a shock emerging from the interface

in order to produce the discontinuity occurring in $\omega$ at $\ms L$ or $\ms R$.

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(205,77.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $ \begin{array}{l}
%( ii) ,\ ( ii)^\prime \\
(2) ,\ (2\text{b}) \\
%\ \ \ \ ( iii)
\ \ \ \ (3)
\end{array}$;

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The different cases of Remark <ref>.

When the connection  is not critical and
$\ms L\doteq \ms L[\omega, f_l] < 0$, $\ms R\doteq \ms R[\omega, f_r] > 0$,
the analysis of
attainable profiles
$\omega\in \mc A^{AB}(T)$
pursued in [2]

only the profiles described in Cases 3 and 4 of Remark <ref>.
In fact,
the characterization of
$\mc A^{AB}(T)$ established in
requires that all profiles
$\omega\in \mc A^{AB}(T)$ satisfy the equalities

\begin{equation*}
\omega(x)=A\qquad\forall~x\in \,]{\ms L}, 0[\,,
\qquad\qquad
\omega(x)=B\qquad\forall~x\in \,]0, \ms R[\,.
\end{equation*}

Therefore, such a characterization
in particular excludes all attainable profiles $\omega$ that either satisfy conditions (<ref>) or (<ref>), of Cases 1 and 2,

\begin{equation*}
\omega(x){\color{blue} \leq} A\qquad\forall~x\in \,]{\ms L}, \widetilde{\ms L}[\,,
\end{equation*}

or satisfy conditions (<ref>), (<ref>),
of Cases 1B and 2B,

\begin{equation*}
\omega(x){\color{blue} \geq} B\qquad\forall~x\in \,] \widetilde{\ms R}, {\ms R}[\,.
\end{equation*}

Whenever $\ms L=0$
or $\ms R=0$,
the characterization
of the profiles
$\omega\in \mc A^{AB}(T)\cap \msc A^{\ms L, \ms R}$ does not involve the constants
$\widetilde{\ms L}$,
$\widetilde{\ms R}$,
as we see in the following

In the same setting of Theorem <ref>,
let $\omega$ be an element of the set $\msc A^{\ms L, \ms R}$ in (<ref>),

let $g, h$ be the functions in (<ref>),
and let $\bs u[\ms R, B, f_r]$, $\bs v[\ms L, A, f_l]$,  be
constants defined as
in (<ref>),
(<ref>) .

Then, if $\ms L < 0$, $\ms R = 0$,
$\omega \in \mc A^{[AB]}(T)$ if and only if there

the following Oleǐnik-type inequalities are satisfied

\begin{equation}\label{eq:1a2}
%     \tag{1.a}
\begin{aligned}
D^+ \omega(x) &\leq \frac{1}{T \cdot f_l^{\second}(\omega(x))} \qquad \forall \; x \in \,]\!-\!\infty,\;  \ms L[\,,\\
\noalign{\smallskip}
D^+ \omega(x) &\leq \frac{1}{T \cdot f_r^{\second}(\omega(x))} \qquad \forall \; x \in \,]0, +\infty[\,,
\end{aligned}
\end{equation}

\begin{equation}\label{eq:1b2}
% \tag{1.b}
D^+\omega(x) \leq g[\omega, f_l, f_r](x) \qquad \forall \; x \in \ ]\ms L,0[\,.\quad\
\end{equation}

(ii) the following pointwise state constraints are satisfied:

\begin{equation}\label{eq:2a2}
% \tag{2.a}
% \omega(\ms L-) \geq \bs v[\ms L, A, f_l], \qquad\ \
\omega(\ms 0+) \leq \pi^l_{r,-}(\omega(0-))\,,
\end{equation}

\begin{equation}\label{eq:2b2}
% \tag{2.b}
\qquad\ \
\omega(x) \leq  A\qquad \forall \; x \in \,]\ms L, 0[\,,
\end{equation}

\begin{align}
\label{eq:2a2b}
\ms L \in \,]T\cdot f_l^{\prime}(A), 0[
%0 > \ms L >  T\cdot f'_l(A)
\qquad  &\Longrightarrow \qquad
\omega(\ms L-) \geq \bs v[\ms L, A, f_l]\geq  \omega(\ms L+)\,,
\\
\noalign{\smallskip}
\label{eq:3a2b}
\ms L~{\leq}~
T\cdot f_l^{\prime}(A)
\qquad  &\Longrightarrow \qquad
\omega(\ms L-) \geq \omega(\ms L+).
\end{align}

Symmetrically, if $\ms L = 0$, $\ms R > 0$, then $\omega \in \mc A^{[AB]}(T)$ if and only if there hold:

(i) $^\prime$
the following Oleǐnik-type inequalities are satisfied

\begin{equation}\label{eq:1a2-prime}
%     \tag{1.a}
\begin{aligned}
D^+ \omega(x) &\leq \frac{1}{T \cdot f_l^{\second}(\omega(x))} \qquad \forall \; x \in \,]\!-\!\infty,\;  0[\,,\\
\noalign{\smallskip}
D^+ \omega(x) &\leq \frac{1}{T \cdot f_r^{\second}(\omega(x))} \qquad \forall \; x \in \,]\ms R, +\infty[\,,
\end{aligned}
\end{equation}

\begin{equation}\label{eq:1b2-prime}
% \tag{1.b}
D^+\omega(x) \leq h[\omega, f_l, f_r](x) \qquad \forall \; x \in \ ]0, \ms R[\,.\quad\
\end{equation}

(ii) $^\prime$ the following pointwise state constraints are satisfied:

\begin{equation}\label{eq:2a2-prime}
% \tag{2.a}
\omega(\ms 0-) \geq \pi^r_{l,+}(\omega(0+))\,,
\end{equation}

\begin{equation}\label{eq:2b2-prime}
% \tag{2.b}
\qquad\ \  \omega(x) \geq  B\qquad \forall \; x \in \,]0, \ms R[\,,
\end{equation}

\begin{align}
\label{eq:2a2b-prime}
\ms R \in \,]0, T \cdot f_r^{\prime}(B)[\,
\qquad  &\Longrightarrow \qquad
\omega(\ms R+) \leq \bs u[\ms R, B, f_r] \leq  \omega(\ms R-)\,,
\\
\noalign{\smallskip}
\label{eq:3a2b-prime}
\ms R ~{\geq}~
T \cdot f_r^{\prime}(B)
\qquad  &\Longrightarrow \qquad
\omega(\ms R+) \leq \omega(\ms R-) .
\end{align}

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Theorem <ref> when $\ms L < 0$, $\ms R = 0$ and $\ms L \in ]T \cdot f_l^{\prime}(A), 0[$.

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Theorem <ref> when $\ms L < 0$, $\ms R = 0$ and $\ms L \leq T \cdot f_l^{\prime}(A)$.

Notice that
the implications (<ref>)-(<ref>),
(<ref>)-(<ref>) can be extended to $\ms L =T\cdot f'_l(A)$ and to $\ms R=T\cdot f'_r(B)$, respectively.
In fact,
by definition (<ref>) of
$\ms L=\ms L[\omega, f_l]$, one has $f'_l(\omega(\ms L-))\geq \ms L/T$.
Hence, if $\ms L =T\cdot f'_l(A)$ it follows that
$f'_l(\omega(\ms L-))\geq f'_l(A)$ which yields
$\omega(\ms L-)\geq A$
by the monotonicity of $f'_l$.
Thus, recalling that by (<ref>)
we have $\bs v[T\cdot f^{\prime}(A), A, f] =
A$, we derive

\begin{equation}
\label{eq:est-tfprimaea-1}
\omega(T\cdot f'_l(A)-) \geq \bs v[T\cdot f'_l(A), A, f_l]\,.
\end{equation}

On the other hand,
since (<ref>) implies $\omega(T\cdot f'_l(A+)\leq A$,
we deduce from (<ref>) that

\begin{equation}
\label{eq:est-tfprimaea-2}
\omega(T\cdot f'_l(A)-) \geq \omega(T\cdot f'_l(A)+)\,.
\end{equation}

With entirely similar arguments one can show that there hold

\begin{align}
\label{eq:est-tfprimaea-3}
\omega(T\cdot f'_l(B)+) &\leq \bs u[T\cdot f'_l(B), B, f_r]\,,
\\
\noalign{\smallskip}
\label{eq:est-tfprimaea-4}
\omega(T\cdot f'_r(B)+)&\leq \omega(T\cdot f'_r(B)-)\,.
\end{align}

Hence, relying on (<ref>), (<ref>), (<ref>),
and on (<ref>), (<ref>),
with the same arguments of Remark <ref> we deduce that the
inequalities $\omega(\ms L-)\geq \omega(\ms L+)$,
$\omega(\ms R-)\geq \omega(\ms R+)$
are always satisfied.

In the same setting of Theorem <ref>,
let $\omega$ be an element of the set $\msc A^{\ms L, \ms R}$ in (<ref>),   with $\ms L = 0$, $\ms R = 0$.
Then $\omega \in \mc A^{[AB]}(T)$ if and only if there hold:

the following Oleǐnik-type inequalities are satisfied

\begin{equation}\label{eq:1a3}
%     \tag{1.a}
\begin{aligned}
D^+ \omega(x) &\leq \frac{1}{T \cdot f_l^{\second}(\omega(x))} \qquad \forall \; x \in \,]-\!\infty,\;  0[\,,\\
\noalign{\smallskip}
D^+ \omega(x) &\leq \frac{1}{T \cdot f_r^{\second}(\omega(x))} \qquad \forall \; x \in \,]0, +\infty[\,.
\end{aligned}
\end{equation}

(ii) the following pointwise state constraints are satisfied:

\begin{equation}\label{eq:2a3-prime}
% \tag{2.a}
\omega(\ms 0-) \geq \pi^r_{l,+}(\omega(0+))\,,
\end{equation}

\begin{equation}\label{eq:2a3}
% \tag{2.a}
\omega(\ms 0-) \geq \overline A, \qquad \omega(\ms 0+) ~{\leq}~ \overline B\,.
\end{equation}

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Structure of profiles described by Theorem <ref>.

Recalling that by (<ref>) we have $\bs v[0, A, f_l] = \overline{A}$, $\bs u[0, B, f_r] = \overline{B}$, we can rephrase the constraint (<ref>) as

\begin{equation}
\label{eq:2a3-bis}
\omega(\ms 0-) \geq~\bs v[0, A, f_l] , \qquad \omega(\ms 0+) \leq ~\bs u[0, B, f_r]\,.
\end{equation}

Combining Theorems <ref>, <ref>, <ref>, with Theorem <ref>, we obtain:

In the same setting of Theorem <ref>, let
$(A,B)$ be
a connection. Then,
for every $T>0$,
and for any $\omega\in  {\bf L^\infty}(\R)$,
the following conditions are equivalent.

* $\omega \in \mc A^{AB}(T)$.

* $\mc \sabpT \circ \sabmT \omega = \omega\,$.

* $\omega$ is an element of
the set $\msc A^{\ms L, \ms R}$ in (<ref>),   with $\ms L \leq 0$, $\ms R \geq 0$, that
satisfies the conditions of Theorem <ref>, <ref>, or <ref>.

Moreover, if $(A,B)$ is a non critical connection, i.e. if $A\neq \theta_l, B\neq \theta_r$, then the conditions (2) and (3) are equivalent to

$\omega \in \mc A_{bv}^{[AB]}(T)$, where

\begin{equation}
\label{eq:attset-bv}
\mc A_{bv}^{[AB]}(T) \doteq  \big\{\mc \sabpT u_0 \; : \; u _0 \in BV_{loc}(\mathbb R)\big\}\,,
\end{equation}

and there holds

\begin{equation}
\label{eq:atteq-bv-inf}
\mc A^{[AB]}(T)=
\mc A_{bv}^{[AB]}(T)\,.
\end{equation}

Theorem <ref>, <ref>, <ref> yield the first  complete

characterization of the attainable set at time $T > 0$
in terms of Oleǐnik-type inequalities and unilateral constraints,
for critical and non critical connections.

Partial results in this direction have been
recently obtained for strict subsets of $\mc A^{AB}(T)$.

In particular, we refer to:

* the work [6], where
it is characterized
only the  subset ${\mc A}_{\rm L}^{AB}(T) \subset \mc A^{AB}(T)$ given by

$$\quad \ \
{\mc A}_{\rm L}^{[AB]}(T) = \{\omega \in \mc A^{[AB]}(T) \; | \; \exists \; \text{$AB$ entropy solution} \; u \in \mr{Lip}_{\mr{loc}}((0,T) \times \mathbb R \setminus\{0\}) \;  : \; u(T, x) = \omega \}.

* the work [2],
in which, for critical connections
$(A, B)$,
the set
$\mc A^{[AB]}(T)$ is fully characterized
in terms

of triples (a monotone function and a pair of points)
related to the
Lax-Oleinik representation formula of solutions (obtained in [5] via
the Hamiton-Jacobi dual formulation).

Instead, for non critical connections $(A, B)$,

when $\ms L < 0$ and $\ms R > 0$,
[2] provides
a characterization of only a subset of the attainable profiles. Namely, in [2]  only the profiles of
Cases 3, 4, discussed in Remark <ref>, are characterized, missing the ones of Cases 1, 2, 1b, 2b. In fact, the profiles constructed in [2] with $\ms L< 0$, $\ms R> 0$,  satisfy always the condition
$\omega(x) = A$ for all $x \in (\ms L, 0)$, and $\omega(x) = B$ for all $x \in (0, \ms R)$,

which is in general not fulfilled by profiles  of Cases 1, 2, 1b, 2b
(cfr. Remark <ref>).

We point out that, as a byproduct of
the characterization
of $\mc A^{AB}(T)$ via Oleǐnik-type estimates, one can establish
uniform BV bounds on solutions to (<ref>), (<ref>)
in the case of non critical connections,
and on the flux of solutions to (<ref>), (<ref>)
for general connections (see. Proposition <ref> in Appendix <ref>).
In turn such bounds yield the
${\mathbf L^1_{loc}}$-Lipschitz continuity
in time of $AB$-entropy solutions
(see the proof of Theorem <ref>-$(v)$)
in Appendix <ref>).

§ PROOF OF THEOREM <REF>

§.§ Proof roadmap

Observe that if $(A,B)$ is a non critical connection, then recalling Definition <ref>,
and relying on Proposition <ref>
in Appendix <ref>,
we deduce that $\sabmT \omega\in BV_{loc}(\R)$
for all $\omega\in {\bf L^\infty}(\R)$.
Hence setting $u_0\doteq \sabmT \omega$,
we deduce immediately the implication
$(2) \Rightarrow (1)'$.
On the other hand, since
$\mc A_{bv}^{[AB]}(T)\subset \mc A^{[AB]}(T)$, from the implication
$(1) \Rightarrow (3)$, one deduces that
$(1)' \Rightarrow (3)$ holds as well.

in order to establish Theorem <ref>
it will be sufficient to prove the equivalence of the conditions (1), (2), (3).
We provide here a road map of the proof of $(1) \Leftrightarrow (2)\Leftrightarrow  (3)$.
There are three main parts, which are somewhat independent one from the other.

Part 1. The case of a non critical connection ${\bf (1)}\Rightarrow{\bf (3)}$.
In Sections <ref>-<ref> we prove the implication $(1) \Rightarrow (3)$ of Theorem <ref>
when $(A, B)$ is  a non critical connection.
The proof has a bootsrap-like structure,

and it is divided in two steps.

This is the first fundamental block of our proof.
We prove in  <ref> the implication $(1) \Rightarrow (3)$ for profiles $\omega\in \mc A^{[AB]}(T)$ that satisfy the BV condition:

\begin{equation}\label{eq:Hhyp}
\tag{H}
\quad\exists~u_0\in {\bf L}^{\infty}(\R)\ : \
\omega = \sabpT u_0\,, \quad\ \text{and}\quad\ \sabpt u_0\in BV_{\mr{loc}}(\R)\quad\forall~t>0\,.
%,\quad \omega = \sabpT u_0\,.
% \text{\textit{there exists an $AB$-solution $u \in BV_{\mr{loc}}(\mathbb R\times (0, T])$ such that $u(T, \cdot) = \omega$}.}
\end{equation}

The derivation of
the conditions of Theorem <ref>, <ref>, and <ref>
is obtained exploiting
as in [6] the non crossing property of genuine characteristics
in the domains $\{x>0, \, t>0\}$,
$\{x<0, \, t>0\}$,
together with the non existence of
rarefactions emanating from the interface (cfr. Appendix <ref> and [2]).
Two key novel points of the analysis here are:

- a blowup argument, possible thanks to assumption (<ref>),  to

derive the Oleǐnik-type inequalities

satisfied by $\omega$  in regions comprising points
with characteristics reflected by the interface $x=0$, and points with characteristics refracted by

- a comparison arguments
(based on the duality of forward and backward shocks of  <ref>, and on the property of the states  $\bs u[\ms R, B, f_r]$, $\bs v[\ms L, A, f_l]$, defined in  <ref>, <ref>)
to establish the unilateral inequalities satisfied by $\omega$
at points of discontinuity generated by shocks that isolate the interface
from the semiaxes $\{x<0\}$, $\{x>0\}$ (cfr. Remark <ref>).

We prove in  <ref>
the implication $(1) \Rightarrow (3)$

for every $\omega \in \mc A^{AB}(T)$
by showing that
every $\omega \in \mc A^{AB}(T)$ actually
satisfy condition (H), and then the conclusion follows
by Part 1.a.
This is achieved: considering a sequence of functions $u_{n,0}\in BV(\R)$ that ${\bf L^1}$-converge to $u_0\in {\bf L^\infty}(\R)$;
observing that $\sabpt u_{n,0}\in BV(\R)$ (see [1, 25]);
deriving uniform BV bounds on $\sabpT u_{n,0}$ based on the Oleǐnik-type inequalities enjoyed by $\sabpT u_{n,0}$ because of Part 1.a;
relying on the ${\bf L^1}$-stability
of the semigroup map $u_0\mapsto \sabpT u_0$ (see Theorem <ref>-(iii)) and on the lower semicontinuity
of the total variation with respect to ${\bf L^1}$-convergence.

Part 2. The case of a non critical connection ${\bf (3)} \Rightarrow {\bf (2)}\Rightarrow {\bf (1)}$.
The implication
$(2) \Rightarrow (1)$of Theorem <ref>
immediately follows observing that,
by virtue of (2), one has
$\omega = \sabpT u_0\in \mc A^{[AB]}(T)$, with $u_0\doteq \sabmT \omega$.

Hence, in Sections <ref>-<ref> we  prove
only the implication $(3) \Rightarrow (2)$ of Theorem <ref>, in the case of a non critical connection $(A,B)$.
This is the second fundamental block of our proof,
which consists in first showing
as an intermediate step that $(3) \Rightarrow (1)$, and next in proving that $(3) \Rightarrow (2)$.

