Consider a scalar conservation law
with a spatially discontinuous flux at a single point $x=0$, and assume that the flux is uniformly convex when $x\neq 0$.
Given an interface connection $(A,B)$,
we define a backward solution operator consistent with the concept of $AB$-entropy solution [4, 13, 16]. We then analyze the
family $\mc A^{[AB]}(T)$ of profiles that
can be attained at time $T>0$ by $AB$-entropy solutions
with ${\bf L^\infty}$-initial data.
We provide a  characterization of
$\mc A^{[AB]}(T)$ as fixed points of
the backward-forward solution operator.
As an intermediate step we establish a full characterization of $\mc A^{[AB]}(T)$
in terms of
unilateral constraints and Oleǐnik-type estimates, valid for all connections.

Building on such a characterization we derive uniform $BV$
bounds on the flux of $AB$-entropy solutions, which in turn yield the
${\bf L^1_{loc}}$-Lipschitz continuity
in time of these solutions.

§ INTRODUCTION

Consider the initial value problem for the scalar conservation law in one space dimension

\begin{align}
\label{conslaw}
%u_t(x,t)+f(x,u(x,t))_x = 0,
\quad x \in \mathbb{R}, \quad t \geq 0,
\\
\noalign{\smallskip}
\label{initdat}
&u(x,0) = u_0(x), \qquad x \in \mathbb{R},
\end{align}

where $u = u(x,t)$ is the state variable and the flux $f$ is a space discontinuous function given by

\begin{equation}\label{discflux}
f(x,u) =\begin{cases}
f_l(u), \qquad x < 0, \\
f_r(u), \qquad x > 0\,.
\end{cases}
\end{equation}

We assume that $f_l, f_r: \R\to \R$
are twice continuously differentiable, uniformly  convex maps that satisfy

\begin{equation}
\label{eq:flux-assumption-1}
f_l''(u),\ f_r''(u) \geq a >0\,,
\end{equation}

and (up to a reparametrization)

\begin{equation}
\label{eq:flux-assumption-2}
f_l(0) = f_r(0), \quad f_l(1) = f_r(1)\,.
\end{equation}

We assume also that
the unique critical points
$\theta_l$, $\theta_r$ of $f_l, f_r$, respectively,

\begin{equation}
\label{eq:flux-assumption-3}
\theta_l \geq 0, \quad \theta_r \leq 1\,.
\end{equation}

laws with discontinuous flux
serve as mathematical models
oil reservoir simulation [26, 27];
traffic flow dynamics
with roads of varying amplitudes or surface conditions [33];
radar shape-from-shading problems [34];
blood flow in endovascular treatments [24, 17];
and for many other different applications
(see [6] and references therein).

We recall that problems of this type
do not  posses classical solutions globally defined in time (even in the continuous flux case when $f_l=f_r$), since, regardless of how smooth the initial data are, they can develop discontinuities (shocks) in finite time
because of the nonlinearity of the equation. To achieve existence results, one has to look for
weak distributional
that, for sake of uniqueness, satisfy
the classical Kružkov entropy inequalities
away from the point of flux discontinuity, and a further interface entropy condition
at the flux-discontinuity interface $x=0$.

Various type
of interface-entropy conditions have
been introduced in the literature
according with the different physical phenomena modelled by (<ref>) (see [11, 12]).
as in [6],
for modellization and control treatment reasons
we employ an admissibility criterion involving the so-called interface connection $(A,B)$,
which yields the Definition <ref> of $AB$-entropy solution
(cfr.[4, 16]).
A connection $(A, B)$ is a pair of states connected by a stationary weak solution
of (<ref>),
taking values $A$ for $x<0$, and $B$ for $x>0$, which has characteristics diverging
from (or parallel to) the flux-discontinuity interface $x=0$

(see Definition <ref>).
The admissibility criterion
for an $AB$-entropy solution
can be equivalently formulated in terms of an interface entropy condition
or of Kružkov-type entropy inequalities adapted to the particular connection $(A, B)$ taken into account
(cfr. [4, 13, 16]). Relying on this extended entropy inequalities
and using  an adapted version of the Kružkov doubling of variables argument, one can establish ${\bf L^1}$-stability and uniqueness of $AB$-entropy solutions to the Cauchy problem (<ref>)-(<ref>)
(see [16, 25]). We shall adopt the semigroup notation
$u(x,t)\doteq \sabpt u_0 (x)$ for the unique solution of (<ref>)-(<ref>).

In this paper we are concerned as in [2, 6]
with a controllability problem
for (<ref>)
where one regards the initial data as controls and study
the corresponding attainable set
at a fixed time $T>0$:

\begin{equation}
\label{eq:attset}
\mc A^{[AB]}(T) \doteq  \big\{\mc \sabpT u_0 \; : \; u _0 \in {\bf L}^{\infty}(\mathbb R)\big\}\,.
\end{equation}

In the same spirit of [22, 23, 28, 32]
we  introduce a backward solution operator
(see Definition <ref>)

\begin{equation}
\sabmT: {\bf L^\infty}(\R) \to
{\bf L^\infty}(\R),
\qquad\quad \omega\mapsto
\sabmT \omega\,,
\end{equation}

and we
characterize the attainable targets for (<ref>) at a time horizon $T>0$
as fixed-points of the composition backward-forward operator $\sabpT \circ \sabmT$, as stated in  our first main result:

Let $f$ be a flux as in (<ref>)
satisfying the assumptions (<ref>)-(<ref>),
and let
be a connection. Then,
for every $T>0$,
and for any $\omega\in  {\bf L^\infty}(\R)$,
the following conditions are equivalent.

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

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

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

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

\begin{equation}
\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}

Clearly the main content of Theorem <ref> are the implication (1) $\Longrightarrow$ (2)
and (1)' $\Longrightarrow$ (2),
since the reverse implications
are straightforward once we define the backward operator
$\sabmT$ and verify that, in the case
of a non critical connection, one has $ \sabmT u_0\in BV_{loc}(\R)$
for all $u_0\in {\bf L^\infty}(\R)$.

The proof of (1) $\Longrightarrow$ (2)
and (1)' $\Longrightarrow$ (2)
are obtained in two steps:

First, we show that any attainable profile $\omega \in \mc A^{[AB]}(T)$
to a class of functions
$\msc A\subset BV_{\mr{loc}}(\mathbb R \setminus \{0\})$
which satisfy suitable Oleǐnik-type inequalities and pointwise constraints
related to the $(A,B)$-connection
in intervals containing the origin
(see Theorem <ref>, <ref>, <ref>).

Next, we prove that any element of
$\msc A$ is a fixed point of
the composition  backward-forward operator $\sabpT \circ \sabmT$.
Namely, for any given $\omega\in \msc A$
we construct an initial datum $u_0\in{\bf L^\infty(\R)}$ such that $\omega=\sabpT u_0$, and then we show that indeed $u_0=\sabmT\omega$.

These two steps are firstly carried out
in the case of a non critical
connection $(A,B)$
and of attainable profiles
$\omega\in \mc A^{[AB]}(T)\cap BV_{loc}(\mathbb R)$.

The proofs are obtained
exploiting as in [6]  the theory of generalized characteristics
by Dafermos [20],
applied to the setting of discontinuous flux, and relying on the duality property of the backward and forward solution operators. Next, we address the case of a critical connection and
of attainable profiles $\omega\in \mc A^{[AB]}(T)$ relying on the ${\bf L^1}$-stability of the map
$(A,B,u_0)\mapsto \sabpt u_0$
(see Theorem <ref>).

Some remarks are here in order.

* The results of Theorem <ref> extend to the present setting of space discontinuous fluxes the similar characterization of attainable profiles in terms of the backward solution operator obtained in <cit.> and <cit.>

for conservation laws with strictly convex flux independent on the space variable.

* The characterization of $\mc A^{[AB]}(T)$
obtained in this paper  unveils the presence of a classs of attainable states
for not critical connections
that were not detected in [2, 6],  see Remark <ref>.

* The characterization of
attainable profiles for (<ref>),
(<ref>) in terms of
unilateral constraints and Oleǐnik-type estimates provides a powerful tool to investigate regularity properties
of the solutions to  (<ref>),
(<ref>). In particular,
we build on such a characterization to
derive uniform BV bounds
on $AB$-entropy solutions
with initial datum in ${\bf L^\infty}$ (in the case of
non critical connections), and on the flux
of $AB$-entropy solutions (for general connections).
This is a fairly non-trivial result since it is well known [1] that the total variation of $AB$-entropy solutions
may well blow up in a neighborhood of the
flux-discontinuity interface $x=0$.
Thanks to these uniform BV bounds,
we can then establish the
${\bf L^1_{loc}}$-Lipschitz continuity
in time of $AB$-entropy solutions.

* The proof that Theorem <ref>
holds for critical connections once we know that Theorem <ref>
is verified by non critical connections
relies on a perturbation argument
for attainable profiles.
This construction yields an approximate controllability result since it
provides a general explicit procedure to  approximate an attainable profile for a critical connection by attainable profiles for non critical connections.

Note furthermore that, by the non-backward uniqueness of (<ref>)
(due to the possible presence of shocks
in its solutions),
there may exist in general multiple initial data $u_0$
that are stirred by (<ref>) to
$\omega \in \mc A^{[AB]}(T)$.

In fact, an important control problem related to the one considered in this paper is the inverse design,
which has the goal to reconstruct

the set of initial data $u_0$ evolving to a given
attainable target $\omega$
(see [19, 28, 32]
for conservation laws with  convex flux independent on the space variable, and [23]
for Hamilton-Jacobi equations with convex Hamiltonian).
On the other hand, when
a target state $\omega$ is not attainable at time $T>0$, the image of $\omega$
through the backward-forward operator $\sabpT \circ \sabmT$ represents a natural candidate to construct a reachable function which is “as
close as possible" (in an appropriate sense) to the the observed state $\omega$ (see [22]
in the case of Hamilton-Jacobi
and Burgers equations).

The results of the present paper
provide a key building block
to address both of these problems, namely
the characterization of the aforementioned set of initial data leading to a given
attainable target $\omega$
for (<ref>),
and  the analysis of the properties
of the backward-forward operator $\sabpT \circ \sabmT$
related to optimization problems for unattainable target profiles, which are
pursued in the
forthcoming paper [10].

The paper is organized as follows. In  <ref>
we recall the definitions of interface connection $(A,B)$, of $AB$-entropy solution and of $AB$-backward solution operator.
We also collect the stability properties
of the ${\bf L^1}$-contractive semigroup of $AB$-entropy solutions.
In  <ref>
we establish the duality property
of the backward and forward solution operators, which constitutes a fundamental ingredient

of the proof of Theorem <ref>.
collects the precise statements of the
results on the characterization
of the attainable set $\mc A^{[AB]}(T)$
Oleǐnik-type inequalities and state constraints.
In  <ref>
we carry out the rather technical and involved proof of Theorem <ref>.
At the beginning of the section,
for reader's convenience,
we  provide a roadmap of the proof of Theorem <ref>,
where we also highlight the key innovative parts of the paper.

In Appendix <ref>
we establish the ${\bf L^1}$-stability
properties of the semigroup of $AB$-entropy solutions with respect to time and with respect to the connections.
We also derive uniform BV bounds
on $AB$-entropy solutions
in the case of non critical connections,
and on the flux of $AB$-entropy solutions
for general connections.
In Appendix <ref>
we provide, for sake of completeness, a simple proof
of the non existence of rarefactions
emanating from the interface $x=0$,
which is a distinctive feature of
$AB$-entropy solutions.
Finally, in Appendix <ref>
we derive some lower/upper ${\bf L^1}$-semicontinuity property for solutions to conservation laws, used to recover the
proof of Theorem <ref>
in the case of critical connections
once we know the validity of Theorem <ref>
for non critical connections.

§ BASIC DEFINITIONS AND GENERAL SETTING

§.§ Connections and $AB$-entropy solutions

We recall here the definitions and properties of interface connection and of entropy admissible solution introduced in [4].

[Interface Connection]

Let $f$ be a flux as in (<ref>)
satisfying the assumptions (<ref>)-(<ref>).
A pair of values $(A,B)\in \mathbb R^2$ is called a connection if

* $f_l(A) = f_r(B)$,

* $A\leq \theta_l$ and $B \geq \theta_r$.

We will say that connection $(A,B)$ is  critical if $A=\theta_l$
or $B=\theta_r$.

[scale = 0.7]
[->] (-6,0)–(4,0)node[right]$u$;
[scale = 0.5, domain=-1:5, smooth, variable=] plot (, (-2)*(-2)+1);
[scale = 0.5, domain=-9:3, smooth, variable=] plot (, (0.5*+1)*(0.5*+2)+1);

(-5,5) node$f_l$;
(3,5) node$f_r$;

[very thick, blue] (-3.8,3)–(2.13,3);
[dotted] (-3.8,3)–(-3.8,0)node[below]$A$;
[dotted] (2.13,3)–(2.13,0)node[below]$B$;

[dotted] (-1.5,0.3)–(-1.5,0)node[below]$\theta_l$;
[dotted] (1,0.5)–(1,0)node[below]$\theta_r$;

An example of connection $(A,B)$ with $f_l, f_r$ strictly convex fluxes

every picture/.style=line width=0.75pt

[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, scale =0.9]

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[dash pattern=on 3pt off 3pt]    (180.5,170.75) – (222,212) ;

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[dash pattern=on 3pt off 3pt]   (350,90.17) – (299.67,110.17) ;

[dash pattern=on 3pt off 3pt]    (419.67,136.5) – (301.67,189.17) ;

[dash pattern=on 3pt off 3pt]    (390.67,90.83) – (300,130.5) ;

[dash pattern=on 3pt off 3pt]    (420,95.5) – (299.67,150.17) ;

[dash pattern=on 3pt off 3pt]    (419.67,157.5) – (300.5,210.83) ;

[dash pattern=on 3pt off 3pt]    (418.67,194.5) – (379.67,210.5) ;

(232.67,216.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $A$;

(355,215.9) node [anchor=north west][inner sep=0.75pt]  [font=]  $B$;

(419,216.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x$;

(305.5,74.9) node [anchor=north west][inner sep=0.75pt]  [font=]  $t$;

The stationary undercompressive solution $c^{AB}$.

Observe that condition (2) is equivalent
to: $f'_l(A)\leq 0$,  $f'_r(B)\geq 0$.
Therefore, if $(A,B)$ is a connection,
the function

\begin{equation}\label{eq:cABdef}
c^{AB}(x) \stackrel{\cdot}{=}
\begin{cases}
A, & x \leq 0, \\
B, & x \geq 0
\end{cases}
\end{equation}

is a weak stationary undercompressive (or marginally undercompressive) solution of (<ref>), since the characteristics diverge from, or are parallel to, the flux-discontinuity interface (see Figure <ref>).
In relation to the function $c^{AB}$ it is introduced  in [16] the   adapted entropy
$\eta_{AB}(x,u)=\big|u-c^{AB}(x)\big|$. Then, in the spirit of [14], it is employed in [16] the entropy $\eta_{AB}$ to select a unique
solution of the Cauchy problem (<ref>)-(<ref>) that satisfies the interface entropy inequality

\begin{equation}\label{adaptedABentropy}
\left|u-c^{AB}\right|_t + \left[\mathrm{sgn}(u-c^{AB})(f(x,u)-f(x,c^{AB}))\right]_x \leq 0, \quad \textrm{in}\ \ \mathcal{D}^{\prime}\,,
\end{equation}

in the sense of distributions, which leads to the following definition.

