# Nonlocal traffic models with general kernels:
singular limit, entropy admissibility, and convergence rate

Maria Colombo  M.C. EPFL B, Station 8, CH-1015 Lausanne, Switzerland.
<EMAIL_ADDRESS>,  Gianluca Crippa  G.C. Departement Mathematik und
Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland.
<EMAIL_ADDRESS>,  Elio Marconi  E.M. EPFL B, Station 8, CH-1015
Lausanne, Switzerland<EMAIL_ADDRESS>and  Laura V. Spinolo  L.V.S.
IMATI-CNR, via Ferrata 5, I-27100 Pavia, Italy<EMAIL_ADDRESS>
Nonlocal conservation laws (the signature feature being that the flux function
depends on the solution through the convolution with a given kernel) are
extensively used in the modeling of vehicular traffic. In this work we discuss
the singular local limit, namely the convergence of the nonlocal solutions to
the entropy admissible solution of the conservation law obtained by replacing
the convolution kernel with a Dirac delta. Albeit recent counter-examples rule
out convergence in the general case, in the specific framework of traffic
models (with anisotropic convolution kernels) the singular limit has been
established under rigid assumptions, i.e. in the case of the exponential
kernel (which entails algebraic identities between the kernel and its
derivatives) or under fairly restrictive requirements on the initial datum. In
this work we obtain general convergence results under assumptions that are
entirely natural in view of applications to traffic models, plus a convexity
requirement on the convolution kernels. We then provide a general criterion
for entropy admissibility of the limit and a convergence rate. We also exhibit
a counter-example showing that the convexity assumption is necessary for our
main compactness estimate.

Keywords: nonlocal-to-local limit, nonlocal conservation laws, singular local
limit, traffic models.

MSC (2020): 35L65

##  1\. Introduction

Consider a family of Cauchy problems for nonlocal conservation laws in the
form

(1.1)
$\left\\{\begin{array}[]{ll}\partial_{t}u_{\varepsilon}+\partial_{x}[V(u_{\varepsilon}\ast\eta_{\varepsilon})u_{\varepsilon}]=0\\\
u_{\varepsilon}(0,\cdot)=u_{0}\\\ \end{array}\right.\qquad\text{with
$\eta_{\varepsilon}(x):=\frac{1}{\varepsilon}\eta\left(\frac{x}{\varepsilon}\right).$}$

In the previous expression,
$u_{\varepsilon}:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ is the unknown,
$\eta:\mathbb{R}\to\mathbb{R}_{+}$ is a given convolution kernel, the symbol
$\ast$ represents the convolution with respect to the $x$ variable only,
$\varepsilon>0$ is a parameter and $V:\mathbb{R}\to\mathbb{R}$ a Lipschitz
continuous function. We specify in the following the precise assumptions
imposed on $V$, $\eta$ and $u_{0}$, for the time being we just mention that
existence and uniqueness results for (1.1) can be established under fairly
general assumptions, see for instance [16]. In the present work we are
concerned with the singular local limit $\varepsilon\to 0+$: when
$\eta_{\varepsilon}$ converges weakly∗ in the sense of measures to the Dirac
delta, (1.1) formally boils down to the conservation law Cauchy problem

(1.2) $\left\\{\begin{array}[]{ll}\partial_{t}u+\partial_{x}[V(u)u]=0\\\
u(0,\cdot)=u_{0}.\\\ \end{array}\right.$

Existence and uniqueness results for so-called entropy admissible solutions of
(1.2) date back to Kružkov [18]. The nonlocal-to-local limit was first
addressed by Zumbrun in [23] and Amorim, R. Colombo and Teixeira in [2].
Zumbrun [23] showed that when $\varepsilon\to 0^{+}$ the family
$u_{\varepsilon}$ converges to the entropy admissible solution $u$ of (1.2)
provided $V(u)=u$, $\eta$ is an even function and the limit solution $u$ is
regular. In [2] the authors posed the general convergence question and
exhibited numerical experiments suggesting convergence of $u_{\varepsilon}$ to
the entropy admissible solution $u$. In [12] the authors provided some
counter-examples showing that, in general, the family $u_{\varepsilon}$ does
not converge to the entropy admissible solution of (1.2).