Given $\omega\in \msc A^{\ms L, \ms R}$
satisfying the condition of Theorem <ref>,

we construct explicitly
in  <ref>
an $AB$-entropy admissible solution $u(x,t)\doteq \mc S_t^{[AB]+}u_0 (x)$, $u_0\in {\bf L^\infty}(\R)$, such that $u( \cdot,T) = \omega$.
The case where $\omega\in \msc A^{\ms L, \ms R}$
satisfies the condition of Theorem <ref>, or <ref> is entirely similar or simpler. The construction of $u_0$ and $u$ follows  a  by now  standard procedure (see [6], [8])
in regions of $\{x<0\}$ or of $\{x>0\}$
that are not influenced by waves reflected or refracted by the interface $x=0$.
Namely, in these regions, one
construct the solution $u$ along two type of lines that correspond to its characteristics:
genuine characteristics $\vartheta_y$
ending at points $(y,T)$, where
$u=\omega(y)$, in the case
$\omega$ is continuous at $y$;
compression fronts $\eta_{y,z}$
connecting points $(z,0)$ and $(y,T)$,
where $u=(f'_l)^{-1}(\frac{y-x}{T})$,
if $y<0$, and $u=(f'_r)^{-1}(\frac{y-x}{T})$,
if $y>0$,
in the case
$\omega$ is discontinuous at $y$.
A key novel point of the analysis here is the construction of $u$ in two polygonal regions around the interface $x=0$,
which relies on the properties of
the shock-rarefaction/rarefaction-shock wave patterns established in  <ref>-<ref>,
which in turn are based on the
the duality properties of forward/backward
shocks derived in <ref>.
Thanks to this construction, one can in
particular explicitly produce $AB$-entropy solutions that attain at time $T$
the profiles of Cases 1, 2, 1b, 2b discussed in Remark <ref>, that are not present in [2]
(cfr. Remark <ref>).

Given $\omega\in \msc A^{\ms L, \ms R}$
satisfying the conditions of Theorem <ref>, we show in  <ref> that $\omega$
is a fixed point of the
backward-forward operator $\sabpT \circ \sabmT$.
The case where $\omega\in \msc A^{\ms L, \ms R}$
satisfies the condition of Theorem <ref>, or <ref> is entirely similar.
Building on the analysis
pursued in the previous part,
in order to prove that
$\omega = \mc \sabpT \circ \sabmT \omega$
it is sufficient to show that, if $u_0$ is the initial datum of the $AB$-entropy solution $u(x,t)$ constructed in Part 2.a,
then  one has $u_0=\mc S^{[AB]-}_T \omega$.

This is again achieved
exploiting the duality properties of forward/backward shocks derived in <ref>,
and the structural properties of
the shock-rarefaction/rarefaction-shock wave patterns established in  <ref>-<ref>.

Part 3. The case of a critical connection
${\bf (1) \Leftrightarrow (2)\Leftrightarrow  (3)}$.
In Sections <ref>, <ref>, <ref> we recover the equivalence of the conditions (1), (2), (3) of Theorem <ref>  in the case of critical
connections, invoking the truth of this equivalence for non critical connections established in Parts 1-2.
The proof is divided in three steps.

In  <ref>
we prove the implication $(1)\Rightarrow (2)$,
relying on the ${\bf L^1}$-stability
of the maps $(A,B, u_0)\mapsto \sabpT u_0$, $(A,B, u_0)\mapsto \sabmT u_0$ (see Theorem <ref>-(iv)
and Definition <ref>).
The reverse implication  is immediate
as observed in Part 2.

In  <ref>
we prove the implication $(1)\Rightarrow (3)$, relying on the ${\bf L^1}$-stability
of the maps

$(A,B)\mapsto f_l(u_l)$,
$(A,B)\mapsto f_r(u_r)$,
where $u_l, u_r$ denote, respectively the left and right states of $u(x,t)\doteq \sabpt u_0(x)$ at $x=0$ (see Corollary <ref>), and on the lower/upper ${\bf L^1}$-semicontinuity property of solutions to conservation laws with uniformly convex flux (see Lemma <ref> in Appendix <ref>).

In  <ref>
we prove the implication
$(3)\Rightarrow (1)$exploiting again the ${\bf L^1}$-stability of the semigroup map of Theorem <ref>-(iv), and using a perturbation argument.
Namely, given  $\omega\in \msc A^{\ms L, \ms R}$
satisfying the conditions of Theorem <ref>, or <ref>, we construct a sequence $\{\omega_n\}_{n}$
of perturbations
of $\omega$
with the property that $\omega_n\stackrel{{\bf L^1}}{\to}\omega$,
$\omega_n \in \mc A^{[A_nB_n]}(T)$, for
a sequence of non critical connections $\{(A_n,B_n)\}_n$.
This is another key point of our analysis, since it provides a general explicit procedure to approximate an attainable profile for a critical connection by attainable profiles for non critical connections.

§.§ Part 1.a -

$(1) \Rightarrow (3)$ for non critical connections assuming (<ref>)

In this Subsection,
given an element $\omega$
of the set $\mc A^{[AB]}(T)$
for a noncritical connection
$(A, B)$, assuming

$\omega$ satisfies (<ref>),
we will show that $\omega$ fulfills condition (3) of Theorem <ref>.

Recalling (<ref>),
this is equivalent to show that,

\begin{equation}
\label{eq:LR-def-23}
\ms L \doteq \ms L[\omega,f_l]\,,
\qquad\quad
\ms R \doteq  \ms R[\omega, f_r],
\end{equation}

be quantities defined as in (<ref>),

there hold:

2a-i) If $\ms L < 0$, $\ms R > 0$, and if $\omega$ satisfies (<ref>), then $\omega$ satisfies the conditions of Theorem <ref>;

2a-ii) If $\ms L = 0$, $\ms R > 0$ or viceversa, and if $\omega$ satisfies (<ref>), then $\omega$ satisfies the conditionsof Theorem <ref>;

2a-iii) If $\ms L = 0$, $\ms R = 0$, then $\omega$ satisfies the conditions of Theorem <ref>.

We will  prove 2a-i) in  <ref>-<ref>,

while 2a-ii) is proven in  <ref>,
and 2a-iii) is discussed in  <ref>.
The further assumption that $\omega$ satisfies (<ref>) is needed only to show that $\omega$ satisfies (<ref>)-(<ref>) in case 2a-i), and
(<ref>), (<ref>) in case 2a-ii).

Throughout the subsection we will assume that

\begin{equation}
\label{eq:att-prof-23}
\omega = \mc S_T^{[AB]+}u_0, \qquad u_0 \in \mathbf L^{\infty}(\mathbb R)\,,
\end{equation}

and we set $u(x,t)\doteq S_t^{[AB]+}u_0(x)$, $x\in\R$, $t\geq 0$.
Moreover, we let $u_l(t), u_r(t)$ denote the left and right traces
at $x=0$ of $u(x,t)$.

($\ms L<0$, $\ms R>0$, proof of (<ref>))

The inequalities $\omega(\ms L-)\geq \omega(\ms L+)$,
$\omega(\ms R-)\geq \omega(\ms R+)$
are the Lax conditions
which are satisfied since $u$ is an entropy admissible solution of
the conservation law
$u_t+f_l(u)_x=0$, on $x<0$,
and of
$u_t+f_r(u)_x=0$, on $x>0$,
and the fluxes $f_l, f_r$
are convex.

§.§.§ ($\ms L<0$, $\ms R>0$, proof of (<ref>))

By definition (<ref>), (<ref>) of $\ms L$,
$\ms R$,
it follows that backward characteristics for $u$
starting at $(x,T)$,
with $x  \in ]-\infty, 0[\, \cup\, ]\ms R, +\infty[$,
never crosses the interface $x=0$.
Thus, we  recover the Oleǐnik estimates (<ref>)
as a classical property
of solutions
to conservation laws with strictly convex flux, which
follows from the fact that genuine characteristics never intersect in the interior of the domain
(e.g. see <cit.>).

§.§.§ ($\ms L<0$, $\ms R>0$, first part of the proof of (<ref>))

Letting $\bs u[\ms R, B, f_r]$  be the
constant defined as
in (<ref>) with $f=f_r$,
we will prove
the implication

\begin{equation}
\label{eq:2aproof}
\ms R \in ]0, T\cdot f^{\prime}(B)[ \quad \Longrightarrow \quad \omega(\ms R+) \leq \bs u[\ms R, B, f_r],
\end{equation}

\begin{equation}\label{eq:contraip}
\ms R \in ]0, T\cdot f^{\prime}(B)[\, ,\qquad\quad  \omega(\ms R+) > \bs u[\ms R, B, f_r]\,,
\end{equation}

and showing that (<ref>) leads to a contradiction. To complete the proof of (<ref>) we will show

in  <ref>

\begin{equation}
\label{eq:2aproof-2}
\ms R \in ]0, T\cdot f^{\prime}(B)[ \quad \Longrightarrow \quad \bs u[\ms R, B, f_r]\leq \omega(\ms R-).
\end{equation}

The proof of the first implication in (<ref>) is obtained in entirely similar way.

We divide the proof of (<ref>) in two steps.
In the first step we construct
the leftmost characteristic curve $\xi_{\strut {\ms R}}$ that
starts on the interface $x=0$
and reaches the point
$(\ms R,T)$, remaining
in the region $\{x>0\}$,
with the property
that all maximal backward characteristics
starting on $\xi_{\strut {\ms R}}$ don't cross the interface $x=0$.
In the second step,
thanks to the assumption (<ref>)
we show that
$\xi_{\strut {\ms R}}$ is located on the left
of the shock curve $\bs x$ constructed as in  <ref>
that emanates from the interface $x=0$ and reaches the point
$(\ms R,T)$.
This leads to a contradiction in accordance
with the characterizing
property of  $\bs u[\ms R, B, f_r]$ discussed in Remark <ref>.

Step 1
Consider the map
$\xi_{\strut {\ms R}}: [\tau_{\strut {\ms R}},T]\to [0,+\infty[$
defined by setting

\begin{equation}\label{eq:hr}
\begin{aligned}
\xi_{\strut {\ms R}}(t) &\doteq  \inf \big\{ R > 0 \; : \; x-t \cdot
f_r^{\prime}(u(t, x))
%   f_r^{\prime}(u(t, x+))
\geq 0  \quad \forall \; x \geq R\big\},
\quad t\geq 0\,,
\\
\noalign{\medskip}
\tau_{\strut {\ms R}}&\doteq\inf\big\{ t \in [0, T] \; : \; \xi_{\strut {\ms R}}(s) > 0 \quad \forall \; s \in [t, T]\big\}\,.
\end{aligned}
\end{equation}

Notice that by definition (<ref>)
we have

\begin{equation}
\label{eq:bc-cond-5}
\xi_{\strut {\ms R}}(\tau_{\strut {\ms R}}) = 0,\qquad \xi_{\strut {\ms R}}(T) = \ms R,\qquad
\xi_{\strut {\ms R}}(t)>0
\quad\ \forall~t\in\, ]\tau_{\strut {\ms R}} , T],
\end{equation}

and that $\xi_{\strut {\ms R}}$ is a backward characteristic for $u$ starting at $(\ms R,T)$, so that
there holds
(e.g. see [20])

\begin{equation}\label{eq:hRH}
\xi'_{\strut {\ms R}}(t)=
\begin{cases}
f'_r(u(\xi_{\strut {\ms R}}(t)\pm,t)
\qquad &\text{if}\quad u(\xi_{\strut {\ms R}}(t)-,t)=u(\xi_{\strut {\ms R}}(t)+,t),
\\
\noalign{\smallskip}
\lambda_r\big(u(\xi_{\strut {\ms R}}(t)-,t),\, u(\xi_{\strut {\ms R}}(t)+,t))\big)
\qquad &\text{if}\quad u(\xi_{\strut {\ms R}}(t)-,t)\neq u(\xi_{\strut {\ms R}}(t)+,t)\,,
\end{cases}
% \begin{aligned}
% & \\
% & f_r(u(t,h_r(t)-)) - f_r(u(t, h_r(t)+)) = \dot h_r(t) \cdot \big(u(t,h_r(t)-)- u(t, h_r(t)+)\big), \qquad \text{for $\msc L^1$-a.e. $t \in [\bar t, T]$}.\\
% &
% \end{aligned}
\end{equation}

\begin{equation}
\label{eq:lambdar-def}
\lambda_r(u,v)\doteq
\frac{f_r(v)-f_r(u)}{v-u},\qquad u,v\in\R, \ u\neq v\,.
\end{equation}

We shall provide now a lower bound on the slope of $\xi_{\strut {\ms R}}$.
Let $t_0 \in \,]\tau_{\strut {\ms R}}, T]$, and observe that
by  definition (<ref>)
it follows that
the minimal backward characteristic starting at $(\xi_{\strut {\ms R}}(t_0), t_0)$ must cross the interface $x=0$
at some non-negative time.
Since such a characteristic is genuine and
has slope $f'_r(u(\xi_{\strut {\ms R}}(t_0)-,t_0))\geq 0$,
and because of the
$AB$-entropy condition (<ref>),
it follows that $f_r(u(\xi_{\strut {\ms R}}(t_0)-, t_0))\geq f_r(B)$
and $u(\xi_{\strut {\ms R}}(t_0)-, t_0) \geq\theta_r$.
Hence, there holds

\begin{equation}
\label{eq:geqBineq}
u(\xi_{\strut {\ms R}}(t_0)-,  t_0) \geq  B\,.
\end{equation}

On the other hand, by definition (<ref>) we have

\begin{equation}
\label{eq:leqineq-uxiR}
f'_r(u(\xi_{\strut {\ms R}}(t_0)+,t_0))\leq \xi_{\strut {\ms R}}(t_0)/t_0\,.
\end{equation}

Thus, letting $\vartheta_{\xi_{{\ms R}}(t_0),+}$ denote the maximal backward characteristic
starting at
$(\xi_{\strut {\ms R}}(t_0),  t_0)$,
because of (<ref>) there holds

\begin{equation}
\label{eq:char1-bound1}
\vartheta_{\xi_{{\ms R}}(t_0),+}(0)=
\xi_{\strut {\ms R}}(t_0) -  t_0 \cdot  f_r^{\prime}(u(\xi_{\strut {\ms R}}(t_0)+, t_0))\geq 0\,,
\end{equation}

and (<ref>) implies

\begin{equation}
\begin{aligned}
\vartheta_{\xi_{{\ms R}}(t_0),+}(t)&>0\qquad
\forall~t\in\,]0, t_0]\,.
% %
% \\
% \vartheta_{\ms R,+}(t)&>0\qquad \forall~t\in\,]0, T]\,,
\end{aligned}
\end{equation}

observe that by
the properties of backward characteristics,
and by definition (<ref>),

the maximal backward characteristics $\vartheta_{\ms R,+}$
starting at $(\ms R, T)$

\begin{equation*}
\xi_{\strut {\ms R}}(t)\leq
\vartheta_{\ms R, +}(t)\qquad \forall~t\in [\tau_{\strut {\ms R}}, T]\,,
\end{equation*}

and, in particular, one has

\begin{equation}
\label{eq:char2-bound2}
\xi_{\strut {\ms R}}(t_0)\leq
\vartheta_{\ms R, +}(t_0)\,.
\end{equation}

Since maximal  genuine? backward characteristics
cannot intersect in the interior of the domain,
it follows from (<ref>) that

\begin{equation}
\label{eq:char2-bound3}
\xi_{\strut {\ms R}}(t_0) -  t_0 \cdot  f_r^{\prime}(u(\xi_{\strut {\ms R}}(t_0)+, t_0))=\vartheta_{\xi_{{\ms R}}(t_0),+}(0) \leq \vartheta_{\ms R, +}(0) =\ms R- T \cdot f_r^{\prime}
(\omega(\ms R+))\,.
\end{equation}

In turn, (<ref>)

\begin{equation}
\label{eq:xietar-est-3}
\xi_{\strut {\ms R}}(t)-\ms R + T \cdot f_r^{\prime}(\omega(\ms R+))\leq
t_0 \cdot  f_r^{\prime}(u(\xi_{\strut {\ms R}}(t_0)+, t_0))\,.
\end{equation}

one has

\begin{equation}
\label{eq:xietar-est-1}
\frac{\xi_{\strut {\ms R}}(t_0)-
\vartheta_{\ms R, +}(0)}{t_0}\leq
\vartheta'_{\ms R, +}=f'_r(\omega(\ms R+))\,.
%    \leq\frac{\ms R}{T}<f'_r(B)\,,
\end{equation}

Since the definition (<ref>) of $\ms R$ and  (<ref>) imply
$f'_r(\omega(\ms R+))
\leq{\ms R}/{T}<f'_r(B)$,
we deduce from (<ref>) that

\begin{equation}
\label{eq:xietar-est-2}
\frac{\xi_{\strut {\ms R}}(t_0)-\ms R + T \cdot f_r^{\prime}(\omega(\ms R+))}{t_0}<
\end{equation}

By the monotonicity of $f_r^{\prime}$, in turn
the estimates (<ref>), (<ref>)

\begin{equation}
\label{eq:xiR-ineq2}
\begin{aligned}
(f_r^{\prime})^{-1} \left( \frac{\xi_{\strut {\ms R}}(t_0)-\ms R + T \cdot f_r^{\prime}(\omega(\ms R+))}{t_0}   \right)
&\leq u(\xi_{\strut {\ms R}}(t_0)+,  t_0),
\\
\noalign{\smallskip}
(f_r^{\prime})^{-1} \left( \frac{\xi_{\strut {\ms R}}(t_0)-\ms R + T \cdot f_r^{\prime}(\omega(\ms R+))}{t_0}   \right)&<B.
\end{aligned}
\end{equation}

Therefore, recalling (<ref>), (<ref>),
and because of the convexity of $f_r$,
we derive
from (<ref>),(<ref>),
(<ref>), that

\begin{equation}
\label{eq:fullspeedineqh}
\xi_{\strut {\ms R}}'(t_0)>
% \lambda_r \!\!\left(\!\!B,     (f_r^{\prime})^{-1} \!\!\left( \frac{\xi_{\strut {\ms R}}(\,\overline t\,)-\ms R + T \cdot f_r^{\prime}(\omega(\ms R+))}{\overline t}   \right) \!\!\right)
\lambda_r \!\left(\!     (f_r^{\prime})^{-1} \!\!\left( \frac{\xi_{\strut {\ms R}}(t_0)-\ms R + T \cdot f_r^{\prime}(\bs u[\ms R, B, f_r])}{t_0}   \right)\!, B \!\right)
\qquad\forall~t_0\in\,]\tau_{\strut {\ms R}}, T]\,.
\end{equation}

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(242.67,17.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $t$;

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Illustration of the proof in  <ref>. The black lines are characteristics of the solution $u$, that cross inside the domain and therefore lead to a contradiction. The blue lines are the comparison curves.

Step 2 (Comparison with an extremal shock).
Let $\bs y[\ms R, B, f_r](\cdot)$ be the function defined in
with $f=f_r$,

\begin{equation}
\label{eq:def_l_32}
\ms L \doteq \bs y[\ms R, B, f_r](T),
\end{equation}

and consider the function

\begin{equation}
\bs x[\ms L, \overline B, f_r](t),\qquad\quad
t \in \big[ \bs s[\ms L, \overline B, f_r] , T\big]\,,
\end{equation}

defined as in  <ref>,
with $A = \overline B$ ($\overline B$ as in (<ref>)),
and $f=f_r$.

