v}_{\varepsilon_{m}}$ with vanishing
speed that as $m\to+\infty$ accumulate at the domain boundary. This heuristic
interpretation is consistent with Lemma 9.5, which dictates that if
$t\notin\widetilde{E}$, the limit of the traces of
$\mathbf{v}_{\varepsilon_{m}}$ coincides with the trace of the limit
$\mathbf{v}$.

In §9.4 we show that, for a.e. $t\notin\widetilde{E}$, the trace
$\mathbf{\bar{v}}(t)=\lim_{x\to 0^{+}}\mathbf{v}(t,x)$ satisfies
$\mathbf{\bar{v}}(t)\sim_{\mathbf{D}}\mathbf{v}_{b}(t)$ in the sense of
Definition 1.1, whereas in §9.5 by using (9.1) we establish the same property
for a.e. $t\in\widetilde{E}$. Finally, in §9.6 we establish the proof of
Theorem 9.1 in the case where the $k$-th vector field is linearly degenerate.

###  9.2. Proof of (9.1)

The proof of (9.1) relies on the following total variation bound.

######  Lemma 9.2.

Under the same assumptions as in Theorem 1.1 we have

(9.3)
$\mathrm{TotVar}\,\mathbf{f}\circ\mathbf{v}_{\varepsilon_{m}}(\cdot,0)\leq
C_{13}\delta,\quad\text{for every $m\in\mathbb{N}$}$

and for a suitable constant $C_{13}>0$ only depending on $\mathbf{g}$,
$\mathbf{f}$, $\mathbf{D}$ and on the change of variables $\mathbf{u}$, and on
the value $\mathbf{u}^{\ast}$. In the above expression, the constant $\delta$
is the same as in (7.1).

The proof of the above theorem uses the following result.

######  Lemma 9.3.

Under the same assumptions as in Theorem 1.1, if the $k$-the vector field is
genuinely nonlinear (2.8) we have

(9.4)
$|\mathbf{f}\circ\mathbf{v}(\mathbf{\widetilde{u}})-\mathbf{f}\circ\mathbf{v}(\mathbf{t}_{k}(\mathbf{\widetilde{u}},s))|\leq\left\\{\begin{array}[]{ll}\mathcal{O}(1)|s||\sigma_{k}(\mathbf{\widetilde{u}},s)|&s>0\\\
\mathcal{O}(1)|s|\max\\{|\lambda_{k}(\mathbf{t}_{k}(\mathbf{\widetilde{u}},s))|,|\lambda_{k}(\mathbf{\widetilde{u}})|\\}&s<0.\\\
\end{array}\right.$

###### Proof of Lemma 9.3.

We separately consider the following cases.
Case 1: $s\geq 0$. We have

$|\mathbf{f}\circ\mathbf{v}(\mathbf{\widetilde{u}})-\mathbf{f}\circ\mathbf{v}(\mathbf{t}_{k}(\mathbf{\widetilde{u}},s))|\stackrel{{\scriptstyle\eqref{e:rh}}}{{=}}|\sigma_{k}(\mathbf{\widetilde{u}},s)||\mathbf{g}\circ\mathbf{v}(\mathbf{\widetilde{u}})-\mathbf{g}\circ\mathbf{v}(\mathbf{t}_{k}(\mathbf{\widetilde{u}},s))|\leq\mathcal{O}(1)|s||\sigma_{k}(\mathbf{\widetilde{u}},s)|$

and this establishes (9.4).
Case 2: $s<0$. We have

$\begin{split}|\mathbf{f}\circ\mathbf{v}(\mathbf{\widetilde{u}})&-\mathbf{f}\circ\mathbf{v}(\mathbf{t}_{k}(\mathbf{\widetilde{u}},s))|\stackrel{{\scriptstyle\eqref{e:laxc}}}{{=}}|\mathbf{f}\circ\mathbf{v}(\mathbf{\widetilde{u}})-\mathbf{f}\circ\mathbf{v}(\mathbf{i}_{k}(\mathbf{\widetilde{u}},s))|\stackrel{{\scriptstyle\eqref{e:raref}}}{{=}}\left|\int_{s}^{0}\mathbf{D}\mathbf{f}\,\mathbf{D}\mathbf{v}\
\mathbf{r}_{k}(\mathbf{i}_{k}(\mathbf{\widetilde{u}},z))dz\right|\\\
&=\left|\int_{s}^{0}\lambda_{k}(\mathbf{i}_{k}(\mathbf{\widetilde{u}},z))\mathbf{D}\mathbf{g}\,\mathbf{D}\mathbf{v}\
\mathbf{r}_{k}(\mathbf{i}_{k}(\mathbf{\widetilde{u}},z))dz\right|\leq\mathcal{O}(1)\int_{s}^{0}|\lambda_{k}(\mathbf{i}_{k}(\mathbf{\widetilde{u}},z))|dz\\\
&\leq\mathcal{O}(1)|s|\max\\{|\lambda_{k}(\mathbf{i}_{k}(\mathbf{\widetilde{u}},s))|,|\lambda_{k}(\mathbf{\widetilde{u}})|\\}\stackrel{{\scriptstyle\eqref{e:laxc}}}{{=}}\mathcal{O}(1)|s|\max\\{|\lambda_{k}(\mathbf{t}_{k}(\mathbf{\widetilde{u}},s))|,|\lambda_{k}(\mathbf{\widetilde{u}})|\\},\end{split}$

i.e. (9.4). To establish the last inequality at the second line of the above
expression we have used the fact that, owing to (2.8), the function
$z\mapsto\lambda_{k}(\mathbf{i}_{k}(\mathbf{\widetilde{u}},z))$ is monotone
non-decreasing and hence the maximum of its modulus on an interval is attained
at one the two extrema. ∎

We are now ready to give the

###### Proof of Lemma 9.2 (genuinely nonlinear case).

We assume that the $k$-th vector field is genuinely nonlinear (2.8) and set

(9.5) $\Lambda(t):=F(t)+K_{4}\Upsilon(t)$

where

(9.6)
$F(t)=\mathrm{TotVar}_{[0,t[}\mathbf{f}\circ\mathbf{v}_{\varepsilon_{m}}(\cdot,0),$

$\Upsilon$ is the same as in (7.4) and $K_{4}$ is a suitable constant, to be
determined in the following. Assume for a moment that we have shown that the
functional $\Lambda$ is monotone non-increasing; then

$\mathrm{TotVar}\;\mathbf{f}\circ\mathbf{v}_{\varepsilon_{m}}(\cdot,0)\leq\lim_{t\to+\infty}\Lambda(t)\leq\Lambda(0)\leq\mathcal{O}(1)\delta$

and this establishes (9.3). To show that $\Lambda$ is monotone non-increasing,
we first of all recall that owing to the analysis in §7 the term $\Upsilon$ is
monotone non-increasing. We now separately analyze the three instances where
the term $F$ can vary.
Case I: at time $\tau$ a wave front $\alpha$ of the family $j_{\alpha}<k$ with
size $s_{j}$ hits the boundary. Assume for a moment that we have established
the inequality

(9.7) $\Delta F(\tau)\leq\mathcal{O}(1)|s_{j}|.$

Then owing to (7.17) we can choose the constant $K_{4}$ in (9.5) in such a way
that $\Delta\Lambda(\tau)\leq 0$. To establish (9.7) we first have to
introduce some notation. We term $\mathbf{u}^{-}$ and $\mathbf{u}^{+}$ the
left and the right state of $\alpha$, and denote by $\mathbf{\hat{u}}$ the
same state as in (5.9). We recall the construction in §5.2.2 and conclude that
if we solve the boundary Riemann problem (1.1),(4.3) by using the simplified
boundary Riemann problem then $\Delta F(\tau)=0$ and (9.7) is trivially
satisfied. We are left to consider the case where the boundary Riemann problem
(1.1),(4.3) is solved by using the accurate boundary Riemann solver: we recall
the construction in §4.6.1 and §5.3.1 and we separately consider cases
i),$\dots$, vi) in there.

  * i)

we have

(9.8) $\begin{split}\Delta
F(\tau)&=|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\leq|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))|+|\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))-\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\leq\mathcal{O}(1)|\mathbf{u}^{-}-\mathbf{u}^{+}|+\mathcal{O}(1)|\mathbf{u}^{+}-\mathbf{\hat{u}}|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\stackrel{{\scriptstyle\eqref{e:nc}}}{{\leq}}\mathcal{O}(1)|s_{j}|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\end{split}$

To control the second term at the right-hand side of the above formula we
point out that (recalling that $\sigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})>0$
by definition of case i))

(9.9)
$\begin{split}|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))&-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\stackrel{{\scriptstyle\eqref{e:rh}}}{{\leq}}[\sigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}|\mathbf{g}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{g}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\leq\mathcal{O}(1)|s^{\prime}_{k}|[\sigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}\leq\mathcal{O}(1)\delta[\sigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}\\\
&\leq\mathcal{O}(1)\delta|[\sigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}-[\sigma_{k}(\mathbf{u}^{+},s^{\prime}_{k})]^{+}|+\mathcal{O}(1)\delta|[\sigma_{k}(\mathbf{u}^{+},s^{\prime}_{k})]^{+}-[\sigma_{k}(\mathbf{u}^{-},s^{\prime}_{k})]^{+}|\\\
&\quad+\mathcal{O}(1)\delta|[\sigma_{k}(\mathbf{u}^{-},s^{\prime}_{k})]^{+}-[\sigma_{k}(\mathbf{u}^{-},\xi_{k})]^{+}|\\\
&\leq\mathcal{O}(1)\delta|\mathbf{\hat{u}}-\mathbf{u}^{+}|+\mathcal{O}(1)\delta|\mathbf{u}^{+}-\mathbf{u}^{-}|+\mathcal{O}(1)\delta|s^{\prime}_{k}-\xi_{k}|\stackrel{{\scriptstyle\eqref{e:nc}}}{{\leq}}\mathcal{O}(1)\delta|s_{j}|\end{split}$

To establish the third to last inequality we have used the fact that
$[\sigma_{k}(\mathbf{u}^{-},\xi_{k})]^{+}=0$, obtained by applying (4.58) with
$\mathbf{u}^{-}$ and $\xi_{k}$ in place of $\mathbf{\bar{u}}$ and $\xi_{k}$,
respectively. By plugging the previous chains of inequalities into (9.8) we
get (9.7)

  * ii)

we use again (9.8) and to control the second term at the right hand side we
proceed as follows. If $\xi_{k}=0$ then owing to (6.2) we have
$|s^{\prime}_{k}|\leq C_{1}|s_{j}|$ and this yields

$|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\leq\mathcal{O}(1)|s^{\prime}_{k}|\leq\mathcal{O}(1)|s_{j}|.$

If $\xi_{k}>0$ then owing to (6.4) we have $\lambda_{k}(\mathbf{u}^{-})\leq
0$. We then have

$\begin{split}|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|&=|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{t}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&=|\mathbf{g}\circ\mathbf{g}^{-1}\circ\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{g}\circ\mathbf{g}^{-1}\circ\mathbf{f}(\mathbf{v}(\mathbf{t}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\leq\mathcal{O}(1)|\mathbf{g}^{-1}\circ\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{g}^{-1}\circ\mathbf{f}(\mathbf{v}(\mathbf{t}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\leq\left|\int_{s^{\prime}_{k}}^{0}\lambda_{k}\mathbf{r}_{k}(\mathbf{t}_{k}(\mathbf{\hat{u}},s)ds\right|\\\
&\leq\mathcal{O}(1)\int_{s^{\prime}_{k}}^{0}|\lambda_{k}(\mathbf{t}_{k}(\mathbf{\hat{u}},s)|ds\stackrel{{\scriptstyle\eqref{e:gnl}}}{{\leq}}\mathcal{O}(1)\int_{s^{\prime}_{k}}^{0}\lambda_{k}(\mathbf{\hat{u}})ds\leq\mathcal{O}(1)|s^{\prime}_{k}|\lambda_{k}(\mathbf{\hat{u}})\\\
&\stackrel{{\scriptstyle\lambda_{k}(\mathbf{u}^{-})\leq
0}}{{\leq}}\mathcal{O}(1)|s^{\prime}_{k}|[\lambda_{k}(\mathbf{\hat{u}})-\lambda_{k}(\mathbf{u}^{-})]\leq\mathcal{O}(1)|s^{\prime}_{k}||\mathbf{\hat{u}}-\mathbf{u}^{-}|\\\
&\stackrel{{\scriptstyle\eqref{e:nc}}}{{\leq}}\mathcal{O}(1)|s^{\prime}_{k}||s_{j}|\leq\mathcal{O}(1)\delta|s_{j}|\end{split}$

and by plugging the above inequality into (9.8) we arrive at (9.7).