The analysis in [12] left open the possibility that convergence holds under
specific assumptions. In this respect, a natural target for the investigation
of the local limit is the framework of traffic flow models: indeed, in recent
years, nonlocal conservation laws in the form (1.1) have been widely used in
the modeling of traffic, see for instance [3, 6, 13] and the references
therein. In this framework, the local counterpart (1.2) is the by now
classical LWR model introduced in [20, 22]: $u$ in this case represents the
density of cars and $V$ their speed. It is then natural to assume that the
initial datum $u_{0}$ satisfies

(1.3) $u_{0}\in L^{\infty}(\mathbb{R}),\qquad 0\leq u_{0}\leq 1,$

where we have normalized to $1$ the maximum possible car density,
corresponding to bumper-to-bumper packing. Also, note that the LWR model
postulates that drivers choose their speed depending on the car density, and,
since the expected reaction to a traffic congestion is deceleration, the
standard assumptions imposed on the function $V$ are

(1.4) $V\in\mathrm{Lip}(\mathbb{R}),\qquad V^{\prime}\leq 0\;\text{on [0,
1]}.$

The nonlocal convolution term in (1.1) models the fact that drivers choose
their speed based on the density of cars in a suitable neighborhood. The
standard hypotheses imposed on $\eta$ are then

(1.5) $\eta\in L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R}),\quad\mathrm{supp}\,\eta\subseteq\mathbb{R}_{-},\qquad\eta\geq
0,\qquad\eta\text{ non-decreasing on
$\mathbb{R}_{-}$},\qquad\int_{\mathbb{R}_{-}}\eta(x)dx=1.$

The assumption $\mathrm{supp}\,\eta\subseteq\mathbb{R}_{-}$ models the fact
that drivers choose their speed based on the downstream car density only (i.e.
they look forward, not backward), whereas the monotonicity condition takes
into account the fact that they pay more attention to closer vehicles.

We now focus on the analysis of nonlocal Cauchy problems satisfying
(1.3),(1.4) and (1.5). Under (1.4) and (1.5) nonlocal conservation laws (1.1)
enjoy the maximum principle and, under some more assumptions, propagation of
monotonicity, see [3]. Keimer and Pflug [17] used these properties to
establish convergence in the local limit $\varepsilon\to 0^{+}$ for monotone
initial data. Bressan and Shen [4, 5] established convergence in the local
limit in the case $\eta(x)=\mathbbm{1}_{]-\infty,0]}(x)e^{x}$ and under the
assumption that the initial datum has finite total variation and is bounded
away from $0$. A key point of the analysis in both [4, 17] is that the total
variation $\mathrm{TotVar}\ u_{\varepsilon}(t,\cdot)$ is a monotone non-
increasing function of time. Note furthermore that the assumption that the
initial datum has bounded total variation is fairly natural in the
conservation laws framework, see [14]. In our previous work [11] we
established convergence via an Oleĭnik-type estimate under quite general
assumptions on the kernel $\eta$, but requiring that the initial datum $u_{0}$
is bounded away from $0$ and satisfies a fairly restrictive one-sided
Lipschitz condition. In [11] we also exhibit a counter-example showing that,
if the initial datum attains the value $0$ then it may happen that for every
$t>0$ the total variation $\mathrm{TotVar}\ u_{\varepsilon}(t,\cdot)$ blows up
as $\varepsilon\to 0^{+}$: this rules out total variation estimates and
implies that the convergence proofs in [4, 11] cannot extend to general
initial data. A way out this obstruction has been recently found by Coclite,
Coron, De Nitti, Keimer and Pflug in [7]: rather than looking at the total
variation of $u_{\varepsilon}$, they show that the total variation of the
convolution term

(1.6)
$w_{\varepsilon}(t,x):=u_{\varepsilon}\ast\eta_{\varepsilon}(t,x)=\frac{1}{\varepsilon}\int_{x}^{+\infty}\eta\left(\frac{x-y}{\varepsilon}\right)u_{\varepsilon}(t,y)dy$

is a monotone non-increasing function of time. This allows to extend the
results by Bressan and Shen [4, 5] to initial data attaining the value $0$.
Note, however, that the analysis in [4, 5, 7] is restricted to the case of the
exponential kernel $\eta(x)=\mathbbm{1}_{]-\infty,0]}(x)e^{x}$. The proofs
crucially rely on the algebraic identity, with no analogue for general
kernels,

(1.7)
$u_{\varepsilon}=w_{\varepsilon}-\varepsilon\partial_{x}w_{\varepsilon}\qquad\mbox{if
}\eta(x)=\mathbbm{1}_{]-\infty,0]}(x)e^{x},$

which is used to derive an equation for $w_{\varepsilon}$ independent of
$u_{\varepsilon}$ and allows to reformulate (1.1) as a system with relaxation.