By definition (<ref>),
and applying Lemma <ref>,
there holds

\begin{equation}
\label{eq:fwsh-cond23}
\bs x[\ms L, \overline B, f_r](\bs s[\ms L, \overline B, f_r])=0,
\qquad\quad
\bs x[\ms L, \overline B, f_r](T) = \ms R,
\end{equation}

\begin{equation}
\label{eq:fwsh-cond24}
\frac{d}{dt} {\bs x}[\ms L, \overline B, f_r](t) = \lambda_r \!\left(\!(f_r^{\prime})^{-1}\bigg(\frac{\bs x[\ms L, \overline B, f_r](t)+\ms L}{t}\bigg), B\,\right), \qquad t \in \big[ \bs s[\ms L, \bar B, f] , T\big].
\end{equation}

Moreover,  because of (<ref>), (<ref>), we have

\begin{equation}
\ms L= T \cdot f_r^{\prime}(\bs u[\ms R, B, f_r])-\ms R\,.
\end{equation}

Then, by virtue of (<ref>), (<ref>), a  comparison argument yields

\begin{equation}
\label{eq:comparison}
\xi_{\strut {\ms R}}(t) < \bs x[\ms L, \overline B, f_r](t), \qquad \forall \; t \in \big[\max\big\{\!\tau_{\strut {\ms R}},\, \bs s[\ms L, \overline B, f_r]\big\},\, T\big[\,.
\end{equation}

Notice that, if $\bs s[\ms L, \overline B, f_r] \geq \tau_{\strut {\ms R}}$,
then because of (<ref>),
and since $\xi_{\strut {\ms R}}(t)\geq 0$,
we find the contradiction $0\leq \xi_{\strut {\ms R}}(\bs s[\ms L, \overline B, f_r])<0$.
Hence it must be

\begin{equation}
\label{eq:stau-est-2}
\bs s[\ms L, \overline B, f_r] < \tau_{\strut {\ms R}}.
\end{equation}

Next, observe that by definition (<ref>)
and because of (<ref>), we have

$u(0 +, \tau_{\strut {\ms R}}) \leq \theta_r$. Thus, by virtue of the
$AB$-entropy condition (<ref>),
it follows that
$u(0 +, \tau_{\strut {\ms R}}) \leq \overline B$.
Then, letting $\zeta_{0,+}: [0, \tau_{\strut {\ms R}}]\to [0,+\infty[$
the maximal backward characteristic
starting at $(0,\tau_{\strut {\ms R}})$, one has

\begin{equation}\label{eq:secondcontraineq}
\zeta_{0,+}(0) = - \tau_{\strut {\ms R}} \cdot f_r^{\prime}(u(0+,\tau_{\strut {\ms R}}))  \geq - \tau_{\strut {\ms R}} \cdot f_r^{\prime}(\overline B).
\end{equation}

On the other hand,
by virtue of (<ref>),
and recalling the definitions (<ref>), (<ref>) of $\bs u[\ms R, B, f_r]$, $\bs s[\ms L, \overline B, f_r]$, we find that
the maximal backward characteristic
$\vartheta_{\ms R,+}
: [0,T]\to [0,+\infty[$
from $(\ms R, T)$

\begin{equation}
\label{eq:secondcontraineq-2}
\begin{aligned}
\vartheta_{\ms R,+}(0) = \ms R-T \cdot f_r^{\prime}(\omega(\ms R+))  & <
% \stackrel{\text{by \eqref{eq:contraip}}}{<}
\ms R-T \cdot f_r^{\prime}(\bs u[\ms R, B, f_r])
\\
&=-\bs y[\ms R, B, f_r](T)
\\
&=  -\bs s[\ms L, \overline B, f_r] \cdot f_r^{\prime}(\overline B)\
\\
& <
% stackrel{\text{by \eqref{eq:stau-est-2}}}{<}
-\tau_{\strut {\ms R}} \cdot f_r^{\prime}(\overline B) \,.
\end{aligned}
\end{equation}

Thus, we deduce from (<ref>)-(<ref>) that

\begin{equation}
\label{eq:char1-ineq23}
\vartheta_{\ms R,+}(0)< \zeta_{0,+}(0),
\end{equation}

while (<ref>) yield

\begin{equation}
\label{eq:char2-ineq23}
\vartheta_{\ms R,+}(\tau_{\strut {\ms R}})>0 =\zeta_{0,+}(\tau_{\strut {\ms R}}).
\end{equation}

The inequalities (<ref>)-(<ref>)
imply that the genuine characteristics
$\zeta_{0,+}, \vartheta_{\ms R,+}$
intersect each other in the interior of the domain, which gives a contradiction
and thus completes the proof of the implication (<ref>).

§.§.§ ($\ms L<0$, $\ms R>0$, proof of

We will prove only the implication (<ref>), the
proof of (<ref>) being entirely similar.
Let $\widetilde {\ms L}\doteq \widetilde{\ms L}[\omega,f_l,f_r,A,B]$
be the constant
and assume that

\begin{equation}
\label{eq:assumpt-534}
\ms R \in \,]0, T\cdot f_r^{\prime}(B)[\,,\qquad\quad \ms L < \widetilde{\ms L}\,.
\end{equation}

Step 1. (proof of:  $\omega(x) \leq A$ in $]\ms L, 0[\,$).

By definition (<ref>), (<ref>) of $\ms L$,
it follows that backward genuine characteristics starting at points $(x,T)$,
with $x \in \,]\ms L, 0[$ of continuity for $\omega$, must cross the interface $x=0$ at some non-negative time.
Since such characteristics
have slope $f'_l(\omega(x))\leq 0$,
and because of the
$AB$-entropy condition (<ref>),
it follows that $f_l(\omega(x))\geq f_l(A)$
and $\omega(x) \leq\theta_l$
at any point $x \in \,]\ms L, 0[$ of continuity for $\omega$.

Hence, there holds
$\omega(\pm x)\leq A$
for all $x\in \,]\ms L, 0[\,$.

Step 2. (proof of:  $\omega(x) = A$ in $]\widetilde {\ms L}, 0[\,$).

In a similar way to (<ref>), consider
the map
$\xi_{\strut {\ms L}}: [\tau_{\strut {\ms L}},T]\to ]-\infty, 0[$
defined symmetrically by setting

\begin{equation}\label{eq:hl}
\begin{aligned}
\xi_{\strut {\ms L}}(t) &\doteq  \sup \big\{ L < 0 \; : \; x-t \cdot f_l^{\prime}(u(t, x))
%   f_l^{\prime}(u(t, x-))
\leq 0  \quad \forall \; x \leq L\big\},
\quad t\geq 0\,,
\\
\noalign{\medskip}
\tau_{\strut {\ms L}}&\doteq\inf\big\{ t \in [0, T] \; : \; \xi_{\strut {\ms L}}(s) < 0 \quad \forall \; s \in [t, T]\big\}\,.
\end{aligned}
\end{equation}

Notice that by definition (<ref>)
we have

$$\xi_{\strut {\ms L}}(\tau_{\strut {\ms L}}) = 0, \qquad \xi_{\strut {\ms L}}(T) = \ms L, \qquad \xi_{\strut {\ms L}}(t) < 0 \ \quad \forall \; t \in \,]\tau_{\strut {\ms L}},\,T].$$

We claim that

\begin{equation}\label{eq:timeclaim}
\begin{aligned}
\tau_{\strut {\ms L}}\leq
\tau_{\strut {\ms R}}
%     & \\
%     & \bar t_r = \max\{\bar t_l, \bar t_r\}
\quad\ \Longrightarrow \ \quad \tau_{\strut {\ms R}} \leq \bs \tau[\ms R, B, f_r],
\end{aligned}
\end{equation}

where $\bs\tau[\ms R, B, f_r]$ is the constant defined as in (<ref>), with $f=f_r$.
We will prove the implication (<ref>)
with similar arguments to the proof of (<ref>) in  <ref>,

\begin{equation}
\label{eq:tau-ineq-29}
\tau_{\strut {\ms L}}\leq
\tau_{\strut {\ms R}}\,,\qquad\quad
\tau_{\strut {\ms R}}>\bs \tau[\ms R, B, f_r]
\,,
\end{equation}

and showing that (<ref>)
lead to a contradiction.

Since $\tau_{\strut {\ms L}}\leq
\tau_{\strut {\ms R}}$, by definitions (<ref>), (<ref>),
and by virtue of the
$AB$-entropy condition (<ref>),
it follows

\begin{equation}\label{eq:tracesABtr}
(u_l(t), u_r(t)) = (A, B)\qquad \forall \; t \in \,]\tau_{\strut {\ms R}}, T]\,,
\end{equation}

which in turn implies

\begin{equation}
\label{eq:Beq29}
u(\xi_{\strut {\ms R}}(t)-,  t) = B\qquad \forall \; t \in \,]\tau_{\strut {\ms R}}, T]\,.
\end{equation}

$\vartheta_{\xi_{{\ms R}}(t),+}$ ,  be
the maximal backward characteristics
starting at $(0,\tau_{\strut {\ms R}})$, and  at
$(\xi_{\strut {\ms R}}(t),  t)$,
$t\in \,]\tau_{\strut {\ms R}}, T]$,
respectively. Relying on (<ref>), (<ref>),
and since maximal backward characteristics cannot intersect in the interior of the domain, we find

\begin{equation}
\label{eq:xietar-est-7}
- \tau_{\strut {\ms R}} \cdot f_r^{\prime}(\overline B)\leq \zeta_{0,+}(0) \leq \vartheta_{\xi_{{\ms R}}(t),+}(0)=\xi_{\strut {\ms R}}(t) -  t \cdot  f_r^{\prime}(u(\xi_{\strut {\ms R}}(t)+, t))\,.
\end{equation}

In turn, (<ref>)
together with (<ref>), yields

\begin{equation}
\label{eq:xietar-est-9}
t \cdot  f_r^{\prime}(u(\xi_{\strut {\ms R}}(t)+, t))\leq \xi_{\strut {\ms R}}(t)+
\bs\tau[\ms R, B, f_r]
\cdot f_r^{\prime}(\overline B)
\qquad\forall~t\in \,]\tau_{\strut {\ms R}}, T],
\end{equation}

since $f_r^{\prime}(\overline B)<0$.
By the monotonicity of $f'_r$
we deduce from (<ref>) that

\begin{equation}
\label{eq:xiR-ineq5}
u(\xi_{\strut {\ms R}}(t)+, t)\leq
(f_r^{\prime})^{-1} \left( \frac{\xi_{\strut {\ms R}}(t)
+\bs\tau[\ms R, B, f_r]
\cdot f_r^{\prime}(\overline B)
}{t}   \right).
\end{equation}

Therefore, recalling (<ref>), (<ref>),
and because of the convexity of $f_r$,
we derive
from (<ref>), (<ref>)

\begin{equation}
\label{eq:fullspeedineqh-2}
\xi_{\strut {\ms R}}'(t)\leq
\lambda_r \!\left(\!     (f_r^{\prime})^{-1} \!\!\left( \frac{\xi_{\strut {\ms R}}(t)+\bs\tau[\ms R, B, f_r]
\cdot f_r^{\prime}(\overline B)}{t}   \right)\!, B \!\right)
\qquad\forall~t\in\,]\tau_{\strut {\ms R}}, T]\,.
\end{equation}

On the other hand, letting $\bs x[\ms L, \overline B, f_r](\cdot)$
be the function defined in  <ref>,
with $\ms L$ as in (<ref>),
$A = \overline B$,
and $f=f_r$,
there hold (<ref>),
Moreover, because of (<ref>), (<ref>), we have

\begin{equation}
\label{eq:Ltau-eq-2}
\ms L= \bs\tau[\ms R, B, f_r]
\cdot f_r^{\prime}(\overline B),\qquad
\bs s[\ms L, \overline B, f_r]=\bs\tau[\ms R, B, f_r]
\,.
\end{equation}

Then, by virtue of (<ref>), (<ref>),
and because of (<ref>), (<ref>),
with a comparison argument we deduce

\begin{equation}
\label{eq:comparison-2}
\xi_{\strut {\ms R}}(t)\geq
\bs x[\ms L, \overline B, f_r](t)
\qquad\forall~t\in [\tau_{\strut {\ms R}}, T]\,.
\end{equation}

But (<ref>),
together with (<ref>),
(<ref>), (<ref>),
and recalling (<ref>),

\begin{equation}
0=\xi_{\strut {\ms R}}(\tau_{\strut {\ms R}})
\geq \bs x[\ms L, \overline B, f_r](\tau_{\strut {\ms R}})>
x[\ms L, \overline B, f_r](\bs\tau[\ms R, B, f_r])=
\end{equation}

which gives a contradiction,
proving the claim (<ref>).

Relying on the implication (<ref>), we  show now that $\omega(x) = A$ in $]\widetilde {\ms L}, 0[\,$,
considering two cases:

Case 1:
$\tau_{\strut {\ms R}}<\tau_{\strut {\ms L}}$.
Then, by definitions (<ref>), (<ref>),
and by virtue of the
$AB$-entropy condition (<ref>),
it follows

\begin{equation}\label{eq:tracesABtr-2}
(u_l(t), u_r(t)) = (A, B)\qquad \forall \; t \in \,]\tau_{\strut {\ms L}}, T]\,.
\end{equation}

Observe that
the maximal backward characteristic
$\vartheta_{\ms L,+}$
starting at $(\ms L, T)$
crosses the interface $x=0$
at time $T-\ms L/f'_l(\omega(\ms L+)$.
Since $ \xi_{\strut {\ms L}}$ is a
backward characteristic
starting at the same point $(\ms L, T)$ and
crossing the interface $x=0$
at time $\tau_{\strut {\ms L}}$,
one has $\tau_{\strut {\ms L}}\leq T-\ms L/f'_l(\omega(\ms L+)$.

This implies that the backward genuine characteristics from
points $(x,T)$, $x\in\,]\ms L, 0[$, impact the interface $x=0$
at times $t_x \geq T-\ms L/f'_l(\omega(\ms L+)\geq \tau_{\strut {\ms L}}$. Since the value of the solution $u$  is constant along genuine characteristics, we deduce from (<ref>)
that  $\omega(x) = A$ for all $x \in \,]\ms L, 0[$.
Hence, by (<ref>) in particular
it follows that
$\omega(x) = A$
for all
$x\in\, ]\widetilde {\ms L}, 0[\,$.

Case 2:
$\tau_{\strut {\ms L}}\leq \tau_{\strut {\ms R}}$.
Then, because of (<ref>) we have
$\tau_{\strut {\ms R}} \leq \bs \tau[\ms R, B, f_r]$.
Observe that by Step 1 we have $\omega(x)\leq A$ for all $x \in \,]\ms L, 0[$.
Relying on  the monotonicity of $f'_l$,
this implies that  the backward genuine
characteristics starting
points $(x,T)$, $x\in\,]\ms L, 0[$, impacts the interface $x=0$
at times

\begin{equation}
\label{eq:tx-ineq-23}
\tau(x)\doteq  T-\frac{x}{f_l^{\prime}(\omega(x))} \geq T-\frac{x}{f_l^{\prime}(A)} \,.
\end{equation}

On the other hand,
recalling definitions (<ref>),
(<ref>), we have

\begin{equation}
\label{eq:tx-ineq-24}
T-\frac{ (T-\bs \tau[\ms R, B, f_r]) \cdot f_l^{\prime}(A)}{f_l^{\prime}(A)}  = \bs \tau[\ms R, B, f_r]\geq \tau_{\strut {\ms R}}
\,,
\end{equation}

for all
$x\in\, ]\widetilde {\ms L}, 0[\,$.
Combining (<ref>), (<ref>), we deduce that the backward genuine
characteristics starting
points $(x,T)$, $x\in\,]\widetilde {\ms L}, 0[$, cross the interface $x=0$
at times $\tau(x)\geq \tau_{\strut {\ms R}}$.

Hence, relying again on the property that
the solution $u$  is constant along genuine characteristics, we
infer from (<ref>) that $\omega(x) = A$
for all
$x\in\, ]\widetilde {\ms L}, 0[\,$
also in this case, thus completing the proof of Step 2.

Step 3. (proof of:  $\omega(\widetilde{\ms L}-) = A$).

We know by Step 1 and Step 2 that
$\omega(\widetilde{\ms L}-)\leq A$
$\omega(\widetilde{\ms L}+)= A$.

On the other hand the Lax entropy condition (see  <ref>) implies $\omega(\widetilde{\ms L}-)\geq \omega(\widetilde{\ms L}+) = A$.

Therefore one has $A\geq \omega(\widetilde{\ms L}-)\geq \omega(\widetilde{\ms L}+) = A$
which yields $\omega(\tilde{\ms L}-) =A$.
This concludes the proof of (<ref>).

§.§.§ ($\ms L<0$, $\ms R>0$, proof of (<ref>) and completion of the proof of (<ref>))
We will prove only the
second implication in (<ref>), the proof of the first one being entirely symmetric.
Assume that

\begin{equation}
\label{eq:assumptio413}
\Big[\ms R \in \,]0, T\cdot f'_r(B)[\,\;  \ \mathrm{and} \;  \  \widetilde{\ms L}\leq \ms L  \,
\Big]
\quad \mathrm{or} \quad\ \ms R \geq T\cdot f'_r(B),
\end{equation}

and let $\tau_{\strut {\ms L}}, \tau_{\strut {\ms R}}$
be the constants defined in (<ref>), (<ref>),
in connection with the characteristics $\xi_{\strut {\ms L}}, \xi_{\strut {\ms R}}$.
As observed in Step 2 of  <ref>,

the fact that
$\xi_{\strut {\ms L}}$ is a
backward characteristic
starting at $(\ms L, T)$
and crossing the interface $x=0$ at time
$\tau_{\strut {\ms L}}$

\begin{equation}
\label{eq:taul-ineq23}
\tau_{\strut {\ms L}}\leq\,
%    \overline t_{\strut {\ms L}} \
\tau_+(\ms L)\doteq
T-\frac{\ms L}{f'_l(\omega(\ms L+)}\,.
\end{equation}

We claim that (<ref>) implies

\begin{equation}
\label{eq:taul-ineq24}
\tau_{\strut {\ms R}}\leq\,
\tau_+(\ms L)\,.
%   \overline t_{\strut {\ms L}}\,.
%    T-\frac{\ms L}{f'_l(\omega(\ms L+)}\,.
\end{equation}

Since (<ref>) clearly implies (<ref>)
when $\tau_{\strut {\ms R}}\leq \tau_{\strut {\ms L}}$, it will be sufficient to prove the claim under the assumption  $\tau_{\strut {\ms L}}<\tau_{\strut {\ms R}}$.
Let's consider first the case that

\begin{equation}
\label{eq:assumptio413-1}
\ms R \in \,]0, T\cdot f'_r(B)[\,\;  \ \ \mathrm{and} \ \;  \  \widetilde{\ms L}\leq \ms L\,.
\end{equation}

Observe that,

because of (<ref>),
$\tau_{\strut {\ms L}}<\tau_{\strut {\ms R}}$ implies

\begin{equation}
\label{eq:taul-ineq25}
\tau_{\strut {\ms R}} \leq \bs \tau[\ms R, B, f_r].
\end{equation}

Moreover, by Step 1 of  <ref>, one has $\omega(\ms L+)\leq A$.
recalling the definition (<ref>),
and because of the monotonicity
of $f'_l$, we deduce from
$\widetilde{\ms L}\leq \ms L$

\begin{equation}
(T-\bs \tau[\ms R, B, f_r])\cdot f'_l(\omega(\ms L+)
\leq \ms L\,,
\end{equation}

which, together with (<ref>), yields (<ref>), under the assumption (<ref>).
Next, consider the case that

\begin{equation}
\label{eq:assumptio413-2}
\ms R \geq T\cdot f'_r(B)\,.
\end{equation}

Observe that by the analogous argument of Step 1 of  <ref> for (<ref>), one has $\omega(\ms R-)\geq B$. Moreover, if $\omega(\ms R-)=B$,
by definition (<ref>) of $\ms R$ it follows that
$f'_r(B)\geq \ms R/T$, which together with (<ref>), implies
$f'_r(B)=\ms R/T$.
In turn, $f'_r(B)=\ms R/T$ implies that the minimal characteristics starting at $(\ms R,T)$ reaches the interface $x=0$ at time
$t=0$, and by definition (<ref>),
it coincides with $\xi_{\strut {\ms R}}$.
Therefore, one has $\tau_{\strut {\ms R}}=0$,
which proves (<ref>).
Hence, it remains to consider the case (<ref>)
when $\omega(\ms R-)> B$.
Notice that, if

\begin{equation}
\frac{\ms L}{f'_l(\omega(\ms L+))}>\frac{\ms R}{f'_r(\omega(\ms R-))}\,,
\end{equation}

it follows that the minimal backward characteristic $\vartheta_{\ms R,-}$
from $(\ms R,T)$ crosses the interface $x=0$ at a time

\begin{equation}
\label{eq:tr-def-23}
%\overline t_{\strut {\ms R}}
\tau_-(\ms R)\doteq
T-\frac{\ms R}{f'_r(\omega(\ms R-))}
\end{equation}

strictly greater than the time
$\tau_+(\ms L)$

at which the maximal
backward characteristic $\vartheta_{\ms L,+}$
from $(\ms L,T)$ crosses the interface $x=0$.
On the other hand, since $\vartheta_{\ms R,-}$ is a genuine characteristic, it follows that
$u_r(\tau_-(\ms R))
%(\,\overline t_{\ms R}\,)
=\omega(\ms R-)>B$.
Because of  the $AB$-entropy condition (<ref>)
this implies that $u_l(\tau_-(\ms R))
%(\,\overline t_{\ms R}\,)
Thus we can trace the minimal backward characteristic starting at $(0,\,\tau_-(\ms R))$

and lying in $\{x<0\}$, which
has slope $f'_l(u_l(\tau_-(\ms R))
%(\,\overline t_{\ms R}\,))
>0$, and hence it

will intersect  the characteristic $\vartheta_{\ms L,+}$ at a positive time $t^* \geq \tau_+(\ms L)$,

giving a contradiction.
Therefore, $\omega(\ms R-)> B$ implies

\begin{equation}
\label{eq:taul-ineq28}
\frac{\ms L}{f'_l(\omega(\ms L+))}\leq\frac{\ms R}{f'_r(\omega(\ms R-))}\,.
\end{equation}

On the other hand,
since $\xi_{\strut {\ms R}}$ is a
backward characteristic
starting at $(\ms R, T)$
and crossing the interface $x=0$ at time
$\tau_{\strut {\ms R}}$,
there holds

\begin{equation}
\label{eq:taul-ineq27}
\tau_{\strut {\ms R}}\leq
\tau_-(\ms R)\,.
%    \overline t_{\strut {\ms R}}\,.
%    T-\frac{\ms R}{f'_r(\omega(\ms R-))}\,.
\end{equation}

Hence, (<ref>), (<ref>), (<ref>) together yield (<ref>). This completes the proof of the Claim that (<ref>) implies (<ref>).