  * iii)

the proof is the same as at the previous point, it suffices to replace
$s^{\prime}_{k}$ with $\bar{s}(\mathbf{\hat{u}})$ and use (6.45).

  * iv)

the proof of (9.7) is the same as in case i).

  * v), vi)

since there is no wave of the $k$-th family entering the domain after the
interaction we have $\Delta
F(\tau)=|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}})|$
and one can then proceed as in (9.8).

Case II: at time $\tau$ a wave front $\alpha$ of the $k$-th family with
(signed) strength $s$ hits the boundary. Assume for a moment that we have
established the inequality

(9.10) $\Delta
F(\tau)\leq\mathcal{O}(1)|s|([\varsigma_{k}(\mathbf{u}^{+},s)]^{-}+|\xi_{k}|).$

Then owing to (7.26) we can choose the constant $K_{4}$ in (9.5) in such a way
that $\Delta\Lambda(\tau)\leq 0$. To establish (9.10) we first assume that the
boundary Riemann problem (1.1), (4.3) is solved by using the accurate boundary
Riemann solver, we recall the analysis in §5.3 and we separately consider the
following cases. As in Case I, we term $\mathbf{u}^{-}$ and $\mathbf{u}^{+}$
the left and the right state of $\alpha$, respectively, and denote by
$\mathbf{\hat{u}}$ the same state as in (5.9). Note that, owing to (5.5) we
have $\lambda_{k}(\mathbf{u}^{+})<0$. We now separately consider cases
i),$\dots$,vi) in §4.6.1 and §5.3.1.

  * i)

We point out that

(9.11) $\begin{split}\Delta
F(\tau)&=|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\leq|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))|+|\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))-\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\end{split}$

To control first term at the right-hand side of (9.11) we recall (9.4). In the
case $s<0$ we recall that owing to (5.5) the speed of the wave-front is
$\lambda_{k}(\mathbf{u}^{+})$ and that $\lambda_{k}(\mathbf{u}^{+})<0$
(otherwise the wave could not hit the boundary). Since the map
$s\mapsto\lambda_{k}(\mathbf{i}_{k}(\mathbf{u}^{+},s))$ is monotone non-
decreasing owing to (2.8), we conclude that

$\max\\{|\lambda_{k}(\mathbf{t}_{k}(\mathbf{u}^{+},s))|,|\lambda_{k}(\mathbf{u}^{+})|\\}=|\lambda_{k}(\mathbf{t}_{k}(\mathbf{u}^{+},s))|\stackrel{{\scriptstyle\eqref{e:speedwft}}}{{=}}|\varsigma_{k}(\mathbf{u}^{+},s)|=[\varsigma_{k}(\mathbf{u}^{+},s)]^{-}\quad\text{if
$s<0$}.$

Owing to (9.4) and recalling (5.4) we conclude that, regardless the sign of
$s$, we have

(9.12)
$|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))|\leq\mathcal{O}(1)|s|[\varsigma_{k}(\mathbf{u}^{+},s)]^{-}.$

To control the second term at the right-hand side of (9.11) we simply point
out that

(9.13)
$|\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))-\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))|\leq\mathcal{O}(1)|\mathbf{u}^{+}-\mathbf{\hat{u}}|\stackrel{{\scriptstyle\eqref{e:me},\eqref{e:hatu}}}{{\leq}}\mathcal{O}(1)|s|([\varsigma_{k}(\mathbf{u}^{+},s)]^{-}+|\xi_{k}|).$

We now recall (5.4) and control the third term at the right-hand side of
(9.11):

(9.14)
$\begin{split}|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-&\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\stackrel{{\scriptstyle\eqref{e:dflux},s^{\prime}_{k}>0}}{{\leq}}\mathcal{O}(1)|s^{\prime}_{k}|[\varsigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}\\\
&\leq\mathcal{O}(1)\delta[\varsigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}-[\varsigma_{k}(\mathbf{u}^{+},s^{\prime}_{k})]^{+}|+\mathcal{O}(1)\delta|[\varsigma_{k}(\mathbf{u}^{+},s^{\prime}_{k})]^{+}-[\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})]^{+}|\\\
&\quad+\mathcal{O}(1)|s^{\prime}_{k}|[\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})]^{+}\\\
&\stackrel{{\scriptstyle\eqref{e:me}}}{{\leq}}\mathcal{O}(1)\delta|\mathbf{\hat{u}}-\mathbf{u}^{+}|+\mathcal{O}(1)\delta|s|([\varsigma_{k}(\mathbf{u}^{+},s)]^{-}+|\xi_{k}|)+\mathcal{O}(1)|s^{\prime}_{k}|[\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})]^{+}\\\
&\stackrel{{\scriptstyle\eqref{e:hatu},\eqref{e:me},\eqref{e:msigma}}}{{\leq}}\mathcal{O}(1)\delta|s|([\varsigma_{k}(\mathbf{u}^{+},s)]^{-}+|\xi_{k}|)+\mathcal{O}(1)|s+\xi_{k}|[\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})]^{+}\end{split}$

When $\xi_{k}=0$, the above estimate yields (9.10), hence to control the last
term at the right-hand side of the above expression we assume $\xi_{k}\neq 0$
and we separately consider the following cases: if
$\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})\leq 0$ then the term vanishes and the
estimate is complete. We are thus left to consider the case
$\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})\geq 0$: recalling that
$\lambda_{k}(\mathbf{u}^{+})<0$ and since $\varsigma_{k}(\mathbf{u}^{+},s)\leq
0$ then $s\leq\underline{s}(\mathbf{u}^{+})$ and
$\xi_{k}\geq\underline{s}(\mathbf{u}^{+})-s\geq 0$. We also point out that
$\varsigma_{k}(\mathbf{t}_{k}(\mathbf{u}^{+},s),\xi_{k})\leq 0$ owing to
(6.4). We thus have

(9.15)
$[\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})]^{+}=\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k}\leq\varsigma_{k}(\mathbf{u}^{+},s+\xi_{k})-\varsigma_{k}(\mathbf{t}_{k}(\mathbf{u}^{+},s),\xi_{k}):=g(s,\xi_{k})$

We now point out that the function $g$ is of class $C^{2}$ and satisfies

(9.16) $g(0,\xi)=0\;\text{for every $\xi$},\qquad
g(s,0)=\varsigma_{k}(\mathbf{u}^{+},s)-\lambda_{k}(\mathbf{t}_{k}(\mathbf{u}^{+},s))\stackrel{{\scriptstyle\text{Lax}}}{{\leq}}0\;\text{for
$s\geq 0$}.$

If $s\geq 0$ we can then argue as follows: since $g(s,\xi_{k})\geq 0$ and
$g(s,0)\leq 0$ then there must be $\xi(s)\in[0,\xi_{k}]$ such that
$g(s,\xi(s))=0$. By using (9.16) and arguing as in the proof of [10, Lemma
2.5] we then get

$\begin{split}|g(s,\xi_{k})|&\leq\int_{\xi(s)}^{\xi_{k}}\left|\frac{\partial
g}{\partial\xi}(s,z)\right|dz\leq\mathcal{O}(1)\int_{\xi(s)}^{\xi_{k}}|s|dz\leq\mathcal{O}(1)|s||\xi_{k}-\xi(s)|\leq\mathcal{O}(1)|s||\xi_{k}|\;\text{for
$s\geq 0$}.\end{split}$

This allows us to control the last term at the right-hand side of (9.14) and
concludes the proof of (9.10) in the case $s\geq 0$. If $s<0$ we recall that
$s^{\prime}_{k}>0$ because we are in case i). Owing to (6.5), this implies
that, if $s+\xi_{k}\leq 0$, then

$|s_{k}+\xi_{k}|\leq|s|([\varsigma_{k}(\mathbf{u}^{+},s)]^{-}+|\xi_{k}|)$

and by plugging this estimate into (9.14) we arrive at (9.10). We are thus
left to control the last term at the right hand-side of (9.14) in the case
$s<0$, $s+\xi_{k}>0$, which yields $\xi_{k}>-s>0.$ We now consider the
function

$h(s,\xi):=[s+\xi]g(s,\xi),$

where $g$ is the same as in (9.15) and point out that $h$ satisfies

$h(0,\xi)=0,\quad h(s,-s)=0\quad\text{for every $\xi$, $s$}$

and by arguing as in the proof of [10, Lemma 2.5] we then get

$|h(s,\xi)|\leq\mathcal{O}(1)|s||\xi+s|\leq\mathcal{O}(1)|s||\xi|\quad\text{for
$\xi>0>s$}$

and this allows us to control the last term at the right-hand side of (9.14)
and concludes the proof of (9.10).

  * ii)

we use (9.11) and to control the first and second term at the right-hand side
we use (9.12) and (9.13), respectively. To control the third term at the right
hand-side of (9.11) we recall that, since we are in case ii),
$\bar{s}(\mathbf{\hat{u}})\leq s^{\prime}_{k}<0$ and hence

$0\leq\lambda_{k}(\mathbf{t}_{k}(\mathbf{\hat{u}},s^{\prime}_{k}))\leq\lambda_{k}(\mathbf{\hat{u}})$

and by using (9.4) we get

$\begin{split}|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))&-\mathbf{f}(\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\leq\mathcal{O}(1)|s^{\prime}_{k}|[\lambda_{k}(\mathbf{\hat{u}})]^{+}\\\
&=\mathcal{O}(1)|s^{\prime}_{k}|\big{|}[\lambda_{k}(\mathbf{\hat{u}})]^{+}-[\lambda_{k}(\mathbf{u}^{+})]^{+}\big{|}\leq\mathcal{O}(1)|s^{\prime}_{k}||\mathbf{\hat{u}}-\mathbf{u}^{+}|\\\
&\stackrel{{\scriptstyle\eqref{e:hatu},\eqref{e:me}}}{{\leq}}\mathcal{O}(1)\delta|s|([\varsigma_{k}(\mathbf{u}^{+},s)]^{-}+|\xi_{k}|)\end{split}$

In the above expression we have used the equality
$[\lambda_{k}(\mathbf{u}^{+})]^{+}=0$: this follows from the fact that the
wave-front $\alpha$ has negative speed and the Lax entropy condition (if
$s>0$) and the equality (5.5) (if $s<0$).

  * iii)

the proof is the same as at the previous point, it suffices to replace
$s^{\prime}_{k}$ with $\bar{s}(\mathbf{\hat{u}})$ and use (6.45).

  * iv)

we proceed as in point i)

  * v), vi)

in these cases there is no wave front of the $k$-th family entering the domain
after the interaction and hence we have $\Delta
F(\tau)=|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\hat{\mathbf{u}}))|$
and we can argue as in (9.11).

To establish (9.10) in the case where we use the simplified boundary Riemann
solver instead of the accurate Riemann solver we can use the exact same
analysis since the trace of the approximate solution after the interaction is
the same as in the case where we use the accurate Riemann solver.
Case III: at time $\tau$ the datum $\mathbf{u}_{bm}$ has a discontinuity. We
term $\mathbf{u}_{bm}^{+}$ and $\mathbf{u}_{bm}^{-}$ the right and left limits
$\mathbf{u}_{bm}(\tau^{+})$ and $\mathbf{u}_{bm}(\tau^{-})$, respectively,
$\mathbf{u}^{+}$ and $\mathbf{\bar{u}}$ the traces of
$\mathbf{u}_{m}(t,\cdot)$ for $t=\tau^{-}$ and $t=\tau^{+}$, respectively, and
$\mathbf{\hat{u}}$ the same state as in (5.9). Assume we have established the
inequality

(9.17) $\Delta
F(\tau)\leq\mathcal{O}(1)|\mathbf{u}_{bm}^{+}-\mathbf{u}_{bm}^{-}|;$

then owing to (7.37) we can choose the constant $K_{4}$ in (9.5) in such a way
that $\Delta\Upsilon(\tau)\leq 0$. To establish (9.17) we point out that

(9.18) $\begin{split}\Delta
F(\tau)&=|\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))-\mathbf{f}(\mathbf{v}(\mathbf{\bar{u}}))|\leq|\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))-\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{\bar{u}})))|\\\
&\stackrel{{\scriptstyle\eqref{e:jd1}}}{{\leq}}\mathcal{O}(1)|\mathbf{u}_{bm}^{+}-\mathbf{u}_{bm}^{-}|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{\bar{u}}))|\end{split}$

To control the second term at the right-hand side of the above expression we
point out that, by separately considering cases i),$\dots$,vi) in §4.6.1 and
§5.3.1 we conclude that
$\mathbf{\bar{u}}=\mathbf{t}_{k}(\mathbf{\hat{u}},\widetilde{s}_{k})$, where
the value attained by $\widetilde{s}_{k}$ in the various cases is as follows:

  * i)

$\widetilde{s}_{k}=s^{\prime}_{k}>0$;