[$AB$-entropy solution]
Let $(A,B)$ be a connection
and let $c^{AB}$ be the function defined in (<ref>). A function $u \in \mathbf L^{\infty}(\mathbb R \times [0,+\infty[\,)$ is said to be an $AB$-entropy solution of the problem (<ref>),(<ref>) if the following holds:

* $u$ is a distributional solution of (<ref>) on $\R\times \,]0,+\infty[$, that is, for all test functions $\phi \in \mathcal{C}^1_c$

with compact support
contained in $\mathbb{R}\times \,]0,+\infty[$,
there holds

\begin{equation}\label{AB1}
\int_{-\infty}^{\infty}\int_0^{\infty}  \big\{u \phi_t+f(x,u)\phi_x \big\}\dif x \dif t = 0\,.
\end{equation}

* $u$ is a Kružkov entropy weak solution of (<ref>),(<ref>) on $(\mathbb R \setminus \{0\}) \times \,]0,+\infty[$, that is, $t \mapsto u(\cdot,t)$ is a continuous map from $[0,+\infty[\,$ to $\mathbf L^1_{\textup{loc}}(\mathbb R)$,
the initial condition (<ref>) is satisfied almost everywhere,

for any non-negative test function $\phi\in\mathcal{C}^1_c$ with compact support
contained in $]-\infty,0[\,\times $ $\,]0,+\infty[$, there holds

\begin{equation}\label{kruz+}
\int_0^{\infty} \int_{-\infty}^0  \big\{|u-k| \phi_t + \mathrm{sgn}(u-k)\left(f_l(u) - f_l(k)\right) \phi_x \big\}
\dif x \dif t \geq 0, \quad \forall k \in \mathbb R\,;
\end{equation}

for any non-negative test function $\phi\in\mathcal{C}^1_c$ with compact support
contained in $]0,+\infty[\,\times$ $]0,+\infty[$, there holds

\begin{equation}\label{kruz-}
\int_0^{\infty}\int_0^{\infty} \big\{
|u-k| \phi_t + \mathrm{sgn}(u-k)\left(f_r(u) - f_r(k)\right) \phi_x
\big\}\dif x \dif t \geq 0, \quad \forall k \in \mathbb R\,.
\end{equation}

* $u$ satisfies a Kružkov-type entropy inequality relative to the connection $(A,B)$, that is,
for any non-negative test function $\phi\in\mathcal{C}^1_c$ with compact support
contained in $\R\times ]0,+\infty[$, there holds

\begin{equation}\label{adaptedentropyinequality}
\int_{-\infty}^{\infty}\int_0^{\infty} \big\{\left|u-c^{AB}\right| \phi_t + \mathrm{sgn}(u-c^{AB})\left(f(x,u) - f(x,c^{AB})\right)\phi_x \big\}\dif x \dif t \geq 0\,.
\end{equation}

an $AB$-entropy solution $u$ is in particular an entropy weak solution
of a scalar conservation law
with uniformly convex flux,
on $]-\infty, 0[\,\times\,]0,+\infty[$,
and on $]0,+\infty[\,\times\,]0,+\infty[$
(by property (2) of Definition <ref>
and assumption (<ref>)),
it follows that $u(\cdot\,,t)\in {BV}_{\mathrm{loc}}(\mathbb R \setminus \{0\})$ for any $t>0.$
Here $BV_{\mathrm{loc}}(\mathbb R\setminus\{0\})$
denotes the set of functions that have finite total variation on bounded subsets of $\mathbb R$ bounded away from the origin.
On the other hand,
relying on  a result in [35]
(see also [36]),
and because of the $\mathbf L^1_{\textup{loc}}$-continuity of the map $t \mapsto u(\cdot,t)$,
we deduce that
$u$ admits left and right strong traces at $x = 0$ for all $t>0$, i.e. that
there exist the one-sided limits
\begin{equation}\label{traces}
u_l(t)\doteq u(0-,t),
%\lim_{x\to 0-}u(x,t),
\qquad
u_r(t)\doteq  u(0+,t),
%\lim_{x\to 0+}u(x,t),
\qquad\forall~t>0\,.
\end{equation}

since $u$ is also a distributional solution of (<ref>) on $\R\times \,]0,+\infty[$
(by property (1) of Definition <ref>), we deduce that
$u$ must satisfy the Rankine-Hugoniot condition at the interface $x=0:$

\begin{equation}\label{RHtraces}
f_l(u_l(t)) = f_r(u_r(t)), \qquad \text{for a.e. $t>0$}.
\end{equation}

In (<ref>) and throughout the paper, for the one-sided limits of a function $u(x)$ we use the notation

\begin{equation}
u(x\,\pm)\doteq \lim_{y\to x\,\pm} u(y).
\end{equation}

In view of (<ref>),
we can extend the classical concept of genuine characteristic for solutions to conservation laws with continuous fluxes
(see [20])
by considering also characteristics that are refracted by the discontinuity interface $x=0$. Thus, we will say  that a polygonal line $\vartheta : [0,T]\to\R$ is a genuine characteristic for an $AB$-entropy solution $u$ if  one of the following cases occurs:

(i) $\vartheta(t)<0$ for all $t\in\,]0,T[\,$, and $\vartheta$ is a  characteristic for the restriction of $u$ on  $]-\infty, 0[\,\times\,]0, T[\,$;

(ii) $\vartheta(t)>0$ for all $t\in\,]0,T[\,$, and $\vartheta$ is a characteristic for the restriction of $u$ on  $]0,+\infty[\,\times\,]0, T[\,$;

(iii) there exists $\tau\in \,]0, T[$ , such that:

$\vartheta(t)<0$ for all $t\in\,]0,\tau [\,$, and $\vartheta$ is a characteristic for the restriction of $u$ on  $]-\infty, 0[\,\times\,]0, \tau[\,$,

$\vartheta(t)>0$ for all $t\in\,]\tau,T[\,$, and $\vartheta$ is a characteristic for the restriction of $u$ on  $]0,+\infty[\,\times\,]\tau, T[\,$,

or viceversa.

where we are using the term “characteristic” for a classical genuine characteristic of a solution to the conservation law $u_t+f_l(u)_x=0$
on $\{x<0\}$, or to the conservation law $u_t+f_r(u)_x=0$
on $\{x>0\}$.

In relation to a connection $(A,B)$
consider the function

\begin{equation}
\label{IABdef}
I^{AB}(u_l,u_r)  \stackrel{\cdot}{=} \mathrm{sgn}(u_r-B)\left(f_r(u_r)-f_r(B)\right) \\- \mathrm{sgn}(u_l-A)\left(f_l(u_l)-f_l(A)\right),
\qquad u_l,u_r\in\R\,,
\end{equation}

which is useful to characterize
the interface entropy admissibility criterium. In fact, by the analysis in  <cit.> and  <cit.>,
it follows that, because of
condition (1) of Definition <ref>
and assumption (<ref>),
the following holds.

Let $u \in \mathbf L^{\infty}(\mathbb R \times [0,+\infty))$
be a function satisfying conditions (1)-(2) of Definition <ref>.
Then, condition (3) is equivalent to the
$AB$ interface entropy condition

\begin{equation}
\label{interfaceentropy}
I^{AB}(u_l(t),u_r(t)) \leq 0 \qquad\quad  \text{for a.e. $t>0$\,.}
\end{equation}

Let $(A,B)$ be a connection. Then, for any pair $(u_l,u_r)\in\R^2$, the conditions

\begin{equation}
\qquad\quad
I^{AB}(u_l, u_r)\leq 0\,,
\end{equation}

are equivalent to the conditions

\begin{equation}
\label{ABtraces}
\begin{aligned}
&f_l(u_l) = f_r(u_r) \geq f_l(A) = f_r(B), \\
\noalign{\smallskip}
&\big(u_l \leq \theta_l, \quad u_r \geq \theta_r\big) \ \ \Longrightarrow \ \ u_l = A, \ \ u_r = B\,.
\end{aligned}
\end{equation}

The first condition in (<ref>)
tells us that, when we choose
a connection $(A,B)$
and we employ the concept of
$AB$-entropy solution, we are
imposing a constraint (from below) on the flux at the interface $x=0$.
In order to achieve existence, we need to compensate for this constraint with an  additional freedom in the admissibility criteria.
In fact, the second condition in (<ref>)
prescribes the admissibility of exactly one undercompressive wave at the interface, given by $c^{AB}$ in (<ref>).
This rule corresponds to the
$(A,B)$ characteristic condition in <cit.>.

It was proved in [4, 16] (see also [13, 25])

that  $AB$-entropy solutions of (<ref>),(<ref>)
with bounded initial data are unique and form an $\mathbf{L}^1$-contractive semigroup. Moreover, we will show that they are ${\bf L^1}$-stable also with respect to the
values $A,B$ of the connection.

This type of stability, beside
being used to extend our main results
from the case of non-critical connections to the critical one,

an interest in its own.

We will also
prove that $AB$-entropy solutions of (<ref>),(<ref>)
are ${\bf L^1}$-Lipschitz continuous in time. This property is an immediate consequence of the BV regularity
of such solutions in the case of
non critical connections.
in the case of critical connections where
$A=\theta_l$ or $B=\theta_r$, and $f_l(\theta_l)\neq f_r(\theta_r)$, the total variation
of an $AB$-entropy solution
may well blow up in a neighborhood of the
flux-discontinuity interface $x=0$,
as shown in [1].
However, we recover the ${\bf L^1}$-Lipschitz continuity in time also in this case exploiting the BV regularity of the flux of an $AB$-entropy solution, which
is established relying on the analysis pursued in this paper.

We collect all these (old and new) results in the following:

(Semigroup of $\textbf{\textit{AB}}$-Entropy Solutions)
Let $f$ be a flux as in (<ref>)
satisfying the assumptions (<ref>)-(<ref>),
and let
be a connection. Then there exists a map

\begin{equation}
\sabp: [0,+\infty[\,\times\,{\bf L^\infty}(\R) \to
{\bf L^\infty}(\R),
\qquad\quad (t,u_0)\mapsto
\sabpt u_0\,,
\end{equation}

enjoying the following properties:

(i) For each $u_0 \in {\bf L}^{\infty}(\R)$, the function $u(x,t) \doteq \sabpt u_0(x)$ provides the unique bounded, $AB$-entropy  solution of the Cauchy problem (<ref>), (<ref>).

$\sabp_0  u_0 = u_0$,  $\sabp_s \circ \sabp_t u_0= \sabp_{s+t}u_0$,  for all  $t,s \geq 0$, $u_0\in {\bf L^\infty}(\R)$.

(iii) $\big\Vert \sabpt u_0-\sabpt v_0\big\Vert_{{\bf L}^1(\R)} \leq \big\Vert u_0-v_0\big\Vert_{{\bf L}^1(\R)}$,
for all  $t\geq 0$,   $u_0, v_0\in {\bf L^\infty}(\R)$.

(iv) $\big\Vert \sabpt u_0-\mathcal{S}_t^{[A^{\prime}B^{\prime}]+}v_0\big\Vert_{{\bf L}^1(\R)} \leq 2\,t\, \big| f_r(B)-f_r(B^{\prime}) \big|+ \big\Vert u_0-v_0\big\Vert_{{\bf L}^1(\R)}$, for all connections $(A,B)$, $(A^{\prime}, B^{\prime})$, and for every $t \geq 0$,  $u_0, v_0\in {\bf L^\infty}(\R)$.

For each
$u_0\in {\bf L^\infty}(\R)$
and for any bounded set $K$,

there exists a constant $C_{K}>0$  depending only on the flux $f$, on the connection $(A,B)$,
on $\|u_0\|_{\bf L^1}$, and on $K$, such that:

$\big\Vert \sabpt u_0-\mc S^{[AB]+}_s u_0\big\Vert_{{\bf L}^1{(K)} } \leq \dfrac{C_{K}}{t}
%\min\{s, t\}^{-1}
|s-t|$ for all $s> t>0$.

The proof of the new properties (iv)-(v) is given in  Appendix <ref>.

By the analysis in <cit.>
(see also <cit.>)
it follows that, for every $M>0$,
there exists  $C_M>0$ such that, if $\|u_0\|_{{\bf L}^\infty}\leq M$, and $A,B\leq M$, then
$\|\sabpt u_0\|_{{\bf L}^\infty}\leq C_M$, for all $t>0$.

Let $\{(A_n, B_n)\}_n$ be a sequence of connections that converge in $\mathbb R^2$ to a connection $(A, B)$, and let $\{u_{n,0}\}_n$
be a sequence of functions in  ${\bf L}^\infty(\R)$
that converge in ${\bf L}^1$ to $u_0\in {\bf L}^{\infty}(\R)$.
Let $u_{n,l}, u_{n,r}$ denote, respectively,
the left and right traces
at $x=0$ of $u_n(x,t)\doteq \mc S^{[A_nB_n]+}_t u_{n,0}(x)$,
defined as in (<ref>).
Similarly, let $u_l, u_r$ denote the left and right traces
at $x=0$ of $u(x,t)\doteq \mc S^{[AB]+}_t u_0(x)$.
Then, there hold:

\begin{equation}
\label{eq:flux-trace-conv1}
\ \ \rightarrow\ \
\qquad
\ \ \rightarrow\ \
\quad\text{in}\quad
\mathbf L^1([0,T])\,,
\end{equation}

for all $T>0$.

The proof of the Corollary is given in  Appendix <ref>.

We point out that, differently
from (<ref>),
in general
the ${\bf L^1}$-convergence
$u_{n, l} \to u_l$
and $u_{n, r} \to u_r$
fails due to the possible formation of stationary boundary layers
at the interface $x=0$, as one can see in the following

Consider a  noncritical connection

$(A, B)$ and the sequence of initial data

u_{n,0}(x) = \begin{cases}
\overline A, & \text{if $x \leq -n^{-1}$}, \\
A, & \text{if $x \in \,]-n^{-1},0[$}, \\
\overline B, & \text{if $x \geq 0$},
\end{cases}

\begin{equation}
\label{eq:bar-AB-def}
\overline A \doteq ({f_l}_{\mid [\theta_l, +\infty[\,})^{-1}\circ f_l(A)\,,
\qquad\quad
\overline B \doteq
({f_r}_{|\,]-\infty, \theta_r]})^{-1}\circ f_r(B),
\end{equation}

where $f_{|\,I}$
denotes the restriction of the function $f$ to the interval $I$.
One can immediately check that the $AB$-entropy  solution
of (<ref>), (<ref>), with initial datum $u_{n,0}$ is the stationary solution $u_n(t, x) = \mc S_t^{[AB]+}u_{n,0}(x) = u_{n,0}(x)$ and that $u_{n}$ converges in ${\mathbf L^1}$ to

\overline A\ \ &\text{if}\quad x<0,
\\
\noalign{\smallskip}
\overline B\ \ &\text{if}\quad x>0\,.
\end{cases}

Moreover, one has $u_{n,l}(t) = A$ and $u_l(t) = \overline A$ for every $t > 0$.

§.§ Backward
solution operator

In this section we shall first  review quickly the concept of
backward solution

for conservation laws with flux

depending only on the state variable, and then
we will introduce the
definition of
backward solution

operator associated to a connection $(A,B)$,
for spatially discontinuous flux as in (<ref>).

§.§.§ Backward
solution operator for conservation laws with space independent flux

The use of the backward-forward method to characterize
the attainable set for

conservation laws   was first proposed in [28, 32]  (see also [22] in the framework of Hamilton-Jacobi equations).
Because of the regularizing effect of the nonlinear dynamics
of a conservation law

\begin{equation}
\label{eq:convexconslaw}
u_t+f(u)_x = 0,\qquad  x\in\R\,,
\end{equation}

uniformly convex flux $f(u)$,
the only restriction to
controllability of (<ref>) at a fixed time $T>0$, when one regards as controls the initial data

\begin{equation}
\label{eq:indatum}
u(x,0) = u_0(x), \qquad\quad\, x \in \mathbb R\,,
\end{equation}

is the decay of positive waves.
Therefore it is by now well known
the characterization of the
attainable set

\begin{equation}
\label{eq:convvattsett}
\mc A(T) = \big\{\omega \; : \; \omega = u(\cdot,T), \; \text{$u$ entropy weak solution of \eqref{eq:convexconslaw}-\eqref{eq:indatum} with $u_0\in{\bf L^\infty}$}\big\},
\end{equation}

in terms of the
Oleǐnik-type inequality

\begin{equation}\label{eq:convexoleinik}
D^+ \omega(x) \leq \frac{1}{Tf^{\prime \prime}(\omega(x))}, \quad \forall x \in \mathbb R\,,
\end{equation}

where $D^+ \omega$ denotes the upper Dini derivative of $\omega$
(see (<ref>)).
Similar results in the case of boundary controllability were obtained in [3, 8, 9, 29]).

A different perspective to address this controllability problem
was introduced in [28, 32],
and consists in constructing
initial data leading to
attainable targets $\omega$
at a time horizon $T>0$,
through the definition
of an
appropriate concept of backward solution to (<ref>).
Namely, letting $\mc S_t^+u_0(x)$ denote the (forward) entropy weak solution of the Cauchy problem (<ref>)-(<ref>)
evaluated at $(x,t)$,
it was defined in [32] an appropriate backward operator $\mc S_T^-: {\bf L^{\infty}} \to {\bf L^{\infty}}$, and proved that a profile $\omega\in\mc A(T)$

if and only if $\omega = \mc S_{T}^+ \circ \mc S_{T}^- \omega$, i.e. if and only if it is a fixed point of the backward-forward operator $\mc S_{T}^+ \circ \mc S_{T}^-$ (see  <cit.>).
Moreover, for $\omega \in \mc A(T)$, the solution defined as

\begin{equation}
u^*(x,t) \doteq  \mc S_t^+ (\mc S_{T}^- \omega)(x),
\qquad x\in\R,\ t\in [0,T]\,,
\end{equation}

is the unique  solution to (<ref>) that
is locally Lipschitz on $]0,T[$, and yields $\omega$ at time $T$. Equivalently,

\begin{equation}
\label{eq:indatum-lip}
u^*_0 \doteq \mc S_{T}^- \omega
\end{equation}

is the unique initial datum that produces a  solution to (<ref>)
locally Lipschitz
on $]0,T[$,
yielding $\omega$ at time $T$.
The operator $\mc S_{t}^-$, for $t \geq 0$, is defined as follows

\begin{equation}
\label{eq:backw-sol-def}
\mc S_{t}^-\omega(x)  \doteq  \mc S_t^+ \big(\omega(-\ \cdot\,)\big)(-x)\qquad x\in\R,\ t\geq 0\,.
\end{equation}

In words, we use $\omega(-\ \cdot\,)$ as initial datum for the forward operator $\mc S_t^+$, we compute  the (forward) solution
to (<ref>), and then we reverse the space variable.