Then, by definitions (<ref>), (<ref>),
relying on (<ref>), (<ref>),
and by virtue of the
$AB$-entropy condition (<ref>),
we find that

\begin{equation}\label{eq:tracesABtr-4}
(u_l(t), u_r(t)) = (A, B)\qquad \forall \; t \in \,]\tau_+(\ms L)
%    \,\overline t_{\ms L}
,\, T]\,.
\end{equation}

Since backward genuine characteristics starting
points $(x,T)$, $x\in\,]{\ms L}, 0[$, cross the interface $x=0$
at times $t_x\geq \tau_+(\ms L)$,

infer from (<ref>) that $\omega(x) = A$
for all
$x\in\, ]{\ms L}, 0[\,$. This concludes the proof of the
second implication in (<ref>).

we prove now the implication (<ref>).
To this end observe that,
because of (<ref>) and (<ref>) (established in  <ref>), we have

\begin{equation*}
\ms R \in ]0, T\cdot f^{\prime}(B)[ \quad \Longrightarrow \quad
\omega(x)\geq B\qquad\forall~x\in\,]0, \ms R[\,,
\end{equation*}

and hence

\begin{equation}
\label{eq:2aproof-3}
\ms R \in ]0, T\cdot f^{\prime}(B)[ \quad \Longrightarrow \quad
\omega(\ms R -)\geq B\,.
\end{equation}

Thus, relying on (<ref>)
with $f=f_r$,
we deduce (<ref>) from (<ref>), which completes the proof of the
second implication in

§.§.§ ($\ms L<0$, $\ms R>0$, proof of (<ref>)-(<ref>))

We will prove only (<ref>), the proof of (<ref>) being entirely similar.
Then, assume that (<ref>) holds as in  <ref>.

Step 1.
For every point $x\in \,]\ms L, \widetilde {\ms L}[$  where $\omega$
is continuous, consider the map

\begin{equation}
\label{eq:phi3def}
\vartheta_{x}(t) \doteq  \begin{cases}
x- (T-t) \cdot f_l^{\prime}(\omega(x)), & \text{if \ $ \tau(x) \leq t \leq T$},\\
\noalign{\medskip}
(t-\tau(x)) \cdot f_r^{\prime}\circ \pi_{r,-}^l(\omega(x)),
%\big( \pi_{r,-}^l(\omega(x)) \big),
& \text{if\  $0 \leq t < \tau(x)$},
\end{cases}
\end{equation}

\begin{equation}
\label{eq:tau3def}
\tau(x)\doteq  T-\frac{x}{f_l^{\prime}(\omega(x))}\,,
\end{equation}

and set

\begin{equation}
\label{eq:phi4def}
\phi(x)\doteq \vartheta_{x}(0)=-\tau(x)\cdot f'_r\circ  \pi_{r,-}^l(\omega(x))\,.
%\big( \pi_{r,-}^l(\omega(x)) \big)\,.
\end{equation}

Observe that

\begin{equation}
\label{eq:char-rifracted-1}
\begin{aligned}
&\vartheta_{x}|_{\,]\tau(x), T]}\quad \text{is a genuine characteristic for u in the halfplane}\ \ \{x<0\}\,,
\\
\noalign{\smallskip}
&\vartheta_{x}|_{\,]0,\tau(x)[\,}\quad \text{is a genuine characteristic for u in the halfplane}\ \ \{x>0\}\ \ \text{if}\ \ u_r(\tau(x))\leq \overline B\,,
\end{aligned}
\end{equation}

and thus $\vartheta_x$ is a genuine characteristic for $u$ as $AB$-solution (see Remark <ref>) only in the case where $u_r(\tau(x))\leq \overline B$.
Note also
that $\tau(x)$ is the  impact time of
$\vartheta_{x}$ with the interface $x=0$,
and that the function $\tau$ has at most countably many discontinuity points as $\omega$.
Since genuine characteristics cannot intersect in the interior of the domain,
it follows that the right continuous extension of $\tau$ is a nondecreasing map.
On the other hand,
we are assuming that $\omega$ satisfies (<ref>) and that $(A,B)$ is a non critical connection, we know by Proposition <ref> in Appendix <ref> that
no pair of genuine characteristics can meet together on the interface $x=0$. Hence, we deduce that the right continuous extension of
the map $\tau$ is actually increasing
$]\ms L, \widetilde {\ms L}[$ .

We will next show that the right continuous extension of the map $\phi$
is nondecreasing on $]\ms L, \widetilde {\ms L}[$ .

Step 2.
Consider two points
$\ms L< x_1<x_2<\widetilde {\ms L}$ of continuity for $\omega$. By Step 1 we know that $\tau(x_1)<\tau(x_2)$.
Moreover, by (<ref>) (established in  <ref>) we have $\omega(x)\leq A$ for all $x\in \,]\ms L, \widetilde {\ms L}[\,$.
Then, we shall provide a proof of

\begin{equation}
\label{eq:phi-decr}
\phi(x_1)\leq \phi(x_2)
\end{equation}

considering different cases
according to the fact that $\omega(x_i)=A$
or $\omega(x_i)<A$, $i=1,2$.

Case 1: $\omega(x_i)<A$, $i=1,2$.

Since $u$ is constant along genuine characteristics, and because of the $AB$-entropy condition (<ref>),
it follows that $u_r(\tau(x_i))=\pi_{r,-}^l(\omega(x_i))
< \overline B$, $i=1,2$.
Therefore, by (<ref>) $\vartheta_{x_i}|_{\,]0,\tau(x_i)[\,}$, $i=1,2$,
are genuine characteristics
in the half plane $\{x>0\}$
starting at $(0, \tau(x_i))$,
which cannot intersect at positive times.
This implies $\phi(x_1)=\vartheta_{x_1}(0)\leq\vartheta_{x_2}(0)=\phi(x_2)$.

Case 2: $\omega(x_i)=A$, $i=1,2$.

definition (<ref>)
we know that $\vartheta_{x_i}|_{\,]0,\tau(x_i)[\,}$, $i=1,2$, are parallel lines
(possibly not characteristics for $u$)
with slope $f'_r\big( \pi_{r,-}^l(A) \big)=f'_r(\overline B)$,
starting at $(0, \tau(x_i))$.
Hence, $\tau(x_1)<\tau(x_2)$ implies

Case 3: $\omega(x_1) = A$, $\omega(x_2) < A$.

Notice that,

by the monotonicity of $f'_l, f'_r$, the map

$$\big]-\infty, (f_l^{\prime})^{-1}(x_1/T)\big] \ni u \mapsto -\left(T-\frac{x_1}{f_l^{\prime}(u)}\right)\cdot f_r^{\prime}\circ \pi^l_{r,-}(u)$$

is decreasing. Then we have

\begin{equation}
\begin{aligned}
\phi(x_1) & \leq -\left(T-\frac{x_1}{f_l^{\prime}(\omega(x_2))}\right) \cdot f_r^{\prime}\circ   \pi^l_{r,-}(\omega(x_2))\\
& \leq -\left(T-\frac{x_2}{f_l^{\prime}(\omega(x_2))}\right)\cdot f_r^{\prime}\circ  \pi^l_{r,-}(\omega(x_2)) = \phi(x_2)\,.
\end{aligned}
\end{equation}

Case 4: $\omega(x_2) = A$, $\omega(x_1) <A$. Since $\omega(x_2) = A$,
it follows with the same arguments as above that $u_l(\tau(x_2)) = A$  and that either $u_r(\tau(x_2)) = \overline B$
or $u_r(\tau(x_2)) = B$. In the first case,
because of (<ref>)
one can proceed as in Case 1 to deduce that
$\phi(x_1)\leq \phi(x_2)$. Then, assume $u_r(\tau(x_2)) = B$,

and set (see Figure <ref>)

\begin{equation}
\label{eq:bar-t-def-22}
\begin{aligned}
&\overline t \doteq \inf \Big\{t\leq \tau(x_2) \; \big| \; (u_l(s),u_r(s)) = (A,B) \quad \forall s \in [t, \tau(x_2)]\Big\}.
\end{aligned}
\end{equation}

Notice that since $\tau({x_1})< \tau({x_2})$
and because $u_l(\tau(x_1))<A$ implies $u_r(\tau(x_1))<\overline B$, it follows that $\overline  t \in \,]\tau({x_1}),\tau(x_2)]$. We claim that  it must hold

\begin{equation}
\label{eq:urc4}
u_r(\,\overline t\,) = \overline B\,.
\end{equation}

Towards a proof of (<ref>), notice first that, since $\omega(x)\leq A$ for all $x\in \,]\ms L, \widetilde {\ms L}[\,$,
it follows that $u_l(t)\leq A$ for all
$t\in [\tau(x_1), \tau(x_2)]$.
Because of  the $AB$-entropy condition (<ref>)
this implies that $u_r(t)\leq \overline B$ for all
$t\in [\tau(x_1), \tau(x_2)]$.
Hence, in particular we have

\begin{equation}
\label{eq:urc5}
u_r(\,\overline t\,) \leq  \overline B\,.
\end{equation}

Then, consider the blow ups

\begin{equation}
\label{eq:bup-def}
u_\rho(x,t) \doteq u(\rho x, \overline t+\rho(t-\overline t))
\qquad\quad x\in\R, \ t\geq 0\,,
\end{equation}

of $u$ at the point $(0, \overline t\,)$, as in the proof of Proposition <ref>. When $\rho \downarrow 0$, the blow-ups $u_\rho(\cdot, t)$ converge in ${\bf L^1}_{loc}$, up to a subsequence,  to a limiting $AB$ entropy solution $v(\cdot, t)$,
for all $t>0$.

Moreover, there holds

\begin{equation}
\label{eq:indata-blup}
v(x, \overline t\,)=
% \bar u(x), \qquad \bar u(x) =
\begin{cases}
u_l(\,\overline t\,), & \text{if $x < 0$},\\
u_r(\,\overline t\,), & \text{if $x > 0$}.
\end{cases}
\end{equation}

By definitions (<ref>), (<ref>),

letting $u_{p,l}(t), u_{p,r}(t)$ denote the left and right traces
at $x=0$ of $u_p(x,t)$,
we have

\begin{equation}
\label{eq:tracesit-3}
(u_{\rho,l}(t), u_{\rho,r}(t))=(A,B)\quad\qquad\forall~t\in\Big]\overline t, \, \overline t + \frac{\tau(x_2)-\overline t}{\rho}\Big[\,,
\end{equation}

$u_{\rho,l}(t), u_{\rho,r}(t)$ denote the
left and right traces of $u_\rho(\cdot, t)$ at $x=0$.
Taking the limit as $\rho \downarrow 0$
in (<ref>), we deduce that

\begin{equation}
\label{eq:tracesit}
v(0-,t) = A,
\qquad\quad  v(0+, t) = B, \qquad \forall \; t >\overline t,
\end{equation}

while (<ref>), (<ref>) imply

\begin{equation}
\label{eq:tracesit-2}
v(x,\overline t\,)
= u_r(\overline t\,)
\leq \overline B, \qquad \forall~x > 0\,.
\end{equation}

By a direct inspection
we find that, if
an $AB$ entropy solution
of a Riemann problem for (<ref>) with initial datum (<ref>)
at time $\overline t$, enjoys the properties
then the initial datum
on $\{(x,\overline t\,),\ x>0\}$
must be

$v(x,\overline t\,)=u_r(\overline t\,)
= \overline B$, thus proving (<ref>).

[scale = 1.2]

[->, thick] (-3.5,0)–(5,0)node[right]$x$;
[->, thick] (0,0)–(0,5)node[above]$t$;
[very thin] (-3.5,4)–(1.5,4)node[above]$\omega$–(5,4);

[dotted] (-3.5,1.5)node[left]$\overline  t$–(0,1.5);
[dotted] (-3.5,0.7)node[left]$\tau(x_1)$–(0,0.7);
[dotted] (-3.5,2.5)node[left]$\tau(x_2)$–(0,2.5);

[dashed]  (-3,4)node[above]$x_1$–(0,0.7);
[dashed] (-1,4)node[above]$x_2$–(0,2.5);

[smooth, tension = 0.8, thick] plot coordinates(0,1.5)(0.1,2)(0.5,2.5)(0.6,3);
[dashed] (0,1.5)–(2.6,0)node[below]$\textcolor{black}{-\overline t \cdot f_r^{\prime}(\overline B)}$;
[dashed] (0,0.7)–(1.7,0);

[dashed] (0.1,3)–(0,2.9);

[dashed, red] (0, 2.5)–(4,0)node[below]$\phi(x_2)$;

The situation described in  Case 4.

Relying on (<ref>) we can now complete
the proof of (<ref>).
Since $u_r(\tau(x_1))<\overline B$, we know by (<ref>) that $\vartheta_{x_1}$ is a genuine characteristic
in the halfplane $\{x>0\}$ starting at $(0, \tau(x_1))$.
On the other hand, because of (<ref>), we can trace the  maximal backward characteristic from $(0, \overline t\,)$ in $\{x>0\}$,
which has slope $f_r^{\prime}(\overline B)$
and reaches the $x$-axis at the point $-\overline t\cdot f'_r(\overline B)$.
Such a (genuine) characteristic cannot intersect at a positive time
the genuine characteristic $\vartheta_{x_1}$.

Therefore, one has

\begin{equation}
\label{eq:phi-est-45}
\phi(x_1) = \vartheta_{x_1}(0)\leq -\overline t\cdot f_r^{\prime}(\overline B).
\end{equation}

Moreover, since $\overline t \leq  \tau(x_2)$,
and because $\pi_{r,-}^l(\omega(x_2))=\pi_{r,-}^l(A)=\overline B$,
we deduce

$$-\overline t\cdot f_r^{\prime}(\overline B) \leq -\tau(x_2)\cdot f_r^{\prime}(\overline B) = \phi(x_2),$$

which together with (<ref>), yields (<ref>).
This concludes the proof of the nondecreasing monotonicity of $\phi$ on
$]\ms L, \widetilde {\ms L}[$ .
Invoking Lemma 4.4 in [6],
this is equivalent to the inequality

D^+\omega(x) \leq g[\omega, f_l, f_r](x), \qquad \forall \; x \in \,]\ms L, \tilde{\ms L}[\,,

where $g$ is the function in (<ref>).
This concludes the proof of (<ref>), and thus
the proof that $\omega$ satisfies conditions (i)-(ii) of Theorem <ref>
is completed.

§.§.§ ($\ms L < 0$, $\ms R = 0$ or viceversa, proof of conditions (i)-(ii), or (i)$^\prime$-(ii)$^\prime$, of Theorem <ref>)

We consider only the case $\ms L < 0$, $\ms R = 0$, the other case $\ms L = 0$, $\ms R > 0$ being symmetrical. The proofs of (<ref>),
(<ref>), (<ref>), (<ref>), (<ref>) in this case, are
entirely similar to the proofs of (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), respectively,
in the case $\ms L<0$, $\ms R>0$.
We provide here only the proof
of (<ref>), which is
the only new constraint arising
in the case $\ms L < 0$, $\ms R = 0$,
that was not present
in the case $\ms L<0$, $\ms R>0$.

Notice first that
by (<ref>) (established in  <ref>) we know that $\omega(x)\leq A$ for all $x\in \,]\ms L, 0[\,$.
Hence, since the connection $(A,B)$
is non critical, tracing the backward characteristics (with negative slope)
in the half plane $\{x<0\}$
from any sequence of points
$(x_n,T)$, $x_n\in \,]\ms L, 0[\,$,
$x_n\uparrow 0$,
we deduce that there exists the one-sided limit $u_l(T-)$

and there holds

\begin{equation}
\label{eq:ul-omega-A}
u_l(T-)=\omega(0-)\leq A\,.
\end{equation}

Then, we will distinguish two cases.

Case 1: Assume that $u_r(t)\geq B$ for all $t\in \,]\tau, T[$ ,
for some $\tau<T$. Then, by the $AB$-entropy condition (<ref>), and because of (<ref>),
we deduce that $\omega(0-)=u_l(T-)=A$. On the other hand,
since $(A,B)$ is a non critical connection,
by definition (<ref>) it follows that
$\ms R=0$ implies $f'_r(\omega(0+))<0$. Therefore we have
$\omega(0+)\leq\overline B = \pi^l_{r,-}(A)=\pi^l_{r,-}(\omega(0-))$, proving (<ref>).