  * ii)

$\widetilde{s}_{k}=s^{\prime}_{k}\leq 0$;

  * iii)

$\widetilde{s}_{k}=\bar{s}(\mathbf{\hat{u}})\leq 0$, and hence
$s^{\prime}_{k}<\widetilde{s}_{k}\leq 0$;

  * iv)

$\widetilde{s}_{k}=s^{\prime}_{k}>0$;

  * v), vi)

$\widetilde{s}_{k}=0$;

If $\widetilde{s}_{k}=0$ then the second term at the right-hand side of (9.18)
vanishes and this completes the proof of (9.17). If $\widetilde{s}_{k}>0$,
then $\widetilde{s}_{k}=s^{\prime}_{k}$ and by applying Lemma 9.3 we get

(9.19)
$\begin{split}|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-&\mathbf{f}(\mathbf{v}(\bar{\mathbf{u}}))|\leq\mathcal{O}(1)[\sigma_{k}(\mathbf{\hat{u}},\widetilde{s}_{k})]^{+}|\widetilde{s}_{k}|\stackrel{{\scriptstyle\widetilde{s}_{k}=s^{\prime}_{k}}}{{=}}\mathcal{O}(1)[\sigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}|s^{\prime}_{k}|\\\
&\leq\mathcal{O}(1)\delta|[\sigma_{k}(\mathbf{\hat{u}},s^{\prime}_{k})]^{+}-[\sigma_{k}(\mathbf{u}^{+},s^{\prime}_{k})]^{+}|+\mathcal{O}(1)\delta|[\sigma_{k}(\mathbf{u}^{+},s^{\prime}_{k})]^{+}-[\sigma_{k}(\mathbf{u}^{+},\xi_{k})]^{+}|\\\
&\stackrel{{\scriptstyle\eqref{e:jd1}}}{{\leq}}\mathcal{O}(1)\delta|\mathbf{u}_{bm}^{+}-\mathbf{u}_{bm}^{-}|.\end{split}$

To establish the second inequality in the previous expression, we have used
the equality $[\sigma_{k}(\mathbf{u}^{+},\xi_{k})]^{+}=0$, which follows from
(4.57) applied with $\mathbf{u}^{+}$ and $\xi_{k}$ in place of
$\mathbf{\bar{u}}$ and $\xi^{\prime}_{k}$, respectively. By plugging (9.19)
into (9.18) we establish (9.17). If $\widetilde{s}_{k}<0$, then by applying
Lemma 9.3 we get

(9.20)
$|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{\bar{u}}))|\leq\mathcal{O}(1)|\widetilde{s}_{k}|\max\\{|\lambda_{k}(\mathbf{\hat{u}})|,|\lambda_{k}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},\widetilde{s}_{k}))|\\}.$

Next, we recall that the inequality $\widetilde{s}_{k}<0$ implies that we are
in either case ii) or case iii) in §5.2.1, and this in turn yields

(9.21)
$|\widetilde{s}_{k}|\leq|s^{\prime}_{k}|,\quad\max\\{|\lambda_{k}(\mathbf{\hat{u}})|,|\lambda_{k}(\boldsymbol{\zeta}_{k}(\mathbf{\hat{u}},\widetilde{s}_{k}))|\\}=|\lambda_{k}(\mathbf{\hat{u}})|=\lambda_{k}(\mathbf{\hat{u}})>0.$

If $\lambda_{k}(\mathbf{u}^{+})\leq 0$ we can then control the right-hand side
of (9.20) by arguing as follows:

$|s^{\prime}_{k}|[\lambda_{k}(\mathbf{\hat{u}})]^{+}\leq|s^{\prime}_{k}||[\lambda_{k}(\mathbf{\hat{u}})]^{+}-[\lambda_{k}(\mathbf{u}^{+})]^{+}|\leq\mathcal{O}(1)\delta|\hat{\mathbf{u}}-\mathbf{u}^{+}|\stackrel{{\scriptstyle\eqref{e:jd1}}}{{\leq}}\mathcal{O}(1)\delta|\mathbf{u}_{bm}^{+}-\mathbf{u}_{bm}^{-}|.$

If $\lambda_{k}(\mathbf{u}^{+})>0$ then $\xi_{k}=0$ by (4.57) applied with
$\mathbf{u}^{+}$ and $\xi_{k}$ in place of $\mathbf{\bar{u}}$ and
$\xi^{\prime}_{k}$, respectively, and then owing to (7.30) we get
$|s^{\prime}_{k}|\leq C_{6}|\mathbf{u}_{bm}^{+}-\mathbf{u}_{bm}^{-}|$ and by
(9.21) this allows us to control the right hand-side of (9.20). This concludes
the proof of (9.17) and hence of Lemma 9.2 in the genuinely nonlinear case. ∎

######  Corollary 9.4.

Under the same assumptions as in the statement of Theorem 1.1 we have (9.1).

###### Proof.

We point out that that $\mathbf{f}\circ\mathbf{v}_{\varepsilon_{m}}(\cdot,0)$
is uniformly bounded because so is $\mathbf{u}^{\varepsilon}$ by (7.2). Owing
to Helly-Kolmogorov Compactness Theorem, the total variation bound (9.3)
implies that $\mathbf{f}\circ\mathbf{v}_{\varepsilon_{m}}(\cdot,0)$ converges
up to subsequences in $L^{1}_{\mathrm{loc}}(\mathbb{R}_{+})$ to some limit
function $\boldsymbol{\rho}$. By the analysis in [10, p.144-5], we have

$\lim_{m\to+\infty}\left(\int_{0}^{+\infty}\int_{0}^{+\infty}\partial_{t}\varphi\
\mathbf{g}\circ\mathbf{v}_{\varepsilon_{m}}(t,x)+\partial_{x}\varphi\
\mathbf{f}\circ\mathbf{v}_{\varepsilon_{m}}(t,x)dxdt+\int_{0}^{+\infty}\varphi(t,0)\
\mathbf{f}\circ\mathbf{v}_{\varepsilon_{m}}(t,0)dt\right)=0$

for every $\varphi\in C^{\infty}_{c}(]0,+\infty[\times\mathbb{R}_{+})$ and
this yields $\boldsymbol{\rho}=\mathbf{f}\circ\mathbf{v}(\cdot,0)$ using the
fact that $\mathbf{v}$ is of bounded total variation and hence has a trace at
$x=0$, where $\mathbf{v}=\mathbf{v}\circ\mathbf{u}$ and $\mathbf{u}$ is the
same as in (8.1). Also, up to subsequences we have (9.1). ∎

###  9.3. Construction and analysis of the set $\widetilde{E}$ in the
genuinely nonlinear case

The set $\widetilde{E}\subseteq\mathbb{R}_{+}$ is defined as in (9.2). We
recall that the set $D$ is the set where the boundary datum $\mathbf{v}_{b}$
is discontinuous and we now rigorously define the sets $F$, $H$ and $G$.

####  9.3.1. Construction of the set $F$

For every $m\in\mathbb{N}$ we construct a sequence of measures
$\mu_{m}\in\mathcal{M}_{\mathrm{loc}}(\mathbb{R}_{+})$ by proceeding as
follows: first, we term $\tau_{1},\dots,\tau_{n_{m}}$ the times at which a
wave front of the $\varepsilon_{m}$-wave front-tracking approximation
$\mathbf{u}^{\varepsilon_{m}}$ hits the boundary; next, for every
$\tau_{\alpha}$, $\alpha=1,\dots,n_{m}$, we term $j_{\alpha}$ and $s_{\alpha}$
the family and the size of the hitting wave, and set

(9.22)
$\chi_{\alpha}:=\left\\{\begin{array}[]{ll}A|s_{\alpha}|&j_{\alpha}<k\\\
|s_{\alpha}|([\varsigma_{\alpha}(\mathbf{u}_{\alpha},s_{\alpha})]^{-}+|\xi_{k}|)&j_{\alpha}=k.\\\
\end{array}\right.$

Finally, we set

(9.23)
$\mu_{m}:=\sum_{\alpha=1}^{n_{m}}\chi_{\alpha}\delta_{t=\tau_{\alpha}},$

where $\delta_{t=\tau_{\alpha}}$ denotes the Dirac delta concentrated at
$\tau_{\alpha}$. Note that by using the specific piecewise constant structure
of $\mathbf{u}^{\varepsilon}$ and (7.42) we get

$|\mu_{m}|(]0,T[)\leq\mathcal{O}(1)\mathrm{D\mathbf{u}^{\varepsilon}}(]0,T[\times\mathbb{R}_{+})\stackrel{{\scriptstyle\eqref{e:settedue}}}{{\leq}}\mathcal{O}(1)T\delta,\quad\text{for
every $T>0$}.$

This implies that, up to subsequences, $\mu_{m}$ converges weakly-∗ in the
sense of measures on $\mathbb{R}_{+}$ to some limit measure
$\mu\in\mathcal{M}_{\mathrm{loc}}(\mathbb{R}_{+})$ as $m\to+\infty$. We term
$F$ the set of atoms of $\mu$, that is

(9.24) $F:=\\{t\in\mathbb{R}_{+}:\mu(\\{t\\})>0\\}.$

Note that $F$ is at most countable (and henceforth
$\mathcal{L}^{1}$-negligible) by the local finiteness of $\mu$.

####  9.3.2. Construction of the set $G$

We employ the same construction of _maximal $\nu_{n}$-shock fronts_ of the
$k$-th family as in [10, page 219, step 2]. In particular, this yields the
construction of a countable family $\\{y_{n}\\}_{n\in\mathbb{N}}$ of shock
curves for the limit function $\mathbf{u}$, obtained as limits of _maximal
$\nu_{n}$-shock fronts_ of the approximate solution, which has strength larger
of equal than $\nu_{n}/2$. Note that this shock curves are contained in the
_closed_ set $[0,+\infty[\times[0,+\infty[$ and that there may be $0$-speed
shocks exactly located at the origin. We set

(9.25)
$G:=\big{\\{}t\in\mathbb{R}_{+}:\,y_{n}(t)=0,\;y^{\prime}_{n}(t)=0\;\text{for
some $n\in\mathbb{N}$}\big{\\}}.$

####  9.3.3. Construction of the set $H$

We consider the same measure
$\mu^{IC}\in\mathcal{M}_{loc}(\mathbb{R}_{+}\times[0,+\infty[)$ of interaction
and cancellation as in [10, formula (10.68)], constructed by using [10,
formula (7.96)]. We term $H$ the set of atoms of $\mu^{IC}$ on the $x=0$ axis,
namely

(9.26) $H:=\\{t\in\mathbb{R}_{+}:\;\mu^{IC}(\\{(t,0)\\})>0\\}.$

Note that $H$ is at most countable (and henceforth
$\mathcal{L}^{1}$-negligible) by the local finiteness of the measure
$\mu^{IC}$.

####  9.3.4. Analysis of the set $\widetilde{E}$

######  Lemma 9.5.

Let $\widetilde{E}$ be the same as in (9.2) and set

(9.27) $\mathbf{\bar{u}}(t):=\lim_{x\to 0^{+}}\mathbf{u}(t,x),$

where $\mathbf{u}$ is the same as in (8.1), and

(9.28) $\mathbf{\bar{u}}_{m}(t):=\lim_{x\to
0}\mathbf{u}_{\varepsilon_{m}}(t,x).$

If $t\notin\widetilde{E}$ then

(9.29) $\lim_{m\to+\infty}\mathbf{\bar{u}}_{m}(t)=\mathbf{\bar{u}}(t).$

###### Proof.

We fix $t\notin\widetilde{E}$, recall footnote 19 at page 19, argue by
contradiction and assume that there is a constant $\iota$ such that

(9.30) $|\lim_{m\to+\infty}\mathbf{\bar{u}}_{m}(t)-\lim_{x\to
0}\mathbf{u}(t,x)|\geq\iota>0.$

Next, we proceed according to the following steps.
Step 1: we point out that (8.1) implies that, up to subsequences,
$\mathbf{u}_{\varepsilon_{m}}(t,x)$ converges to $\mathbf{u}(t,x)$ for a.e.
$x\in\mathbb{R}_{+}$. Owing to (9.30), this in turn implies that there are two
sequences $\\{x_{m}\\}$ and $\\{y_{m}\\}$ such that
$\mathbf{u}_{\varepsilon_{m}}$ is continuous at both $x_{m}$ and $y_{m}$ and

(9.31) $y_{m}>x_{m}\geq
0,\;|\mathbf{u}_{\varepsilon_{m}}(t,x_{m})-\mathbf{u}_{\varepsilon_{m}}(t,y_{m})|\geq\frac{\iota}{2}\;\text{for
every $m$},\quad\lim_{m\to+\infty}y_{m}=\lim_{m\to+\infty}x_{m}=0.$