Our first main result states that the fact that $\mathrm{TotVar}\
w_{\varepsilon}(t,\cdot)$ is a non-increasing function of time is true under
the sole minimal assumptions (1.4) and (1.5) combined with a convexity
requirement, namely

(1.8) $\eta\mbox{ is convex on }\mathbb{R}_{-},$

that we comment upon in the following.

######  Theorem 1.1.

Assume that $u_{0}$, $V$ and $\eta$ satisfy (1.3), (1.4), (1.5) and (1.8),
respectively, and let $u_{\varepsilon}$ be the solution of the Cauchy problem
(1.1) and $w_{\varepsilon}$ be as in (1.6). If $\mathrm{TotVar}\
u_{0}<+\infty$ then

(1.9)
$\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)\leq\mathrm{TotVar}\,w_{\varepsilon}(0,\cdot)\quad\text{for
every $\varepsilon>0$ and a.e. $t>0$}.$

Note that the right-hand side of (1.9) is then easily controlled by the
inequality

(1.10) $\mathrm{TotVar}\ w_{\varepsilon}(0,\cdot)\leq\mathrm{TotVar}\ u_{0},$

which directly follows from (1.6). The proof of Theorem 1.1 is independent
from the one in [7] since it does not rely on any variant of (1.7), and more
delicate since the equation for $w_{\varepsilon}$ is more involved, see (3.1).
Note furthermore that, by relying on (1.9), it is rather easy (see §5.1) to
see that, up to subsequences,

(1.11) ${w_{\varepsilon}\to u\;\;\text{strongly in
$L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$,}}\qquad
u_{\varepsilon}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}u\quad\text{weakly${}^{\ast}$
in $L^{\infty}(\mathbb{R}_{+}\times\mathbb{R})$, }$

for some function $u$ that is a _distributional_ solution of the Cauchy
problem (1.2). The problem of the entropy admissibility of the limit is highly
nontrivial and, even in the case of the exponential kernel
$\eta(x)=\mathbbm{1}_{]-\infty,0]}(x)e^{x}$ it was partially left open in [4]
and treated specifically in [5, 7] by relying on formula (1.7). In [11] the
entropy admissibility follows from an Oleĭnik-type estimate [21] whose proof
requires the one-sided Lipschitz condition on the initial datum. Our second
main result establishes entropy admissibility of the limit function under very
minimal assumptions.

######  Theorem 1.2.

Assume that $u_{0}$, $V$ and $\eta$ satisfy (1.3) (1.4) and (1.5),
respectively, and let $w_{\varepsilon}$ be as in (1.6), where
$u_{\varepsilon}$ is the solution of the Cauchy problem (1.1). Consider a
sequence $\varepsilon_{k}\to 0^{+}$ and assume that $w_{\varepsilon_{k}}\to u$
in $L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$ for some function
$u\in L^{\infty}(\mathbb{R}_{+}\times\mathbb{R})$; then $u$ is the entropy
admissible solution of (1.2).

We explicitly point out that the convexity assumption (1.8) is not needed in
the statement of Theorem 1.2. Also, Theorem 1.2 only requires strong $L^{1}$
compactness rather than total variation bounds, and as such it provides a
general entropy admissibility criterion for the limit, which replaces the ad
hoc arguments used in specific cases in [5, 7, 17] and might be useful in
contexts where the $L^{1}$ compactness is obtained through weaker a priori
estimates (see [10] for results in the case of the exponential kernel). The
following theorem collects our main convergence results.

######  Theorem 1.3.

Assume that $u_{0}$, $V$ and $\eta$ satisfy (1.3) (1.4) and (1.5),
respectively, and let $w_{\varepsilon}$ be as in (1.6), where
$u_{\varepsilon}$ is the solution of the Cauchy problem (1.1). Let $u$ be the
entropy admissible solution of (1.2). If (1.8) holds and $\mathrm{TotVar}\
u_{0}<+\infty$, then (1.11) holds true.