One can easily verify that
the function $\text{w}(x,t)\doteq \mc S_{t}^-\omega(x)$ is the entropy weak solution of the Cauchy problem

\begin{equation}
\label{eq:backw-cp}
\begin{cases}
\text{w}_t-f(\text{w})_x=0, \quad &x \in \mathbb{R}, \quad t \geq 0,
\\
\noalign{\smallskip}
\text{w}(x,0)=\omega(x), \quad &x \in \mathbb{R}.
\end{cases}
\end{equation}

In fact, by definition (<ref>) it follows that $\text{w}(x,t)$
is a distributional solution
of (<ref>), since it is obtained from the distributional solution $\mc S_t^+ \big(\omega(-\ \cdot\,)\big)(x)$ of (<ref>) by the change of variable $x\mapsto -x$.
On the other hand,
since every shock discontinuity
of $\mc S_t^+ \big(\omega(-\ \cdot\,)\big)(x)$,
connecting a left state $u^-$ with a right state $u^+$, must satisfy the Lax condition $u^->u^+$
(equivalent to the entropy admissibility criterium since the  flux $f(u)$ in (<ref>) is convex, e.g. see [21, 30]),

it follows
that the left and right states
$u^-, u^+$ of
every shock discontinuity in $\text{w}(x,t)$
must satisfy the reverse condition
$u^-<u^+$, which is the Lax admissibility condition for (<ref>), since the flux $-f(\text{w})$ is concave.
Finally, we can observe that
$\text{w}(x,0)=\mc S_0^+ \big(\omega(-\ \cdot\,)\big)(-x)=\omega (x)$, for all $x\in\R$,
which completes the proof of our claim.

This procedure to characterize the attainable profiles is motivated by the following observation.
Given a target profile
$\omega$, if we know that
for any $t\in \,]0,T[\,$,
the map $x\mapsto v(x,t)\doteq \mc S_t^+ \big(\omega(-\ \cdot\,)\big)(x)$
is locally Lipschitz on $\R$,
it would follows that
$u(x,t)\doteq v(-x,T-t) =
\mc S_{T-t}^-\,\omega(x) $
is a classical solution
of (<ref>)
which attains the target profile
$\omega$ at time $t=T$,
and starts with the initial datum $u_0^*$ in (<ref>).
Since classical solutions
of (<ref>)
entropy admissible,
by uniqueness of entropy weak solutions of the Cauchy problem for (<ref>)
it would follow that $u(x,t)=
\mc S_t^+ u_0^*(x)=u^*(x,t)$.

However, if $v$ admits shock
the function $v(-x,T-t)$
fails to be an entropy admissible solution of (<ref>),
despite still being a weak distributional solution of (<ref>).

one-sided Lipschitz condition (<ref>)

is precisely equivalent to the property that the map $x\mapsto v(x,t)\doteq \mc S_t^+ \big(\omega(-\ \cdot\,)\big)(x)$
is locally Lipschitz on $\R$,
for all $t\in \,]0,T[\,$
(e.g. see [7, 8]),

and thus one obtains
the characterization of
the elements of $\mc A(T)$

as fixed points of
the  backward-forward operator.

§.§.§ Backward
solution operator in the spatially-discontinuous flux setting

Given a flux  $f$  as in (<ref>)
satisfying the assumptions (<ref>)-(<ref>),
and a connection
$\sabp$  be the forward semigroup operator associated to the connection $(A,B)$, as in Theorem <ref>.
Observe that,
letting $\overline{A}, \overline{B}$
be as in

the pair $(\overline B, \overline A\,)$
turns out to be a connection for the symmetric flux

\begin{equation}
\label{eq:symm-flux}
\overline{f}(x,u) = \begin{cases}
f_r(u) , & x \leq 0, \\
f_l(u), & x \geq 0\,,
\end{cases}
\end{equation}

(see Figure <ref>).

[scale = 0.7]
[->] (-6,0)–(4,0)node[right]$u$;
[scale = 0.5, domain=-1:5, smooth, variable=] plot (, (-2)*(-2)+1);
[scale = 0.5, domain=-9:3, smooth, variable=] plot (, (0.5*+1)*(0.5*+2)+1);

(-5,5) node$f_l$;
(3,5) node$f_r$;

[very thick, blue] (-3.8,3)–(2.13,3);
[dotted] (-3.8,3)–(-3.8,0)node[below]$A$;
[dotted] (2.1,3)–(2.1,0)node[below]$B$;
[dotted] (0.8,3)–(0.8,0)node[below]$\overline A$;
[dotted] (-0.1,3)–(-0.1,0)node[below]$\overline B$;

The connection $(\overline B, \overline A)$ of the symmetric flux $\overline f(x, u)$ defined in (<ref>).

Then, letting
$\sobapt u_0(x)$ denote
the unique $\overline B\,\overline A$-entropy solution

\begin{equation}\label{eq:invertedproblem}
\begin{cases}
u_t+\overline{f}(x,u)_x = 0 & x \in \mathbb{R}, \quad t \geq 0, \\
\noalign{\smallskip}
u(x,0) = u_0(x) &  x \in \mathbb R,
\end{cases}
\end{equation}

evaluated at $(x,t)$,
we shall define the backward
solution operator associated to the connection  $(A,B)$ in terms of the operator $\sobapt$ as follows.

[$AB$-Backward solution operator]
Given a connection $(A,B)$, the backward solution operator
associated to $(A,B)$ is the map
$\mc S_{(\cdot)}^{\, [A B]-} :
%$\mc S_{(\cdot)}^{AB} : (-\infty,0]
[0,+\infty)\times {\bf L^{\infty}(\R)} \to {\bf L^{\infty}(\R)}$,  defined by

\begin{equation}
\label{eq:backw-conn-sol-def}
\sabmt \omega (x) \doteq  \sobapt
\big(\omega(-\ \cdot\,)\big)
%    (y \mapsto \omega(-y))
(-x)\qquad x\in\R, \ t\geq 0\,.
\end{equation}

One can show that the function
$\text{w}(x,t)\doteq \sabmt \omega (x)$ is the $\overline{A}\,\overline{B}$-entropy solution of the Cauchy problem

\begin{equation}
\label{eq:backw-cp-disc}
\begin{cases}
\text{w}_t-f(x,\text{w})_x=0, \quad &x \in \mathbb{R}, \quad t \geq 0,
\\
\noalign{\smallskip}
\text{w}(x,0)=\omega(x), \quad &x \in \mathbb{R}.
\end{cases}
\end{equation}

Notice that in (<ref>) the flux is $-f(x,\text{w})$,
which is a discontinuous function that
coincides with the
uniformly strictly concave maps $-f_l(\text{w}), -f_r(\text{w})$, on the left and on the right, respectively, of $x=0$.
As observed in <cit.>, in the case
of a two-concave flux as $-f(x,\text{w})$, one replaces
the $\leq$ sign with the $\geq$ sign, and viceversa,
in the Definition <ref> of
interface connection.

Thus, $(\overline{A},\,\overline{B})$
is indeed a connection for the flux
interface entropy admissibility condition for $\text{w}(x,t)$ is formulated
as in (<ref>).
In order to verify the claim that
$\text{w}(x,t)$ is the $\overline{A}\,\overline{B}$-entropy solution of the Cauchy problem (<ref>)
we proceed as in Remark <ref>.
We first observe that
$\text{w}(x,t)$ is a distributional solution of (<ref>),
and that it is entropy admissible
in the regions $\{x<0\}$, $\{x>0\}$.
In fact,  by definition (<ref>), $\text{w}(x,t)$
is obtained from $\sobapt
\big(\omega(-\ \cdot\,)\big)(x)$ with the
change of variable $x\mapsto -x$, and there holds $\text{w}(x,0)=\sobap_0
\big(\omega(-\ \cdot\,)\big)(-x)=\omega(x)$,
for all $x\in\R$.
Next, we check that $\text{w}(x,t)$ satisfies the $\overline A\, \overline B$
entropy  condition (<ref>)
for the two-concave flux $-f(x,\text{w})$, i.e. that,
letting $\text{w}_l(t), \text{w}_r(t)$ denote the left and right traces of $\text{w}(x,t)$ at $x=0$, there holds

\begin{equation}
\label{eq:entr-conc-w}
\mathrm{sgn}(\text{w}_r(t)-\overline{B}\,  )\left(-f_r(\text{w}_r(t))+f_r(\,\overline{B}\,)\right)-
\mathrm{sgn}(\text{w}_l(t)-\overline{A}\,)\left(-f_l(\text{w}_l(t))+f_l(\,\overline{A}\,)\right)
\leq 0\quad  \text{for a.e. $t>0$}\,.
\end{equation}

Observe that
the left and right traces
$u_l(t), u_r(t)$
of $\sobapt
\big(\omega(-\ \cdot\,)\big)(x)$ at $x=0$,
the $\overline{B}\,\overline{A}$ entropy condition (<ref>) for the flux $\overline f$ in (<ref>), that

\begin{equation}
\label{eq:entr-conv-u}
\mathrm{sgn}(u_r(t)-\overline{A}\,  )\left(f_l(u_r(t))-f_l(\,\overline{A}\,)\right)-
\mathrm{sgn}(u_l(t)-\overline{B}\,)\left(f_r(u_l(t))-f_r(\,\overline{B}\,)\right)
\leq 0\qquad  \text{for a.e. $t>0$}\,.
\end{equation}

On the other hand,
since one obtains $\text{w}(x,t)$
from $\sobapt
\big(\omega(-\ \cdot\,)\big)(x)$
reversing the space variables,

we have $u_l(t)=\text{w}_r(t)$,
for all $t>0$.
Hence, we recover (<ref>)
from (<ref>),
thus completing the proof of the claim.

We observe that

if $\omega$ is an attainable state in $\mc A^{[AB]}(T)$, it will follow from our results that
the solution $v(x,t)\doteq \sobapt  \big(\omega(-\ \cdot\,)\big)(x)$
related to the backward solution operator
may well contain a shock discontinuity exiting from the interface $x=0$ at a time $\tau< T$.
As a consequence here,
differently from the space-independent flux setting,
the map
$x\mapsto v(x,t)$

is in general not
locally Lipschitz outside the interface $\{x = 0\}$.
In turn, this implies that,
for $\omega\in \mc A^{[AB]}(T)$,
the (forward) $AB$-entropy solution defined by

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

will be in general different
from $v(-x,T-t)$ on $\R\times [0,T[\,$.
However, exploiting the duality property enjoyed by the forward and backward solution operators
(see  <ref>), one can still prove

that $u^*(x,T)= v(-x,0)=\omega(x)$ for all $x\in\R$, which shows that
$\omega$ is a fixed point
of the backward-forward operator $\sabpT \circ \sabmT$ as stated in Theorem <ref>.

§ TECHNICAL TOOLS FOR CHARACTERIZATION
OF THE NEAR-INTERFACE WAVE STRUCTURE

In this section we

introduce some
technical tools needed to
characterize the pointwise constraints satisfied by
the attainable profiles
of (<ref>)
in intervals containing the origin.
Throughout the section,

$f:\R\to\R$ will be a
twice continuously differentiable, uniformly  convex map,
and we let $\theta$ be its unique critical point, $f'(\theta)=0$.

\begin{equation}
\label{eq:lambda-def}
\lambda(u,v)\doteq
\frac{f(v)-f(u)}{v-u},\qquad u,v\in\R, \ u\neq v\,,
\end{equation}

and observe that, by the convexity of $f$ one has

\begin{equation}
\label{eq:der-mon}
f'(u)<\lambda(u,v)<f'(v)\qquad \forall~ u<v\in\R\,.
\end{equation}

§.§ Left backward shock (Figure <ref>, left)

For every $B >\theta$, $0<\ms R< T\cdot f^{\prime}(B)$,
we define here:

-two constants
$\bs t[\ms R, B, f]$, $\bs u[\ms R,B, f]$;

-a function
$t\mapsto \bs y[\ms R,B, f](t)$,  $t\in [\bs t[\ms R, B, f],\,T]$;

with the following properties.
When $f=f_r$,
the map
$t\mapsto \bs y[\ms R,B, f_r](t)$
identifies the location of a shock curve
in a solution

v(x,t) \doteq  \sobapt
\big(\omega(-\ \cdot\,)\big)
\qquad x\leq 0,\ t\geq 0,

associated to the operator $\sobapt $, as defined in <ref>,
with $f=f_r$,
for some $\omega\in{\bf L^\infty}(\R)$.

This curve starts from the interface $\{x=0\}$
at time $t=\bs t[\ms R, B, f_r]$,
and reaches
the point $x=\bs y[\ms R,B, f_r](T)$
at time $t=T$.
Such a shock discontinuity

left state $v=\bs u[\ms R,B, f_r]$
and right state $v=\overline{B}$
at time $t=T$.
The point $(-\ms R,0)$
is the center of a rarefaction fan of $v(x,t)$
located on the left of $\bs y[\ms R,B, f_r](t)$.

We proceed to introduce these definitions as follows. Set

\begin{equation}
\label{eq:trbf-def}
\bs t[\ms R, B, f] \doteq
\frac{\ms R}{f^{\prime}(B)},
\qquad\quad\overline B \doteq
({f}_{|\,]-\infty, \theta]})^{-1}\circ f(B)\,.
\end{equation}

Then, consider
the Cauchy problem

\begin{equation}\label{eq:yR}
\begin{cases}
{y}'(t) = \lambda \Big(\!(f^{\prime})^{-1}\big(\frac{y(t)+\ms R}{t}\big), \overline{B}\, \Big), \qquad t\geq \bs t[\ms R, B, f]\,,
%\in \big[\bs t[\ms R, B, f] ,\, T\big],
\\
\noalign{\smallskip}
y( \bs t[\ms R, B, f] ) = 0,
\end{cases}
\end{equation}

and observe that, since
$g(t,y)\doteq \lambda\big((f')^{-1}\big(\frac{y+\ms R}{t}\big),\, \overline{B}\,\big)$ is locally
Lipschitz continuous in $y$,
by classical arguments
it admits a unique solution $\bs y(t)$
defined on some maximal interval
$[t[\ms R, B, f],\,\tau[$ .
On the other hand,
because of (<ref>) we have

\begin{equation}
g(t,y)>f'(\overline B)\qquad
\forall~t\in
\big[\bs t[\ms R, B, f],\,\min\{\tau,T\}\big[\,,\ y> - \ms R + T\cdot f'(\overline{B})\,,
\end{equation}

and hence,
since $f'(\overline{B})<0$,
it follows that

\begin{equation}
\label{eq:est-sol-cp-bs}
\begin{aligned}
\bs y(t)&>\big(\min\{\tau,T\}-t[\ms R, B, f]
\big)\cdot f'(\overline B)
\\
&\geq \min\{\tau,T\}\cdot f'(\overline B)
\end{aligned}
\qquad\quad
\forall~t\in
\big[t[\ms R, B, f],\,\min\{\tau,T\}\big[\,.
\end{equation}

In turn, (<ref>) implies that
$\tau>T$. Then, we will
denote by

\begin{equation*}
\bs y[\ms R,B, f] : \big[ \bs t[\ms R, B, f]  ,\, T\big]\ \to\ \,]-\infty,0[\,,
%\mathbb R,
\qquad t\mapsto \bs y[\ms R,B, f](t)\,,
\end{equation*}

the unique solution to (<ref>)
defined on the interval $\big[ \bs t[\ms R, B, f]  ,\, T\big]$,
and we call $\bs y[\ms R,B, f]$
a left backward shock.