Step 2: by recalling the structure of the function
$\mathbf{u}_{\varepsilon_{m}}$ we conclude that the last inequality in (9.31)
implies that the space interval $]x_{m},y_{m}[$ is crossed at time $t$ by an
amount of wave fronts such that their combined strength satisfies

$\sum_{\begin{subarray}{c}\text{$\alpha$ wave in}\\\
\,]x_{m},y_{m}[\end{subarray}}|s_{\alpha}|\geq\frac{L\iota}{2}$

for a suitable constant $L>0$. Up to subsequences, we can then assume that at
least one of the following cases is verified.
Case 1: the wave fronts of the families $1,\dots,k-1$ satisfy

$\sum_{\begin{subarray}{c}\text{$\alpha$ wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=1,\dots,k-1\end{subarray}}|s_{\alpha}|\geq\frac{L\iota}{10}.$

Then there are two possibilities:

  * i)

the combined strength of the wave fronts hitting the boundary in the interval
$]t,t+y_{m}/c[$, where $c$ is the same as in (2.6), is at least $L\iota/20$.
By recalling (9.22) and (9.23) this implies

(9.32)
$\mu_{m}\left(\left[t,t+\frac{y_{m}}{c}\right[\right)\geq\frac{LA\iota}{20}\quad\text{for
every $m\in\mathbb{N}$}.$

We claim that (9.32) yields $\mu(\\{t\\})>0$, that is $t\in
F\subseteq\widetilde{E}$, and thus contradicts the assumption
$t\notin\widetilde{E}$. To show that $\mu(\\{t\\})>0$ we use the outer
regularity of $\mu$ (see [2, Proposition 1.43 (ii)]), which in particular
implies that to show that $\mu(\\{t\\})>0$ it suffices to show that

(9.33) $\mu(\mathcal{O})\geq\frac{LA\iota}{20},\quad\text{for every open set
$\mathcal{O}$ such that $t\in\mathcal{O}$}.$

We now fix $\widetilde{m}$ such that
$[t,t+y_{\widetilde{m}}/c]\subseteq\mathcal{O}$, which implies
$\mu([t,t+y_{\widetilde{m}}/c])\leq\mu(\mathcal{O})$. Owing to the upper
semicontinuity of the measure of compact sets with respect to weak∗
convergence (see [2, formula (1.8)]) we also have that

(9.34)
$\limsup_{m\to+\infty}\mu_{m}([t,t+y_{\widetilde{m}}/c])\leq\mu([t,t+y_{\widetilde{m}}/c])\leq\mu(\mathcal{O}).$

By extracting, if needed, a subsequence from $\\{y_{m}\\}$, we can assume that
$[t,t+y_{m}/c]\subseteq[t,t+y_{\widetilde{m}}/c]$ for every
$m\geq\widetilde{m}$, which implies

$\frac{LA\iota}{20}\stackrel{{\scriptstyle\eqref{e:abbagrande}}}{{\leq}}\limsup_{m\to+\infty}\mu_{m}([t,t+y_{m}/c])\leq\limsup_{m\to+\infty}\mu_{m}([t,t+y_{\widetilde{m}}/c])\stackrel{{\scriptstyle\eqref{e:limsup}}}{{\leq}}\mu(\mathcal{O}),$

which eventually yields (9.33).

  * ii)

an amount of wave fronts of total strength at least $L\iota/20$ is cancelled
by the interactions with other waves on the set
$]t,t+y_{m}/c[\times]0,y_{m}[$. By recalling [10, formula (7.96)] this implies
that

(9.35) $\mu^{IC}_{m}([t,t+y_{m}/c]\times[0,y_{m}])\geq L\iota/20.$

and by arguing as in item i) above this implies $\mu^{IC}(\\{(t,0)\\})>0$ and
hence $t\in H\subseteq\widetilde{E}$, which contradicts the assumption
$t\notin\widetilde{E}$.

Case 2: the wave fronts of the families $k+1,\dots,N$ satisfy

$\sum_{\begin{subarray}{c}\text{$\alpha$ wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=k+1,\dots,N\end{subarray}}|s_{\alpha}|\geq\frac{L\iota}{10}.$

Then there are two possibilities:

  * i)

an amount of waves of combined strength at least $L\iota/20$ is generated on
the interval $]t-y_{m}/c,t[$, where $c$ is the same as in (2.6), either at
discontinuity points of $\mathbf{u}_{bm}$ or when a wave hits the boundary. By
arguing as in Case 1, item i) we conclude that either
$\mu^{IC}(\\{(t,0)\\})>0$ or $t$ is a discontinuity point of $\mathbf{u}_{b}$.
Either way, we find a contradiction with the definition (9.2) of
$\widetilde{E}$ and the assumption $t\notin\widetilde{E}$.

  * ii)

an amount of wave fronts of total strength at least $L\iota/20$ is generated
by the interactions with other waves on the set
$]t-y_{m}/c,t[\times]0,y_{m}[$. By recalling [10, formula (7.96)] this implies
that $\mu^{IC}_{m}([t-y_{m}/c,t]\times[0,y_{m}])\geq L\iota/20$ and by arguing
as in item i) of Step 1 this in turn implies $\mu^{IC}(\\{(t,0)\\})>0$, which
contradicts the assumption $t\notin\widetilde{E}$.

Case 3: there is $\upsilon>0$ such that the combined amount of the waves of
the $k$-th family with negative speed $\upsilon$-bounded away from $0$
satisfies

$\sum_{\begin{subarray}{c}\text{$\alpha$ wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=k,\,\varsigma_{\alpha}<-\upsilon\end{subarray}}|s_{\alpha}|\geq\frac{L\iota}{10}.$

Then there are three possibilities:

  * i)

the combined strength of the wave fronts hitting the boundary in the interval
$]t,t+2y_{m}/\upsilon[$ is at least $L\iota/30$. This is ruled out by arguing
as in Case 1, item i).

  * ii)

an amount of wave fronts of total strength at least $L\iota/30$ is cancelled
by the interactions with other waves on the set
$]t,t+2y_{m}/\upsilon[\times]0,y_{m}[$. This is ruled out by arguing as in
Case 1, item i).

  * iii)

in the time interval $]t,t+2y_{m}/\upsilon[$ an amount of waves of total
strength at least $L\iota/30$ increases its speed from at most $-\upsilon$ to
at least $-\upsilon/2$ owing to the interaction with other waves. By the
Lipschitz continuity of the map $\mathbf{u}\mapsto\lambda_{k}(\mathbf{u})$
this implies that
$\mu^{IC}_{m}([t,t+2y_{m}/\upsilon]~{}\times~{}[0,y_{m}])~{}\geq~{}\mathcal{O}(1)L\iota\upsilon~{}>~{}0$
and by arguing as in item i) of Step 1 this in turn implies
$\mu^{IC}(\\{(t,0)\\})>0$, which contradicts the assumption
$t\notin\widetilde{E}$.

Case 4: there is $\upsilon>0$ such that the combined amount of the waves of
the $k$-th family with positive speed $\upsilon$-bounded away from $0$
satisfies

$\sum_{\begin{subarray}{c}\text{$\alpha$ wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=k,\,\varsigma_{\alpha}>\upsilon\end{subarray}}|s_{\alpha}|\geq\frac{L\iota}{10}.$

This case is ruled out by combining the arguments in Case 2 and Case 3.
Case 5: there is a sequence $\eta_{m}$ such that $\eta_{m}>0$ for every $m$,
$\lim_{m\to+\infty}\eta_{m}=0$ and

(9.36) $\sum_{\begin{subarray}{c}\text{$\alpha$ wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=k,\,|\varsigma_{\alpha}|<\eta_{m}\end{subarray}}|s_{\alpha}|\geq\frac{L\iota}{10}.$

Assume for a moment that there are $\eta>0$, $\nu>0$ such that

$\sum_{\begin{subarray}{c}\text{$\alpha$ wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=k+1,\dots,N\end{subarray}}|s_{\alpha}|+\sum_{\begin{subarray}{c}\text{$\alpha$
wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=1,\dots,k-1\end{subarray}}|s_{\alpha}|+\sum_{\begin{subarray}{c}\text{$\alpha$
wave in}\\\ \,]x_{m},y_{m}[\\\
i_{\alpha}=k,|\varsigma_{\alpha}|\geq\eta\end{subarray}}|s_{\alpha}|\geq\nu,$

then we can argue as in Cases $1,\dots,4$ before. Hence, we can assume with no
loss of generality that the strength of all the waves different from the one
in (9.36) vanishes in the $m\to+\infty$ limit. Up to subsequences, we know
that one of the following two possibilities is verified.
Case 5A: there is $\nu>0$ such that for every $m\in\mathbb{N}$ there is at
least one front comprised between $x_{m}$ and $y_{m}$ with strength greater or
equal than $\nu>0$ and speed in modulus smaller or equal than $\eta_{m}$.
Since owing to Lemma 8.2 the maximal strength $C_{9}r_{\varepsilon_{m}}$ of
the rarefaction fronts vanishes in the $m\to+\infty$ limit, this front must be
a shock front. By the construction of maximal $\nu$-shock fronts given in [10,
p. 219] and the definition (9.25) of $G$, this implies $t\in
G\subseteq\widetilde{E}$ and hence contradicts the assumption
$t\notin\widetilde{E}$.
Case 5B: the maximal strength of the wave-fronts confined between $x_{m}$ and
$y_{m}$ vanishes in the $m\to+\infty$ limit. Since the strength of all the
waves of the $j$-th family, $j\neq k$, vanishes in the $m\to+\infty$ limit,
the values $\mathbf{u}_{\varepsilon_{m}}(t,x)$ with $x\in]x_{m},y_{m}[$ get
closer and closer to the curve
$\mathbf{t}_{k}(\mathbf{u}_{\varepsilon_{m}}(t,x_{m}),\cdot)$ in the
$m\to+\infty$ limit. Since also the maximal speed of the wave fronts of
$\mathbf{u}_{\varepsilon_{m}}(t,\cdot)$ comprised between $x_{m}$ and $y_{m}$
vanishes in the $m\to+\infty$, we conclude that

$\lim_{m\to+\infty}|\mathbf{u}_{\varepsilon_{m}}(t,x)-\mathbf{t}_{k}\big{(}\mathbf{u}_{\varepsilon_{m}}(t,x_{m}),\bar{s}(\mathbf{u}_{\varepsilon_{m}}(t,x_{m})\big{)}|=0\quad\text{for
every $x\in[x_{m},y_{m}]$},$

which contradicts (9.31). ∎

###  9.4. Definition 1.1 in the case $t\notin\widetilde{E}$

In this paragraph we verify the boundary condition in the statement of Theorem
9.1 on the set $\mathbb{R}_{+}\setminus\widetilde{E}$, where $\widetilde{E}$
is the same as in (9.2). In other words, we show that for a.e.
$t\in\mathbb{R}_{+}$, $t\notin\widetilde{E}$, we have
$\mathbf{\bar{v}}(t)\sim_{\mathbf{D}}\mathbf{v}_{b}(t)$ in the sense of
Definition 1.1, provided $\mathbf{\bar{v}}(t)$ is the trace of the limit
function $\mathbf{v}$ as in the statement of Theorem 9.1. This is a
consequence of Lemma 9.6 and Lemma 9.7 below.

For simplicity in this paragraph and in the following one we assume (2.11).
The case of assumption C) in Hypothesis 5 is analogous. The case of assumption
B) is slightly more delicate and is explicitly discussed in §9.6.2 since the
most prominent example where assumption B) is satisfied are the Navier-Stokes
and viscous MHD equations in Eulerian coordinates when the fluid velocity
vanishes, and in that case the $k$-th characteristic field is linearly
degenerate.

Let $\mathbf{\bar{u}}_{m}$ be the same as in (9.28) then owing to (4.53) we
have

$\boldsymbol{\beta}(\boldsymbol{\phi}(\mathbf{\bar{u}}_{m}(t),\xi^{m}_{\ell+1}(t),\dots,\xi^{m}_{k}(t)),\mathbf{u}_{b}^{\varepsilon_{m}}(t))=\mathbf{0}_{N}\quad\text{for
a.e. $t\in\mathbb{R}_{+}$}$

and suitable functions $\xi^{m}_{\ell+1}(t),\dots,\xi^{m}_{k}(t)$ satisfying
$|\xi^{m}_{i}|\leq\delta$ for every $i=\ell+1,\dots,k$ and $m\in\mathbb{N}$.
Note that owing to (5.1) we have that, up to subsequences,

(9.37) $\mathbf{u}_{b}^{\varepsilon_{m}}(t)\to\mathbf{u}_{b}(t),\quad\text{for
a.e. $t\in\mathbb{R}_{+}$}.$

By combining (9.37) and (9.29) with the Implicit Function Theorem we conclude
that for a.e. $t\notin\widetilde{E}$ we have that
$\xi^{m}_{\ell+1}(t),\dots,\xi^{m}_{k}(t)$ converge as $m\to+\infty$ to
suitable functions $\xi_{\ell+1}(t),\dots,\xi_{k}(t)$ satisfying

(9.38)
$\boldsymbol{\beta}(\boldsymbol{\phi}(\mathbf{\bar{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k}(t)),\mathbf{u}_{b}(t))=\mathbf{0}_{N}\quad\text{for
a.e. $t\in\mathbb{R}_{+}$},$

provided $\mathbf{\bar{u}}$ is the same as in (9.27).