If furthermore $\eta(\xi)\xi\in L^{1}(\mathbb{R})$ then we have the
convergence rate

(1.12) $\|u(t,\cdot)-w_{\varepsilon}(t,\cdot)\|_{L^{1}(\mathbb{R})}\leq
C(\eta,V)\big{[}\varepsilon+\sqrt{\varepsilon t}\big{]}\mathrm{TotVar}\
u_{0}\quad\text{for every $\varepsilon>0$ and a.e. $t>0$}$

for a suitable constant $C(\eta,V)$ only depending on $\eta$ and $V$.

The convergence result in the above statement is a consequence of Theorem 1.1
and Theorem 1.2, whereas to establish the convergence rate (1.12) we provide a
fairly precise estimate on the entropy dissipation rate of nonlocal solutions
(see Proposition 5.1) and then rely on an argument due to Kuznetsov [19].

To conclude we are left to discuss the role of the convexity assumption (1.8),
which is the sole hypothesis in our convergence results Theorem 1.1 and
Theorem 1.3 that is not entirely standard in traffic models. It turns out
that, if the convexity assumption (1.8) is violated then the key estimate
(1.9) fails in general.

######  Theorem 1.4.

Assume $V(w):=1-w$ and $\eta:=\mathbbm{1}_{]-1,0[}$; then for every sequence
$\\{\varepsilon_{n}\\}$ satisfying

(1.13)
$\varepsilon_{n}>0,\quad\varepsilon_{n+1}\leq\frac{1}{16}\varepsilon_{n}\quad\text{
for every $n\in\mathbb{N}$}$

the following holds. There are an initial datum $u_{0}$ satisfying (1.3) and
such that $\mathrm{TotVar}\ u_{0}~{}<~{}+\infty$ and a sequence $\\{t_{n}\\}$
such that for every $n\in\mathbb{N}$

(1.14)
$t_{n}>0,\qquad\mathrm{TotVar}\,w_{\varepsilon_{n}}(t,\cdot)>\mathrm{TotVar}\,w_{\varepsilon_{n}}(0,\cdot)\quad\text{for
a.e. $t\in\ ]0,t_{n}[$}.$

Some remarks are here in order. First, the function $V(w)=1-w$ satisfies (1.4)
and the function $\eta~{}=~{}\mathbbm{1}_{]-1,0[}$ satisfies (1.5), but does
not satisfy the convexity condition (1.8). Second, the counter-example given
in Theorem 1.4 is in contrast with numerical evidence provided in [7, §5] and
[17, p. 1949] which suggested that, when $V(w):=1-w$ and
$\eta:=\mathbbm{1}_{]-1,0[}$, $\mathrm{TotVar}\,w_{\varepsilon_{n}}(t,\cdot)$
is a monotone non-increasing function of time. The numerical elusiveness is
mainly due to the need of a specific choice of initial datum to enforce (1.14)
rather than to the numerical viscosity issue discussed in [9]. Our choice of
initial datum is through an explicit formula, see (6.5), which depends on the
chosen sequence $\\{\varepsilon_{n}\\}$. Third, we are reasonably confident
that the basic ideas of our counter-example are not restricted to the case
$\eta=\mathbbm{1}_{]-1,0[}$ and could be extended to a rather general class of
kernels violating the convexity condition (1.8) but, for simplicity, we do not
pursue this direction here. Finally, we refer to §6.1 for an heuristic
presentation of the counter-example.

### Paper outline

In §2 we overview some known preliminary results, in §3, §4, §5 and §6 we
establish Theorems 1.1, 1.2, 1.3 and 1.4, respectively. For the reader’s
convenience we conclude the introduction by collecting the main notation used
in paper.

### Notation

  * •

$\mathbbm{1}_{E}:$ the characteristic function of the measurable set $E$, i.e.
$\mathbbm{1}_{E}(x)=1$ if $x\in E$, $\mathbbm{1}_{E}(x)=0$ if $x\notin E$;

  * •

$u\ast\eta$: the convolution of $u$ and $\eta$, computed with respect to the
$x$ variable only, in other words
$[u\ast\eta](t,x)~{}=~{}\int_{\mathbb{R}}\eta(x-y)u(t,y)dy$;

  * •

$\mathrm{TotVar}\,u_{0}$: the total variation of the function $u_{0}$, see [1,
§3.2];

  * •

$\mathbb{R}_{-}$, $\mathbb{R}_{+}$: the negative and the positive real axes,
respectively, i.e. $\mathbb{R}_{-}=]-\infty,0]$, $\mathbb{R}_{+}=[0,+\infty[$;

  * •

$L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$: the space of functions
$u$ such that $u\in L^{1}(\Omega)$ for every open bounded set
$\Omega\subseteq\mathbb{R}_{+}\times\mathbb{R}$;

  * •

$\mathrm{Lip}$: the space of Lipschitz continuous functions;

  * •

$C^{1,1}$: the space of continuously differentiable functions with Lipschitz
continuous derivatives;

  * •

$o(t)$ as $t\to 0$: the Bachmann-Landau notation for any function $g$ such
that $\lim_{t\to 0}g(t)/t=0$.