Case 2:
Assume that there exist a sequence of times $t_n \uparrow T$ such that $u_r(t_n)\leq \overline B$ for all $n$, and such that
$\lim_n u_r(t_n) = u^*$, for some $u^*\leq \overline B$.
By the $AB$-entropy condition (<ref>)
this implies that $u_r(t_n)=
\pi^l_{r,-}(u_l(t_n))$ for all $n$.
Therefore, relying on (<ref>), we find
$u^*=\lim_n \pi^l_{r,-}(u_l(t_n))
On the other hand we have
$\omega(0+)\leq u^*$, since
backward genuine characteristics
issuing from points $(x_n,T)$,
$x_n \downarrow 0$, would eventually cross backward genuine characteristics
in the half plane $\{x>0\}$
starting from
points $(0, t_n)$.
In fact, if $\omega(0+)> u^*$
we can find points $(x_n,T)$,
$x_n>0$ ($x_n$ of continuity for $\omega$), and
$(0,t_n)$, $t_n<T$ ($t_n$ of continuity for $u_r$), such that $\omega(x_n)>u_r(t_n)$,
which would imply that the backward characteristic
starting from
$(x_n,T)$ with negative slope
intersect the backward characteristic
starting from $(0,t_n)$
with slope $f'_r(u_r(t_n))<f'_r(\omega(x_n,T))$.
Therefore it must be
$\omega(0+)\leq u^*$, which
together with $u^*=\pi^l_{r,-}(\omega(0-))$, yields (<ref>).

This concludes the proof of (<ref>), and thus
the proof that $\omega$ satisfies conditions (i)-(ii)
(or (i)$^\prime$-(ii)$^\prime$) of Theorem <ref>
is completed.

§.§.§ ($\ms L = 0$, $\ms R = 0$, proof of conditions (i)-(ii) of Theorem <ref>)

The proofs of (<ref>),
(<ref>), (<ref>), are entirely similar to the proofs of (<ref>),
in the case $\ms L<0$, $\ms R>0$,
and of
in the case $\ms L=\ms R=0$,

Instead, (<ref>)
can be established with the same arguments of
the proof of (<ref>) in the case $\ms L<0$, $\ms R>0$,
Remark <ref>.

This completes the proof that $\omega$ satisfies conditions (i)-(ii) of Theorem <ref>.

§.§ Part 1.b - $(1) \Rightarrow (3)$ for non critical connections  without assuming (<ref>)

In this Subsection,
given an element $\omega$
of the set $\mc A^{[AB]}(T)$
for a noncritical connection
$(A, B)$,
we will show that
$\omega$ satisfies (<ref>).
In view of the analysis in  <ref>,
this will imply that
$\omega$ fulfills condition (3) of Theorem <ref>,
thus completing the proof of the implication $(1) \Rightarrow (3)$
of Theorem <ref>.

Then, given $\omega\in \mc A^{[AB]}(T)$

\begin{equation}
\label{eq:att-prof-25}
\omega = \mc S_T^{[AB]+}u_0, \qquad u_0 \in \mathbf L^{\infty}(\mathbb R)\,,
\end{equation}

set $u(x,t)\doteq S_t^{[AB]+}u_0(x)$, $x\in\R$, $t\geq 0$.

Next, let $\{u_{n,0}\}_n$ be a sequence of
functions in $BV(\R)$
such that

u_{n,0} \ \ \to \ \ u_0 \quad \text{in}\quad  \mathbf L^1(\R)\,,
%\qquad u_0^n \in BV(\mathbb R)\,,

and define $u_n(x,t) \doteq \mc S_t^{[AB]+} u_{n,0}(x)$.

Then, by Theorem <ref>-(iii)

it follows

\begin{equation}
\label{eq:untconv}
\begin{aligned}
u_n(\cdot, t) \  &\rightarrow\ \ u(\cdot, t)\qquad\text{in}\quad
\mathbf L^1(\R)\qquad \forall~t\geq 0\,.
% \\
% \noalign{\smallskip}
% f_l(u_{n,l})
%  \ &\rightarrow\ \ f_l(u_l)\qquad\, \text{in}\quad
% \mathbf L^1_{\mr{loc}}(\R)\,,
% \\
% \noalign{\smallskip}
% f_r(u_{n,r})
%  \ &\rightarrow\ \ f_r(u_r)\qquad \text{in}\quad
% \mathbf L^1_{\mr{loc}}(\R)\,.
\end{aligned}
\end{equation}

Since $(A, B)$ is a non critical connection
and because the initial data $u_{n,0}$
are in $BV$, invoking the BV bounds on $AB$-entropy solutions
provided in <cit.>
(see also
we deduce that $u_n(\cdot, t)\in BV(\R)$
for all $t>0$, and for all $n$.

\begin{equation*}
u_n(\cdot, t)\in \mc A^{[AB]}(t)\,,\quad \text{and \, satisfies~~\eqref{eq:Hhyp}}\quad\forall~t>0\,,\ \ \forall~n\,.
\end{equation*}

Hence, relying on the analysis in  <ref>,
and recalling (<ref>),
we know that,

\begin{equation}
\label{eq:LR-n-def-33}
\ms L_n(t) \doteq \ms L[u_n(\cdot, t),f_l]\,,
\quad\qquad \ms R_n(t) \doteq  \ms R[u_n(\cdot, t), f_r]\,,
\end{equation}

$u_n(\cdot, t)$ satisfies the
conditions stated in:

- Theorem <ref>  if  $\ms L_n(t)<0, \ \ms R_n(t)>0$;

- Theorem <ref>  if  $\ms L_n(t)<0, \ \ms R_n(t)= 0$  or viceversa;

- Theorem <ref>  if  $\ms L_n(t)=0$,  $R_n(t)= 0$.

Thus, in particular, $u_n(\cdot, t)$
satisfies the Oleǐnik-type inequalities

\begin{equation}
\label{eq:ol-est-un-1}
\begin{aligned}
D^+ u_n(x, t) &\leq \frac{1}{t\cdot f''_l(u_n(x,t))}  \qquad\quad
\text{in}\quad  \,]\!-\!\infty, \ms L_n(t)[\,,
\\
\noalign{\smallskip}
D^+ u_n(x, t) &\leq g[u_n(\cdot, t), f_l, f_r](x)  \quad \ \,
\text{in} \quad \,]\ms L_n(t), 0[\,,
\quad\text{if}\quad \ms L_n(t)<0\,,
\\
\noalign{\medskip}
D^+ u_n(x, t) &\leq h[u_n(\cdot, t), f_l, f_r](x)   \quad\ \,
\text{in}\quad \,]0,\ms R_n(t)[\,,
\quad\text{if}\quad \ms R_n(t)>
\\
\noalign{\smallskip}
D^+ u_n(x, t) &\leq \frac{1}{t\cdot f''_r(u_n(x,t))}  \qquad \quad  \,\text{in}\quad  \,]\ms R_n(t), +\infty[\,,
\end{aligned}
\end{equation}

and the constraints

\begin{equation}
\label{eq:constr-n-2}
\begin{aligned}
u_n(x, t)&\leq A\qquad \forall~x\in \,]\ms L_n(t), 0[\,,
\\
\noalign{\medskip}
u_n(x, t)&\geq B\qquad \forall~x\in \,]0, \ms R_n(t)[\,,
\end{aligned}
\end{equation}

for all $t>0$.
Since (<ref>) implies $f'_r(u_n(x,t))\geq f'_r(B)$
for all $x\in\,]0,\ms R_n(t)[$ ,
by the monotonicity of $f'_r$,
we find

\begin{equation}
\label{eq:lwhn-1}
\begin{aligned}
t\cdot f'_r(u_n(x,t))-x
%&\geq  t\cdot f'_r(B)- \delta_n(t)
\geq  \frac{t\cdot f'_r(B)}{2}\qquad\  \forall~x\in \left[0,\, \min\bigg\{\ms R_n(t),\, \frac{t\cdot f'_r(B)}{2}
\bigg\}\right[\,.
\end{aligned}
\end{equation}

recalling definition (<ref>),
$\overline \Lambda\doteq \sup_{|z|\leq M}  \max \{|f'_l(z)|, |f'_r(z)|\}$, with
$M$ being a uniform ${\bf L^\infty}$ bound
for $u_n$,
and letting $a$ be the lower bound on $f''_l, f''_r$
given in (<ref>),
we deduce from (<ref>),

\begin{equation*}
\ms R_n(t)\leq \frac{t\cdot f'_r(B)}{2}\,,
\end{equation*}

then for all $n$ there hold

\begin{equation}
\label{eq:ol-est-un-212}
\qquad  h[u_n(\cdot, t), f_l, f_r](x)\leq
\dfrac{{\overline \Lambda}^{\,2}}{a\, f'_r(B)\big(t\cdot f'_r(u_n(x,t))-x\big)}
\leq
\dfrac{2}{a\, t}\cdot \bigg(\dfrac{\overline \Lambda }{f'_r(B)}\bigg)^{\!2}\qquad \forall~x\in [0, \ms R_n(t)[\,,
\end{equation}

while if

\begin{equation*}
\ms R_n(t)> \frac{t\cdot f'_r(B)}{2}\,,
\end{equation*}

then for all $n$ there hold

\begin{equation}
\label{eq:ol-est-un-21}
h[u_n(\cdot, t), f_l, f_r](x)\leq
\begin{cases}
\dfrac{{\overline \Lambda}^{\,2}}{a\, f'_r(B)\big(t\cdot f'_r(u_n(x,t))-x\big)}
\leq
\dfrac{2}{a\, t}\cdot \bigg(\dfrac{\overline \Lambda}{f'_r(B)}\bigg)^{\!2}\qquad \forall~x\in \left[0,\, \frac{t\cdot f'_r(B)}{2}\right]\,,
\\
\noalign{\bigskip}
\dfrac{\overline \Lambda}{x\, a}~\leq~
\dfrac{2\, \overline \Lambda}{a\, t\!\cdot\! f'_r(B)}
\qquad \qquad
\forall~x\in \left[\frac{t\cdot f'_r(B)}{2},\, \ms R_n(t)\right[\,.
\end{cases}
\end{equation}

we derive from (<ref>), (<ref>), (<ref>), the uniform bounds

\begin{equation}
\label{eq:ol-est-un-32}
\begin{aligned}
D^+ u_n(x, t) &\leq
\dfrac{2\, \overline \Lambda}{a\, t\!\cdot\! f'_r(B)}\cdot\max\bigg\{
1,\, \dfrac{\overline \Lambda}{f'_r(B)}
\bigg\}
\quad
\text{in}\quad \,]0,\ms R_n(t)[\,,
\qquad\text{if}\quad \ms R_n(t)>
\\
\noalign{\smallskip}
D^+ u_n(x, t) &\leq \frac{1}{t\cdot a}  \quad   \text{in}\quad  \,]\ms R_n(t), +\infty[\,,
\end{aligned}
\end{equation}

for all n.
Since $(A,B)$ is a non critical connection,

the one-sided uniform upper bounds
provided by (<ref>)

yield  uniform bounds on the total increasing variation (and hence on the total variation as well) of $u_n(t)$ on bounded subsets of $[0, +\infty[$. In turn, this uniform bounds

together with (<ref>),
by the lower-semicontinuity of the total variation with respect to  the $\bf L^1$ convergence
imply that

\begin{equation}
\label{eq:bound-tv-nc-2}
u(\cdot, t)\in BV_{\mr{loc}}([0, +\infty[\,)\,.
%\qquad\text{if}\qquad \ms R(t)>0\,.
\end{equation}

With the same type of arguments, relying on (<ref>), (<ref>), we can show that

\begin{equation}
\label{eq:bound-tv-nc-3}
u(\cdot, t)\in BV_{\mr{loc}}(\,]\!-\infty, 0])\,.
%\qquad\text{if}\qquad \ms L(t)<0\,.
\end{equation}

Therefore, we deduce from

(<ref>), that

\begin{equation}
\label{eq:bound-tv-nc-4}
u(\cdot, t)\in BV_{\mr{loc}}(\R)\qquad\forall~t>0\,,
\end{equation}

which shows that the function $\omega$ in (<ref>)
satisfies condition (<ref>), thus completing the proof of
the implication $(1) \Rightarrow (3)$  of Theorem <ref> in the case of a non critical connection.

§.§ Part 2.a - $(3) \Rightarrow (1)$ for non critical connections.

In this Subsection, given

\begin{equation}
\label{eq:LR-def-24}
\omega\in \msc A^{\ms L, \ms R},\qquad\quad
\ms L \doteq \ms L[\omega, f_l]< 0, \quad \ms R \doteq \ms R[\omega, f_r] > 0\,,
\end{equation}

($\msc A^{\ms L, \ms R}$ being the set in (<ref>)),

assuming that

\begin{equation}
\label{eq:hyp-3a2}
\omega \quad  \text{satisfies conditions (i)-(ii) of Theorem \ref{thm:attprofiles}},
\end{equation}

we will
show that $\omega \in \mc A^{AB}(T)$ by explicitly constructing an $AB$-entropy solution attaining $\omega$ at time $T$.

With entirely similar arguments one can show that
the same conclusion hold assuming that  $\omega\in \msc A^{\ms L, \ms R}$:

- satisfies the conditions of Theorem <ref>, if $\ms L = 0$, $\ms R > 0$ or viceversa;

- satisfies the conditions of Theorem <ref>, if $\ms L = 0$, $\ms R = 0$.

Then, consider $\omega$ satisfying (<ref>),
By Remark <ref>
we can distinguish six cases
of pointwise constraints on $\omega$, prescribed by condition (ii)
of Theorem <ref>,
which depend on the reciprocal positions of the points
$\ms L$,  $\ms R$,
$\widetilde{\ms L}$,
$\widetilde{\ms R}$, defined in (<ref>)-(<ref>).
We shall consider here only the Cases 1 and 2 discussed in Remark <ref>. The Cases 1b, 2b are symmetrical to Cases 1, 2, up to a change of variables $x \mapsto -x$,
while the Cases 3, 4 are entirely similar or simpler.

Notice that, by Remark <ref>,
in Case 1 there hold (<ref>), (<ref>),
and in particular we shall assume that

\begin{equation}
\label{eq:ur-constr}
\omega(\ms R+) < \bs u[\ms R, B, f_r]\,,
%    \leq \omega(\ms R-)\,,
\end{equation}

while in Case 2 there hold (<ref>), (<ref>),
and we shall assume that (<ref>) is
verified together with

\begin{equation}
\label{eq:vl-constr}
\omega(\ms L-) > \bs v[\ms L, A, f_l]\,.
%\geq \omega(\ms L+)\,.
\end{equation}

The cases in which $\omega(\ms R+) = \bs u[\ms R, B, f_r]$ or $\omega(\ms L-) = \bs v[\ms L, A, f_l]$ can be treated with entirely similar or simpler arguments.
Moreover, in both Cases 1 and 2 we have

\begin{equation}
\label{eq:hyp-cases-1-2}
% \text{(i) $\vee$ (ii) $\Leftrightarrow$ $\tilde {\ms L} > \ms L$}
\widetilde {\ms L} > \ms L\,,
\qquad\quad \omega(\,\widetilde{\ms L} -)=\omega(\,\widetilde{\ms L} +)\,.
\end{equation}

The construction of the initial datum $u_0$
so that the corresponding $AB$-entropy solution solution $u(x,t)\doteq \mc S_t^{[AB]+}u_0 (x)$ attains the value $\omega$
at time $T$ follows a  by now  standard procedure (see [6], [8]), that we describe in  <ref>-<ref> below. To this end we first introduce some technical notations
in  <ref>-<ref>.

§.§.§ Characteristics of compression waves

We introduce a class of

curves connecting two points
$(z,0)$, $(y,T)$,
that will be treated as characteristics
of compression waves
generating a shock
at the point $(y,T)$.
In particular, in the case $y< 0 < z$,
such curves will be
characteristics of a compression wave
that starts at time $t=0$ on the half plane $\{z\geq 0\}$,
and generates a shock
at time $t=T$
after being refracted at
the discontinuity interface.
Given any  $y<0$,

consider the
continuous function

]-\infty, (f_l^{\prime})^{-1}(y/T)] \ni u \mapsto h_y(u) \doteq  -\Big(T-\frac{y}{f_l^{\prime}(u)}\Big)\cdot f_r^{\prime}\circ \pi^{l}_{r,-}(u).

Notice that, by definition (<ref>) and since $f'_l, f'_r$ are increasing functions,
it follows that $u\mapsto -(T-y/f'_l(u))$, $u\mapsto f_r^{\prime}\circ \pi^{l}_{r,-}(u)$ are decreasing maps, and hence the map $h_y$ is decreasing as well. On the other hand we have
$\displaystyle{\lim_{u\to-\infty}}h_y(u) = +\infty$, $h_y((f_l^{\prime})^{-1}(y/T)-) = 0$.
Therefore by a continuity and monotonicity argument, it follows that,
for every $z>0$, there exists a unique state $u_{y,z}\leq (f_l^{\prime})^{-1}(y/T)$, such that

\begin{equation}
\label{eq:hyz-def}
h_y(u_{y,z}) = z\,.
\end{equation}

Moreover, the map $z\mapsto u_{y,z}$, $z>0$
is continuous.

Then, for every pair
$y< 0 < z$,
we denote by $\eta_{y,z}:[0,T] \mapsto \mathbb R$ the polygonal line given by

\begin{equation}
\label{eq:poly-line-def1}
\eta_{y,z}(t) \doteq \begin{cases}
y-(T-t)\cdot f_l^{\prime}(u_{y,z}), & \  \ \text{if \ $\tau(y,z) < t\leq  T$},\\
\noalign{\smallskip}
(t-\tau({y,z}))\cdot f_r^{\prime}\circ \pi^l_{r,-}(u_{y,z}), & \ \ \text{if \ $0 \leq t\leq  \tau(y,z)$},
%           t < T-\dfrac{x
%          {f_l^{\prime}(u_{x,y})}},
\end{cases}
\end{equation}

\begin{equation}
\label{eq:tau-def-23}
%    t_{x,y} =
\tau(y,z)\doteq T-\frac{y}{f_l^{\prime}(u_{y,z})}.
\end{equation}

Instead, for every pair
$y,z < 0$, or $y,z > 0$, we
denote by $\eta_{y,z}:[0,T] \to \mathbb R$ the segment

\begin{equation}
\label{eq:poly-line-def2}
\eta_{y,z}(t) \doteq  y-(T-t)\cdot \dfrac {(y-z)}{T}\qquad \ \forall~0\leq t\leq T\,.
\end{equation}

Notice that,
in the case $y< 0 < z$,
if we consider a function $u(x,t)$ that
assumes the values

\begin{equation*}
\begin{aligned}
u= u_{y,z}\qquad &\text{on the segment}\quad
\eta_{y,z}(t), \ \tau(y,z) < t\leq  T,
\\
%        \noalign{\smallskip}
\qquad &\text{on the segment}\quad
\eta_{y,z}(t), \ 0\leq t\leq \tau(y,z),
\end{aligned}
\end{equation*}

then the states $u_l=u_{y,z}$, $u_r=\pi^l_{r,-}(u_{y,z})$ satisfy the interface entropy condition (<ref>)
at time $t=\tau(y,z)$, and $\eta_{y,z}$
enjoys the properties of a (genuine) characteristic for $u$ as
an $AB$-entropy solution
(see Remark <ref>).
Similar observations hold for $\eta_{y,z}$
in the case $y,z < 0$, considering a function
$u(x,t)$ that
assumes the value $(f'_l)^{-1}((y-z)/T)
= u_{y,z}$
along the segment $\eta_{y,z}$,
and in the case $y,z > 0$, considering a function
$u(x,t)$ that
assumes the value $(f'_r)^{-1}((y-z)/T)= u_{y,z}$
along the segment $\eta_{y,z}$.