Notice that $t\mapsto  \frac{d}{dt}{\bs y}[\ms R,B, f](t)$ is strictly decreasing, and $\frac{d}{dt}{\bs y}[\ms R,B, f](t)\leq 0$
for all $t\in [ \bs t[\ms R, B, f]  ,\, T]$.
Hence, by (<ref>)
with $\min\{\tau,T\}=T$,

the terminal point satisfies $\bs y[\ms R,B, f](T)\in\,]T\cdot f'(\overline{B}), 0[\,.$

Next, we set

\begin{equation}
\label{eq:urbf-def}
\bs u[\ms R,B, f]\doteq
(f^{\prime})^{-1}\left(\frac{\ms R+\bs y[\ms R, B, f](T)}{T}\right)\,.
\end{equation}

Observe that, by construction, $\bs u[\ms R,B, f]$ depends continuously on the
parameters $\ms R$ and $B$,
and that there holds

\begin{equation}
\label{eq:uB-ineq}
\bs u[\ms R,B, f]< B\,.
\end{equation}

§.§ Right backward shock (Figure <ref>, right)

Symmetrically to  <ref>,
for every $A <\theta$, $T\cdot f^{\prime}(A)<\ms L< 0$,
we define here:

-two constants
$\bs s[\ms L, A, f]$, $\bs v[\ms L,A, f]$;

-a function
$t\mapsto \bs x[\ms L,A, f](t)$,  $t\in [\bs s[\ms L, A, f],\,T]$;

with the following properties.
When $f=f_l$,
the map
$t\mapsto \bs y[\ms R,B, f_r](t)$
identifies the location of a shock curve
in a solution

v(x,t) \doteq  \sobapt
\big(\omega(-\ \cdot\,)\big)
\qquad x\leq 0,\ t\geq 0,

associated to the operator $\sobapt $, as defined in <ref>,
with $f=f_l$,
for some $\omega\in{\bf L^\infty}(\R)$.

This curves starts from the interface $\{x=0\}$
at time $t=\bs s[\ms L, A, f_l]$,
and reaches
the point $x=\bs x[\ms L,A, f_l](T)$
at time $t=T$.
Such a shock discontinuity
has left state $v=\overline{A}$ and
right state
$v=\bs v[\ms L,A, f_l]$

at time $t=T$.
The point $(-L,0)$
is the center of a rarefaction fan of $v(x,t)$
located on the right of $\bs x[\ms L,A, f_l](t)$.

We proceed to introduce these definitions as follows. Set

\begin{equation}
\label{eq:slaf-def}
\bs s[\ms L, A, f] \doteq
\frac{\ms L}{f^{\prime}(A)},
\qquad\quad\overline A \doteq
({f}_{|\,[\theta,+\infty[})^{-1}\circ f(A)\,.
\end{equation}

Then, let
$\bs x[\ms L,A, f] : [ \bs s[\ms L, A, f]  , T] \ \to\ \,]0,+\infty[$

denote the unique solution to the Cauchy problem

\begin{equation}\label{eq:xL}
\begin{cases}
{x}'(t) = \lambda \left(\!(f^{\prime})^{-1}\big(\frac{x(t)+\ms L}{t}\big), \overline{A}\,\right), \qquad t \in \big[ \bs s[\ms L, A, f] , T\big], \\
\noalign{\smallskip}
x( \bs s[\ms L, A, f] ) = 0.
\end{cases}
\end{equation}

We call $\bs x[\ms L,A, f]$ a
right backward shock.
Its terminal point satisfies $\bs x[\ms L, A, f](T) \in \big]0,T\cdot f^{\prime}(\overline{A})\big[$ ,
and the map $t\mapsto \frac{d}{dt}{\bs x}[\ms L,A, f](t)$ is strictly increasing. Next, we define the quantity
\begin{equation}
\label{eq:vlaf-def}
\bs v[\ms L,A, f]\doteq
(f^{\prime})^{-1}\left(\frac{\ms L+\bs x[\ms L,A, f](T)}{T}\right)\,,
\end{equation}

which depends continuously on $\ms L$
and $A$, and satisfies

\begin{equation}
\label{eq:vA-ineq}
\bs v[\ms L,A, f]>A\,.
\end{equation}

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(22.91,54.37) node [anchor=north west][inner sep=0.75pt]  [font=]  $t\ =\ T$;

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(613.76,253.54) node [anchor=north west][inner sep=0.75pt]  [font=]  $x$;

(403.36,20.01) node [anchor=north west][inner sep=0.75pt]  [font=]  $t$;

(457.56,51.24) node [anchor=north west][inner sep=0.75pt]  [font=]  $B$;

(537.31,275.09) node [anchor=north west][inner sep=0.75pt]  [font=]  $-\mathbf{y}[\mathsf{R} ,\ B,\ f]$;

(567.74,175.14) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathbf{\tau }[\mathsf{R} ,\ B,\ f]$;

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(507.45,43.95) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathsf{R}$;

The dual solutions $\bs y[\ms R, B, f](\cdot)$ (left) and $\bs x[\bs y[\ms R, B, f](T), \overline{B}, f](\cdot)$ (right) of the Cauchy problems (<ref>), (<ref>). This represents the statement of Lemma <ref>

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(224.29,44.83) node [anchor=north west][inner sep=0.75pt]  [font=]  $A$;

(133.25,44.83) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathsf{L}$;

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(180.03,264.13) node [anchor=north west][inner sep=0.75pt]  [font=]  $-\mathbf{x}[\mathsf{L} ,\ A,\ f]( T)$;

(597.66,157.29) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathbf{s}[\mathsf{L} ,\ A,\ f]$;

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The dual solutions $\bs x[\ms L, A, f](\cdot)$ (right) and $\bs y[\bs x[\ms L, A, f](T), \overline{A}, f](\cdot)$ (left) of the Cauchy problems (<ref>), (<ref>). This represents the statement of
Lemma <ref>

§.§ Duality of forward and

backward shocks

The definitions of backward shocks given in
<ref>-<ref> turn out to  be dual one of the other, as clarified by the following:

With the notations introduced in  <ref>-<ref>,
for every $B>\theta$, the following holds.
The maps

\begin{equation}
\label{eq;yx-def-37}
\begin{aligned}
&\bs y[\cdot\,, B, f](T)
: \ ]0, T\cdot f^{\prime}(B)[\,\ \to\ ]T \cdot f^{\prime}(\overline{B}), 0[\,,\qquad
\ms R \mapsto \bs y[\ms R, B, f](T),
\\
&\bs x[\cdot\, ,\overline{B}, f](T)
: \ ]T \cdot f^{\prime}(\overline{B}), 0[\, \ \to\ ]0, T\cdot f^{\prime}(B)[\,,\qquad
\ms L  \mapsto \bs x[\ms L, \overline{B}, f](T)
\end{aligned}
\end{equation}

are strictly increasing,
and one is the inverse of the other, i.e.
there holds

\begin{equation}\label{eq:yxinvertible}
\begin{aligned}
\ms R &= \bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T), \qquad \forall~\ms R \in  \,]0, T\cdot f^{\prime}(B)[, \\
\noalign{\smallskip}
\ms L &= \bs y \big[ \bs x[\ms L, \overline B, f](T), B, f\big](T), \qquad \forall~\ms L \in \,]T \cdot f^{\prime}(\overline{B}), 0[\,.
\end{aligned}
\end{equation}

Moreover, one has

\begin{equation}
\label{eq:xy-1sdlim}
\begin{aligned}
\lim_{\ms R \to 0+}
\,\bs y[R, B, f](T) &= T\cdot f^{\prime}(\overline{B}), \qquad  \lim_{\ms R \to T\cdot f^{\prime}(B)-}
\bs y[\ms R, B, f](T) = 0,
\\
\noalign{\smallskip}
\lim_{\ms L \to 0-}
\bs x[\ms L, \overline{B}, f](T) &= T\cdot f^{\prime}( B),  \qquad
\lim_{\ms L \to T\cdot f^{\prime}(\overline{B})+}
\bs x[\ms L, \overline{B}, f](T) = 0.
\end{aligned}
\end{equation}

1. We will prove only the first equality in (<ref>),
the proof of the second one being entirely similar.
Fix $\ms R \in \,]0, T \cdot f^{\prime}(B)[$, and consider the polygonal
region  (the pink set in Figure <ref>) defined by

\begin{equation}
\label{eq:gamma12def-0}
\begin{aligned}
\Gamma&\doteq \Gamma_1\cup\Gamma_2,
\\
\noalign{\medskip}
\Gamma_1&\doteq
%      [\ms R, B, f]&\doteq
\Big\{(x, t) \in\,]0,+\infty[\, \times \,]0, T[\, \; : \; \ms R\!-\!(T-t)\cdot f^{\prime}(B) < x < \ms R \!-\!(T-t) \cdot f^{\prime}(\bs u),
%      u[\ms R, B, f]),
\ \bs \tau <t < T \Big\},
%   \\
% \noalign{\smallskip}
%   &\hspace{3in}
%   \bs \tau[\ms R, B, f] < t < T \Big\},
\\
\noalign{\bigskip}
\Gamma_2&\doteq
% [\ms R, B, f]&\doteq
\Big\{(x, t) \in
\,]0,+\infty[\, \times \,]0, T[\,  \; : \;
%      -(\bs \tau[\ms R, B, f]\!-t)
(t-\bs \tau)\cdot\! f^{\prime}(\,\overline{B}\,) < x < \ms R \!-\!(T-t) \cdot f^{\prime}(\bs u),
%       \bs u[\ms R, B, f]),
%      \\
%   \noalign{\smallskip}
%      &\hspace{3in}
\ 0 \leq t \leq \bs \tau
%      \bs \tau[\ms R, B, f]
\Big\},
\end{aligned}
\end{equation}

where $\bs u\doteq \bs u[\ms R, B, f]$
is the constant in (<ref>),
and $\bs \tau\doteq \bs s \big[\bs y[\ms R,B,f](T), \; \overline{B}, \; f\big]$
is defined as in (<ref>)
with $\ms L=\bs y[\ms R, B, f](T)$ and $A=\overline{B}$.
Observe that the function $u : \Gamma \to\R$ defined by

\begin{equation}
\begin{cases}
B  &\text{if}\qquad  0< x < \bs x\big[\bs y[\ms R, B, f](T), \overline{B}, f\big](t),
\\
\noalign{\smallskip}
(f^{\prime})^{\strut -1}\bigg(\dfrac{x-\ms R+T \cdot f^{\prime}(\bs u)
%\bs u[\ms R, B, f])
}{t}\bigg) \quad &\text{otherwise,}
\end{cases}
\end{equation}

is locally Lipschitz continuous
and satisfies the equation (<ref>)
at every point  $(x,t)\in\Gamma$ outside the curve $\gamma(\cdot )\doteq \bs x\big[\bs y[\ms R, B, f](T), \overline{B}, f\big](\cdot)$.
Moreover, since by virtue of  (<ref>)
one has
$-\ms R+T \cdot f^{\prime}(\bs u)=\bs y[\ms R, B, f](T)$,
because of the construction of $\bs x[\ms L,A,f]$ in  <ref>,
with $\ms L=\bs y[\ms R, B, f](T)$ and $A=\overline{B}$,

it follows that
$u$ satisfies the Rankine-Hugoniot conditions along
the curve $\gamma$.

Therefore $u(x,t)$ is a distributional solution

of (<ref>) on $\Gamma$.

Hence, applying the divergence theorem
to the piecewise smooth vector field
$(u,f(u))$ on $\Gamma$,
and setting $\tau_1\doteq T-\ms R/f'(B)$, we find

\begin{equation*}
\begin{aligned}
\big(\!-\bs u\, f'(\bs u) + f(\bs u)\big)T+
\\&+\big(\overline{B}f'(\,\overline{B}\,)-f(\,\overline{B}\,)\big)\, \bs\tau- f(B)(\tau_1-\bs\tau)\,.
\end{aligned}
\end{equation*}

Then, observing that $f(B)=f(\overline B)$ and that $f'(\,\overline{B}\,) \bs\tau = \bs y[\ms R, B, f](T)$, we find

\begin{equation}
\label{eq:cons-gamma}
\overline{B}\,\bs y[\ms R, B, f](T)+ B\, \ms R
-\big(\bs u\, f'(\bs u) - f(\bs u)\big) T - f(B) T=0\,.
\end{equation}

Since $f'(\bs u)=(\bs y[\ms R, B, f](T)+\ms R)/T$,
and because the Legendre transform $f^*$ of $f$
satisfies the identity
f^*(f'(u))=u\,f'(u) - f(u)\qquad\forall~u\,,

(e.g. see [A.2][18]), we derive
from (<ref>) the identity

\begin{equation}
\label{eq:massconsyR}
\qquad
\overline{B}\, \bs y[\ms R, B, f](T)+ B\, \ms R  -f^{*}\left(\frac{\bs y[\ms R, B, f](T)+\ms R}{T}\right)\,T-f(B)\,T=0\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

Next, consider the polygonal
region  (the blue set in Figure <ref> with $A=\overline{B}$ and $\overline{A}=B$) defined by

\begin{equation}
\label{eq:delta12def-0}
\begin{aligned}
\Delta&\doteq \Delta_1\cup\Delta_2,
\\
\noalign{\medskip}
\Delta_1&\doteq
\Big\{(x, t) \in\,]-\infty,0[\, \times \,]0, T[\,   \; : \;
\ms L \!-\!(T-t) \cdot f^{\prime}(\bs v)<x
<\ms L\!-\!(T-t)\cdot f^{\prime}(A),
\ \bs \sigma <t < T \Big\},
\\
\noalign{\bigskip}
\Delta_2&\doteq
\Big\{(x, t) \in\,]-\infty,0[\, \times \,]0, T[\,   \; : \;
\ms L \!-\!(T-t) \cdot f^{\prime}(\bs v)
<x< (t-\bs \sigma)\cdot\! f^{\prime}(\,\overline{A}\,),
\ 0 \leq t \leq \bs \sigma
\Big\},
\end{aligned}
\end{equation}

where $\bs v\doteq \bs v\big[\bs y[\ms R, B, f](T), \overline{B}, f\big]$
is the constant defined as in (<ref>), with
$\ms L=\bs y[\ms R, B, f](T)$ and $A=\overline{B}$, while
$\bs \sigma\doteq \bs t \Big[\bs x\big[\bs y[\ms R, B, f](T),\,\overline{B},f\big](T), \; B, \; f\Big]$
is defined as in (<ref>)
with $\ms R=\bs x\big[\bs y[\ms R, B, f](T),\,\overline{B},f\big](T)$.
Observe that the function $v : \Delta\to\R$ defined by

\begin{equation}
\begin{cases}
\overline{B}  &\text{if}\qquad  0< x < \gamma(t),
\\
\noalign{\smallskip}
(f^{\prime})^{\strut -1}\bigg(\dfrac{x-\bs y[\ms R, B, f](T)+T \cdot f^{\prime}(\bs v)}{t}\bigg) \quad &\text{otherwise,}
\end{cases}
\end{equation}

with $\gamma(t)\doteq \bs y\Big[\bs x\big[\bs  y[\ms R, B, f](T), \overline{B}, f\big](T), B, f\Big](t)$,
is a distributional solution
of (<ref>) on $\Delta$
for the symmetric arguments of the previous point.
Then, repeating the same type of
analysis of above for the piecewise smooth vector field
$(v,f(v))$ on $\Delta$, one finds
the identity

\begin{equation}
\label{eq:massconsyR2}
\begin{aligned}
&B\, \bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T) +\overline{B}\,\bs y[\ms R, B, f](T) +
\\
\noalign{\smallskip}
&\quad -f^*\left(\frac{\bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T)+\bs y[\ms R, B, f](T)}{T}\right)T- f(B)\, T =0
\end{aligned}
\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

Notice that,  by
definition of the function
$\bs y[\ms R, B, f](\cdot)$
in  <ref>,
the terminal value satisfies$\bs y[\ms R, B, f](T) \in \,]T\cdot f^{\prime}(\overline{B}),0[$ ,
for all $\ms R \in \,]0, T\cdot f^{\prime}(B))[$ . In turn, from the
definition of
$\bs x[\ms L, A, f]$
in <ref>,
with $A = \overline{B}$,
and $\ms L = \bs y[\ms R, B, f](T)$,
it follows that

\begin{equation}
\label{eq:xyinint}
\bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T) \in \,]0 ,T \cdot f^{\prime}(B)[\,,\
\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

We fix now $R \in \,]0, T\cdot f^{\prime}(B))[\,$, and we
consider the map
$\Upsilon : \ ]0, T \cdot f^{\prime}(B)[\ \to \R$, defined by

\begin{equation}
\Upsilon(x) \doteq   \overline{B}\, \bs y[\ms R, B, f](T) + B\, x -f^*\left(\frac{\bs y[\ms R, B, f](T)+x}{T}\right)T  - f(B)\,T\,.
\end{equation}

Observe that, by (<ref>), (<ref>), (<ref>), one has

\begin{equation}
\Upsilon(\ms R)=
\Upsilon\Big(\bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T)\Big)=0\,.
\end{equation}

Hence,  it is sufficient to show that $\Upsilon$ admits only one zero in the interval $]0, T\cdot f^{\prime}(B)[$ to conclude the proof of the first equality in (<ref>).