##  2\. Preliminary results

Existence and uniqueness results for nonlocal conservation laws with
anisotropic convolution kernels have been obtained in several works under
different assumptions, see for instance [3, 4, 6, 8]. We now very slightly
extend [8, Corollary 2.1]

######  Proposition 2.1.

Assume that $u_{0}$, $V$ and $\eta$ satisfy (1.3), (1.4) and (1.5),
respectively; then

  * i)

there is a unique distributional solution $u_{\varepsilon}\in
L^{\infty}(\mathbb{R}_{+}\times\mathbb{R})$ of the Cauchy problem (1.1). Also,

(2.1) $0\leq u_{\varepsilon}\leq 1.$

  * ii)

$u_{\varepsilon}\in C^{0}(\mathbb{R}_{+},L^{1}_{\mathrm{loc}}(\mathbb{R}))$,
namely the function $u_{\varepsilon}$ admits a representative such that the
map $t\mapsto u_{\varepsilon}(t,\cdot)$ is continuous from $\mathbb{R}_{+}$ to
$L^{1}_{\mathrm{loc}}(\mathbb{R})$ endowed with the strong topology.

  * iii)

If $u_{0}\in\mathrm{Lip}(\mathbb{R})$, then
$u_{\varepsilon}\in\mathrm{Lip}(\mathbb{R}_{+}\times\mathbb{R})$.

###### Proof.

Concerning items i) and ii), the only difference with respect to [8, Corollary
2.1] is that in that paper uniqueness is established in the more restrictive
class $u_{\varepsilon}\in L^{\infty}(\mathbb{R}_{+}\times\mathbb{R})\cap
C^{0}(\mathbb{R}_{+},L^{1}_{\mathrm{loc}}(\mathbb{R}))$. However, one can use
the same argument as in the proof of [15, Corollary 3.14] and [12, Proposition
2.3] and show that, owing to renormalization, any bounded distributional
solution of (1.1) satisfies ii). Item iii) is a straightforward consequence of
the analysis in [8] and of classical results concerning the method of
characteristics. ∎

As an easy consequence of Proposition 2.1 we get

######  Corollary 2.2.

Assume that $u_{0}$, $V$ and $\eta$ satisfy (1.3), (1.4) and (1.5),
respectively, and that $w_{\varepsilon}$ is given by (1.6), where
$u_{\varepsilon}$ is the solution of the Cauchy problem (1.1). Then

  * i)

$w_{\varepsilon}\in\mathrm{Lip}(\mathbb{R}_{+}\times\mathbb{R})$ and

(2.2) $0\leq w_{\varepsilon}\leq 1;$

  * ii)

$w_{\varepsilon}\in C^{0}(\mathbb{R}_{+},L^{1}_{\mathrm{loc}}(\mathbb{R}))$;

  * iii)

if $u_{0}\in\mathrm{Lip}(\mathbb{R})$, then $w_{\varepsilon}\in
C^{1,1}(\mathbb{R}_{+}\times\mathbb{R})$.

###### Proof.

By combining (1.5), (1.6) and (2.1) we get the maximum principle (2.2). By
differentiating (1.6) we get

(2.3) $\frac{\partial w_{\varepsilon}}{\partial
x}=\frac{1}{\varepsilon}\left[\frac{1}{\varepsilon}\int_{x}^{+\infty}\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)u_{\varepsilon}(t,y)dy-\eta(0^{-})u_{\varepsilon}(t,x)\right]$

and

(2.4) $\begin{split}\frac{\partial w_{\varepsilon}}{\partial
t}&=-\frac{1}{\varepsilon}\int_{x}^{+\infty}\eta\left(\frac{x-y}{\varepsilon}\right)\partial_{y}[u_{\varepsilon}V(w_{\varepsilon})](t,y)dy\\\
&=\frac{1}{\varepsilon}\left[\eta(0^{-})u_{\varepsilon}V(w_{\varepsilon})(t,x)-\frac{1}{\varepsilon}\int_{x}^{+\infty}\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)u_{\varepsilon}V(w_{\varepsilon})(t,y)dy\right].\end{split}$