######  Lemma 9.6.

Let $\mathbf{\bar{u}}(t)$ be the same as in (9.27) and $\xi_{k}(t)$ the same
as in (9.38). For a.e. $t\in\mathbb{R}_{+}$, $t\notin\widetilde{E}$,
satisfying $\xi_{k}(t)\neq\underline{s}(\mathbf{\bar{u}}(t))$, we have that
$\mathbf{\bar{v}}(t):=\mathbf{v}(\mathbf{\bar{u}}(t))$ satisfies
$\mathbf{\bar{v}}(t)\sim_{\mathbf{D}}\mathbf{v}_{b}(t)$ in the sense of
Definition 1.1.In particular, property ii) is satisfied provided
$\mathbf{\underline{v}}=\mathbf{\bar{v}}(t)$.

###### Proof.

We recall Remark 5.2 and in particular that the approximation
$\mathbf{u}_{\varepsilon_{m}}$ satisfies (4.57) with
$\mathbf{u}^{+}=\mathbf{\bar{u}}_{m}(t)$ and $\xi_{k}=\xi_{k}^{m}(t)$
respectively. Up to subsequences we can assume that one of the following
condition holds true:

  * a)

$\lambda_{k}(\mathbf{\bar{u}}_{m}(t))>0$ and $\xi^{m}_{k}(t)=0$ for every $m$;

  * b)

$\lambda_{k}(\mathbf{\bar{u}}_{m}(t))\leq 0$ and
$\xi^{m}_{k}(t)\leq\underline{s}(\mathbf{\bar{u}}_{m})(t)$ or every $m$.

Condition a) above implies $\xi_{k}(t)=0$, which by (4.50) yields
$\boldsymbol{\psi}_{p}(\mathbf{\bar{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k-1}(t),0)=\mathbf{0}_{N}$
and by plugging this equality into (4.49) and recalling (9.38) we arrive at

(9.39)
$\boldsymbol{\beta}(\boldsymbol{\psi}_{s}(\mathbf{\bar{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k-1}(t)),\mathbf{u}_{b}(t))=\mathbf{0}_{N},$

where $\boldsymbol{\psi}_{s}$ is defined in (4.40). By the construction of
$\boldsymbol{\psi}_{s}$, there is a solution of (3.1) with initial datum
$\boldsymbol{\psi}_{s}(\mathbf{\bar{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k-1}(t))$
which converges exponentially fast to $\mathbf{\bar{u}}(t)$ as $y\to+\infty$.
Owing to (9.39), this implies that there is a boundary layer $\mathbf{w}$
satisfying (4.7) with $\mathbf{\underline{u}}=\mathbf{\bar{u}}(t)$ and going
back to the variable $\mathbf{v}$ we conclude that
$\mathbf{\bar{v}}(t):=\mathbf{v}(\mathbf{\bar{u}}(t))$ satisfies property ii)
in Definition 1.1 with $\mathbf{\underline{v}}=\mathbf{\bar{v}}(t)$.

Owing to (9.29) condition b) above yields
$\lambda_{k}(\mathbf{\bar{u}}(t))\leq 0$ and
$\xi_{k}(t)\leq\underline{s}(\mathbf{\bar{u}}(t))$. Since
$\xi_{k}(t)\neq\underline{s}(\mathbf{\bar{u}}(t))$ by assumption, we conclude
that $\xi_{k}(t)<\underline{s}(\mathbf{\bar{u}}(t))$. If
$\lambda_{k}(\mathbf{\bar{u}}(t))<0$ then by recalling case vi) in §4.6.1 we
conclude that $\mathbf{\bar{v}}(t):=\mathbf{v}(\mathbf{\bar{u}}(t))$ satisfies
property ii) in Definition 1.1 with
$\mathbf{\underline{v}}=\mathbf{\bar{v}}(t)$. If
$\lambda_{k}(\mathbf{\bar{u}}(t))=0$ then
$\underline{s}(\mathbf{\bar{u}}(t))=\bar{s}(\mathbf{\bar{u}}(t))=0$ by (4.13)
and (4.15), and by recalling case iii) in §4.6.1 we conclude the proof of the
lemma. ∎

######  Lemma 9.7.

Let $\mathbf{\bar{u}}(t)$ be the same as in (9.27) and $\xi_{k}(t)$ the same
as in (9.38). For a.e. $t\in\mathbb{R}_{+}$, $t\notin\widetilde{E}$,
satisfying $\xi_{k}(t)=\underline{s}(\mathbf{\bar{u}}(t))$ we have that
$\mathbf{\bar{v}}(t):=\mathbf{v}(\mathbf{\bar{u}}(t))$ satisfies
$\mathbf{\bar{v}}(t)\sim_{\mathbf{D}}\mathbf{v}_{b}(t)$ in the sense of
Definition 1.1.In particular, property ii) is satisfied provided
$\mathbf{\underline{v}}(t)=\mathbf{v}(\boldsymbol{\zeta}_{k}(\mathbf{\bar{u}}(t),\underline{s}(\mathbf{\bar{u}}(t))))$.

###### Proof.

We recall cases a) and b) in the proof of Lemma 9.6. In case a), we obtain
$\xi_{k}(t)=0$, which implies
$\underline{s}(\mathbf{\bar{u}}(t))=\xi_{k}(t)=0$ and hence
$\boldsymbol{\zeta}_{k}(\mathbf{\bar{u}}(t),\underline{s}(\mathbf{\bar{u}}(t))))=\mathbf{\bar{u}}(t)$.
Owing to (9.39), and arguing as in the proof of Lemma 9.6, this yields the
desired conclusion.

We now focus on case b). If $\lambda_{k}(\mathbf{\bar{u}}(t))=0$ then
$\underline{s}(\mathbf{\bar{u}})=0$, which yields on the one hand the equality
$\boldsymbol{\zeta}_{k}(\mathbf{\bar{u}}(t),\underline{s}(\mathbf{\bar{u}}(t)))=\mathbf{\bar{u}}(t)$,
on the other the equalities $\xi_{k}(t)=\underline{s}(\mathbf{\bar{u}}(t))=0$
and hence (by arguing as in the proof of Lemma 9.6) the equality (9.39) and
from this we arrive at the desired conclusion. Conversely, if
$\lambda_{k}(\mathbf{\bar{u}}(t))<0$ we can combine (9.38) with the same
argument as in case v) in §4.6.1 and conclude the proof of the lemma. ∎

###  9.5. Definition 1.1 in the case $t\in\widetilde{E}$

In this paragraph we verify the boundary condition in the statement of Theorem
9.1 on the set $\widetilde{E}$. Namely, we show that for a.e.
$t\in\widetilde{E}$, we have
$\mathbf{\bar{v}}(t)\sim_{\mathbf{D}}\mathbf{v}_{b}(t)$ in the sense of
Definition 1.1, provided $\mathbf{\bar{v}}(t)$ is the same as in the statement
of Theorem 9.1. Since the sets $D,F$ and $H$ in the definition (9.2) of
$\widetilde{E}$ are negligible, this is a consequence of Lemma 9.10 below. We
recall that in this paragraph, as in the previous one, to ease the exposition
we assume (2.11).

Owing to the construction in §9.3.2 we have that if $t\in G$ then there is
$\nu>0$ and a sequence of maximal $\nu$-shock fronts $\\{y_{m}\\}$ such that
$y_{m}\to y$ in $C^{0}$ as $m\to+\infty$ for some Lipschitz continuous curve
$y$ such that $y(t)=0$, $y^{\prime}(t)=0$. We now fix $t\in G$ and term
$\mathbf{u}^{-}_{m}(t)$ and $\mathbf{u}^{+}_{m}(t)$ the left and right state
of the shock curve $y_{m}$ at time $t$. We recall (9.27) and by arguing as in
the proof of [10, formula (10.80)] and of [10, formula (10.66)] we get

(9.40)
$\lim_{m\to+\infty}\mathbf{u}^{+}_{m}(t)=\mathbf{\bar{u}}(t),\quad\lambda_{k}(\mathbf{\bar{u}}(t))\leq
0\;\text{for a.e. $t\in G$.}$

Owing to the second condition in (9.40), for a.e. $t\in G$ by setting

(9.41)
$\mathbf{\underline{u}}(t)=\mathbf{t}_{k}(\mathbf{\bar{u}}(t),\underline{s}(\mathbf{\bar{u}}(t)))$

we get that $\mathbf{\underline{u}}(t)$ and $\mathbf{\bar{u}}(t)$ are the left
and the right state, respectively, of a $0$-speed Lax admissible shock.

######  Lemma 9.8.

Let $\mathbf{\underline{u}}(t)$ be the same as in (9.41), then

(9.42)
$\mathbf{\underline{u}}(t)=\lim_{m\to+\infty}\mathbf{u}^{-}_{m}(t)\;\text{for
a.e. $t\in G$.}$

###### Proof.

By the definition of $\mathbf{u}^{+}_{m}(t)$ and $\mathbf{u}^{-}_{m}(t)$ we
have

(9.43)
$\mathbf{u}^{-}_{m}(t)=\mathbf{t}_{k}(\mathbf{u}^{+}_{m}(t),s_{m}(t))\;\text{for
a.e. $t\in G$}$

and for some $s_{m}(t)\in[-\delta,\delta].$ Note that, owing to the definition
of $G$ and of $\nu$-shock front, $s_{m}(t)\geq\nu(t)/2$ for every
$m\in\mathbb{N}$ and some $\nu(t)>0$. We now fix any such $t\in G$ and point
out that, up to subsequences (which to ease the notation we do not relabel),
$s_{m}(t)$ converges to to some limit $s(t)\geq\nu(t)/2>0$. By the continuity
of the function $\sigma_{k}$ defined by (4.11) and by recalling (9.40) we
arrive at

$\lim_{m\to+\infty}\sigma_{k}(\mathbf{u}^{+}_{m}(t),s_{m}(t))=\sigma_{k}(\mathbf{\bar{u}}(t),s(t)).$

On the other hand, by the analysis in [10, p.227] we get that
$\lim_{m\to+\infty}\sigma_{k}(\mathbf{u}^{+}_{m}(t),s_{m}(t))=0$, which
combined with (9.40) yields $s(t)=\underline{s}(\mathbf{\bar{u}}(t))$. By
passing to the limit in (9.43) and recalling (9.41) we eventually arrive at
(9.42). ∎

######  Lemma 9.9.

Let $\mathbf{\bar{u}}_{m}(t)$ and $\mathbf{\underline{u}}(t)$ be the same as
in (9.28) and in (9.41), respectively; then

(9.44)
$\mathbf{\underline{u}}(t)=\lim_{m\to+\infty}\mathbf{\bar{u}}_{m}(t)\quad\text{for
a.e. $t\in G$}.$

###### Proof.

We fix $t\in G\setminus F\cup H\cup D$ satisfying (9.42) and establish (9.44).
Since the sets $F$, $H$ and $D$ are all negligible, this provides the proof of
the lemma. We argue by contradiction and assume that there is $\iota>0$ such
that

$|\lim_{m\to+\infty}\mathbf{\bar{u}}_{m}(t)-\lim_{m\to+\infty}\mathbf{u}^{-}_{m}(t)|\geq\iota.$

In particular, this implies that there is a sequence of points
$\\{\widetilde{y}_{m}\\}$ (to ease the notation, we do not indicate the
dependence on $t$) such that $\lim_{m\to+\infty}\widetilde{y}_{m}=0$,
$0<\widetilde{y}_{m}<y_{m}$, $\mathbf{u}_{\varepsilon_{m}}$ is continuous at
$(t,\widetilde{y}_{m})$,
$|\mathbf{u}_{\varepsilon_{m}}(t,\widetilde{y}_{m})-\mathbf{u}^{-}_{m}|$
vanishes in the $m\to+\infty$ limit, and therefore

$|\mathbf{u}_{\varepsilon_{m}}(t,\widetilde{y}_{m})-\lim_{m\to+\infty}\mathbf{\bar{u}}_{m}(t)|\geq\iota/2\quad\text{for
every $m$}.$

We can then construct a sequence of points $\\{x_{m}\\}$ such that
$0<x_{m}<\widetilde{y}_{m}$, $\mathbf{u}_{\varepsilon_{m}}(t,\cdot)$ is
continuous at $x_{m}$ and $\mathbf{u}_{\varepsilon_{m}}(t,x_{m})$ is
arbitrarily close to $\mathbf{\bar{u}}_{m}(t)$, and hence

$|\mathbf{u}_{\varepsilon_{m}}(t,\widetilde{y}_{m})-\mathbf{u}_{\varepsilon_{m}}(t,x_{m})|\geq\iota/4\quad\text{for
every $m$.}$

To show that this implies $t\in F\cup H\cup D$ and hence find a contradiction
with the assumptions of the present lemma we can then argue as in the proof of
Lemma 9.5, the only point that requires a different argument is the handling
of Case 5A. Assume therefore that we are in Case 5, that all the waves
different from the one in (9.36) vanish in the $m\to+\infty$ limit and that
there is $\widetilde{\nu}>0$ such that for every $m\in\mathbb{N}$ there is at
least one shock-front of the $k$-th family comprised between $x_{m}$ (on the
left) and $\widetilde{y}_{m}$ (on the right) and with strength at least
$\widetilde{\nu}$ and with speed vanishing in the $m\to+\infty$ limit. Up to
subsequences, we can then assume that one of the following conditions is
satisfied.