To this end, differentiating $\Upsilon$
and recalling the well known property of the Legendre transform (e.g. see [A.2][18])

(f^*)^{\prime}(p) = (f^{\prime})^{-1}(p)
\qquad\forall~p\,,

we find

\begin{equation}
\label{eq:diffups}
\begin{aligned}
\Upsilon'(x)&=B-(f^{\prime})^{-1}\left(\frac{ \bs y[\ms R, B, f](T) +x}{T}\right) \\
& = (f^{\prime})^{-1}\left(\frac{0+Tf^{\prime}(B)}{T}\right)-(f^{\prime})^{-1}\left(\frac{ \bs y[\ms R, B, f](T) +x}{T}\right)\,.
\end{aligned}
\end{equation}

Since $y[\ms R, B, f](T)  < 0$, $ x < T\cdot f^{\prime}(B)$, and because $f^{\prime}$ is strictly increasing
as $f'$, we deduce from (<ref>)
that $\Upsilon'(x)>0$ for all $x\in \,]0, T\cdot f^{\prime}(B))[\,$.
Therefore $\Upsilon$ is strictly increasing in the interval $\,]0, T\cdot f^{\prime}(B))[\,$, completing the proof of of the first equality in (<ref>).

We show now that the map $\ms R \mapsto \bs y(\ms R)\doteq y[\ms R, B, f](T)$ is strictly increasing in the interval $\,]0, T\cdot f^{\prime}(B)[$ . Differentiating (<ref>) with respect to $\ms R$, we obtain

\begin{equation}\label{eq:consdiffereq}
\left[\overline{B}-(f^{\prime})^{-1}\Big( \frac{y(\ms R)+\ms R}{T} \Big) \right] y'(\ms R)
= (f^{\prime})^{-1}\Big( \frac{y(\ms R)+\ms R}{T} \Big)-B\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

Since $T\cdot f'(\,\overline{B}\,)<y(\ms R) <0$ and $0<\ms R <T\cdot f'(B)$,
because $f'$ is strictly increasing we deduce

\overline{B} < (f')^{-1}\Big( \frac{y(\ms R)}{T} \Big)<
(f^{\prime})^{-1}\Big( \frac{y(\ms R)+\ms R}{T} \Big) <
(f')^{-1}\Big( \frac{\ms R}{T} \Big)

which, together with (<ref>), implies that $y'(\ms R) >0$
for all $R\in \,]0, T\cdot f^{\prime}(B))[$ , as wanted.
In turn, since $\ms L \mapsto x[\ms L, \overline{B}, f](T)$ is the inverse of $\ms R \mapsto \bs y[\ms R, B, f](T)$, this implies that $\ms L \mapsto x[\ms L, \overline{B}, f](T)$ is strictly increasing as well in its domain, and
that the image of the maps
$\bs y[\cdot\,, B, f](T)$, $\bs x[\cdot\,, \overline{B}, f](T)$, in (<ref>) are the sets $\,]0, T\cdot f^{\prime}(B))[\,$ and $\,]0, T\cdot f^{\prime}(B)[$, respectively.
This, together with the monotonicity of the maps $\bs y[\cdot\,, B, f](T)$, $\bs x[\cdot\,, \overline{B}, f](T)$,
in particular implies the one-sided limits in (<ref>),
thus concluding the proof of the Lemma.

1. We will prove only the first equality in (<ref>),
the proof of the second one being entirely similar.
Fix $\ms R \in \,]0, T \cdot f^{\prime}(B)[$, and consider the polygonal
region  (the blue set in Figure <ref>) defined by

\begin{equation}
\label{eq:delta12def-0}
\begin{aligned}
\Delta &\doteq \Delta_1\cup\Delta_2,
\\
\noalign{\medskip}
\Delta_1&\doteq
\Big\{(x, t) \in\,]-\infty,0[\, \times \,]0, T[\,   \; : \;
\ms L \!-\!(T-t) \cdot f^{\prime}(\bs u)<x
<\ms L\!-\!(T-t)\cdot f^{\prime}(\bar B),
\ \bs t<t < T \Big\},
\\
\noalign{\bigskip}
\Delta_2&\doteq
\Big\{(x, t) \in\,]-\infty,0[\, \times \,]0, T[\,   \; : \;
\ms L \!-\!(T-t) \cdot f^{\prime}(\bs u)
<x< (t-\bs t)\cdot\! f^{\prime}(\,B\,),
\ 0 \leq t \leq \bs t
\Big\},
\end{aligned}
\end{equation}

where $\bs u\doteq \bs u[\ms R, B, f]$
is the constant in (<ref>), $\bs t \doteq \bs t[\ms R, B, f]$
is defined as in (<ref>) and $\ms L \doteq  \bs y[\ms R, B, f](T)$.
Observe that the function $v :  \Delta \to\R$ defined by

\begin{equation}
\begin{cases}
\overline{B}  &\text{if}\qquad   \gamma(t)< x < 0,
\\
\noalign{\smallskip}
(f^{\prime})^{\strut -1}\bigg(\dfrac{x+\ms R}{t}\bigg) \quad &\text{otherwise,}
\end{cases}
\end{equation}

is locally Lipschitz continuous
and satisfies the equation (<ref>)
at every point  $(x,t)\in\Delta$ outside the curve $\gamma(\cdot )\doteq  \bs y[\ms R, B, f](\cdot)$.
Moreover, because of the construction of $\bs y[\ms R, B, f ](\cdot)$,
$u$ satisfies the Rankine-Hugoniot conditions along
the curve $\gamma$.

Therefore $v(x,t)$ is a distributional solution

of (<ref>) on $\Delta$.

Hence, applying the divergence theorem
to the piecewise smooth vector field
$(v,f(v))$ on $\Delta$,
and setting $\tau_1\doteq T-\bs y[\ms R, B, f]/f'(\overline B)$, we find

\begin{equation*}
\begin{aligned}
0&=\big(f(\overline B )-\overline  Bf'(\overline B)\big)(T-\tau_1)+
\big(\bs u\, f'(\bs u) -f(\bs u)\big)T+
\\&\big(f(\,B\,)-Bf'(\,B\,)\big)\, \bs t + f(B)(\tau_1-\bs t)\,.
\end{aligned}
\end{equation*}

Then, observing that $f(B)=f(\overline B)$ and that $f'(\,B\,) \bs t = \ms R$, we find

\begin{equation}
\label{eq:cons-gamma}
\overline{B}\,\bs y[\ms R, B, f](T)+ B\, \ms R
-\big(\bs u\, f'(\bs u) - f(\bs u)\big) T - f(B) T=0\,.
\end{equation}

Since $f'(\bs u)=(\bs y[\ms R, B, f](T)+\ms R)/T$,
and because the Legendre transform $f^*$ of $f$
satisfies the identity
f^*(f'(u))=u\,f'(u) - f(u)\qquad\forall~u\,,

(e.g. see [A.2][18]), we derive
from (<ref>) the identity

\begin{equation}
\label{eq:massconsyR}
\qquad
\overline{B}\, \bs y[\ms R, B, f](T)+ B\, \ms R  -f^{*}\left(\frac{\bs y[\ms R, B, f](T)+\ms R}{T}\right)\,T-f(B)\,T=0\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

Next, consider the polygonal
region  (the red set in Figure <ref> with $A=\overline{B}$ and $\overline{A}=B$) defined by

\begin{equation}
\label{eq:gamma12def-0}
\begin{aligned}
\Gamma &\doteq  \Gamma_1\cup  \Gamma_2,
\\
\noalign{\medskip}
\Gamma_1&\doteq
%      [\ms R, B, f]&\doteq
\Big\{(x, t) \in\,]0,+\infty[\, \times \,]0, T[\, \; : \;
\bs x\big[\bs  y[\ms R, B, f](T), \overline{B}, f\big](T)\!-\!(T-t)\cdot f^{\prime}(B) < x <
\\
&\hspace{2.4in} < \bs x\big[\bs  y[\ms R, B, f](T), \overline{B}, f\big](T) \!-\!(T-t) \cdot f^{\prime}(\bs v),
%      u[\ms R, B, f]),
\ \bs s <t < T \Big\},
%   \\
% \noalign{\smallskip}
%   &\hspace{3in}
%   \bs \tau[\ms R, B, f] < t < T \Big\},
\\
\noalign{\bigskip}
\Gamma_2&\doteq
% [\ms R, B, f]&\doteq
\Big\{(x, t) \in
\,]0,+\infty[\, \times \,]0, T[\,  \; : \;
%      -(\bs \tau[\ms R, B, f]\!-t)
(t-\bs s)\cdot\! f^{\prime}(\,\overline{B}\,) < x <
\\
&\hspace{2.4in}< \bs x\big[\bs  y[\ms R, B, f](T), \overline{B}, f\big](T) \!-\!(T-t) \cdot f^{\prime}(\bs v),
%       \bs u[\ms R, B, f]),
%      \\
%   \noalign{\smallskip}
%      &\hspace{3in}
\ 0 \leq t \leq \bs s
%      \bs \tau[\ms R, B, f]
\Big\},
\end{aligned}
\end{equation}

where $\bs v\doteq \bs v\big[\bs y[\ms R, B, f](T), \overline{B}, f\big]$
is the constant defined as in (<ref>), with
$\ms L=\bs y[\ms R, B, f](T)$, $A=\overline{B}$ and $\bs s = \bs s[\ms L, A, f]$.
Observe that the function $u :  \Gamma\to\R$ defined by

\begin{equation}
\begin{cases}
B  &\text{if}\qquad  0< x < \gamma(t),
\\
\noalign{\smallskip}
(f^{\prime})^{\strut -1}\bigg(\dfrac{x+\bs y[\ms R, B, f](T)}{t}\bigg) \quad &\text{otherwise,}
\end{cases}
\end{equation}

with $\gamma(t)\doteq \bs x\big[\bs  y[\ms R, B, f](T), \overline{B}, f\big](t)$,
is a distributional solution
of (<ref>) on $ \Gamma$
for the symmetric arguments of the previous point.
Then, repeating the same type of
analysis of above for the piecewise smooth vector field
$(u,f(u))$ on $ \Gamma$, one finds
the identity

\begin{equation}
\label{eq:massconsyR2}
\begin{aligned}
&B\, \bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T) +\overline{B}\,\bs y[\ms R, B, f](T) +
\\
\noalign{\smallskip}
&\quad -f^*\left(\frac{\bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T)+\bs y[\ms R, B, f](T)}{T}\right)T- f(B)\, T =0
\end{aligned}
\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

Notice that,  by
definition of the function
$\bs y[\ms R, B, f](\cdot)$
in  <ref>,
the terminal value satisfies$\bs y[\ms R, B, f](T) \in \,]T\cdot f^{\prime}(\overline{B}),0[$ ,
for all $\ms R \in \,]0, T\cdot f^{\prime}(B))[$ . In turn, from the
definition of
$\bs x[\ms L, A, f]$
in <ref>,
with $A = \overline{B}$,
and $\ms L = \bs y[\ms R, B, f](T)$,
it follows that

\begin{equation}
\label{eq:xyinint}
\bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T) \in \,]0 ,T \cdot f^{\prime}(B)[\,,\
\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

We fix now $\ms R \in \,]0, T\cdot f^{\prime}(B))[\,$, and we
consider the map
$\Upsilon : \ ]0, T \cdot f^{\prime}(B)[\ \to \R$, defined by

\begin{equation}
\Upsilon(x) \doteq   \overline{B}\, \bs y[\ms R, B, f](T) + B\, x -f^*\left(\frac{\bs y[\ms R, B, f](T)+x}{T}\right)T  - f(B)\,T\,.
\end{equation}

Observe that, by (<ref>), (<ref>), (<ref>), one has

\begin{equation}
\Upsilon(\ms R)=
\Upsilon\Big(\bs x \big[ \bs y[\ms R, B, f](T), \overline{B}, f\big](T)\Big)=0\,.
\end{equation}

Hence,  it is sufficient to show that $\Upsilon$ admits only one zero in the interval $]0, T\cdot f^{\prime}(B)[$ to conclude the proof of the first equality in (<ref>).

To this end, differentiating $\Upsilon$
and recalling the well known property of the Legendre transform (e.g. see [A.2][18])

(f^*)^{\prime}(p) = (f^{\prime})^{-1}(p)
\qquad\forall~p\,,

we find

\begin{equation}
\label{eq:diffups}
\begin{aligned}
\Upsilon'(x)&=B-(f^{\prime})^{-1}\left(\frac{ \bs y[\ms R, B, f](T) +x}{T}\right) \\
& = (f^{\prime})^{-1}\left(\frac{0+Tf^{\prime}(B)}{T}\right)-(f^{\prime})^{-1}\left(\frac{ \bs y[\ms R, B, f](T) +x}{T}\right)\,.
\end{aligned}
\end{equation}

Since $\bs y[\ms R, B, f](T)  < 0$, $ x < T\cdot f^{\prime}(B)$, and because $f^{\prime}$ is strictly increasing
as $f'$, we deduce from (<ref>)
that $\Upsilon'(x)>0$ for all $x\in \,]0, T\cdot f^{\prime}(B))[\,$.
Therefore $\Upsilon$ is strictly increasing in the interval $\,]0, T\cdot f^{\prime}(B))[\,$, completing the proof of of the first equality in (<ref>).

We show now that the map $\ms R \mapsto \bs y(\ms R)\doteq \bs y[\ms R, B, f](T)$ is strictly increasing in the interval $\,]0, T\cdot f^{\prime}(B)[$ . Differentiating (<ref>) with respect to $\ms R$, we obtain

\begin{equation}\label{eq:consdiffereq}
\left[\overline{B}-(f^{\prime})^{-1}\Big( \frac{y(\ms R)+\ms R}{T} \Big) \right] y'(\ms R)
= (f^{\prime})^{-1}\Big( \frac{y(\ms R)+\ms R}{T} \Big)-B\qquad \forall~\ms R \in \,]0, T\cdot f^{\prime}(B))[\,.
\end{equation}

Since $T\cdot f'(\,\overline{B}\,)<y(\ms R) <0$ and $0<\ms R <T\cdot f'(B)$,
because $f'$ is strictly increasing we deduce

\overline{B} < (f')^{-1}\Big( \frac{y(\ms R)}{T} \Big)<
(f^{\prime})^{-1}\Big( \frac{y(\ms R)+\ms R}{T} \Big) <
(f')^{-1}\Big( \frac{\ms R}{T} \Big)

which, together with (<ref>), implies that $y'(\ms R) >0$
for all $R\in \,]0, T\cdot f^{\prime}(B))[$ , as wanted.
In turn, since $\ms L \mapsto x[\ms L, \overline{B}, f](T)$ is the inverse of $\ms R \mapsto \bs y[\ms R, B, f](T)$, this implies that $\ms L \mapsto x[\ms L, \overline{B}, f](T)$ is strictly increasing as well in its domain, and
that the image of the maps
$\bs y[\cdot\,, B, f](T)$, $\bs x[\cdot\,, \overline{B}, f](T)$, in (<ref>) are the sets $\,]0, T\cdot f^{\prime}(B))[\,$ and $\,]0, T\cdot f^{\prime}(B)[$, respectively.
This, together with the monotonicity of the maps $\bs y[\cdot\,, B, f](T)$, $\bs x[\cdot\,, \overline{B}, f](T)$,
in particular implies the one-sided limits in (<ref>),
thus concluding the proof of the Lemma.