In the previous expression, $\eta(0^{-})$ denotes the left limit of $\eta$ at
$0$, which is well-defined since the restriction of $\eta$ to $\mathbb{R}_{-}$
is a monotone function. By combining (2.3) and (2.4) and using (2.1) and (2.2)
we get

(2.5)
$|\partial_{x}w_{\varepsilon}(t,x)|\leq\frac{1}{\varepsilon}\eta(0^{-}),\quad|\partial_{t}w_{\varepsilon}(t,x)|\leq\frac{1}{\varepsilon}\eta(0^{-})\max_{w\in[0,1]}|V(w)|\quad\text{for
every $(t,x)\in\mathbb{R}_{+}\times\mathbb{R}$},$

that is $w_{\varepsilon}\in\mathrm{Lip}(\mathbb{R}_{+}\times\mathbb{R})$. Item
ii) in the statement of Corollary 2.2 follows from item ii) of Proposition
2.1. To establish item iii) we recall item iii) in the statement of
Proposition 2.1 and use again (2.3) and (2.4). ∎

We now recall a couple of well-known facts concering functions of bounded
total variation that we need in the following. Assume $v\in
L^{\infty}(\mathbb{R})$ satisfies $\mathrm{TotVar}\ v<+\infty$; then

(2.6) $\int_{\mathbb{R}}|v(x-\xi)-v(x)|dx\leq|\xi|\mathrm{TotVar}\
v\quad\text{for every $\xi\in\mathbb{R}$}.$

Also, let $\rho\in L^{1}(\mathbb{R})$ satisfy

(2.7)
$C_{\rho}:=\int_{\mathbb{R}}|\rho(\xi)\xi|d\xi<+\infty,\qquad\int_{\mathbb{R}}\rho(\xi)d\xi=1$

and set $\rho_{h}(\xi):=h^{-1}\rho(h^{-1}\xi)$; then

(2.8) $\|v\ast\rho_{h}-v\|_{L^{1}(\mathbb{R})}\leq h\ C_{\rho}\mathrm{TotVar}\
v\quad\text{for every $h>0$}.$

To conclude, we point out that, if $\eta$ satisfies (1.5) then

(2.9)
$\begin{split}\int_{\mathbb{R}}|\eta^{\prime}(\xi)\xi|d\xi&=\lim_{R\to+\infty}\int_{-R}^{0}|\eta^{\prime}(\xi)\xi|d\xi\stackrel{{\scriptstyle\eta^{\prime}\geq
0}}{{=}}-\lim_{R\to+\infty}\int_{-R}^{0}\eta^{\prime}(\xi)\xi d\xi\\\
&\stackrel{{\scriptstyle\text{integration by
parts}}}{{=}}-\lim_{R\to+\infty}R\eta(-R)+\lim_{R\to+\infty}\int_{-R}^{0}\eta(\xi)d\xi\leq\lim_{R\to+\infty}\int_{-R}^{0}\eta(\xi)d\xi\stackrel{{\scriptstyle\eqref{e:eta}}}{{\leq}}1.\end{split}$

##  3\. Proof of Theorem 1.1

By combining (2.3) and (2.4) we get that the material derivative of
$w_{\varepsilon}$ is given by

(3.1)
$\partial_{t}w_{\varepsilon}+V(w_{\varepsilon})\partial_{x}w_{\varepsilon}=\frac{1}{\varepsilon^{2}}\int_{x}^{+\infty}\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)[V(w_{\varepsilon}(t,x))-V(w_{\varepsilon}(t,y))]u_{\varepsilon}(t,y)dy.$

To avoid some technicalities, we first provide the proof of Theorem 1.1 under
some further assumptions on the initial datum $u_{0}$ and the convolution
kernel $\eta$. We then remove these assumptions by relying on a fairly
standard approximation argument.
Step 1: we impose the further assumptions $u_{0}\in\mathrm{Lip}(\mathbb{R})$,
$\eta\in C^{2}(]-\infty,0[)$, $\eta^{\prime\prime}\in L^{1}(]-\infty,0[)$.
Owing to iii) in the statement of Corollary 2.2, this implies that
$w_{\varepsilon}\in C^{1,1}(\mathbb{R}_{+}\times\mathbb{R})$. We set
$z_{\varepsilon}:=\partial_{x}w_{\varepsilon}$ and point out that
$z_{\varepsilon}\in\mathrm{Lip}(\mathbb{R}_{+}\times\mathbb{R})$. By
differentiating (3.1) we get