§.§.§ Maximal/minimal backward characteristics

We introduce a class of
curves with end point $(y,T)$
that will be treated as
maximal and minimal
backward characteristics
starting at $(y,T)$.
For every $y \in \,]-\infty, \widetilde{\ms L}\,]\,\cup \,]\ms R, +\infty[$ , we denote by  $\vartheta_{y, \pm}:[0,T] \to \mathbb R$ the segments or polygonal

\begin{equation}
\label{eq:pollines}
\vartheta_{y, \pm}(t) \doteq \!\begin{cases}
y-(T-t) \cdot f_l^{\prime}\big(\omega(y\pm)\big), & \text{ if \ $y < \ms L, \quad 0\leq t \leq T$},\\
\noalign{\smallskip}
y-(T-t) \cdot f_l^{\prime}\big(\omega(y\pm)\big), & \text{ if \ $\ms L \leq y \leq \widetilde {\ms L}, \quad \tau_{\pm}(y)
% T-\dfrac{x}{f_l^{\prim\bige}(\omega(x\pm))}
\leq t \leq  T$},\\
\noalign{\smallskip}
\big(t-\tau_{\pm}(y)\big)
% \bigg(T-\dfrac{x}{f_l^{\prime}(\omega(x\pm))}\bigg)\!\!\bigg) \!\cdot\!
\cdot f_r^{\prime}\circ\pi^l_{r,-}(\omega(y\pm)), & \text{ if \ $\ms L \leq y \leq \widetilde {\ms L}, \quad 0 \leq t < \tau_{\pm}(y),
% T-\dfrac{x}{f_l^{\prime}(\omega(x\pm))}
\noalign{\smallskip}
y-(T-t) \cdot f_r^{\prime}\big(\omega(y\pm)\big), & \text{ if \ $y> \ms R, \quad 0\leq t \leq T$},
\end{cases}
\end{equation}

\begin{equation}
\label{eq:tau-def-24}
\tau_{\pm}(y)\doteq T-\frac{y}{f_l^{\prime}(\omega(y\pm))}.
\end{equation}

We will write $\vartheta_{y}(t)\doteq \vartheta_{y, \pm}(t)$ for all $t\in [0,T]$, whenever $\omega(y-)=\omega(y+)$.
In particular, because o (<ref>), we have $\vartheta_{\widetilde{\ms L}}(t)\doteq \vartheta_{\widetilde{\ms L}, \pm}(t)$.
Instead, for $y=\ms R$, we denote
by  $\vartheta_{\ms R, +}:[0,T] \to \mathbb R$ the segment

\begin{equation}
\label{eq:pollines-R}
\vartheta_{\ms R, +}(t) \doteq
\ms R -(T-t) \cdot f_r^{\prime}\big(\omega(\ms R+)\big)
\qquad\forall~0\leq t \leq T\,.
\end{equation}

Notice that, because of definition (<ref>), (<ref>), whenever $y\in \,]-\infty, {\ms L}\,[\,\cup \,]\ms R, +\infty[$, the curves $\vartheta_{y, \pm}$ are segments that never cross the interface
instead for all $y\in  \,]\ms L,\, \widetilde {\ms L}]$, $\vartheta_{y, \pm}$
are polygonal lines that are refracted
at $\{x=0\}$ .
Moreover, at every point of discontinity $y\in \,]-\infty, {\ms L}\,[\,\cup \,]\ms R, +\infty[$ of $\omega$, conditions (<ref>), (<ref>)
imply the Lax condition $\omega(y-)>\omega(y+)$, which in turn,
by the monotonicity of $f'_l, f'_r$, implies

\begin{equation}
\label{eq:lax-start-char}
\vartheta_{y, -}(0)<
\vartheta_{y, +}(0)\qquad\quad
\forall~y\in \,]-\infty, {\ms L}\,[\,\cup \,]\ms R, +\infty[\,.
\end{equation}

As in  <ref>,
observe that in the case $\ms L < y \leq \widetilde {\ms L}$,
if we consider a function $u(x,t)$ that
assumes the values

\begin{equation*}
\begin{aligned}
u= \omega(y\pm)\qquad &\text{on the segment}\quad
\vartheta_{y, \pm}(t), \ \tau_\pm(y) < t\leq  T,
\\
%        \noalign{\smallskip}
\qquad &\text{on the segment}\quad
\vartheta_{y, \pm}(t), \ 0\leq t\leq \tau_\pm(y),
\end{aligned}
\end{equation*}

than $\vartheta_{y, \pm}$
enjoys the properties of a maximal/minimal backward characteristic for $u$ as
an $AB$-entropy solution that attains the value $\omega$ at time $T$.
Similar observations hold for $\vartheta_{y, \pm}$
in the case $y<\ms L$ or $y\geq \ms R$,
considering a function $u(x,t)$ that
assumes the value $\omega(y\pm)$ along $\vartheta_{y, \pm}$.

§.§.§ Partition of $\R$

The initial datum will be defined in a different way on different intervals of
the following partition of $\mathbb R$ (see Figure <ref>):

\begin{equation}
\label{eq:partition}
\begin{aligned}
\mc I_{{\ms L}} &\doteq  \Big\{ x \in \mathbb R \; \big| \;  \vartheta_{\ms L,-}(0) < x < \vartheta_{\ms L,+}(0)\Big\},
\\
\mc I_{{\ms R}} &\doteq  \Big\{x \in \mathbb R \; \big| \;  -\bs y[\ms R, B, f_r](T)< x < \ms R-T \cdot f_r^{\prime}(\omega(\ms R+))\Big\},
\\
\mc I_{\mc C} &\doteq  \Big\{x \in \mathbb R \setminus  (\mc I_{\ms L} \cup \mc I_{\ms R} )\; \big| \; \nexists \; y \in \mathbb R   \; : \; \vartheta_{y,+}(0) = x \; \text{or}\;\vartheta_{y,-}(0) = x \Big\},
\\
\mc I_{\mc Ra} &\doteq  \Big\{x \in \mathbb R \setminus  (\mc I_{\ms L} \cup \mc I_{\ms R} )\; \big| \; \exists \; y< z   \; : \; \vartheta_{y,+}(0) = \vartheta_{z,-} (0) = x\Big\},
\\
\mc I_{\mc W} &\doteq  \Big\{x \in \mathbb R \setminus  (\mc I_{\ms L} \cup \mc I_{\ms R} )\; \big| \; \exists ! \; y \in \mathbb R   \; : \; \vartheta_{y,+}(0) = x \; \text{or}\;\vartheta_{y,-}(0) = x \Big\},
\end{aligned}
\end{equation}

where $\bs y[\ms R, B, f_r](T)$
is defined as in  <ref>
with $f=f_r$. Notice that the set $\mc I_{\ms R}$ is non empty because
the increasing monotonicity of $f'_r$,
together with (<ref>), (<ref>),

\begin{equation*}
f'_r(\omega(\ms R+))<f'_r\big(\bs u[\ms R, B, f_r]\big)=\frac{\ms R+\bs y[\ms R, B, f_r](T)}{T}\,.
\end{equation*}

[scale = 0.8]

[->] (-6.5,0)–(6.5,0)node[above]$x$;
[->] (0,0)–(0,4.5);
[ultra thin] (-6.5,4)–(6.5,4);

(-3.5,4)node[above]$\ms L$–(0,1.5)–(1.5,0);
(2,4)node[above]$\ms R$–(4,0);

(-2,0)node[below]$\mc I_{\ms L}$;
(3,0)node[below]$\mc I_{\ms R}$;

[dashed] (-3.5,4)–(0,0.25);
[dashed] (-3.5,4)–(0,0);
[dashed] (-3.5,4)–(-0.2,0);
[dashed] (-3.5,4)–(-0.4,0);
[dashed] (-3.5,4)–(-0.6,0);
[dashed] (-3.5,4)–(-0.8,0);
[dashed] (-3.5,4)–(-1,0);
[dashed] (-3.5,4)–(-1.2,0);
[dashed] (-3.5,4)–(-1.4,0);
[dashed] (-3.5,4)–(-1.6,0);
[dashed] (-3.5,4)–(-1.8,0);
[dashed] (-3.5,4)–(-2,0);
[dashed] (-3.5,4)–(-2.2,0);
[dashed] (-3.5,4)–(-2.4,0);
[dashed] (-3.5,4)–(-2.6,0);
[dashed] (-3.5,4)–(-2.8,0);
[dashed] (-3.5,4)–(-3,0);
[dashed] (-3.5,4)–(-3.2,0);
[dashed] (-3.5,4)–(-3.4,0);
[dashed] (-3.5,4)–(-3.6,0);
[dashed] (-3.5,4)–(-3.8,0);

[dashed] (-3.5,4)–(0,0.5)–(0.2,0);
[dashed] (-3.5,4)–(0,0.7)–(0.5,0);
[dashed] (-3.5,4)–(0,0.9)–(0.8,0);
[dashed] (-3.5,4)–(0,1.1)– (1.1,0);
[dashed] (-3.5,4)–(0,1.3)– (1.3,0);

[dashed] (2.2,4)–(4.2,0);
[dashed] (2.4,4)–(4.4,0);
[dashed] (2.6,4)–(4.8,0);
[dashed] (2.6,4)–(5,0);
[dashed] (2.6,4)–(5.2,0);
[dashed] (2.6,4)–(5.4,0);
(5.2,0) node[below]$\mc I_{\mc C}$;

[dashed] (3, 4)–(5.8, 0);
[dashed] (3.4, 4)–(6, 0);
[dashed] (3.8, 4)–(6.2, 0);

[dashed] (-3.7, 4)–(-4.2, 0);
[dashed] (-3.9, 4)–(-4.4, 0);
[dashed] (-4.1, 4)–(-4.6, 0);
[dashed] (-4.3, 4)–(-4.85, 0);
[dashed] (-4.5, 4)–(-5.1, 0);

(-4.6, 0) node[below]$\mc I_{\mc W}$;

(-1.5,4)node[above]$\tilde{\ms L}$–(0,2.5)–(2,0);
[dashed] (-2,4)–(0,2.3)–(1.9,0);
[dashed](-2.5,4)–(0,2.1)– (1.8,0);
[dashed] (-3,4)–(0,1.9)– (1.7,0);
[dashed] (-3.5,4)–(0,1.7)– (1.6,0);

[dashed] (-1.25,4)–(0,2.75);
[dashed] (-1,4)–(0,3);
[dashed] (-0.75,4)–(0,3.25);
[dashed] (-0.5,4)–(0,3.5);
[dashed] (-0.25,4)–(0,3.75);

[dashed] (1,4)–(0,3.75);

[smooth, thick] plot coordinates(0,2.5)(0.2,3)(1,3.6)(2,4);

[dashed] (0,2.75)–(0.1,2.85);
[dashed] (0,3)–(0.4,3.15);
[dashed] (0,3.25)–(0.8,3.55);

[dashed] (0.2,3)–(2,0);
[dashed] (1.5,3.8)–(2,0);
[dashed] (0.5,3.3)–(2,0);

[dashed] (1,3.6)–(2,0);

[dashed] (2,4)–(2.25,0);
[dashed] (2,4)–(2.5,0);
[dashed] (2,4)–(2.75,0);
[dashed] (2,4)–(3.5,0);
[dashed] (2,4)–(3.75,0);

(6,4) node[above]$\omega$;

Partition of $\R$ in  Case  1

[scale = 0.8]

[->] (-6.5,0)–(6.5,0)node[right]$x$;
[->] (0,0)–(0,4.5);
[ultra thin] (-6.5,4)–(6.5,4);

(-3.5,4)node[above]$\ms L$–(0,1.5)–(1.5,0);
(2,4)node[above]$\ms R$–(4,0);

(-1,0)node[below]$\mc I_{\ms L}$;
(3,0)node[below]$\mc I_{\ms R}$;

(-3.5,4)–(0,1.1)– (1.1,0);
[dashed] (-3.5,4)–(0,1.3)– (1.3,0);

[smooth,  thick]  plot coordinates(0, 0.5) (-2, 2.6) (-3.5,4);

(-3.5, 4)–(-0.5, 0)–(0,0.5);
[dashed] (-0.5, 0)–(-3, 3.5);
[dashed] (-0.5, 0)–(-2, 2.6)–(0,0.8)–(0.75,0);
[dashed] (0, 0.4)–(0. 4, 0);
[dashed] (-0.5, 0)–(-1, 1.5);
[dashed] (-0.5, 0)–(-0.5, 1);
[dashed] (-0.5, 0)–(-0.2, 0.7);

[dashed] (-0.2,0)–(0,0.2);

[dashed] (-3.5,4)–(-0.8,0);
[dashed] (-3.5,4)–(-1,0);
[dashed] (-3.5,4)–(-1.2,0);
[dashed] (-3.5,4)–(-1.4,0);
[dashed] (-3.5,4)–(-1.6,0);
[dashed] (-3.5,4)–(-1.8,0);
[dashed] (-3.5,4)–(-2,0);
[dashed] (-3.5,4)–(-2.2,0);
[dashed] (-3.5,4)–(-2.4,0);
[dashed] (-3.5,4)–(-2.6,0);
[dashed] (-3.5,4)–(-2.8,0);
[dashed] (-3.5,4)–(-3,0);
[dashed] (-3.5,4)–(-3.2,0);
[dashed] (-3.5,4)–(-3.4,0);
[dashed] (-3.5,4)–(-3.6,0);
[dashed] (-3.5,4)–(-3.8,0);

(-1.5,4)node[above]$\tilde{\ms L}$–(0,2.5)–(2,0);
[dashed] (-2,4)–(0,2.3)–(1.9,0);
[dashed](-2.5,4)–(0,2.1)– (1.8,0);
[dashed] (-3,4)–(0,1.9)– (1.7,0);
[dashed] (-3.5,4)–(0,1.7)– (1.6,0);

[dashed] (-1.25,4)–(0,2.75);
[dashed] (-1,4)–(0,3);
[dashed] (-0.75,4)–(0,3.25);
[dashed] (-0.5,4)–(0,3.5);
[dashed] (-0.25,4)–(0,3.75);

[dashed] (1,4)–(0,3.75);

[smooth,  thick] plot coordinates(0,2.5)(0.2,3)(1,3.6)(2,4);

[dashed] (0,2.75)–(0.1,2.85);
[dashed] (0,3)–(0.4,3.15);
[dashed] (0,3.25)–(0.8,3.55);

[dashed] (0.2,3)–(2,0);
[dashed] (1.5,3.8)–(2,0);
[dashed] (0.5,3.3)–(2,0);

[dashed] (1,3.6)–(2,0);

[dashed] (2,4)–(2.25,0);
[dashed] (2,4)–(2.5,0);
[dashed] (2,4)–(2.75,0);
[dashed] (2,4)–(3.5,0);
[dashed] (2,4)–(3.75,0);

[dashed] (6,4) node[above]$\omega$;

[dashed] (-3.7, 4)–(-4.2, 0);
[dashed] (-3.9, 4)–(-4.4, 0);
[dashed] (-4.1, 4)–(-4.6, 0);
[dashed] (-4.3, 4)–(-4.85, 0);
[dashed] (-4.5, 4)–(-5.1, 0);

(-4.6, 0) node[below]$\mc I_{\mc W}$;

[dashed] (2.2,4)–(4.2,0);
[dashed] (2.4,4)–(4.4,0);
[dashed] (2.6,4)–(4.8,0);
[dashed] (2.6,4)–(5,0);
[dashed] (2.6,4)–(5.2,0);
[dashed] (2.6,4)–(5.4,0);
(5.2,0) node[below]$\mc I_{\mc C}$;

Partition of $\R$ in  Case  2.

The elements of this partition
enjoy the following properties.

- In the Case 1, the set $\mc I_{\ms L}$ consists of the starting points of a compression wave
that is partly refracted by the interface, and that generates a shock at the point $(\ms L, T)$.
In the Case 2, only the subsets of $\mc  I_{\ms L}$
given by $\,]\vartheta_{\ms L,-}(0) ,\, -\bs \sigma[\ms L, A, f_l]\cdot f_l^{\prime}(\overline A)[$,   and   $](\ms L/f'_l(A)-T)\cdot f'_r(\overline B),\,\vartheta_{\ms L,+}(0)[$,
consist of the starting points of compression waves with center at the point $(\ms L, T)$.
In the complementary sets of $\mc  I_{\ms L}$:
$]-\bs \sigma[\ms L, A, f_l]\cdot f_l^{\prime}(\overline A),\, 0[$   and  $]0,\, (\ms L/f'_l(A)-T)\cdot f'_r(\overline B)[$,
the initial datum will assume the constant values $\overline A$ and $\overline B$, respectively.
Here $\bs\sigma[\ms L, A, f_l]$ is the
constant defined as in  <ref>,
with $f=f_l$.

- The set $\mc I_{\ms R}$ consists of the starting points of a compression wave that generates a shock at the point $(\ms R, T)$.

- The set $\mathcal{I}_{\mc C}$ consists of the
starting points of compression
waves that generate a shock at points $(y,T)$,
$y\in \,]-\infty, \ms L[\, \cup\,]\ms L, \widetilde {\ms L}[\, \cup\,]\ms R, +\infty[\,$.
The set $\mathcal{I}_{\mc C}$ is a disjoint union of at most countably many open intervals

of the form

\begin{equation}
\begin{aligned}
\mathcal{I}^n_{\ms L}&=\,]x_n^-,x_n^+[\,, \quad x_n^\pm=\vartheta_{y_n,\pm}(0),
\qquad y_n\in \,]-\infty, \ms L[\,,
\\
\mathcal{I}^n_{\ms R}&=\,]x_n^-,x_n^+[\,, \quad x_n^\pm=\vartheta_{y_n,\pm}(0),
\qquad y_n\in  \,]\ms R,+\infty[\,,
\\
\widetilde{\mathcal{I}}^n_{\ms L}
&=\,]x_n^-,x_n^+[\,, \quad x_n^\pm=\vartheta_{y_n,\pm}(0),
\qquad y_n\in \,]\ms L, \tilde{\ms L}[\,,
\end{aligned}
\end{equation}

which are non empty because of (<ref>).

- The set $\mathcal{I}_{\mc Ra}$ consists of at most countably many points that are the
centers of rarefaction waves originated at time $t=0$.

- The set $\mathcal{I}_{\mc W}$
consists of the
starting points of all genuine characteristics reaching points $(y,T)$,
$y\in \,]-\infty, \ms L[\, \cup\,]\ms L, \widetilde {\ms L}[\, \cup\,]\ms R, +\infty[\,$.