  * •

The speed of the shock located at $y_{m}$ is bounded away from $0$, uniformly
in $m$. Since the strength of this shock is at least $\nu/2$ by the
construction of maximal $\nu$-shock front in [10, p. 219] we can handle this
case by arguing as in Case 3 and Case 4 in the proof of Lemma 9.5 to conclude
that in this case $t\in F\cup H\cup D$, which yields a contradiction.

  * •

The speed of the shock located at $y_{m}$ vanishes in the $m\to+\infty$ limit.
Since $\mathbf{u}^{-}_{m}$ is the left state of a shock with strength at least
$\nu/2$ and speed vanishing in the $m\to+\infty$ limit, the genuine
nonlinearity of the $k$-th characteristic field implies that
$\lambda_{k}(\mathbf{u}^{-}_{m})\geq\mathcal{O}(1)\nu>0$ for $m$ sufficiently
large and this in turn yields
$\lambda_{k}(\mathbf{u}_{\varepsilon_{m}}(t,\widetilde{y}_{m}))\geq\mathcal{O}(1)\nu>0$.
On the other side, by assumption between $x_{m}$ and $\widetilde{y}_{m}$ there
is a shock with strength at least $\widetilde{\nu}>0$ and speed vanishing in
the $m\to+\infty$ limit. By the Lax entropy condition, this implies that there
is a sequence $\\{z_{m}\\}\subseteq]x_{m},\widetilde{y}_{m}[$ such that
$z_{m}$ is just at the right of the shock and
$\lambda_{k}(\mathbf{u}_{\varepsilon_{m}}(t,z_{m})\leq-\mathcal{O}(1)\widetilde{\nu}<0$
for every $m\in\mathbb{N}$. We claim that this contradicts the assumption that
the total strength of all the waves different from the ones in (9.36) vanishes
in the $m\to+\infty$ limit. Indeed, since $z_{m}<\widetilde{y}_{m}$,
$\lambda_{k}(\mathbf{u}_{\varepsilon_{m}}(t,z_{m})\leq-\mathcal{O}(1)\widetilde{\nu}<0$,
$\lambda_{k}(\mathbf{u}_{\varepsilon_{m}}(t,\widetilde{y}_{m}))\geq\mathcal{O}(1)\nu>0$
and the total strength of the wave-fronts of the $j$-th family, $j\neq k$, in
$]z_{m},\widetilde{y}_{m}[$ vanishes in the in the $m\to+\infty$ limit then
there is $\mathcal{O}(1)\nu>0$ and a group $\mathcal{I}_{m}$ of rarefaction
wave-fronts of the $k$-th family such that

$\sum_{\begin{subarray}{c}\alpha\in\mathcal{I}_{m}\\\
i_{\alpha}=k,s_{\alpha}<0\end{subarray}}|s_{\alpha}|\geq\mathcal{O}(1)\nu,\quad\lambda_{k}(\mathbf{u}^{+}_{m\alpha})\geq\mathcal{O}(1)\nu>0\;\text{for
every $\alpha\in\mathcal{I}_{m}$ },$

where $\mathbf{u}^{+}_{m\alpha}$ represents the state at the right-hand side
of the wave-front $\alpha$. This however contradicts the assumption that the
total strength of all the waves different from the ones in (9.36) vanishes in
the $m\to+\infty$ limit and hence concludes the proof of the lemma. ∎

######  Lemma 9.10.

For a.e. $t\in G$ the trace $\mathbf{\bar{v}}(t):=\lim_{x\to
0^{+}}\mathbf{v}(t,x)$ satisfies
$\mathbf{\bar{v}}(t)\sim_{\mathbf{D}}\mathbf{v}_{b}(t)$, in the sense of
Definition 1.1. In particular, conditions i) and ii) are satisfied provided
$\mathbf{\underline{v}}(t):=\mathbf{v}(\mathbf{\underline{u}}(t))$ and
$\mathbf{\underline{u}}(t)$ is the same as in (9.41) and satisfies (9.42) and
(9.44).

###### Proof.

We proceed according to the following steps.
Step 1: we establish property ii) in Definition 1.1, provided
$\mathbf{\underline{v}}:=\mathbf{v}(\mathbf{\underline{u}}(t))$ and
$\mathbf{\underline{u}}(t)$ is the same as in (9.41). By construction, we have
that202020We recall that in §9.5, as in §9.4, we assume (2.11) and hence focus
on case A in Hypothesis 5 for a.e. $t\in\mathbb{R}_{+}$

(9.45)
$\boldsymbol{\beta}(\boldsymbol{\phi}(\mathbf{\bar{u}}_{m}(t),\xi^{m}_{\ell+1}(t),\dots,\xi^{m}_{k}(t)),\mathbf{u}^{\varepsilon_{m}}_{b}(t))=\mathbf{0}_{N},$

with $\mathbf{\bar{u}}_{m}(t)$ and $\xi^{m}_{k}(t)$ satisfying (4.57). Owing
to (9.37) we have that
$\lim_{m\to+\infty}\mathbf{u}_{bm}(t)=\mathbf{u}_{b}(t)$ for a.e.
$t\in\mathbb{R}_{+}$ and by (9.44) we can pass to the limit in (9.45) and
conclude that for a.e. $t\in G$ we have that
$\xi^{m}_{1}(t),\dots,\xi^{m}_{k}(t)$ converge as $m\to+\infty$ to some
$\xi_{1}(t),\dots,\xi_{k}(t)$ satisfying

(9.46)
$\boldsymbol{\beta}(\boldsymbol{\phi}(\mathbf{\underline{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k}(t)),\mathbf{u}_{b}(t))=\mathbf{0}_{N}.$

Owing to (4.57) for a.e. $t\in G$ we can assume that up to subsequences one of
the following conditions is satisfied for every $m$.

  * •

$\lambda_{k}(\mathbf{\bar{u}}_{m}(t))>0$ and $\xi^{m}_{k}(t)=0$, which by
(9.44) yields $\lambda_{k}(\mathbf{\underline{u}}(t))\geq 0$ and
$\xi_{k}(t)=0$. Owing to (4.50) and (4.49), this implies

$\boldsymbol{\phi}(\mathbf{\underline{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k}(t))=\boldsymbol{\psi}_{s}(\mathbf{\underline{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k-1}(t))$

and by plugging this equality into (9.46), recalling the definition (4.40) of
$\boldsymbol{\psi}_{s}$ and arguing as in the proof of Lemma 9.6 this yields
the existence of a boundary layer $\mathbf{w}$ satisfying (4.7).

  * •

$\lambda_{k}(\mathbf{\bar{u}}_{m}(t))\leq 0$ and
$\xi^{m}_{k}(t)\leq\underline{s}(\mathbf{\bar{u}}_{m}(t))$, which by (9.44)
yields $\lambda_{k}(\mathbf{\underline{u}}(t))\leq 0$ and
$\xi_{k}(t)\leq\underline{s}(\mathbf{\underline{u}}(t))$. Since
$\mathbf{\underline{u}}(t)$ is defined as in (9.41), with
$\lambda_{k}(\mathbf{\bar{u}}(t))\leq 0$ by (9.40), we have
$\lambda_{k}(\mathbf{\underline{u}}(t))\geq 0$. We conclude that
$\lambda_{k}(\mathbf{\underline{u}}(t))=0$, which implies
$\underline{s}(\mathbf{\underline{u}}(t))=0$ and hence $\xi_{k}\leq 0$ and, by
(9.41), $\mathbf{\underline{u}}(t)=\mathbf{\bar{u}}(t)$. We conclude that

$\begin{split}\boldsymbol{\phi}(\mathbf{\underline{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k}(t))&\stackrel{{\scriptstyle\eqref{e:bl10}}}{{=}}\boldsymbol{\psi}_{s}(\underbrace{\boldsymbol{\zeta}_{k}(\mathbf{\underline{u}}(t)}_{=\boldsymbol{\pi}_{\mathbf{u}}\circ\mathbf{b}_{k}(\mathbf{\underline{u}}(t))\
\text{by~{}\eqref{e:uguale1}}},\xi_{k}(t)),\xi_{\ell+1}(t),\dots,\xi_{k-1}(t))\\\
&+\boldsymbol{\psi}_{p}(\mathbf{\underline{u}}(t),\xi_{\ell+1}(t),\dots,\xi_{k-1}(t),\xi_{k}(t)).\\\
\end{split}$

We now recall the construction of the Slaving Manifold given in §4.4. In
particular, we recall (4.48) and that the solution of the Cauchy problem
obtained by coupling (3.2) and the initial datum $\boldsymbol{\omega}(0)$ in
(4.48) satisfies (4.46), where $\boldsymbol{\omega}_{0}$ is by (4.45) the
solution of (3.2) with initial datum
$\boldsymbol{\gamma}_{k}(\mathbf{\underline{u}}(t),\xi_{k}(t))=\mathbf{b}_{k}(\mathbf{\underline{u}}(t),\xi_{k}(t))$.
Owing to Lemma 4.5, the orbit $\boldsymbol{\omega}_{0}$ converges to
$(\mathbf{\underline{u}}(t),\mathbf{0}_{N-h})$ as $y\to+\infty$. Wrapping up,
we conclude that the solution of the Cauchy problem obtained by coupling (3.2)
and $\boldsymbol{\omega}(0)$ in (4.48) converges to
$(\mathbf{\underline{u}}(t),\mathbf{0}_{N-h})$ as $y\to+\infty$. Recalling
that the function $\mathbf{a}$ in (3.2) is given by (3.3), this yields the
existence of a boundary layer $\mathbf{w}$ satisfying (4.7) with asymptotic
state $\mathbf{\underline{u}}(t)$.

Step 2: we establish property i) in Definition (1.1). We have

$\mathbf{f}(\mathbf{\underline{v}}(t))=\mathbf{f}\circ\mathbf{v}\circ\mathbf{\underline{u}}(t)\stackrel{{\scriptstyle\eqref{e:tracciainterna}}}{{=}}\lim_{m\to+\infty}\mathbf{f}\circ\mathbf{v}\circ\mathbf{\bar{u}}_{m}(t)\stackrel{{\scriptstyle\eqref{e:flupointwise},\eqref{e:baruemme}}}{{=}}\mathbf{f}(\mathbf{\bar{v}}(t)),$

and since $\lambda_{k}(\mathbf{\bar{u}}(t))\leq 0$ by (9.40) the $0$-speed
shock between $\mathbf{\underline{u}}$ (on the left) and $\mathbf{\bar{u}}(t)$
(on the right) is Lax admissible. This concludes the proof of Lemma 9.10. ∎

###  9.6. Case of a linearly degenerate case characteristic field

We now establish the proof of Theorem 9.1 in the case of a linearly degenerate
boundary characteristic field (2.9). The proof follows the same outline as in
the case of a genuinely nonlinear characteristic field, so we only discuss the
points where the argument is different.