(3.2)
$\begin{split}\partial_{t}z_{\varepsilon}+\partial_{x}(V(w_{\varepsilon})z_{\varepsilon})=\frac{1}{\varepsilon^{2}}\int_{x}^{+\infty}\biggl{[}&\frac{1}{\varepsilon}\eta^{\prime\prime}\left(\frac{x-y}{\varepsilon}\right)\big{(}V(w_{\varepsilon}(t,x))-V(w_{\varepsilon}(t,y))\big{)}\\\
&+\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)V^{\prime}(w_{\varepsilon}(t,x))z_{\varepsilon}(t,x)\biggr{]}u_{\varepsilon}(t,y)dy.\end{split}$

Note that the right-hand side of the above expression is finite since
$w_{\varepsilon}$ and $u_{\varepsilon}$ are both bounded functions, and both
$\eta^{\prime\prime}$ and $\eta^{\prime}$ are summable. Assume for a moment we
have shown that

(3.3)
$\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)=\int_{\mathbb{R}}|z_{\varepsilon}(t,x)|<+\infty\quad\text{for
every $t>0$ and $\varepsilon>0$};$

then by multiplying (3.2) by $s(t,x):=\mathrm{sign}(z_{\varepsilon}(t,x))$ and
$x$-integrating over $\mathbb{R}$ we arrive at

$\begin{split}\frac{d}{dt}\int_{\mathbb{R}}\\!\\!|z_{\varepsilon}(t,x)|dx\\!+\\!\int_{\mathbb{R}}\\!\partial_{x}(V(w_{\varepsilon})|z_{\varepsilon}|)dx&=\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}}s(t,x)\int_{x}^{+\infty}\biggl{[}\frac{1}{\varepsilon}\eta^{\prime\prime}\left(\frac{x-y}{\varepsilon}\right)\big{(}V(w_{\varepsilon}(t,x))\\!-\\!V(w_{\varepsilon}(t,y))\big{)}\\\
&\qquad+\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)V^{\prime}(w_{\varepsilon}(t,x))z_{\varepsilon}(t,x)\biggr{]}u_{\varepsilon}(t,y)dydx.\end{split}$

By relying on Fubini’s theorem we rewrite the above equality as

(3.4)
$\begin{split}\frac{d}{dt}\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)=\frac{d}{dt}\int_{\mathbb{R}}\\!\\!|z_{\varepsilon}(t,x)|dx&=\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}}u_{\varepsilon}(t,y)\sigma(t,y)dy,\end{split}$

where

$\sigma(t,y):=\int_{-\infty}^{y}\\!\\!\frac{1}{\varepsilon}\eta^{\prime\prime}\left(\frac{x-y}{\varepsilon}\right)\big{(}V(w_{\varepsilon}(t,x))-V(w_{\varepsilon}(t,y))\big{)}s(t,x)dx+\int_{-\infty}^{y}\\!\\!\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)V^{\prime}(w_{\varepsilon}(t,x))|z_{\varepsilon}(t,x)|dx.$

By using the integration by parts formula we arrive at

(3.5)
$\sigma(t,y):=\int_{-\infty}^{y}\\!\\!\frac{1}{\varepsilon}\eta^{\prime\prime}\left(\frac{x-y}{\varepsilon}\right)\Big{[}\big{(}V(w_{\varepsilon}(t,x))-V(w_{\varepsilon}(t,y))\big{)}s(t,x)-\gamma(t,x,y)\Big{]}dx,$

where

$\gamma(t,x,y):=-\int_{x}^{y}V^{\prime}(w_{\varepsilon}(t,\xi))|z_{\varepsilon}(t,\xi)|d\xi.$