As a consequence of Lemma <ref>
and of the monotonicity of $f'$, we find that
the maps

\begin{equation}
\ms R \mapsto \bs u[\ms R, B, f],\qquad\quad
\ms L \mapsto \bs v[\ms L, A, f],
\end{equation}

defined as in (<ref>)
and (<ref>),
are strictly increasing, and that there holds

\begin{equation}\label{eq:limitsuv}
\begin{aligned}
% \ms R \mapsto \bs u[\ms R, B, f] \qquad \text{is increasing \ and} \qquad
\lim_{\ms R \to 0+}  \bs u[\ms R, B, f]  = \overline{B}, \qquad
\lim_{\ms R \to T\cdot f^{\prime}(B)-}\bs u[\ms R, B, f] = B,\\
\noalign{\smallskip}
% \ms L \mapsto \bs v[\ms L, A, f] \qquad\ \text{is increasing \ and} \qquad
\lim_{\ms L \to 0-}  \bs v[\ms L, A, f] = \overline{A}, \qquad\  \lim_{\ms L \to T\cdot f^{\prime}(A)+}\bs v[\ms L, A, f] = A.
\end{aligned}
\end{equation}

This implies that
the functions

\begin{equation*}
\begin{aligned}
&\bs u[\cdot, \cdot, f]: \ ]0, T\cdot f'(B)[\,
\times \,]\theta,+\infty[\, \to \mathbb R,
\\
% \bs v[\cdot, \cdot, f]: (-\infty, 0] \times [\theta,+\infty) \to \mathbb R
&\bs v[\cdot, \cdot, f]: \ ]T\cdot f'(A),0[\, \times \,]\theta,+\infty[\, \to \mathbb R
\end{aligned}
\end{equation*}

can be extended to continuous function on $[0, T\cdot f'(B)]\times$ $\,]\theta,+\infty[$  and
$[T\cdot f'(A),0] \times \,]\theta,+\infty[$, setting

\begin{equation}
\label{eq:u-v-cont-ext}
\begin{aligned}
\bs u[0, B, f]  &= \overline{B},\qquad\quad \bs u[T\cdot f^{\prime}(B), B, f]  =B,
\\
\noalign{\smallskip}
\bs v[0, A, f] &= \overline{A},
\qquad\quad\
\bs v[T\cdot f^{\prime}(A), A, f] =
\end{aligned}
\end{equation}

Moreover, one has

\begin{equation}
\label{eq:uv-ba-ineq}
\begin{aligned}
\bs u[\ms R, B, f] < B\qquad\quad &\forall~\ms R\in \,]0, T\cdot f'(B)[\,,
\\
\noalign{\smallskip}
\bs v[\ms L, A, f] > A\qquad\quad &\forall~\ms L\in\,]T\cdot f'(A),0[\,.
\end{aligned}
\end{equation}

§.§ Right forward shock-rarefaction wave pattern (Figure <ref>, right)

For every $B >\theta$, $0<\ms R< T\cdot f^{\prime}(B)$,
we define now:

-a constant $\bs \tau[\ms R, B, f]$;

-a function
$(x,t)\mapsto u[\ms R, B, f](x,t),$ $(x,t)\in \Gamma[\ms R, B, f]$;

with the following properties.
When $f=f_r$, the function
$u[\ms R, B, f](x,t)$ defines a (forward) solution
associated to the operator
$\sabp$, which contains
a shock
from the interface $\{x=0\}$ at time
$t=\bs \tau[\ms R, B, f_r]$.

The  location of such a shock is given by the map $t\mapsto \bs x\big[\bs y[\ms R, B, f_r](T), \overline{B}, f_r\big](t)$,
$\bs y[\ms R,B, f_r]$ and $\bs x[\ms L,\overline{B}, f_r]$
with $\ms L=\bs y[\ms R, B, f_r](T)$,
are the backward shocks
of a backward solution
associated to the operator

introduced in  <ref>-<ref>.
Because of Lemma <ref>,
the shock
$t\mapsto \bs x\big[\bs y[\ms R, B, f_r](T), \overline{B}, f_r\big](t)$
reaches the point $x=R$
at time $t=T$.
We can regard
$\bs x\big[\bs y[\ms R, B, f_r](T), \overline{B}, f_r\big]$ as
the “dual shock”
of the backward shock $\bs y[\ms R,B, f_r]$.

We proceed to introduce these definitions as follows.
With the same notations of  <ref>-<ref>,
for every $B >\theta$, $0<\ms R< T\cdot f^{\prime}(B)$,
we set

\begin{equation}
\label{eq:tau-rbf-def}
\bs \tau[\ms R, B, f]\doteq
\bs s \big[\bs y[\ms R,B,f](T), \; \overline{B}, \; f\big]=\frac{\bs y[\ms R,B,f](T)}{f'(\,\overline{B}\,)}\,,
\end{equation}

and  we denote by $ \Gamma[\ms R, B, f] \subset (0, T) \times \mathbb R$ the polygonal set (the pink set in Figure <ref>)

\begin{equation}
\label{eq:gammadef}
\Gamma[\ms R, B, f]\doteq \Gamma_1[\ms R, B, f]\cup\Gamma_2[\ms R, B, f]\,,
\end{equation}

\begin{equation}
\label{eq:gamma12def}
\begin{aligned}
\Gamma_1[\ms R, B, f]&\doteq
\Big\{(x, t) \in\,]0,+\infty[\, \times \,]0, T[\,   \; : \; \ms R-(T-t)\cdot f^{\prime}(B) < x < \ms R -(T-t) \cdot f^{\prime}(\bs u[\ms R, B, f]),
\\
\noalign{\smallskip}
\bs \tau[\ms R, B, f] < t < T \Big\},
\\
\noalign{\bigskip}
\Gamma_2[\ms R, B, f]&\doteq
\Big\{(x, t) \in\,]0,+\infty[\, \times \,]0, T[\,  \; : \; -(\bs \tau[\ms R, B, f]\!-t) \cdot\! f^{\prime}(\,\overline{B}\,) < x < \ms R \!-\!(T-t) \cdot f^{\prime}(\bs u[\ms R, B, f]),
\\
\noalign{\smallskip}
0 \leq t \leq \bs \tau[\ms R, B, f]\Big\}.
\end{aligned}
\end{equation}

Then, set $\gamma(t)\doteq \bs x\big[\bs y[\ms R, B, f](T), \overline{B}, f\big](t),$
and denote by $\ms u[\ms R, B, f]: \Gamma[\ms R, B, f] \to \mathbb R$ the function defined by

\begin{equation}
\label{eq:urbff-def}
\ms u[\ms R, B, f] (x,t)\doteq
\begin{cases}
\!B  &\text{if}\qquad  0< x < \gamma(t),
% \bs x\big[\bs y[\ms R, B, f](T), \overline{B}, f\big](t),
%\ (x,t)\in\Gamma_1,
% and $\bs \tau[\ms R, B, f] < t < T$},\\
\\
\noalign{\smallskip}
\!(f^{\prime})^{\strut -1}\bigg(\dfrac{x-\ms R+T \cdot f^{\prime}(\bs u[\ms R, B, f])}{t}\bigg) &\text{otherwise.}
\end{cases}
\end{equation}

Notice that, by (<ref>) and (<ref>),
one has $\gamma(\bs \tau[\ms R, B, f])=0$,
$\gamma(T)=\ms R$.
Moreover, by the same arguments of the
proof of Lemma <ref> it follows that
$u[\ms R, B, f] (x,t)$
is a distributional solution of (<ref>)
on $\Gamma[\ms R, B, f]$. Furthermore,
$t\mapsto \gamma'(t)$
is strictly increasing
as observed in  <ref>, it follows
that also the

\dfrac{\gamma(t)-\ms R+T \cdot f^{\prime}(\bs u[\ms R, B, f])}{t}

is strictly increasing.
by virtue of (<ref>),
and relying on (<ref>), we find

\begin{equation}
\label{eq:lax-rfs-cond}
\begin{aligned}
% \lim_{x\to \bs x[\bs y[\ms R, B, f](T), \overline{B}, f](t)+}
\lim_{x\to\gamma(t)+}
\ms u[\ms R, B, f] (x,t)&\leq
\lim_{x\to\gamma(T)+}
\ms u[\ms R, B, f] (x,T)
\\
\lim_{x\to \ms R+}\ms u[\ms R, B, f] (x,T)
\\
&=\bs u[\ms R, B, f]<B\\
\ms u[\ms R, B, f] (x,t)
\qquad \forall~t\in
[\bs \tau[\ms R, B, f],T]\,,
\end{aligned}
\end{equation}

which shows that the
Lax entropy condition

is satisfied along the curve $(t,\gamma(t))$, $t\in [\bs\tau[\ms R, B, f],T]$.
Since the flux in (<ref>) is strictly convex,
this proves that
$u[\ms R, B, f] (x,t)$
provides an entropy weak solution of (<ref>)
on the region $\Gamma[\ms R, B, f]$.
Notice that, by (<ref>), from (<ref>) we deduce in particular that $f'(B)>\lambda\big(\bs u[\ms R, B, f], B\big)=\gamma'(T)$,
which in turn, by the strict monotonicity of
$\dot\gamma(t)$, yields

\begin{equation}
\label{eq:lax-rfs-cond-2}
f'(B)>\gamma'(t)\qquad \forall~t\in
[\bs \tau[\ms R, B, f],T]\,.
\end{equation}

Hence, relying on (<ref>), we find

\begin{equation}
\label{eq:lax-rfs-cond-3}
f'(B)>\frac{\gamma(T)-\gamma(\bs\tau[\ms R, B, f])}{T-\bs\tau[\ms R, B, f]}=
\frac{\ms R}{T-\bs\tau[\ms R, B, f]}\,.
\end{equation}

§.§ Left forward rarefaction-shock wave pattern (Figure <ref>, left)

Symmetrically to  <ref>,
for every $A <\theta$, $T\cdot f^{\prime}(A)<\ms L< 0$,
we define here:

-a constant $\bs \sigma[\ms L, A, f]$;

-a function
$(x,t)\mapsto \ms v[\ms L, A, f](x,t),$ $(x,t)\in \Delta[\ms L, A, f]$;

with the following properties.
When $f=f_l$, the function
$\ms v[\ms L, A, f](x,t)$ defines a (forward) solution
associated to the operator
$\sabp$, which contains
a shock
from the interface $\{x=0\}$ at time
$t=\bs \sigma[\ms L, A, f]$.
The  location of such a shock is given by the map $t\mapsto \bs y \big[\bs x[\ms L,A,f_l](T), \; \overline{A}, \; f_l\big](t)$,
$\bs x[\ms L,A, f_l]$ and $\bs y[\ms R,\overline{A}, f_l]$
with $\ms R=\bs x[\ms L,A,f_l](T)$,
are the backward shocks
of a backward solution
associated to the operator

introduced in  <ref>-<ref>.
Because of Lemma <ref>,
the shock
$t\mapsto \bs y \big[\bs x[\ms L,A,f_l](T), \; \overline{A}, \; f_l\big](t)$
reaches the point $x=L$
at time $t=T$.
We can regard $\bs y \big[\bs x[\ms L,A,f_l](T), \; \overline{A}, \; f_l\big]$ as
the “dual shock”
of the backward shock $\bs x[\ms L,A,f_l]$.

We proceed to introduce these definitions as follows.
With the same notations of  <ref>-<ref>,
for every $A <\theta$, $T\cdot f^{\prime}(A)<\ms L< 0$
we set

\begin{equation}
\bs \sigma[\ms L, A, f]\doteq
\bs t \big[\bs x[\ms L,A,f](T), \; \overline{A}, \; f\big]=
\frac{\bs x[\ms L,A,f](T)}{f'(\,\overline{A}\,)}\,,
\end{equation}

and  we denote by $\Delta[\ms L, A, f] \subset (0, T) \times \mathbb R$ the polygonal set (the blue set in Figure <ref>)

\begin{equation}
\label{eq:deltadef}
\Delta[\ms L, A, f]\doteq \Delta_1[\ms L, A, f]\cup\Delta_2[\ms L, A, f]\,,
\end{equation}

\begin{equation}
\label{eq:delta12def}
\begin{aligned}
\Delta_1[\ms L, A, f]&\doteq
\Big\{(x, t) \in\,]-\infty,0[\, \times \,]0, T[\,   \; : \; \ms L -(T-t) \cdot f^{\prime}(\bs v[\ms L, A, f])<x<\ms L-(T-t)\cdot f^{\prime}(A),
\\
\noalign{\smallskip}
\bs \sigma[\ms L, A, f] < t < T \Big\},
\\
\noalign{\bigskip}
\Delta_2[\ms L, A, f]&\doteq
\Big\{(x, t) \in\,]-\infty,0[\, \times \,]0, T[\,   \; : \; \ms L\! -\!(T-t) \cdot f^{\prime}(\bs v[\ms L, A, f])< x < -(\bs \sigma[\ms L, A, f]\!-t) \cdot\! f^{\prime}(\,\overline{A}\,),
\\
\noalign{\smallskip}
0 \leq t \leq \bs \sigma[\ms L, A, f]\Big\}.
\end{aligned}
\end{equation}

Then, set $\gamma(t)\doteq \bs y\big[\bs x[\ms L, A, f](T), \overline{A}, f\big](t)$,
and denote by $\ms v[\ms L, A, f]: \Delta[\ms L, A, f]\to \mathbb R$ the function defined by

\begin{equation}
\label{eq:vlaff-def}
\ms v[\ms L, A, f] (x,t)\doteq
\begin{cases}
\!A  &\text{if}\qquad  \gamma(t)< x < 0,
%     \bs y\big[\bs x[\ms L, A, f](T), %\overline{A}, f\big](t)< x < 0,
\\
\noalign{\smallskip}
\!(f^{\prime})^{\strut -1}\bigg(\dfrac{x-\ms L+T \cdot f^{\prime}(\bs v[\ms L, A, f])}{t}\bigg) &\text{otherwise.}
\end{cases}
\end{equation}

Observe that, by (<ref>) and (<ref>),
one has $\gamma(\bs \sigma[\ms L, A, f])=0$,
$\gamma(T)=\ms L$.
With the same arguments of  <ref>, it follows that $v[\ms L, A, f] (x,t)$ provides an entropy weak solution of (<ref>)
on the region $\Delta[\ms L, A, f]$,
and that there holds

\begin{equation}
\label{eq:lax-rfs-cond-4}
f'(A)<\frac{\gamma(T)-\gamma(\bs \sigma[\ms L, A, f])}{T-\bs \sigma[\ms L, A, f]}=
\frac{\ms L}{T-\bs \sigma[\ms L, A, f]}\,.
\end{equation}

The constant $\bs u[\ms R, B, f]$ defined in  <ref>
is crucial to characterize the
jump of an attainable profile
$\omega\in \mc A^{[AB]}(T)$
at the point

\begin{equation*}
\ms R \doteq  \inf \big\{ R > 0 \; : \; x-T \cdot f_r^{\prime}(\omega(x+)) \geq 0  \quad \forall \; x \geq R\big\},
\end{equation*}

when $\ms R \in \,]0, T \cdot f_r^{\prime}(B)$.

The state $\bs u[\ms R, B, f]$ is constructed so to be the largest right state that one can achieve
at $(\ms R,T)$ with a shock that isolates the interface $\{x=0\}$ from the semiaxis $\{x>0\}$. In fact,

the constant $\bs u[\ms R, B, f]$ with $f=f_r$, identifies
a unique state $\bs u$  $< B$ that has the

- If $\omega =\sabp_T u_0$,  and $u(t,x)=\sabp_t u_0(x)$ admits a shock
generated in $\{x\geq 0\}$
at some time $t=\tau$,
and reaching the point $(\ms R,T)$, then letting
$\gamma(t)$, $t\in [\tau, T]$, denote the location of such a shock, one has

\begin{equation}
\label{eq:char-u-1}
\lim_{t\to T-} u(t,\gamma(t)+)\leq (f_r^{\prime})^{-1}(\mathsf R/T) \quad \Longrightarrow\quad  u_{\gamma}  \leq  \bs u\,.
\end{equation}

In particular, one has $u_\gamma=\bs u$
in (<ref>) only in the case
where $\sabp_t u_0$
coincides in the polygonal region $\Gamma[\ms R, B, f]$ with the right forward shock-rarefaction pattern described in section <ref>.

By definition of $u_\gamma$ it follows that either
$\omega(\ms R+)=u_\gamma$,

or else there is another jump
connecting $u_\gamma$

with $\omega(\ms R+)$ which
must satisfy the Lax entropy condition $\omega(\ms R+)< u_\gamma$.

Therefore, as a consequence of (<ref>)
we find a necessary condition
for the attainability of $\omega$ at time $T$ given by

\begin{equation}
\label{eq:lax-om-r-cond}
\omega(\ms R+)\leq  \bs u[\ms R, B, f_r]\,,
\end{equation}

(see (<ref>) of Theorem <ref>
and the proof in  <ref>).
The interesting fact is that,
in the case $\ms R \in \,]0, T \cdot f_r^{\prime}(B)$,
condition (<ref>),
together with the condition

\begin{equation}
\label{eq:omega-cond-1}
\omega(x)\geq B \qquad \forall~x\in \,]0, \ms R[\,,
\end{equation}

(see (<ref>), (<ref>),
of Theorem <ref>),
is also sufficient to
guarantee the existence of
an $AB$-entropy solution $u(x,t)$
that satisfies

\begin{equation}
\forall~x\in ]0, \ms R],
\qquad
u(\ms R,T)=\omega(\ms R+).
\end{equation}

To see this, assuming (<ref>)-(<ref>),
in view of the definitions introduced in the previous sections one can proceed as follows.