####  9.6.1. Proof of Lemma 9.2 (linearly degenerate $k$-th characteristic
field)

The argument is very similar to the one in the case of a genuinely nonlinear
characteristic field, so we only provide a sketch. First, we recall §5.3.2 and
also we point out that instead of (9.4) we have the inequality

(9.47)
$|\mathbf{f}\circ\mathbf{v}(\mathbf{\widetilde{u}})-\mathbf{f}\circ\mathbf{v}(\mathbf{t}_{k}(\mathbf{\widetilde{u}},s))|\leq\mathcal{O}(1)|s||\lambda_{k}(\mathbf{\widetilde{u}})|.$

Next, we consider the same functional $\Lambda$ as in (9.5). We recall that
now the functional $S$ in $\Upsilon$ is given by the second line of (7.8). We
now discuss Cases I,$\dots$,III in the proof of Lemma 9.2.
Case I: it suffices to establish (9.7). To this end, we term $\mathbf{u}^{-}$
and $\mathbf{u}^{+}$ the left and right state of the hitting wave of the
$j$-th family, $j<k$, and by $\mathbf{\hat{u}}$ the same state as in (5.9).
Next, we point out that there are two possibilities:

  * •

if $\lambda_{k}(\mathbf{\hat{u}})>0$ by following the same computations done
in (9.8) for cases i),$\dots$, iii) in §5.3, we arrive at

$\Delta
F(\tau)\leq\mathcal{O}(1)|s_{j}|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{t}_{k}(\mathbf{\hat{u}},s_{k})))|$

and the second term at the right-hand side of the above expression can be
controlled by arguing as in (9.9).

  * •

if $\lambda_{k}(\mathbf{\hat{u}})\leq 0$ there is no wave front of the $k$-th
family entering the domain after the interaction and hence the last term at
the right hand side of (9.8) is not there, and in this way we arrive at (9.7).

Case II: we term $\mathbf{u}^{-}$ and $\mathbf{u}^{+}$ the left and right
state of the hitting wave front of the $k$-th family, and $\mathbf{\hat{u}}$
the same state as in (5.9). Assume that we have shown that

(9.48) $\Delta
F(\tau)\leq\mathcal{O}(1)|s_{k}|\big{(}|\xi_{k}|+[\lambda_{k}(\mathbf{u}^{+})]^{-}\big{)};$

then we can choose the constant $K_{5}$ in such a way that the function
$\Lambda$ is decreasing at $\tau$. To establish (9.48) we first assume that
$\lambda_{k}(\mathbf{\hat{u}})>0$ and we point out that

$\begin{split}\Delta
F(\tau)&\leq|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\mathbf{t}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\leq|\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))-\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))|+|\mathbf{f}(\mathbf{v}(\mathbf{u}^{+}))-\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{t}_{k}(\mathbf{\hat{u}},s^{\prime}_{k})))|\\\
&\stackrel{{\scriptstyle\eqref{e:melindeg},\eqref{e:dflux2}}}{{\leq}}[\lambda_{k}(\mathbf{u}^{+})]^{-}|s_{k}|+\mathcal{O}(1)|s_{k}||\xi_{k}|+|\lambda_{k}(\mathbf{\hat{u}})||s^{\prime}_{k}|\\\
&\leq[\lambda_{k}(\mathbf{u}^{+})]^{-}|s_{k}|+\mathcal{O}(1)|s_{k}||\xi_{k}|+\delta([\lambda_{k}(\mathbf{\hat{u}})]^{+}-[\lambda_{k}(\mathbf{u}^{+})]^{+})\\\
&\leq[\lambda_{k}(\mathbf{u}^{+})]^{-}|s_{k}|+\mathcal{O}(1)|s_{k}||\xi_{k}|+\delta\mathcal{O}(1)|\mathbf{\hat{u}}-\mathbf{u}^{+}|\stackrel{{\scriptstyle\eqref{e:melindeg}}}{{\leq}}[\lambda_{k}(\mathbf{u}^{+})]^{-}|s_{k}|+\mathcal{O}(1)|s_{k}||\xi_{k}|+\delta|s_{k}||\xi_{k}|,\end{split}$

that is (9.48). If $\lambda_{k}(\mathbf{\hat{u}})\leq 0$ the estimate is
actually simpler because we do not have the last term at the right hand side
of the above expression since there is no $k$-th wave front entering the
domain after the interaction.
Case III: as in the proof of Lemma 9.2 we use the following notation:

  * •

$\mathbf{u}_{bm}^{+}$ and $\mathbf{u}_{bm}^{-}$ are the right and left limits
$\mathbf{u}^{\varepsilon_{m}}_{b}(\tau^{+})$ and
$\mathbf{u}^{\varepsilon_{m}}_{b}(\tau^{-})$, respectively;

  * •

$\mathbf{\bar{u}}^{-}$ and $\mathbf{\bar{u}}^{+}$ are the traces of
$\mathbf{u}_{m}(t,\cdot)$ for $t=\tau^{-}$ and $t=\tau^{+}$, respectively;

  * •

$\mathbf{\hat{u}}$ is the same state as in (5.9).

As in the genuinely nonlinear case, it suffices to establish (9.17). We recall
that if $\lambda_{k}(\mathbf{\hat{u}})\leq 0$ then there is no wave front of
the $k$-th family entering the domain immediately after time $\tau$. This
yields $\mathbf{\bar{u}}^{+}=\mathbf{\hat{u}}$ and hence

$\Delta
F(\tau)\leq|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{u}^{-}))|\leq\mathcal{O}(1)\sum_{j=k+1}^{N}|s^{\prime}_{j}|\stackrel{{\scriptstyle\eqref{e:jd1}}}{{\leq}}\mathcal{O}(1)|\mathbf{u}_{bm}^{+}-\mathbf{u}^{-}_{bm}|,$

that is (9.7). We now consider the case $\lambda_{k}(\mathbf{\hat{u}})>0$ and
assume that there is a wave-front of the $k$-th family with strength $s_{k}$
entering the domain after the interaction. We point out that

(9.49) $\begin{split}\Delta
F(\tau)&\leq|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\mathbf{\bar{u}}^{-}))|+|\mathbf{f}(\mathbf{v}(\mathbf{\hat{u}}))-\mathbf{f}(\mathbf{v}(\underbrace{\mathbf{t}_{k}(\mathbf{\hat{u}},s_{k})}_{=\mathbf{\bar{u}}^{+}})|\\!\\!\stackrel{{\scriptstyle\eqref{e:jd1},\eqref{e:dflux2}}}{{\leq}}\\!\\!\mathcal{O}(1)|\mathbf{u}_{bm}^{+}-\mathbf{u}^{-}_{bm}|+\mathcal{O}(1)|s_{k}||\lambda_{k}(\mathbf{\hat{u}})|\end{split}$

If $\lambda_{k}(\mathbf{\bar{u}}^{-})\leq 0$ then we can control the last term
at the right-hand side of (9.49) as follows:

$\begin{split}|s_{k}||\lambda_{k}(\mathbf{\hat{u}})|=|s_{k}|[\lambda_{k}(\mathbf{\hat{u}})]^{+}\leq|s_{k}||\big{(}[\lambda_{k}(\mathbf{\hat{u}})]^{+}-[\lambda_{k}(\mathbf{\bar{u}}^{-})]^{+}|\big{)}\leq\mathcal{O}(1)|s_{k}||\mathbf{\hat{u}}-\mathbf{\bar{u}}^{-}|\stackrel{{\scriptstyle\eqref{e:jd1}}}{{\leq}}\mathcal{O}(1)|s_{k}||\mathbf{u}_{bm}^{+}-\mathbf{u}^{-}_{bm}|\end{split}$

and by plugging the above inequality into (9.49) we arrive at (9.7). If
$\lambda_{k}(\mathbf{\bar{u}}^{-})>0$ then we can apply (4.60) with $\xi_{k}$
and $\mathbf{\bar{u}}^{-}$ in place of $\xi^{\prime}_{k}$ and
$\mathbf{\bar{u}}$, respectively, and conclude that $\xi_{k}=0$, which in turn
owing to (7.30) (with $s_{k}$ in place of $s^{\prime}_{k}$) yields
$|s_{k}|\leq\mathcal{O}(1)|\mathbf{u}_{bm}^{+}-\mathbf{u}^{-}_{bm}|$ and by
plugging this inequality into (9.49) we eventually arrive at (9.7). This
concludes the proof of Lemma 9.2 in the linearly degenerate case.

####  9.6.2. Definition 1.1 in the linearly degenerate case

We now show that the limit function $\mathbf{v}$ in (1.8) satisfies the
boundary condition in the sense of Definition 1.1. We recall that if the
$k$-th characteristic field is linearly degenerate the Lax admissibility
requirement in condition i) is redundant because contact discontinuities are
always admissible. To ease the exposition, we focus on case B) in Hypothesis
5, the analysis in the other cases being similar but easier. We consider the
sequence $\\{\mathbf{\bar{u}}_{m}(t)\\}$ defined as in (9.28) and point out
that

(9.50)
$\boldsymbol{\beta}(\boldsymbol{\phi}(\mathbf{\bar{u}}_{m}(t),\xi^{m}_{\ell(\mathbf{u}^{\varepsilon_{m}}_{b}(t))+1}(t),\dots,\xi^{m}_{k}(t)),\mathbf{u}^{\varepsilon_{m}}_{b}(t))=\mathbf{0}_{N}\quad\text{for
a.e. $t\in\mathbb{R}_{+}$ and every $m\in\mathbb{N}$}.$

for suitable functions
$\xi^{m}_{\ell(\mathbf{u}^{\varepsilon_{m}}_{b}(t))+1}(t),\dots,\xi^{m}_{k}(t))$.
We now fix $t\in\mathbb{R}_{+}$ such that both (9.50) and the pointwise
convergence in (9.1) hold true, and such that
$\lim_{m\to+\infty}\mathbf{u}^{\varepsilon_{m}}_{b}(t)=\mathbf{u}_{b}(t)$. Up
to subsequences212121We explicitly point out that, in general, the particular
subsequence such that we have convergence _depends_ on the point $t$, so we
are not stating that the whole function $\mathbf{\bar{u}}_{m}$ converges at
almost every point., we have

(9.51) $\lim_{m\to+\infty}\mathbf{\bar{u}}_{m}(t)=\grave{\mathbf{u}}(t)$

for some limit value $\grave{\mathbf{u}}(t)\in\mathbb{R}^{N}$. Owing to (9.1),
we have

(9.52)
$\mathbf{f}(\mathbf{v}(\mathbf{\bar{u}}(t)))=\mathbf{f}(\mathbf{v}(\grave{\mathbf{u}}(t))),$

where $\mathbf{\bar{u}}$ is the same as in (9.27). By recalling (4.11),
formula (9.52) implies that either

(9.53) $\mathbf{\bar{u}}(t)=\grave{\mathbf{u}}(t)$

or

(9.54)
$\lambda_{k}(\grave{\mathbf{u}}(t))=\lambda_{k}(\bar{\mathbf{u}}(t))=0\;\text{and}\;\grave{\mathbf{u}}(t)=\mathbf{t}_{k}(\mathbf{\bar{u}}(t),s_{k}(t))\;\text{for
some $s_{k}(t)\in[-\delta,\delta]$}.$

We now turn our attention to (9.50), recall the analysis in [9, §2.2], and in
particular [9, formula (2.17)], and we conclude that
$\ell(\mathbf{u}^{\varepsilon_{m}}_{b}(t))$ depends on the sign of a certain
continuous function $\alpha$, and that the sign of $\alpha$ only depends on
the last $N-h$ components of $\mathbf{u}^{\varepsilon_{m}}_{b}(t)$, where we
recall that $h$ denotes the dimension of the kernel of the matrix
$\mathbf{B}$, see (2.1). Since
$\lim_{t\to+\infty}\mathbf{u}^{\varepsilon_{m}}_{b}(t)=\mathbf{u}_{b}(t)$, up
to subsequences one of the following conditions is satisfied:
Case 1: $\alpha(\mathbf{u}^{\varepsilon_{m}}_{b}(t))\leq 0$ for every $m$,
which implies $\alpha(\mathbf{u}_{b}(t))\leq 0$. Owing to [9, formula (2.17)]
in this case we have $\ell(\mathbf{u}_{bm})=h=\ell(\mathbf{u}_{b})$ for every
$m$. We can then pass to the limit in (9.50) and arrive at

(9.55)
$\boldsymbol{\beta}(\boldsymbol{\phi}(\mathbf{\grave{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k}(t)),\mathbf{u}_{b}(t))=\mathbf{0}_{N}.$

for suitable values $\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k}(t)$.
Case 1a: we have (9.53). In this case formula (9.55) boils down to

(9.56)
$\boldsymbol{\beta}(\boldsymbol{\phi}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k}(t)),\mathbf{u}_{b}(t))=\mathbf{0}_{N}.$

We separately consider the following cases.