§.§.§ Construction of  $AB$-entropy solution on two regions with  vertexes
at $(\ms L, T)$ and at $({\ms R}, T)$

Consider the two polygonal regions

\begin{equation}
\label{eq:delta-L- gamma-R-def}
\begin{aligned}
\Delta_{\ms L}&\doteq
\Big\{ (x, t) \in\R \times [0, T] \; : \; \vartheta_{\ms L,-}(t) < x < \vartheta_{\ms L,+}(t)\Big\},
%, \ 0<t<T \Big\},
\\
\noalign{\smallskip}
\Gamma_{\ms R}&\doteq
\Big\{ (x, t) \in\R \times [0, T] \; : \; \vartheta_{\, \widetilde{\ms L}}(t) < x < \vartheta_{\ms R,+}(t)\Big\}.
%, \ 0<t<T \Big\},
\end{aligned}
\end{equation}

In the Case 2
(see Figure <ref>),
letting $\Delta[\ms L, A, f_l]$ be the region defined as in  <ref>, with  $f=f_l$,
we can express $\Delta_{\ms L}$ as

\begin{equation}
\label{eq:delta-L-def}
\Delta_{\ms L} =
\Delta[\ms L, A, f_l]\,\cup\, \bigcup_{i=1}^4 \Delta_{\ms L, i},
\end{equation}

\begin{equation}
\label{eq:delta-i-def}
\begin{aligned}
\Delta_{\ms L, 1}&\doteq
\Big\{ (x, t) \in\,]-\infty, 0[\,\times [0,T] \; : \; \vartheta_{\ms L,-}(t) < x \leq  \ms L -(T-t) \cdot f_l^{\prime}(\bs v[\ms L, A, f_l])\Big\},
%, \ 0<t<T \Big\},
\\
\noalign{\smallskip}
\Delta_{\ms L, 2}&\doteq
\Big\{ (x, t) \in\,]-\infty, 0]\,\times \, [0,T] \; : \;  x\geq  (t-\bs \sigma[\ms L, A, f_l])\cdot f'_l(\,\overline{A}\,)\Big\},
\\
\noalign{\smallskip}
\Delta_{\ms L, 3}&\doteq
\Big\{ (x, t) \in\,]0, +\infty[\,\times \,[0,T] \; : \; x< \eta_{\ms L,\,  x({A,\overline{B}})}(t)\Big\},
\\
\noalign{\smallskip}
\Delta_{\ms L, 4}&\doteq
% \Big\{ (x, t) \in\,]-\infty, 0[\,\times\,]0,T[\, \; : \; \ms L -(T-t) \cdot f_l^{\prime}(A)< x < \vartheta_{\ms L,+}(t)\Big\},
\Big\{ (x, t) \in\R\, \times\, [0,T] \; : \;
\eta_{\ms L,\,  x({A,\overline{B}})}(t)\leq x < \vartheta_{\ms L,+}(t)\Big\},
\end{aligned}
\end{equation}

with $\bs v[\ms L, A, f_l]$ as in (<ref>) taking $f=f_l$,

\begin{equation}
\label{eq:def-}
x({A,\overline{B}})\doteq \big(\ms L/f'_l(A)-T\big)\cdot f'_r(\overline{B})>0\,.
\end{equation}

Similarly, in both  Cases 1, 2
(see Figures <ref>-<ref>),
letting $\Gamma[\ms R, B, f_r]$,  be the region defined as in  <ref>, with $f=f_r$,
we can express $\Gamma_{\ms R}$ as

\begin{equation}
\label{eq:gamma-r-def}
\Gamma_{\ms R}=\Gamma[\ms R, B, f_r]\,\cup\, \bigcup_{i=1}^3 \Gamma_{\ms R, i},
\end{equation}

\begin{equation}
\label{eq:gamma-i-def}
\begin{aligned}
\Gamma_{\ms R, 1}&\doteq
\Big\{ (x, t) \in\,]-\infty, 0]\,\times\,[0,T] \; : \; \vartheta_{\, \widetilde{\ms L}}(t) < x\Big\},
\\
\noalign{\smallskip}
\Gamma_{\ms R, 2}&\doteq  \Big\{ (x, t) \in\,]0, +\infty[\,\times\,[0,T] \; : \; x\leq \ms R - (T-t)\cdot f'_r(B)\Big\},
\\
\noalign{\smallskip}
\Gamma_{\ms R, 3}&\doteq \Big\{ (x, t) \in\,]0, +\infty[\,\times\,[0,T]\, \; : \;
\ms R -(T-t) \cdot f_r^{\prime}(\bs u[\ms R, B, f_r])\leq x<\vartheta_{\ms R,+}(t)\Big\},
\end{aligned}
\end{equation}

with $\bs u[\ms R, B, f_l]$ as in (<ref>), taking $f=f_r$.

Now,  consider the function $ u_{\ms L} : \Delta_{\ms L} \to \R$ defined by
setting for every $(x,t)\in \Delta_{\ms L}$ :

in  Case 1:

\begin{equation}
\label{eq:uL-sol-def-1}
u_{\ms L}(x, t)  = \begin{cases}
(f_l^{\prime})^{-1}\big(\frac{\ms L-x}{T-t}\big), & \text{if $x \leq  0$}, \\
\noalign{\smallskip}
\pi^l_{r,-}(u_{\ms L,z}),
% (f_r^{\prime})^{-1}(\eta^{\prime}_{\ms L, \bar x}(t)),
& \text{if $x = \eta_{\ms L, z}(t)$, \
for some \ $z>0$},
\end{cases}
\end{equation}

where $u_{\ms L,z}$ is defined as in  <ref>, with $y =\ms L$;

in  Case 2:

\begin{equation}
\label{eq:uL-sol-def-2}
u_{\ms L}(x, t)  = \begin{cases}            (f_l^{\prime})^{-1}\big(\frac{\ms L-x}{T-t}\big), & \text{if
$(x,t)\in \Delta_{\ms L, 1}\cup \Delta_{\ms L, 4},\ \ x\leq 0,$}
% \\
%     (f_l^{\prime})^{-1}((\ms L-x)/t), & \text{if $\ms L -(T-t) \cdot f_l^{\prime}(A)< x <{ \min\{0,\,\vartheta_{\ms L,+}(t)\}}$},
\\
\noalign{\smallskip}
\pi^l_{r,-}(u_{\ms L,z}),
& \text{if $(x,t)\in \Delta_{\ms L, 4}$, \ $x = \eta_{\ms L, z}(t)$, \
for some \ $z>0$ }\\
\noalign{\smallskip}
\ms v[\ms L, A, f_l](x,t), & \text{if $(x,t) \in \Delta[\ms L, A, f_l],$}\\
\overline A, & \text{if $(x,t)\in \Delta_{\ms L, 2}$,}\\
% $-(\bs \sigma[\ms L, A, f_l]-t)\cdot f_l^{\prime}(\bar A) < x < 0$, $0 < t < \bs \sigma[\ms L, A, f_l]$},\\
\overline B, & \text{if $(x,t)\in \Delta_{\ms L, 3}$.}\\
% $0 < x < -(T-\ms L /f_l^{\prime}(A)-t)\cdot f_r^{\prime}(\bar B)$},
\end{cases}
\end{equation}

where $\ms v[\ms L, A, f_l]$ denotes the function defined in (<ref>), with $f=f_l$.

By construction, because of (<ref>), and relying on the analysis in <ref>, it follows that
in both  Cases 1, 2, the function $u_{\ms L}(x, t)$:

- is locally Lipschitz continuous on
$\big(\Delta_{\ms L} \setminus
\overline{\Delta[\ms L, A, f_l]}\,\big)\cap \big((\R\setminus\{0\})\,\times\,]0,T[\,\big)$, and it is continuous on the boundary $\partial\Delta[\ms L, A, f_l]\setminus \big(\{0\}\,\times\,]0,T[\big)$;

- is a classical solution of $u_t+f_l(u)_x=0$ on
$\big(\Delta_{\ms L} \setminus
\overline{\Delta[\ms L, A, f_l]}\,\big)\cap \big(\,]-\infty, 0[\,\times\,]0,T[\,\big)$, and of $u_t+f_r(u)_x=0$ on

$\Delta_{\ms L} \cap \big(\,]0, +\infty[\,\times\,]0,T[\,\big)$;

- is an entropy weak solution of $u_t+f_l(u)_x=0$ on
$\Delta[\ms L, A, f_l]$;

- satisfies
the interface entropy condition (<ref>)
at any point $(0,t)$, $t\leq \tau_+(\ms L)$.

Therefore, by Definition <ref>,
we deduce that $u_{\ms L}$ is an $AB$-entropy solution of (<ref>) on $\Delta_{\ms L}$.

Next, consider (for both  Cases 1, 2) the function $ u_{\ms R} : \Gamma_{\ms R} \to \R$ defined by
setting for every $(x,t)\in \Gamma_{\ms R}$ :

\begin{equation}
\label{eq:uR-sol-def}
u_{\ms R}(x, t)  = \begin{cases}
A, & \text{if $(x,t)\in \Gamma_{\ms R,1},$} \\
\noalign{\smallskip}
B, & \text{if $(x,t)\in \Gamma_{\ms R,2},$} \\
\noalign{\smallskip}
\ms u[\ms R, A, f_r](x, t), & \text{if $(x,t) \in \Gamma[\ms R, B, f_r]$},
\\
\noalign{\smallskip}
(f_r^{\prime})^{-1}\big(\frac{\ms R-x}{T-t}\big), & \text{if $(x,t)\in \Gamma_{\ms R,3},$}
\end{cases}
\end{equation}

where $\ms u[\ms R, A, f_r]$ denotes the function defined in (<ref>), with $f=f_r$.
By construction and relying on the analysis in
we deduce as above that $u_{\ms R}$
provides an $AB$-entropy solution of (<ref>)
on $\Gamma_{\ms R}$.
Moreover, because of (<ref>), (<ref>), (<ref>),
we have

\begin{equation}
\label{eq:term-cond-23}
u_{\ms R}(x,T)=\omega(x)
\qquad \forall~x\in \,]\widetilde {\ms L}, \ms R[\,.
\end{equation}

§.§.§ Construction of

$AB$-entropy solution on whole $\R\times [0,T]$

Observing that, because of (<ref>), (<ref>), we have  $\Delta_{\ms L} \cap \{x=0\}= \mc I_{\ms L}$, $\Gamma_{\ms R} \cap \{x=0\}= \mc I_{\ms R}$,
we define the initial datum on $\mc I_{\ms L}\cup \mc I_{\ms R}$ as

\begin{equation}
\label{eq:idatum-22}
u_{\ms L}(x, 0)\ \ &\text{if}\quad x\in
\mc I_{\ms L},
\\
\noalign{\smallskip}
u_{\ms R}(x, 0)\ \ &\text{if}\quad x\in
\mc I_{\ms R}.
\end{cases}
\end{equation}

where, in  Case 1,

\begin{equation}
\label{eq:init-datum-L-1}
u_{\ms L}(x, 0)= \begin{cases}
(f_l^{\prime})^{-1}\big(\frac{\ms L-x}{T}\big), & \text{if $x\in
\mc I_{\ms L},$ \ $x \leq  0$}, \\
\noalign{\smallskip}
\pi^l_{r,-}(u_{\ms L,z}),
% (f_r^{\prime})^{-1}(\eta^{\prime}_{\ms L, \bar x}(t)),
& \text{if $x\in
\mc I_{\ms L}$, \ $x = \eta_{\ms L, z}(0)$, \
for some \ $z>0$},
\end{cases}
\end{equation}

while, in  Case 2,

\begin{equation}
u_{\ms L}(x, 0)  = \begin{cases}            (f_l^{\prime})^{-1}\big(\frac{\ms L-x}{T}\big), & \text{if \
$x\in \,]\vartheta_{\ms L,-}(0),\, \ms L -T \cdot f_l^{\prime}(\bs v[\ms L, A, f_l])[$,}
% $(x,t)\in \Delta_{\ms L, 1}\cup \Delta_{\ms L, 4},\ \ x\leq 0,$}
% \\
%     (f_l^{\prime})^{-1}((\ms L-x)/t), & \text{if $\ms L -(T-t) \cdot f_l^{\prime}(A)< x <{ \min\{0,\,\vartheta_{\ms L,+}(t)\}}$},
%  \\
%  \noalign{\smallskip}
% v[\ms L, A, f_l](x,t), & \text{if $(x,t) \in \Delta[\ms L, A, f_l],$}
\\
\noalign{\smallskip}
\overline A, & \text{if \ $x\in
\,]\ms L -T \cdot f_l^{\prime}(\bs v[\ms L, A, f_l]),\, 0[\,$,}\\
% $-(\bs \sigma[\ms L, A, f_l]-t)\cdot f_l^{\prime}(\bar A) < x < 0$, $0 < t < \bs \sigma[\ms L, A, f_l]$},\\
\noalign{\smallskip}
\overline B, & \text{if \ $x\in\,]0,\, \eta_{\ms L,\,  x({A,\overline{B}})}(0)[$,}
% $0 < x < -(T-\ms L /f_l^{\prime}(A)-t)\cdot f_r^{\prime}(\bar B)$},
\\
\noalign{\smallskip}
\pi^l_{r,-}(u_{\ms L,z}),
& \text{if \ $x\in [\eta_{\ms L,\,  x({A,\overline{B}})}(0),\, \vartheta_{\ms L,+}(0)[\,$, \ $x = \eta_{\ms L, z}(t)$, \
for some \ $z>0$,}
\end{cases}
\end{equation}

and, in both  Cases 1, 2,

\begin{equation}
\label{eq:init-datum-R}
u_{\ms R}(x, 0)  =
\textstyle{\frac{\ms R-x}{T}}\big), \quad \text{if \ $x\in \mc I_{\ms R}.$}
\end{equation}

In view of the observations in  <ref>-<ref>,
the construction of the $AB$ entropy solution on
$\big(\R\times [0,T]\big)\setminus \big(\Delta_{\ms L}\cup \Gamma_{\ms R}\big)$, and the corresponding definition of the initial datum on $\R\setminus\big(\mc I_{\ms L}\cup \mc I_{\ms R}\big)$, proceed as follows:

- For any $y\in \,]-\infty, \ms L[\, \cup\,]\ms L, \widetilde {\ms L}[\, \cup\,]\ms R, +\infty[\,$, we trace the lines $\vartheta_{y, \pm}$ starting at $(y,T)$ until they reach the $x$-axis at the point $\phi_{\pm}(y)\doteq  \vartheta_{y, \pm}(0)$.
Since conditions (<ref>), (<ref>) of Theorem <ref> is equivalent to the monotonicity of the map $\phi(y)\doteq \vartheta_{y}(0)$ (see <cit.>), it follows that $\vartheta_{y, \pm}$ never intersect each other in the region
$\R\times \,]0, T[\,$.
Then, if $y\in \,]-\infty, \ms L[\,  \cup\,]\ms R, +\infty[\,$, we define a function $u(x,t)$ that is
equal to $\omega(y\pm)$ along the segment $\vartheta_{y, \pm}$. Instead if
$y\in \,]\ms L, \widetilde {\ms L}[\,$
we define $u$ to be equal to $\omega(y\pm)$
along the segment $\vartheta_{y, \pm}(t)$, $\tau_\pm(y)\leq t\leq T$, and to be equal
to $\pi^l_{r,-}(\omega(y\pm))$ along the segment $\vartheta_{y, \pm}(t)$, $0\leq t\leq \tau_\pm(y)$.

- For any $z\in \mathcal{I}^n_{\ms L}\,\cup\, \mathcal{I}^n_{\ms R}$, we trace the line $\eta_{y_n,z}$, $y_n\in \,]-\infty, \ms L[\,
%\cup\,]\ms L, \widetilde {\ms L}[\,
\cup\,]\ms R, +\infty[\,$.
By construction the lines $\eta_{y_n,z}$ never cross each other in the region
$\R\times \,]0, T[\,$.
Then, if $y_n\in \,]-\infty, \ms L[$ , we define $u(x,t)$ to be equal to
= u_{y,z}$ along the segment $\eta_{y_n,z}$, instead  if $y_n\in \,]\ms R, +\infty[$ , we define $u(x,t)$ to be equal to
= u_{y,z}$ along the segment $\eta_{y_n,z}$.

-  For any $z\in \widetilde{\mathcal{I}}^n_{\ms L}$, we trace the polygonal line $\eta_{y_n,z}$, $y_n\in\,]\ms L, \widetilde {\ms L}[\,$. By construction the lines $\eta_{y_n,z}$ never cross each other in the region
$\R\times \,]0, T[\,$.
Then, we define $u(x,t)$ to be equal to
= u_{y,z}$ along the segment $\eta_{y_n,z}$,
$\tau(y,z)< t \leq T$, and to be equal to
$\pi^l_{r,-}(u_{y,z})$ along the segment
$0\leq t\leq \tau(y,z)$.

Therefore, we define the
the function

\begin{equation}
\label{ab-entr-sol-def-21}
u(x, t)  \doteq \begin{cases}
\omega(y\pm), & \text{if \ $x=\vartheta_{y,\pm}(t)$ \ for some \ $y \in \,]-\infty, \ms L[\, \cup \,]\ms R, +\infty[\,$},\\
\noalign{\smallskip}
\omega(y\pm), & \text{if \
$x=\vartheta_{y,\pm}(t)<0$ \ for some \ $y \in \,]\ms L, \widetilde{\ms L}[ \,$},\\
\noalign{\smallskip}
\pi^l_{r,-}(\omega(y\pm)), & \text{if \
$x=\vartheta_{y,\pm}(t)>0$ \ for some \ $y \in \,]\ms L, \widetilde{\ms L}[ \,$},\\
\noalign{\smallskip}
(f_l^{\prime})^{-1}\big(\frac{y_n-z}{T}\big), & \text{if \ $ x = \eta_{y_n, z}(t)$ \ for some \ $z \in \mc I^n_{\ms L}$},\\
\noalign{\smallskip}
(f_r^{\prime})^{-1}\big(\frac{y_n-z}{T}\big), & \text{if \ $ x = \eta_{y_n, z}(t)$ \ for some \ $z \in \mc I^n_{\ms R}$},\\
\noalign{\smallskip}
(f_l^{\prime})^{-1}\big(\frac{\ms L-x}{T-t}\big), & \text{if \ $ x = \eta_{y_n, z}(t)<0$ \ for some \ $z \in \widetilde{\mathcal{I}}^n_{\ms L}$},\\
\noalign{\smallskip}
\pi^l_{r,-}(u_{y_n,z}), & \text{if \ $ x = \eta_{y_n, z}(t)>0$ \ for some \ $z \in \widetilde{\mathcal{I}}^n_{\ms L}$},\\
\noalign{\smallskip}
% A, & \text{if \ $\vartheta_{\, \widetilde{\ms L}}(t)  < x \leq  0$},\\
% %$\bs \tau < t < T$},\\
% B, & \text{if \ $0 < x \leq \ms R-(T-t) \cdot f_r^{\prime}(B)$},\\
% %$T-\ms R/f_r^{\prime}(B) < t < T$},\\
u_{\ms L}(x, t), & \text{if \ $(t, x) \in \Delta_{\ms L}$},\\
\noalign{\smallskip}
u_{\ms R}(x, t), & \text{if \ $(t, x) \in \Gamma_{\ms R}$},
\end{cases}
\end{equation}

and the initial datum

\begin{equation}
\label{eq:ab-initial-datum}
u_0(x)  \doteq \begin{cases}
\omega(y\pm), & \text{if \ $x \in \mc I_{\mc W}$, $x = \theta_{y,\pm}(0)$ \ for some \ $y \in\,]-\infty,\ms L[\, \cup \,]\ms R, +\infty[\,$},\\
\noalign{\smallskip}
\pi^l_{r,-}(\omega(y\pm)), & \text{if \ $x \in \mc I_{\mc W}$, $x = \theta_{y,\pm}(0)$, $y \in \,]\ms L, \tilde{ \ms L}[\,$},\\
\noalign{\smallskip}
(f_l^{\prime})^{-1}\big(\frac{y_n-x}{T}\big), & \text{if \ $x \in \mc I^n_{\ms L}$},
%(x_n^-,x_n^+)$, $y_n \in (-\infty, \ms L)$},
\\
\noalign{\smallskip}
(f_r^{\prime})^{-1}\big(\frac{y_n-x}{T}\big), & \text{if \ $x \in \mc I^n_{\ms R}$},
%(x_n^-,x_n^+)$, $y_n \in (\ms R,  +\infty)$},
\\
\noalign{\smallskip}
% (f_r^{\prime})^{-1}(\eta^{\prime}_{y_n, x}(0)),
\pi^l_{r,-}(u_{y_n,x}), & \text{if \ $x \in \widetilde{\mathcal{I}}^n_{\ms L}$},
%(x_n^-,x_n^+)$, $y_n \in (\ms L,  \tilde{\ms L})$},
\\
\noalign{\smallskip}
u_{\ms L}(x,0), & \text{if \ $x \in \mc I_{{\ms L}}$},
\\
\noalign{\smallskip}
u_{\ms R}(x,0), & \text{if \ $x \in \mc I_{\ms R}$}.
\end{cases}
\end{equation}

Notice that $u_0$ is not defined on the countable set $\mathcal{I}_{\mc Ra}$ which is  of measure zero, and clearly $u_0\in {\bf L}^\infty(\R)$.
By construction, the function $u(x,t)$:

- is locally Lipschitz continuous on
$\big(\R\times \,]0,T[\,\big)\setminus
\big(\,\overline{\Delta_{\ms L}\cup \Gamma_{\ms R}}
\,\cup\, (\{0\}\,\times\,]0,T[\,)\big)$,

and it is continuous on the boundary $\partial \big(\Delta_{\ms L}\cup \Gamma_{\ms R}\big) \setminus \big(\{0\}\,\times\,]0,T[\big)$;

is a classical solution of $u_t+f_l(u)_x=0$ on
$\big(\,]-\infty, 0[\,\times\,]0,T[\,\big)\setminus \,\overline{\Delta_{\ms L}\cup \Gamma_{\ms R}}$ ;

is a classical solution of $u_t+f_r(u)_x=0$ on
$\big(\,]0, +\infty[\,\times\,]0,T[\,\big)\setminus \,\overline{\Delta_{\ms L}\cup \Gamma_{\ms R}}$ ;

- is an $AB$-entropy solution of (<ref>) on $\Delta_{\ms L}\cup\Gamma_{\ms R}$;

the interface entropy condition (<ref>)
at any point $(0,t)$, $t\in \,]0,T[\,$.