By  solving (<ref>) one determines the end point $\bs y[\ms R, B, f_r](T)$ of a “left backward shock” (Figure <ref>, left). The map $t \mapsto y[\ms R, B, f_r](t)$ represents the position of a shock in a

solution associated to the backward operator

(see  <ref>);

- given the final position $\bs y[\ms R, B, f_r]$
of the “backward shock" , one considers the solution $t  \mapsto \gamma(t)\doteq\bs x[\bs y[\ms R, B, f_r], \overline{B}, f_r](t)$ to (<ref>), when $\ms L =\bs y[\ms R, B, f_r]$, $A = \overline{B}$, $f = f_r$ (see Figure <ref>, right). This map represents the position of a shock in a “forward solution", i.e. in an $AB$-entropy solution associated to the operator
$\sabpt$. Actually, it can be shown that $(t, \gamma(t))$

is the location
of a shock of $\sabpt u_0$, with $u_0=\sabmt\omega$.

- once determined the point $\bs y[\ms R, B, f_r](T)$, one defines $\bs u[\ms R, B, f_r]$ as the state realizing the slope $(\bs y[\ms, B, f_r](T)+\ms R)/T$ (see (<ref>)):

f_r^{\prime}(\bs u[\ms R, B, f_r]) = \frac{\bs y[\ms R, B, f_r](T)+\ms R}{T};

- thanks to Lemma <ref>, we know that the final position at time $T$ of the shock $\gamma(t)$

\gamma(T)=
\ms R.
% = \bs x[\bs y[\ms R, B, f_r], \bar B, f_r](T)

Using this procedure, if a profile $\omega$ satisfies the conditions(<ref>)-(<ref>),

one can construct admissible $AB$-shocks
that produce at time $T$ the given jump in the profile $\omega$ at position $\ms R$.

Entirely symmetric considerations hold for the  state $\bs v[\ms L, A, f]$ defined in  <ref>.
As a byproduct of this analysis we will obtain that attainable profiles are fixed points of the backward forward solution operator, as stated in Theorem <ref>.

§ STATEMENT OF THE MAIN RESULTS

Conditions (1), (2) of Theorem <ref> will be shown to be equivalent by proving

that they are both equivalent to a  characterization of the
attainable set $\mc A^{[AB]}(T)$ in (<ref>) via
Oleǐnik-type inequalities and state constraints.

To present these results we need
to introduce some furhter notations.

Given a flux $f(x,u)$ as in (<ref>), we
will use the notations

({f_l}_{\mid (-\infty,\theta_l]})^{-1}$,
({f_r}_{\mid (-\infty,\theta_r]})^{-1}$, for the inverse of the restriction of
$f_l$, $f_r$  to their decreasing part, respectively,
$f_{l,+}^{-1}\doteq ({f_l}_{\mid [\theta_l,+\infty)})^{-1}$,
$f_{r,+}^{-1}\doteq ({f_r}_{\mid [\theta_{r},+\infty)})^{-1}$, for the inverse of the restriction of
$f_l$, $f_r$ to their increasing part, respectively.
Then, we set

\begin{equation}
\label{pimap-def}
\pi_{l,\pm}\doteq f_{l,\pm}^{-1} \circ f_l\,,
\qquad\quad
\pi_{r,\pm}\doteq f_{r,\pm}^{-1} \circ f_r\,,
\qquad\quad
\pi_{l,\pm}^{r}\doteq f_{l,\pm}^{-1} \circ f_r\,,
\qquad\quad
\pi_{r,\pm}^{l}\doteq f_{r,\pm}^{-1} \circ f_l\,,
\end{equation}

where $\pi_{l,\pm}^{r}$ are
defined on

\begin{equation}
I_l^r\doteq f_{r,-}^{-1}\big([f_l(\theta_l), +\infty[\,)\cup f_{r,+}^{-1}\big([f_l(\theta_l), +\infty[\,),
\end{equation}

and $\pi_{r,\pm}^{l}$
defined on

\begin{equation}
I_r^l\doteq f_{l,-}^{-1}\big([f_r(\theta_r), +\infty[\,)\cup f_{l,+}^{-1}\big([f_r(\theta_r), +\infty[\,)\,.
\end{equation}

Moreover, in connection with a
function $\omega:\R\to\R$

we define the quantities

\begin{equation}
\label{eq:LR-def}
\begin{aligned}
\ms R[\omega, f_r] &\doteq  \inf \big\{ R > 0 \; : \; x-T \cdot f_r^{\prime}(\omega(x)) \geq 0  \quad \forall \; x \geq R\big\},
\\
\noalign{\smallskip}
\ms L[\omega, f_l] &\doteq  \sup \big\{ L < 0 \; : \; x-T \cdot f_l^{\prime}(\omega(x)) \leq 0 \quad \forall \; x \leq L\big\},
\end{aligned}
\end{equation}

and, if $\ms L[\omega, f_l] \in \,]T\cdot f_l^{\prime}(A), 0[$ ,
we set

\begin{equation}
\label{eq:Rtildedef}
\widetilde{\ms R}
\doteq  \big(T-\bs \sigma\big[\ms L[\omega,f_l], A, f_l\big]\big) \cdot f_r^{\prime}(B),
\end{equation}

while, if $\ms R[\omega, f_r] \in \,]0, T\cdot f^{\prime}_r(B)[\,$, we set

\begin{equation}
\label{eq:Ltildedef}
%            \widetilde{\ms L} \doteq
\widetilde{\ms L}[\omega,f_l,f_r,A,B]
\doteq \big(T-\bs \tau\big[\ms R[\omega,f_r], B, f_r\big]\big) \cdot f_l^{\prime}(A).
\end{equation}

where $\bs \sigma[\ms L, A, f_l]$, $\bs \tau[\ms R, B, f_r]$, denote the shock starting
times introduced in

Such quantities are used to express the pointwise constraints satisfied by $\omega$  in intervals containing the origin
whenever $\omega$ is attainable.
Instead, to express the Oleǐnik-type inequalities satisfied by
the attainable profiles it is useful to define the functions

\begin{equation}
\label{eq:ghdef}
\begin{aligned}
\!\dfrac{f'_l(\omega(x))\left[  f'_r \circ
\pi_{r,-}^{l}
%        f_{r,-}^{-1}\circ f_l
(\omega(x)) \right]^2 }
{  \left[f''_r\circ
\pi_{r,-}^{l}
%      f_{r,-}^{-1}\circ f_l
(\omega(x))\right]\!\left[f'_l(\omega(x))\right]^2\!\big(T\!\cdot\! f'_l(\omega(x))-x \big)+x\big[f'_r\circ
\pi_{r,-}^{l}
%       f_{r,-}^{-1}\circ f_l
(\omega(x))\big] ^2\!  f''_l(\omega(x)) },
\\
\noalign{\medskip}
h[\omega,f_l,f_r](x)&\doteq\!\dfrac{f'_r(\omega(x))\big[  f'_l \circ
\pi_{l,+}^{r}
%        f_{l,+}^{-1}\circ f_r
(\omega(x)) \big]^2 }
{  \big[f''_l\circ
\pi_{l,+}^{r}
%     f_{l,+}^{-1}\circ f_r
(\omega(x))\big]\!\left[f'_r(\omega(x))\right]^2\big( T\!\cdot\!f'_r(\omega(x))-x \big)+x\big[f'_l\circ
\pi_{l,+}^{r}
%     f_{l,+}^{-1}\circ f_r
(\omega(x))\big] ^2\!  f''_r(\omega(x)) }.
\end{aligned}
\end{equation}

Since by Remark <ref> we know that
$\mc A^{[AB]}(T) \subset BV_{\mr{loc}}(\R\setminus\{0\})$,
we can partition the attainable set as

\begin{equation}
\label{eq:att-set-decomp}
\mc A^{[AB]}(T)= \bigcup_{\ms L \leq 0,\, \ms R \geq 0} \big( \mc A^{[AB]}(T) \cap \msc A^{\ms L, \ms R}\big),
\end{equation}

\begin{equation}
\label{eq:ALR-def1}
\msc A^{\ms L, \ms R} \doteq  \big\{ \omega \in
%    BV_{\mr{loc}}
(\mathbf{L^\infty}\cap BV_{\mr{loc}})
(\mathbb R\setminus\{0\}) \; : \; \
\exists~\omega(0\pm),\quad
\omega(0-) \in I_r^l, \quad \omega(0+) \in I_l^r,\quad
\ms L[\omega, f_l] = \ms L, \quad  \ms R[\omega, f_r] = \ms R\big\}\,.
\end{equation}

The characterization of the attainable profiles
in $\msc A^{\ms L, \ms R}$ will be given in:

-Theorem <ref>,  if
$\ms L < 0$, $\ms R > 0$ ;

-Theorem <ref>,  if  $\ms L < 0$, $\ms R = 0$  or  $\ms L = 0$, $\ms R > 0$;

-Theorem <ref>,  if  $\ms L = 0$, $\ms R = 0$.

Any element of $\msc A^{\ms L, \ms R}$
is an equivalence class of functions that
admit one-sided limit at any point $x\in\R$,
and that have at most countably many discontinuities.
Therefore, for any element of $\msc A^{\ms L, \ms R}$, we can always choose a representative  which is left or right continuous. For sake of uniqueness, throughout the paper we will
consider a representative of $\omega$ that is right continuous.

Throughout the following

\begin{equation}
\label{eq:dini_der_def}
D^- \omega(x)= \liminf_{h\rightarrow0}   \frac{\omega(x+h)-\omega(x)}{h} ,\quad \quad
D^+ \omega(x)= \limsup_{h\rightarrow0}   \frac{\omega(x+h)-\omega(x)}{h},
\end{equation}

will denote, respectively, the lower and
the  upper Dini derivative of a function $\omega$ at $x$.

In the same setting of Theorem <ref>,
let $(A, B)$ be a  connection,
let $\mc A^{[AB]}(T)$, $T>0$, be the set
in (<ref>), and
let $\omega$ be an element of the
set $ \msc A^{\ms L, \ms R}$
in (<ref>),
with $\ms L < 0$, $\ms R > 0$.

$\omega \in \mc A^{[AB]}(T)$ if and only if the connection $(A, B)$ is not critical and there

the following Oleǐnik-type inequalities are satisfied

\begin{equation}\label{eq:1a}
%     \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 \,]\ms R, +\infty[\,.
\end{aligned}
\end{equation}

letting $g, h$ be the functions in (<ref>),
and letting
$\widetilde {\ms L}\doteq \widetilde{\ms L}[\omega,f_l,f_r,A,B]$, $\widetilde {\ms R}\doteq \widetilde{\ms R}
be the constants in (<ref>),
if $\ms R \in \,]0, T \cdot f_r^{\prime}(B)[$ , and if   $ \widetilde{\ms L}>{\ms L}$,  then one has

\begin{equation}
\label{eq:1b1}
D^+\omega(x) \leq g[\omega, f_l, f_r](x) \qquad \forall \; x \in\,]\ms L,\, \widetilde{\ms L}\,[\,,
\end{equation}

while, if $\ms L \in \,]T\cdot f_l^{\prime}(A), 0[$ , and if  $ \widetilde{\ms R}<{\ms R}$,  then one has

\begin{equation}
\label{eq:1b12}
D^+\omega(x) \leq h[\omega, f_l, f_r](x) \qquad \forall \; x \in \,]\widetilde{\ms R }, \; \ms R[\,.
\end{equation}

(ii) letting $\bs u[\ms R, B, f_r]$, $\bs v[\ms L, A, f_l]$,  be
constants defined as
in (<ref>),
the following pointwise state constraints are satisfied

\begin{equation}
\label{eq:2a}
% \tag{2.a}
\begin{aligned}
\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{\medskip}
% 0<\ms R< T\cdot f'_r(B)
\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-)\,.
\end{aligned}
\qquad
\end{equation}

\begin{equation}
\label{eq:2bl-r1}
\begin{aligned}
\Big[\ms L \in \,]T\cdot f_l^{\prime}(A), 0[\,\;  \ \mathrm{and} \;  \ \ms R \leq  \widetilde{\ms R}\,
\Big]
\quad \mathrm{or} \quad\ \ms L \leq T\!\cdot\!f_l^{\prime}(A)
\qquad
\omega(x) = B\qquad \forall~x\in\,]0, \ms R[\,,
\qquad
\\
\noalign{\smallskip}
\Big[\ms R \in \,]0, T\cdot f_r^{\prime}(B)[\,\; \ \mathrm{and} \; \  \widetilde{\ms L}\leq \ms L\,\Big] \quad \mathrm{or} \quad \ms R \geq T\!\cdot\!f_r^{\prime}(B)
\qquad&\Longrightarrow\qquad
\omega(x) = A\qquad \forall~x\in\,]\ms L, 0[\,,
\qquad
\end{aligned}
\end{equation}

\begin{align}
\label{eq:2b-lr2}
% &\hspace{0,1in}
\hspace{0.6in}\ms L \in \,]T\cdot f_l^{\prime}(A), 0[\; \  \mathrm{and}
\; \ \widetilde{\ms R}<\ms R\qquad
\left\{
\begin{aligned}
\omega(x)&=B\qquad\forall~x\in\,]0,
\widetilde{\ms R}\,],
\\
\omega( \widetilde{\ms R}+)&= B,
\\
\omega(x)&\geq B\qquad\forall~x\in\,]\,\widetilde{\ms R},\,\ms R[\,,
\end{aligned}
\right.
\\
\noalign{\medskip}
\label{eq:2b-lr2-2}
\hspace{0.6in}\ms R \in \,]0, T\cdot f_r^{\prime}(B)[\, \; \ \mathrm{and}
\; \ \ms L <
\widetilde{\ms L}\qquad
\left\{
\begin{aligned}
\omega(x)&=A\qquad\forall~x\in [\,
\widetilde{\ms L},\,0[\,,
\\
\omega( \widetilde{\ms L}-)&= A,
\\
\omega(x)&\leq A\qquad\forall~x\in\,]\ms L, \widetilde{\ms L}\,[\,,
\end{aligned}
\right.
\end{align}

\begin{equation}
\label{eq:3b-lr2}
\quad\ \begin{aligned}
\ms L \leq T\cdot f'_l(A)
\qquad
&\Longrightarrow\qquad   \omega(\ms L-)\geq \omega(\ms L+),
\\
\noalign{\smallskip}
\ms R \geq T\cdot f'_r(B)
\qquad
&\Longrightarrow\qquad   \omega(\ms R-)\geq \omega(\ms R+).
\end{aligned}
\end{equation}

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[line width=1.5]    (90.8,239.53) – (590.13,238.3) ;
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(335.67,184) – (373.33,238.33) ;

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(138.67,52) – (166.67,236.18) ;

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[dash pattern=on 4.5pt off 4.5pt]  (138.67,52) – (182.67,238.85) ;

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(138.67,52) – (334,179.51) ;

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[dash pattern=on 4.5pt off 4.5pt]  (367.2,103.2) – (411.87,237.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (343.2,129.2) – (411.87,237.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (343.87,125.87) – (333.87,129.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (353.87,113.87) – (411.87,237.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (425.87,66.53) – (411.87,237.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (410.53,73.2) – (411.87,237.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (395.2,81.2) – (411.87,237.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (380.53,91.87) – (411.87,237.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (441.2,59.2) – (411.87,237.87) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (335.87,142.53) .. controls (348.53,105.2) and (393.2,72.53) .. (455.87,52.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (353.87,113.87) – (335.43,118.6) ;

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(472.97,53.52) – (499.2,238.53) ;

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(472.97,53.52) – (450.53,238.53) ;

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[dash pattern=on 4.5pt off 4.5pt]  (465.89,54.04) – (433.2,238.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (472.97,53.52) – (471.2,238.53) ;

[dash pattern=on 0.84pt off 2.51pt]  (335.87,142.53) – (593.87,142.53) ;

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(569.33,34.07) node [anchor=north west][inner sep=0.75pt]  [font=]  $\omega $;

(600,231.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x$;

(346,14.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $t$;

(451.33,28.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathsf{R}$;

(584,120.07) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathbf{\tau }[\mathsf{R} ,\ B,\ f_{r}]$;

(255.33,28.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $A$;

(376,28.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $B$;

(161.33,24.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $\tilde{\mathsf{L}}$;

Case 1.

every picture/.style=line width=0.75pt

[fill=rgb, 255:red, 74; green, 144; blue, 226   ,fill opacity=0.49 ] (315.2,177.53) – (118.67,64.33) – (118.67,64.33) – (295.33,250.51) – (315.67,231.67) – cycle ;

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(68.13,63.46) – (570.2,64.5) ;

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(146.53,64.2) – (315.87,154.87) ;

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[dash pattern=on 4.5pt off 4.5pt]  (107.27,63.01) – (132,251.18) ;

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[dash pattern=on 4.5pt off 4.5pt]  (103.33,64.51) – (118,250.51) ;