  * •

$\lambda_{k}(\mathbf{\bar{u}}(t)))>0$. First, we point out that

(9.57) $\xi_{k}(t)=0\;\text{if $\lambda_{k}(\mathbf{\bar{u}}(t)))>0$}.$

To establish (9.57), it suffices to recall (4.60), infer that
$\lambda_{k}(\mathbf{\bar{u}}_{m}(t))>0$ and thus $\xi^{m}_{k}(t)=0$, and then
pass to the limit. We then have

$\begin{split}\boldsymbol{\phi}&(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k}(t))\stackrel{{\scriptstyle\eqref{e:giusto}}}{{=}}\boldsymbol{\phi}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t),0)\\\
&\stackrel{{\scriptstyle\eqref{e:bl10}}}{{=}}\boldsymbol{\psi}_{s}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t))+\boldsymbol{\psi}_{p}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t),0)\\\
&\stackrel{{\scriptstyle\eqref{e:p2}}}{{=}}\boldsymbol{\psi}_{s}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t))\end{split}$

By the definition (4.40) there is a solution of (3.1) with initial datum given
by the value
$\boldsymbol{\psi}_{s}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t))$
which converges to $\mathbf{\bar{u}}(t)$ as $y\to+\infty$ and owing to (9.56)
we conclude that there is a solution of (4.7) with
$\mathbf{\underline{u}}=\mathbf{\bar{u}}(t)$, which in turn yields condition
i) (without the redundant Lax admissibility requirement) and ii) in Definition
1.1 with $\mathbf{\underline{u}}=\mathbf{\bar{u}}(t)$.

  * •

$\lambda_{k}(\mathbf{\bar{u}}(t)))=0$. We have

$\begin{split}\boldsymbol{\phi}&(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k}(t))=\stackrel{{\scriptstyle\eqref{e:bl10}}}{{=}}\boldsymbol{\psi}_{s}(\boldsymbol{\zeta}_{k}(\mathbf{\bar{u}}(t),\xi_{k}(t)),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t))\\\
&+\boldsymbol{\psi}_{p}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t),\xi_{k}(t))\\\
&\stackrel{{\scriptstyle\eqref{e:p2},\
\lambda_{k}(\mathbf{\bar{u}}(t)))=0}}{{=}}\boldsymbol{\psi}_{s}(\boldsymbol{\zeta}_{k}(\mathbf{\bar{u}}(t),\xi_{k}(t)),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t))\end{split}$

and by definition of $\boldsymbol{\psi}_{s}$ the above equality combined with
(9.56) dictates that there is a solution of (4.7) which converges to
$\boldsymbol{\zeta}_{k}(\mathbf{\bar{u}}(t),\xi_{k}(t))$ as $y\to+\infty$.
This yields conditions i) (without the redundant Lax admissibility
requirement) and ii) in Definition 1.1 with
$\mathbf{\underline{u}}=\boldsymbol{\zeta}_{k}(\mathbf{\bar{u}}(t),\xi_{k}(t))$.

  * •

$\lambda_{k}(\mathbf{\bar{u}}(t)))<0$. In this case we recall (4.48) and that
$\boldsymbol{\omega}(0)$ and $\boldsymbol{\omega}_{0}(0)$ are as in (4.45) and
(4.44), respectively. We also use the equality
$\boldsymbol{\gamma}_{k}(\mathbf{\bar{u}}(t),\xi_{k}(t))=\mathbf{b}_{k}(\mathbf{\bar{u}}(t),\xi_{k}(t))$,
which follows from (4.43). Note that, by construction,
$\boldsymbol{\omega}(0)$ is the initial datum for an orbit of (3.1) that has
the same asymptotic state as the orbit with initial datum
$\boldsymbol{\omega}_{0}(0)$. Owing to (4.38), the
$\lim_{y\to+\infty}\boldsymbol{\pi}_{\mathbf{u}}(\boldsymbol{\omega}_{0}(y))=\mathbf{\bar{u}}(t)$.
Wrapping up, we conclude that there is a solution of (3.1) with initial datum
given by the value
$\boldsymbol{\phi}(\mathbf{\bar{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k}(t))$
converging to $\mathbf{\bar{u}}(t)$ as $y\to+\infty$. This yields conditions
i) (without the redundant Lax admissibility requirement) and ii) in Definition
1.1 with $\mathbf{\underline{u}}=\mathbf{\bar{u}}(t)$.

Case 1b: we have (9.54). By arguing as in the second item of Case 1a this
yields

$\begin{split}\boldsymbol{\phi}&(\mathbf{\grave{u}}(t),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k}(t))=\stackrel{{\scriptstyle\eqref{e:bl10}\eqref{e:p2},\
\lambda_{k}(\mathbf{\grave{u}}(t)))=0}}{{=}}\boldsymbol{\psi}_{s}(\mathbf{t}_{k}(\mathbf{\grave{u}}(t),\xi_{k}(t)),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t))\\\
&\stackrel{{\scriptstyle\eqref{e:951}}}{{=}}\boldsymbol{\psi}_{s}(\mathbf{t}_{k}(\mathbf{\bar{u}}(t),s_{k}(t)+\xi_{k}(t)),\xi_{\ell(\mathbf{u}_{b}(t))+1}(t),\dots,\xi_{k-1}(t))\end{split}$

and owing to (9.55) this yields conditions i) (without the redundant Lax
admissibility requirement) and ii) in Definition 1.1 with
$\mathbf{\underline{u}}=\mathbf{t}_{k}(\mathbf{\bar{u}}(t),s_{k}(t)+\xi_{k}(t))$.
Case 2: $\alpha(\mathbf{u}_{bm}(t))>0$ for every $m$ and
$\alpha(\mathbf{u}_{b}(t))>0$. In this case
$\ell(\mathbf{u}_{bm}(t))=0=\ell(\mathbf{u}_{b}(t))$ for every $m$ and by
arguing as in the previous case we get Definition 1.1 without the redundant
Lax admissibility requirement in condition i).
Case 3: $\alpha(\mathbf{u}_{bm}(t))>0$ for every $m$ and
$\alpha(\mathbf{u}_{b}(t))=0$. In this case $\ell(\mathbf{u}_{bm}(t))=0$ for
every $m$, but $\ell(\mathbf{u}_{b}(t))=h$. This means that the number of
boundary condition imposed on the boundary layers drops in the limit. To
handle this case we have to recall some facts from the analysis in [9].

  * •

First, we recall the explicit expression of the function $\boldsymbol{\beta}$
(see [9, (2.17)]), and that $h$ denotes the dimension of the kernel of the
matrix $\mathbf{B}$ in (2.1). We conclude that since
$\alpha(\mathbf{u}^{\varepsilon_{m}}_{b}(t))>0$ formula (9.50) boils down to

(9.58)
$\boldsymbol{\phi}(\mathbf{\bar{u}}_{m}(t),\xi^{m}_{1}(t),\dots,\xi^{m}_{k}(t))=\mathbf{u}^{\varepsilon_{m}}_{b}(t)\quad\text{for
every $m\in\mathbb{N}$}.$

We extract a converging subsequence
$\\{\xi^{m}_{1}(t),\dots,\xi^{m}_{k}(t)\\}_{m\in\mathbb{N}}$ and by passing to
the limit in (9.58) and recalling (9.51) we arrive at

(9.59)
$\boldsymbol{\phi}(\mathbf{\grave{u}}(t),\xi_{1}(t),\dots,\xi_{k}(t))=\mathbf{u}_{b}(t).$

  * •

Next, we briefly overwiew the construction of the map $\boldsymbol{\phi}$
given in the proof of [9, Lemma 6.1]222222Note that the map
$\boldsymbol{\phi}$ in (9.59) is given by the projection onto the first $N$
components of the map $\boldsymbol{\widetilde{\psi}}_{b}$ defined in the
statement of [9, Lemma 6.1]. To construct $\boldsymbol{\phi}$ one considers
system [9, (1.14)], where $\mathbf{h}$ is given by [9, (8.1)], and poses
$\sigma=0$, which yields232323Note that in [9, §6] one assumes $h=1$, and that
the case $h>1$ is handled in [9, §8].

(9.60)
$\left\\{\begin{array}[]{ll}\dot{\mathbf{u}}_{1}=-\mathbf{E}_{11}^{-1}\mathbf{A}_{21}^{t}\mathbf{z}_{2}\\\
\dot{\mathbf{u}}_{2}=\alpha\mathbf{z}_{2}\\\
\dot{\mathbf{z}}_{2}=\mathbf{B}^{-1}_{22}\displaystyle{\big{[}(\mathbf{G}_{1}-\mathbf{A}_{21})\mathbf{E}_{11}^{-1}\mathbf{A}_{21}^{t}+\alpha[\mathbf{A}_{22}-\mathbf{G}_{2}]\big{]}\mathbf{z}_{2}}.\\\
\end{array}\right.\quad\mathbf{u}_{1}\in\mathbb{R}^{h},\;\mathbf{u}_{2}\in\mathbb{R}^{N-h},\;\mathbf{z}_{2}\in\mathbb{R}^{N-h},$

where the functions $\mathbf{E}_{11}$, $\mathbf{A}_{21}$, $\mathbf{B}_{22}$,
$\mathbf{G}_{1}$, $\mathbf{A}_{22}$, $\mathbf{G}_{2}$ are the same as in
Hypothesis 1. In what follows, we will systemtically use the notation

$\mathbf{u}=\left(\begin{array}[]{cc}\mathbf{u}_{1}\\\ \mathbf{u}_{2}\\\
\end{array}\right)\quad\mathbf{u}_{1}\in\mathbb{R}^{h},\;\mathbf{u}_{2}\in\mathbb{R}^{N-h}.$

In the proof of [9, Lemma 6.1] one applies the Slaving Manifold Lemma (see [9,
§11]) and constructs the manifold $\boldsymbol{\phi}$ in such a way that,
under (9.59), there is a solution
$(\mathbf{u}_{1}(y),\mathbf{u}_{2}(y),\mathbf{z}_{2}(y))$ of (9.60) such that
$(\mathbf{u}_{1}(0),\mathbf{u}_{2}(0))=\mathbf{u}_{b}$, and

(9.61)
$\lim_{y\to+\infty}|(\mathbf{u}_{1}(y),\mathbf{u}_{2}(y))-(\mathbf{w}_{1}(y),\mathbf{w}_{2}(y))|=0,$

where $(\mathbf{w}_{1}(y),\mathbf{w}_{2}(y))$ denote the first $N$ components
of a suitable orbit of (9.60), lying on the manifold $\mathcal{M}^{0}$ defined
in [9, §3] and satisfying
$(\mathbf{w}_{1}(0),\mathbf{w}_{2}(0))=\boldsymbol{\psi}_{sl}(\mathbf{\grave{u}}(t),\xi_{h}(t),\dots,\xi_{k}(t))$,
where the function $\boldsymbol{\psi}_{sl}$ is defined in [9, Theorem 5.2].

  * •

We now recall that, owing to [9, (2.12)], if $\alpha(\mathbf{u}(0))=0$ and
$(\mathbf{u}(y),\mathbf{z}_{2}(y))$ is a solution of (9.60) then
$\alpha(\mathbf{u}(y))=0$ for every $y$. Since $\alpha(\mathbf{u}_{b}(t))=0$
by assumption, then by combining the second line of (9.60) with (9.61) and the
identities $(\mathbf{u}_{1}(0),\mathbf{u}_{2}(0))=\mathbf{u}_{b}$ and
$(\mathbf{w}_{1}(0),\mathbf{w}_{2}(0))=\boldsymbol{\psi}_{sl}(\mathbf{\grave{u}}(t),\xi_{h}(t),\dots,\xi_{k}(t))$
we eventually arrive at

(9.62)
$\mathbf{p}_{\mathbf{u}_{2}}(\boldsymbol{\psi}_{sl}(\mathbf{\grave{u}}(t),\xi_{h}(t),\dots,\xi_{k}(t)))=\mathbf{p}_{\mathbf{u}_{2}}(\mathbf{u}_{b}(t)),$

where $\mathbf{p}_{\mathbf{u}_{2}}(\mathbf{u})$ represents the projection of
$\mathbf{u}$ onto the last $N-h$ components.

  * •

We now use again [9, (2.17)] and conclude that, since
$\alpha(\mathbf{u}_{b}(t))=0$, then (9.62) is equivalent to

(9.63)
$\boldsymbol{\beta}(\boldsymbol{\psi}_{sl}(\mathbf{\grave{u}}(t),\xi_{h}(t),\dots,\xi_{k}(t)),\mathbf{u}_{b}(t))=\mathbf{0}_{N}.$

By using the properties of the function $\boldsymbol{\psi}_{sl}$ given in the
statement of [9, Theorem 5.2] we can then argue as in Case 1 above and
conclude the proof of Theorem 9.1.

## Acknowledgments

All the authors are members of the GNAMPA group of INDAM and of the PRIN
Project 20204NT8W4 (PI Stefano Bianchini). FA and LVS are also members of the
PRIN 2022 PNRR Project P2022XJ9SX (PI Roberta Bianchini). LVS is also member
of the PRIN Project 2022YXWSLR (PI Paolo Antonelli). Part of this work was
done when LVS was visiting the Department of Mathematics of the University of
Padua: its kind hospitality is gratefully acknowledged.

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