Thus, by Definition <ref>,
it follows that the function $u(x,t)$ in (<ref>)
provides an
$AB$-entropy solution to (<ref>)
on $\R\times [0,T]$. Moreover,
because of (<ref>),  (<ref>), (<ref>),
we have

\begin{equation}
\label{eq:indatum-finalprpfile}
u(x,0) = u_0(x),\qquad\quad u(x,T)=\omega(x)\qquad\ \text{for a.e.}\quad x\in\R\,.
\end{equation}

This proves that

\begin{equation}
\label{eq:att-prof-sol}
\omega= \mc S_T^{[AB]+}u_0,
\end{equation}

and thus
$\omega \in \mc A^{AB}(T)$, which completes
the proof of
the implication $(3) \Rightarrow (1)$  of Theorem <ref> in the case of a non critical connection.

§.§ Part 2.b - $(3) \Rightarrow (2)$ for non critical connections.

As a byproduct of the construction described in  <ref>,
we show in this Subsection  that,
if $\omega$ satisfies (<ref>), (<ref>),
then $\omega$ verifies condition (2) of
Theorem <ref>,
i.e. $\omega$ is a fixed point
of the map $\omega\mapsto \mc \sabpT \circ \sabmT \omega$.

We shall assume that $\omega$ satisfies the pointwise constraints of Case 1 discussed in Remark <ref>, the other cases being simmetric, or entirely similar,
or simpler.

In order to verify that $\mc \sabpT \circ \sabmT \omega = \omega$,
because of (<ref>) it is sufficient to prove that,
letting $u_0$ be the function defined
by (<ref>), (<ref>), (<ref>), there hold

\begin{equation}
\label{eq:indatum-back-1}
u_0= \sabmT \omega\,.
\end{equation}

In turn, recalling the definition  (<ref>) of $AB$ backward solution operator, the equality (<ref>)
is equivalent to the equality

\begin{equation}
\label{eq:indatum-back-2}
\big(\omega(-\ \cdot\,)\big)
(x)\qquad x\in\R,
\end{equation}

\begin{equation}
%    v(x,t)\doteq
(x,t)\mapsto \sobapt
\big(\omega(-\ \cdot\,)\big)
\end{equation}

the unique $\overline B\,\overline A$-entropy solution

\begin{equation}
\label{eq:invertedproblem-2}
\begin{cases}
v_t+\overline{f}(x,v)_x = 0 & x \in \mathbb{R}, \quad t \geq 0, \\
\noalign{\smallskip}
v(x,0) = \omega(-x) &  x \in \mathbb R,
\end{cases}
\end{equation}

$\overline{f}(x,v)$ being the symmetric flux in (<ref>).

Towards a proof of (<ref>), we will determine the solution of (<ref>) on $\R\times [0,T]$ relying on the construction in  <ref> and on the properties of the
left forward rarefaction-shock wave pattern
derived in  <ref>.

Observe that  the function $u(x,t)$ defined by (<ref>)
for  Case 1,
with $u_{\ms L}$, $u_{\ms R}$ defined by (<ref>), (<ref>), respectively, is:

- locally Lipschitz continuous in the region

\begin{equation*}
\msc L\doteq \big(\,\R\times \,]0,T[\,\big) \setminus \big( (\{0\}\,\times\,]0,T[\,) \cup \Gamma[\ms R, B, f_r] \big)
\end{equation*}

where $\Gamma[\ms R, B, f_r]$,  is defined as in  <ref>, with $f=f_r$
(the region $\msc L$ is the complement of the pink region and of the axis $\{x = 0\}$ in Figure <ref>);

a classical solution of $u_t+f_l(u)_x=0$ on
$\,]-\infty, 0[\,\times\,]0,T[\,$;

a classical solution of $u_t+f_r(u)_x=0$ on
$\big(\,]0, +\infty[\,\times\,]0,T[\,\big)\setminus \overline{\Gamma[\ms R, B, f_r]}$ ;

the interface entropy condition (<ref>)
at any point $(0,t)$, $t\in \,]0,T[\,$.

Therefore, if we define the transformation $(x, t) \mapsto \alpha( x,t)\doteq  (-x, T-t)$,  the function

\begin{equation}
\label{eq:ualpha-def}
%    u_{\alpha(\msc L)}
v(x,t) \doteq u(-x, T-t),
%    u(\alpha(x,t)),
\qquad (x,t) \in \alpha(\,\overline{\msc L})\setminus (\{0\}\,\times\,]0,T[\,),
%\alpha(\msc L),
\end{equation}

- an entropy weak solution of $v_t + f_r(v)_x = 0$ in the open set $\alpha(\msc L) \cap \{x < 0\}$;

- an entropy weak solution of
$v_t + f_l(v)_x = 0$ in the open set  $\alpha(\msc L) \cap \{x > 0\}$.

On the other hand,

letting $\Delta[\bs y[\ms R, B, f_r], \overline B, f_r]$ denote the region defined in (<ref>) with $\ms L =y[\ms R, B, f_r]$, $A=\overline B$, and $f=f_r$.
one can directly verify that

\begin{equation}
\label{eq:imgamma}
\alpha\big(\Gamma[\ms R, B,f_r]\big)=\Delta[\bs y[\ms R, B, f_r], \overline B, f_r]
\subset \,]-\inf, 0[\times\,]0,T[\,.
\end{equation}

Notice that $(\R\times [0,T])\setminus (\{0\}\,\times\,]0,T[\,)$ is the disjoint union of
$\alpha(\,\overline{\msc L})\setminus (\{0\}\,\times\,]0,T[\,)$ and of $\alpha\big(\Gamma[\ms R, B,f_r]\big)$.
Then, letting $\ms v [\bs y[\ms R, B, f_r], \overline B, f_r](x,t)$ denote the function defined in (<ref>), with $\ms L =y[\ms R, B, f_r]$, $A=\overline B$, and $f=f_r$,
consider the function $v:\R\times [0,T]\to\R$ defined by setting

\begin{equation}
\label{eq:sol-invertedproblem-2}
\begin{cases}
u(-x, T-t), & \text{if \ \ $(x,t) \in \alpha(\,\overline{\msc L})\setminus (\{0\}\,\times\,]0,T[\,)$},\\
\noalign{\smallskip}
%\alpha(\msc L)$},\\
\ms v[\bs y[\ms R, B, f_r], \overline B, f_r](x,t), & \text{if \ \ $(x,t) \in \Delta[\bs y[\ms R, B, f_r], \overline B, f_r]$}\,.
\end{cases}
\end{equation}

By construction and because of the analysis in
<ref>, the function $v(x,t)$ :

- is locally Lipschitz continuous on
$\big(\R\times \,]0,T[\,\big)\setminus
\big(\,\overline{\Delta[\bs y[\ms R, B, f_r], \overline B, f_r]}\,
\cup \, (\{0\}\,\times\,]0,T[\,)\big)$,
and it is continuous on the boundary $\partial \big(\Delta[\bs y[\ms R, B, f_r], \overline B, f_r]\big) \setminus \big(\{0\}\,\times\,]0,T[\big)$;

is a classical solution of $v_t+f_r(v)_x=0$ on
$\big(\,]-\infty, 0[\,\times\,]0,T[\,\big)\setminus \,\overline{\Delta[\bs y[\ms R, B, f_r], \overline B, f_r]}$ ;

-is an entropy weak solution of $v_t+f_r(v)_x=0$ on $\Delta[\bs y[\ms R, B, f_r], \overline B, f_r]$;

is a classical solution of $v_t+f_l(v)_x=0$ on
$\,]0, +\infty[\,\times\,]0,T[\,$ ;

- satisfies
the $\overline{B} \overline{A}$ interface entropy condition,

namely, setting $v_l(t)\doteq v(0-,t)$, $v_r(t)\doteq v(0+,t)$, and considering the function

\begin{equation*}
\label{IBAdef}
I^{\overline B \overline A}(v_l,v_r)  \stackrel{\cdot}{=} \mathrm{sgn}(v_r-\overline A\,)\left(f_l(v_r)-f_l(\,\overline A\,)\right) \\- \mathrm{sgn}(v_l-\overline B\,)\left(f_r(v_l)-f_r(\,\overline B\,)\right),
\end{equation*}

there hold

\begin{equation}
I^{\overline B \overline A}\big(v_l(t), v_r(t)\big)\leq 0,\qquad
\text{for a.e. $t\in\,]0,T[$\,.}
\end{equation}

Notice also that,

because of (<ref>), (<ref>), (<ref>), it follows

\begin{equation}
\qquad\ \text{for a.e.}\quad x\in\R\,.
\end{equation}

Therefore, by Definition <ref>,
we deduce that the function $v(x,t)$ in (<ref>)
provides the $\overline B\,\overline A$-entropy solution to (<ref>)
on $\R\times [0,T]$, and hence
we have

\begin{equation}
\big(\omega(-\ \cdot\,)\big)
(x)\qquad x\in\R,\ t\in [0,T]\,.
\end{equation}

Moreover, by (<ref>), (<ref>), there hold

\begin{equation*}
\sobapT
\big(\omega(-\ \cdot\,)\big)
= u(-x,0)=u_0(-x)\qquad x\in\R\,,
\end{equation*}

which proves (<ref>), and thus concludes the proof of the implication $(3) \Rightarrow (2)$  of Theorem <ref> in the case of a non critical connection.

§.§ Part 3.a - $(1) \Leftrightarrow  (2)$ for critical connections

In this Subsection we
assume that
the equivalence of conditions (1), (2) of Theorem <ref> holds for connections which are non critical,
and we will show that it remains true also for critical connections
To fix the ideas, throughout this and the following subsections we shall assume that

the connection $(A, B)$ is critical at the left, i.e. that

\begin{equation}
\label{eq:critical-ip}
A = \theta_l\,,
\end{equation}

the case where one assumes that
$B = \theta_r$ being symmetric.
Notice that the assumption  $A = \theta_l$ does not prevent the connection to be critical also at the right, i.e. $B = \theta_r$: it might or might not happen.
We shall consider a
$\{A_n,B_n\}_n$  of non critical connections that satisfy

\begin{equation}
\label{eq:AnBnconv}
\lim_n (A_n,B_n) = (A,B)\,.
\end{equation}

We will show only that
$(1)\Rightarrow (2)$, since
the reverse implication  is clear
(see  <ref>).
Then, given
$\omega\in \mc A^{[AB]}(T)$

\begin{equation}
\label{eq:att-prof-21}
\omega = \mc S_T^{[AB]+}u_0, \qquad u_0 \in \mathbf L^{\infty}(\mathbb R)\,,
\end{equation}

\begin{equation}
\label{eq:att-prof--approx}
\omega_n \doteq \mc S^{[A_nB_n]+}_T u_0\,.
\end{equation}

Hence, since $\omega_n\in  \mc A^{[A_nB_n]}(T)$,
by the validity of Theorem <ref> in the non critical case it holds

\begin{equation}\label{eq:noncrifixedn}
\omega_n=  \mc S^{[A_nB_n]+}_T \circ \mc S^{[A_nB_n]-}_T \omega_n \qquad \forall \; n\,.
\end{equation}

Notice that by definition (<ref>)
it follows that the
${\bf L^1}$ stability property (iv) of Theorem <ref> holds also for the
$AB$-backward solution operator $\mc S^{[AB]-}_T$, so that we have

\begin{equation}
\label{eq:backelleunostab}
\mc S^{[A_nB_n]-}_T \omega_n
\ \ \rightarrow\ \
\mc S^{[AB]-}_T\omega
\qquad\text{in}\quad
\mathbf L^1(\R)\,.
% \mc S^{[AB]-}_T \omega=
% \mathbf L^1\text{-}\lim_n
% \mc S^{[A_nB_n]-}_T \omega_n\,.
\end{equation}

Hence, we deduce that

\omega \stackrel{[\text{Thm \ref{theoremsemigroup}-(iv)]}}{=} \lim_n \omega_n \stackrel{[\text{\eqref{eq:noncrifixedn}]}}{=} \lim_n  \mc S^{[A_nB_n]+}_T \circ \mc S^{[A_nB_n]-}_T \omega_n \stackrel{[\text{Thm \ref{theoremsemigroup}-(iv) and~\eqref{eq:backelleunostab}]}}{=}  \mc S^{[AB]+}_T \circ \mc S^{[AB]-}_T \omega\,,

which proves $(1)\Rightarrow (2)$.

§.§ Part 3.b - $(1) \Rightarrow (3)$ for critical connections

In this Subsection we
assume that
the implication $(1)\Rightarrow (3)$ of Theorem <ref> holds for connections which are non critical, and in particular we assume that
Theorems <ref>, <ref>, <ref>,
are verified
for non critical connections.

We will prove that
any element $\omega\in \mc A^{AB}(T)$
satisfies the conditions of
Theorem <ref>, or of Theorem <ref>,  also for a critical connection
We divide the proof
in eight steps.

Step 1.
Let $\{A_n,B_n\}_n$  be a sequence of non critical connections as in Part 3.a.
Given $\omega\in \mc A^{AB}(T)$
as in (<ref>), and

as in (<ref>),

\begin{equation}
\label{eq:u-def-34}
u(x,t) \doteq \mc S^{[AB]+}_t u_0(x), \qquad t \geq 0, \quad x \in \mathbb R\,,
\end{equation}

and consider the sequence $u_n$ of $A_nB_n$-entropy weak solutions  defined by

\begin{equation}
\label{eq:un-def-35}
u_n(x,t) \doteq \mc S^{[A_nB_n]+}_t u_0(x), \qquad t \geq 0, \quad x \in \mathbb R\,.
\end{equation}

Let $u_{n,l}, u_{n,r}$ denote, respectively, the left and right traces of $u_n$ at $x=0$ defined as in (<ref>), and let $u_l, u_r$ be the left and right traces of $u$ at $x=0$.
Then, by Theorem <ref>
and Corollary <ref>,
and because of (<ref>),
it follows

\begin{align}
\label{eq:eq:un-41}
&u_n(\cdot, t) \ \ \rightarrow\ \ u(\cdot, t)\qquad\text{in}\quad
\mathbf L^1(\R)\qquad \forall~t\in [0,T]\,,
\\
\noalign{\smallskip}
\label{eq:eq:flun-41}
\ \rightarrow\ \ f_l(u_l)\qquad\ \text{in}\quad
\mathbf L^1([0,T])\,\,,
\\
\noalign{\smallskip}
\label{eq:eq:flrun-41}
\ \rightarrow\ \ f_r(u_r)\quad\ \ \, \text{in}\quad
\mathbf L^1([0,T])\,\,,
\end{align}

and hence, in particular, we have

\begin{equation}
\label{eq:omegan-conv-35}
\omega_n
\ \ \rightarrow\ \
\omega
\qquad\text{in}\quad
\mathbf L^1(\R)\,.
% \omega = \mathbf L^1-\lim_{n \to \infty} \omega_n\,.
\end{equation}

In order to prove that $\omega$ satisfies condition (3) of Theorem <ref>,

\begin{equation}
\label{eq:LR-def-21}
\ms L \doteq \ms L[\omega,f_l]\,,
\qquad\quad
\ms R \doteq  \ms R[\omega, f_r],
\end{equation}

be quantities defined as in (<ref>),
we need to show that:

- If $\ms L = 0$, $\ms R > 0$ or viceversa, then $\omega$ satisfies the conditions of Theorem <ref>;

- If $\ms L = 0$, $\ms R = 0$, then $\omega$ satisfies the conditions of Theorem <ref>.

Notice that, since we are assuming the validity of
Theorem <ref>,
know  that the case $\ms L< 0, \ms R > 0$ is excluded because the connection $(A, B)$ is critical.

Throughout the subsection we let  $\ms L_n$,. $\ms R_n$, denote the objects defined as in (<ref>) for $\omega_n$:

\begin{equation}
\label{eq:LR-n-def-21}
\ms L_n \doteq \ms L[\omega_n,f_l]\,,
\quad\qquad \ms R_n \doteq  \ms R[\omega_n, f_r]\,.
\end{equation}

Observe that by Remark <ref>, and because of (<ref>), the functions $\omega_n$
have a uniform bound

\begin{equation}
\label{linf-bound-omega-n}
\|\omega_n\|_{{\bf L}^\infty}\leq \overline C
\qquad\quad\forall~n,
\end{equation}

for constant $\overline C>0$.
Hence, by definition  (<ref>),
the constant $|\ms L_n|$
are bounded by $T\cdot \sup_{|u|\leq \overline C} |f'_l(u)|$, and
the constant $\ms R_n$
are bounded by $T\cdot \sup_{|u|\leq \overline C} |f'_r(u)|$.
Thus, up to a subsequence, we can
define the limits

\begin{equation}
\label{eq:RLn-lim-24}
\widehat {\ms L}\doteq
\lim_{n \to \infty} \ms L_n\,,
\qquad\quad
\widehat{\ms R}\doteq
\lim_{n \to \infty} \ms R_n\,.
\end{equation}

We claim that

\begin{equation}\label{eq:Rsemic}
\widehat {\ms L}\leq {\ms L},
\qquad\quad
\widehat {\ms R}\geq \ms R\,.
% \bar {\ms R}:=  \lim_{n \to \infty} \ms R^n  \geq  \ms R , \qquad \bar {\ms L}:=  \lim_{n \to \infty} \ms L^n  \leq  \ms L
\end{equation}

By definition (<ref>), (<ref>) of $\ms R$,
in order to prove the second inequality in (<ref>),
it is sufficient to show that

\begin{equation}
\label{eq:Rsemic2}
\widehat {\ms R}
- T\cdot f'_r(\omega(\widehat {\ms R}+))\geq 0\,.
\end{equation}

Observe that by Definition <ref>
$u_n$ and $u$ are entropy weak solutions of

\begin{equation}
\label{eq:conlaw-n-qp}
\quad\ x >0, \quad t \in [0,T],