[dash pattern=on 4.5pt off 4.5pt]  (343.2,114.87) – (315.87,120.87) ;

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[dash pattern=on 4.5pt off 4.5pt]  (365.87,97.53) – (315.2,109.53) ;

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[dash pattern=on 4.5pt off 4.5pt]  (323.87,138.2) – (313.87,141.53) ;

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[dash pattern=on 4.5pt off 4.5pt]  (421.2,71.53) – (391.87,250.2) ;

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(452.97,65.85) – (479.2,250.87) ;

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(452.97,65.85) – (430.53,250.87) ;

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[dash pattern=on 4.5pt off 4.5pt]  (307.87,219.87) – (295.33,250.51) ;

[dash pattern=on 4.5pt off 4.5pt]  (192.53,117.87) – (295.33,250.51) ;

[dash pattern=on 4.5pt off 4.5pt]  (279.2,187.87) – (295.33,250.51) ;

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[dash pattern=on 4.5pt off 4.5pt]  (237.2,149.87) – (315.67,196.33) ;

[dash pattern=on 4.5pt off 4.5pt]  (295.2,205.2) – (295.33,250.51) ;

[dash pattern=on 4.5pt off 4.5pt]  (255.2,166.53) – (295.33,250.51) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (118.67,64.33) .. controls (181.87,111.2) and (273.87,165.2) .. (315.67,231.67) ;

[dash pattern=on 4.5pt off 4.5pt]  (295.2,205.2) – (315,217.67) ;

[dash pattern=on 0.84pt off 2.51pt]  (611.87,69.2) – (315.67,231.67) ;

[dash pattern=on 0.84pt off 2.51pt]  (572.53,231.2) – (315.67,231.67) ;

[dash pattern=on 4.5pt off 4.5pt]  (316.53,241.87) – (306,250.33) ;

(108.67,40.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathsf{L}$;

(549.33,46.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $\omega $;

(580,243.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $x$;

(326,26.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $t$;

(431.33,40.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathsf{R}$;

(564,132.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathbf{\tau }[\mathsf{R} ,\ B,\ f_{r}]$;

(235.33,40.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $A$;

(356,41.07) node [anchor=north west][inner sep=0.75pt]  [font=]  $B$;

(141.33,36.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $\tilde{\mathsf{L}}$;

(611.33,43.07) node [anchor=north west][inner sep=0.75pt]  [font=]  $\tilde{\mathsf{R}}$;

(566,210.07) node [anchor=north west][inner sep=0.75pt]  [font=]  $\mathbf{\sigma }[\mathsf{L} ,\ A,\ f_{l}]$;

(299.33,257.73) node [anchor=north west][inner sep=0.75pt]  [font=]  $\overline{A}$;

(334,258.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $\overline{B}$;

Case 2.

every picture/.style=line width=0.75pt

[fill=rgb, 255:red, 208; green, 2; blue, 27   ,fill opacity=0.45 ] (455.87,69.87) – (411.87,255.2) – (335.87,159.87) – (335.87,91.87) – cycle ;

[line width=1.5]    (335.59,254.96) – (335.59,36.2) ;
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[line width=1.5]    (83.92,255.58) – (583.26,254.34) ;
[shift=(587.26,254.34), rotate = 179.86] [fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.08]  [draw opacity=0] (6.97,-3.35) – (0,0) – (6.97,3.35) – cycle    ;

(88.13,68.46) – (590.2,69.5) ;

[dash pattern=on 4.5pt off 4.5pt]  (324.16,69.29) – (336.53,71.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (369.87,69.2) – (332.53,73.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (335.67,236.67) – (349.87,254.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (335.2,170.53) – (399.87,254.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (337.67,220.67) – (365.2,255.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (335.67,201.33) – (377.87,255.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (335.2,187.2) – (390.53,254.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (363.2,119.87) – (335.87,125.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (415.2,69.87) – (337.87,82.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (385.87,102.53) – (335.2,114.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (425.87,83.87) – (334.53,103.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (367.2,120.53) – (411.87,255.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (343.2,146.53) – (411.87,255.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (343.87,143.2) – (333.87,146.53) ;

[dash pattern=on 4.5pt off 4.5pt]  (353.87,131.2) – (411.87,255.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (425.87,83.87) – (411.87,255.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (410.53,90.53) – (411.87,255.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (395.2,98.53) – (411.87,255.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (380.53,109.2) – (411.87,255.2) ;

[dash pattern=on 4.5pt off 4.5pt]  (441.2,76.53) – (411.87,255.2) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (335.87,159.87) .. controls (348.53,122.53) and (393.2,89.87) .. (455.87,69.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (353.87,131.2) – (335.43,135.93) ;

[dash pattern=on 4.5pt off 4.5pt]  (472.97,70.85) – (486.53,253.87) ;

(472.97,70.85) – (499.2,253.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (472.97,70.85) – (457.87,253.2) ;

(472.97,70.85) – (449.87,253.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (480.53,71.2) – (509.87,253.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (465.89,71.38) – (433.2,255.87) ;

[dash pattern=on 4.5pt off 4.5pt]  (472.97,70.85) – (471.2,253.2) ;

[dash pattern=on 0.84pt off 2.51pt]  (335.87,159.87) – (593.87,159.87) ;

[dash pattern=on 0.84pt off 2.51pt]  (578,69.5) – (334.33,200.35) ;

[dash pattern=on 0.84pt off 2.51pt]  (591.2,199.89) – (334.33,200.35) ;

[fill=rgb, 255:red, 74; green, 144; blue, 226   ,fill opacity=0.64 ] (183.67,68.69) – (222,256.02) – (334.33,200.35) – (335.87,118.53) – cycle ;

[dash pattern=on 4.5pt off 4.5pt]  (217.67,89.02) – (221.67,255.69) ;

[dash pattern=on 4.5pt off 4.5pt]  (197.67,77.02) – (221.67,255.69) ;

[dash pattern=on 4.5pt off 4.5pt]  (278.33,133.69) – (221.67,255.69) ;

[dash pattern=on 4.5pt off 4.5pt]  (183.33,68.35) – (189,255.02) ;

[dash pattern=on 4.5pt off 4.5pt]  (263.67,119.69) – (221.67,255.69) ;

[dash pattern=on 4.5pt off 4.5pt]  (183.33,68.35) – (208.33,255.02) ;

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Case 4.

By Theorem <ref>,
if the connection is critical, then the set $\mc A^{[AB]}(T)$ does not contain profiles $\omega$ with $\ms L[\omega, f_l] < 0$ and $\ms R[\omega, f_r] > 0$.
Notice that conditions (<ref>), (<ref>) imply $\omega(\ms R-)\geq B$.
On the other hand, if $\ms R<T\cdot f'_r(B)$,
by virtue of (<ref>),
and because of (<ref>), we have
$\omega(\ms R+)\leq B$.
Hence, because of (<ref>), it follows that
the inequality $\omega(\ms R-)\geq \omega(\ms R+)$ is always
With similar arguments we  deduce that also
the inequality
$\omega(\ms L-)\geq \omega(\ms L+)$ is always verified.

$\ms R[\omega, f_r]\in\,]0,T\cdot f'_r(B)[$ ,
applying (<ref>)
with $f_r$ in place of $f$ and $\ms R=\ms R[\omega, f_r]$,
and recalling (<ref>),
we derive

\begin{equation}
\label{eq:lax-rfs-cond-5}
\frac{\ms R[\omega, f_r]}{f'_r(B)}<T-\bs\tau\big[\ms R[\omega, f_r], B, f_r\big]
=\frac{\widetilde{\ms L}[\omega,f_l,f_r,A,B]}{f'_l(A)}\,.
\end{equation}

Similarly, if $\ms L[\omega, f_l]\in\,]T\cdot f'_l(A),0[$ , applying (<ref>)
with $f_l$ in place of $f$ and $\ms L=\ms L[\omega, f_l]$,
and recalling that $f'_l(A)<0$
we find

\begin{equation}
\label{eq:lax-rfs-cond-6}
\frac{\ms L[\omega, f_l]}{f'_l(A)}<T-\bs\sigma\big[\ms L[\omega, f_l], A, f_l\big]\,.
\end{equation}

Hence, if $\widetilde
{\ms L}[\omega,f_l,f_r,A,B]
\geq \ms L[\omega, f_l]$, combining (<ref>),
(<ref>), we deduce

\begin{equation}
\frac{\ms R[\omega, f_r]}{f'_r(B)}<T-\bs\sigma\big[\ms L[\omega, f_l], A, f_l\big]\,,
\end{equation}

which, in turn, by (<ref>) yields

\begin{equation}
\ms R[\omega, f_r]<\widetilde
{\ms R}[\omega,f_l,f_r,A,B]\,.
\end{equation}

With entirely similar arguments one can show that, if $\widetilde
{\ms R}[\omega,f_l,f_r,A,B]
\leq \ms R[\omega, f_r]$,
then one has

\begin{equation}
\ms L[\omega, f_l]>\widetilde
{\ms L}[\omega,f_l,f_r,A,B]\,.
\end{equation}

Therefore, when $\ms L[\omega, f_l]\in\,]T\cdot f'_l(A),0[$ ,  and $\ms R[\omega, f_r]\in\,]0,T\cdot f'_r(B)[$ ,
there hold

\begin{equation}
\label{eq:LR-prop26}
\begin{aligned}
\widetilde
{\ms L}[\omega,f_l,f_r,A,B]
\geq \ms L[\omega, f_l]
\quad &\Longrightarrow \quad \widetilde {\ms R}[\omega,f_l,f_r,A,B] > \ms R[\omega, f_r]\,,
\\
\noalign{\smallskip}
\widetilde
{\ms R}[\omega,f_l,f_r,A,B]
\leq \ms R[\omega, f_r]
\quad &\Longrightarrow \quad \widetilde {\ms L}[\omega,f_l,f_r,A,B] < \ms L[\omega, f_l]\,.
\end{aligned}
\end{equation}

These implications, in particular, show that it can never occur the case where

\begin{equation}
\widetilde
{\ms L}[\omega,f_l,f_r,A,B]
\geq \ms L[\omega, f_l]
\qquad\text{and}\qquad
\widetilde {\ms R}[\omega,f_l,f_r,A,B]
\leq \ms R[\omega, f_r]\,.
\end{equation}

Notice that by condition (<ref>) in Theorem <ref>,
and because of (<ref>),
it follows that
if $\ms L[\omega, f_l]\in\,]T\cdot f'_l(A),0[\,$, and
$\ms R[\omega, f_r]\leq \widetilde {\ms R}[\omega,f_l,f_r,A,B]$,
then one has

$\ms R[\omega, f_r]< T\cdot f^{\prime}_r(B)$.
Therefore, we have

\begin{equation}
\label{eq:LRtR-impl1}
\Big[\ms L[\omega, f_l]\in\,]T\cdot f'_l(A),0[\,\quad\text{and}\quad
\ms R[\omega, f_r]\geq  T\cdot f^{\prime}_r(B)\Big]
\quad
\Longrightarrow\quad
\ms R[\omega, f_r]> \widetilde {\ms R}[\omega,f_l,f_r,A,B]\,.
\end{equation}

Similarly, one can show that,
by (<ref>), (<ref>), there holds

\begin{equation}
\label{eq:LRtR-impl2}
\Big[\ms R[\omega, f_r]\in\,]0, T\cdot f'_r(B)[\,\quad\text{and}\quad
\ms L[\omega, f_l]\leq  T\cdot f^{\prime}_l(A)\Big]
\quad
\Longrightarrow\quad
\ms L[\omega, f_l]< \widetilde {\ms L}[\omega,f_l,f_r,A,B]\,.
\end{equation}

Then, relying on (<ref>),
(<ref>), (<ref>), we deduce
that we can distinguish
six cases
of pointwise constraints prescribed by condition (ii)
of Theorem <ref>,

which depend on the reciprocal positions of the points
$\ms L=\ms L[\omega, f_l]$,  $\ms R=\ms R[\omega, f_r]$,
$\widetilde{\ms L}$,
$\widetilde{\ms R}$:

Case 1:
If $\ms L \leq T\cdot f_l^{\prime}(A)<0$,  $0 <\ms R< T\cdot f^{\prime}_r(B)$  (Figure <ref>), then    $\widetilde{\ms L} > \ms L$, and
there holds
\begin{equation}
\label{eq:case-i-A}
\omega(x)\leq A
\quad\ \forall~x\in\,]{\ms L}, \widetilde{\ms L}\,[\,,
\qquad\ \ \omega(\,\widetilde{\ms L}-)=A,\qquad \ \
\omega(x)=A\quad\ \forall~x\in\,]\widetilde{\ms L}, 0[\,,
\end{equation}

\begin{equation}
\label{eq:case-i-B}
\omega(x)=B\ \quad\ \forall~x\in\,]0, {\ms R}[\,,
\qquad \ \
\omega(\ms R+) \leq \bs u[\ms R, B, f_r]\leq B\,;
\end{equation}

Case 2: If
$T\cdot f_l^{\prime}(A)< \ms L<0$, $0< \ms R< T\cdot f^{\prime}_r(B)$, and $\widetilde{\ms L} > \ms L$,  $\widetilde{\ms R }> \ms R$  (Figure <ref>), then

hold (<ref>) and

\begin{equation}
\label{eq:case-ii-A}
\omega(\ms L-) \geq \bs v[\ms L, A, f_l]\geq A\,,
\quad\ \ \omega(x)\leq A
\quad \forall~x\in\,]{\ms L}, \widetilde{\ms L}\,[\,,
\quad\ \ \omega(\,\widetilde{\ms L}-)=A,\quad \ \
\omega(x)=A\quad \forall~x\in\,]\widetilde{\ms L}, 0[\,;
\end{equation}

the symmetric ones:

Case 1b:
If $T\cdot f_l^{\prime}(A)< \ms L<0$, $0<T\cdot f^{\prime}_r(B)\leq \ms R$, then $\widetilde{\ms R }< \ms R$ and there holds

\begin{equation}
\label{eq:case-iprime-A}
\omega(x)=A\qquad \forall~x\in\,]{\ms L}, 0[\,,\qquad\ \omega(\ms L-) \geq \bs v[\ms L, A, f_l]\geq A\,,
\end{equation}

\begin{equation}
\label{eq:case-iprime-B}
\omega(x)= B
\quad\ \forall~x\in\,]0, \widetilde{\ms R}\,[\,,
\qquad\ \ \omega(\,\widetilde{\ms R}+)=B,\qquad \ \
\omega(x)\geq B\quad\ \forall~x\in\,]\widetilde{\ms R}, \ms R[\,;
\end{equation}

Case 2b:
If    $T\cdot f_l^{\prime}(A)< \ms L<0$, $0< \ms R< T\cdot f^{\prime}_r(B)$, and $\widetilde{\ms L}< \ms L$,  $\widetilde{\ms R }< \ms R$, then there hold (<ref>),

\begin{equation}
\label{eq:case-iiprime-B}
\omega(x)= B
\quad \forall~x\in\,]0, \widetilde{\ms R}\,[\,,
\quad\ \ \omega(\,\widetilde{\ms R}+)=B,\quad \ \
\omega(x)\geq B\quad \forall~x\in\,]\widetilde{\ms R}, \ms R[\,
\quad \ \
\omega(\ms R+) \leq \bs u[\ms R, B, f_r]\leq B\,;
\end{equation}

and the remaining ones:

Case 3:
$T\cdot f_l^{\prime}(A)<\ms L <0$, $0<\ms R < T\cdot f_r^{\prime}(B)$, and $\widetilde{\ms L} \leq \ms L$, $\widetilde{\ms R} \geq \ms R$ (Figure <ref>),
then there

hold (<ref>), (<ref>);

Case 4:
If $\ms L \leq T\cdot f_l^{\prime}(A)<0$ and $\ms R \geq T \cdot f_r^{\prime}(B)>0$
(Figure <ref>),
then there holds

\begin{equation}
\label{eq:case-iv}
\omega(x)=A\qquad \forall~x\in\,]{\ms L}, 0[\,,
\qquad\quad
\omega(x)=B\ \quad\ \forall~x\in\,]0, {\ms R}[\,.
\end{equation}

The six cases are depicted in Figure <ref>. One can regard the intervals $]T\!\cdot\! f_l^{\prime}(A), 0[$
and $]0, T\!\cdot\! f_r^{\prime}(B)[$   as “active zones"
for the presence of shocks in an $AB$-entropy solution
that attains $\omega$
at time $T$: as soon as $\ms L$
belongs to
$]T\!\cdot\!f_l^{\prime}(A), 0[$  or $\ms R$
belongs to  $]0, T\!\cdot\! f_r^{\prime}(B)[$ ,

it is needed