Note that $\gamma\geq 0$ since $x\leq y$ and $V^{\prime}\leq 0$. Also,

$|V(w_{\varepsilon}(t,x))-V(w_{\varepsilon}(t,y))|\leq\left|\int_{y}^{x}V^{\prime}(w_{\varepsilon}(t,\xi))z_{\varepsilon}(t,\xi)d\xi\right|\leq\gamma(t,x,y)$

and by recalling (3.5) and the inequality $\eta^{\prime\prime}\geq 0$ we
conclude that $\sigma(t,y)\leq 0$. By plugging this inequality into (3.4) and
recalling that $u_{\varepsilon}\geq 0$ owing to (2.1) we conclude that the
right-hand side of (3.4) is nonpositive, which in turn yields (1.9).
Step 2: we establish (3.3) under the assumptions that
$u_{0}\in\mathrm{Lip}(\mathbb{R})$ and $\eta\in C^{2}(]-\infty,0[)$,
$\eta^{\prime\prime}\in L^{1}(]-\infty,0[)$. We recall that
$z_{\varepsilon}=\partial_{x}w_{\varepsilon}$ and it is bounded by the first
inequality in (2.5). Next, we fix $R>0$, multiply (3.2) by
$s(t,x):=\mathrm{sign}(z_{\varepsilon}(t,x))$ and $x$-integrate over the
interval $]-R,R[$. By performing a change of variables and applying Fubini’s
theorem we arrive at

(3.6)
$\begin{split}\frac{d}{dt}\int_{-R}^{R}|z_{\varepsilon}(t,x)|dx&+\int_{-R}^{R}\partial_{x}(V(w_{\varepsilon})|z_{\varepsilon}|)(t,x)dx\leq\underbrace{\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}_{-}}\eta^{\prime\prime}(\xi)\int_{-R}^{R}\big{|}V(w_{\varepsilon}(t,x))\\!-\\!V(w_{\varepsilon}(t,x-\varepsilon\xi))\big{|}dxd\xi}_{:=T_{1}}\\\
&+\underbrace{\frac{1}{\varepsilon}\int_{\mathbb{R}_{-}}\eta^{\prime}(\xi)\int_{-R}^{R}|V^{\prime}(w_{\varepsilon}(t,x))||z_{\varepsilon}(t,x)|u_{\varepsilon}(t,x-\varepsilon\xi)dxd\xi}_{:=T_{2}}.\end{split}$

Note that

(3.7)
$\begin{split}T_{1}&\leq\underbrace{\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}_{-}}\eta^{\prime\prime}(\xi)\int_{-R}^{R+\varepsilon\xi}\big{|}V(w_{\varepsilon}(t,x))\\!-\\!V(w_{\varepsilon}(t,x-\varepsilon
z))\big{|}dxd\xi}_{T_{11}}\\\
&+\underbrace{\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}_{-}}\eta^{\prime\prime}(\xi)\int_{R+\varepsilon\xi}^{R}\big{|}V(w_{\varepsilon}(t,x))\\!-\\!V(w_{\varepsilon}(t,x-\varepsilon
z))\big{|}dxd\xi.}_{T_{12}}\end{split}$

We have

(3.8)
$T_{12}\stackrel{{\scriptstyle\eqref{e:maxpw}}}{{\leq}}-\frac{2}{\varepsilon}\max_{w\in[0,1]}|V(w)|\int_{\mathbb{R}_{-}}\eta^{\prime\prime}(\xi)\xi
d\xi\stackrel{{\scriptstyle\text{integration by
parts},~{}\eqref{e:etader}}}{{=}}\frac{2}{\varepsilon}\max_{w\in[0,1]}|V(w)|\eta(0^{-}).$

To control the term $T_{11}$, we point out that, if $\xi\leq 0$ and
$x\in]-R,R+\varepsilon\xi[$, then both $x$ and $x-\varepsilon\xi$ belong to
the interval $]-R,R[$. This implies that

(3.9)
$T_{11}\leq\frac{1}{\varepsilon}\|V^{\prime}\|_{L^{\infty}}\left(\int_{\mathbb{R}_{-}}\eta^{\prime\prime}(\xi)|\xi|d\xi\right)\left(\int_{-R}^{R}|z_{\varepsilon}(t,x)|dx\right)=\frac{1}{\varepsilon}\|V^{\prime}\|_{L^{\infty}}\eta(0^{-})\int_{-R}^{R}|z_{\varepsilon}(t,x)|dx.$

We also have

(3.10)
$T_{2}\stackrel{{\scriptstyle\eqref{e:maxpu}}}{{\leq}}\frac{1}{\varepsilon}\eta(0^{-})\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\int_{-R}^{R}|z_{\varepsilon}(t,x)|dx.$

We plug (3.8) and (3.9) into (3.7), combine them with (3.6) and (3.10), recall
(2.5) and apply Gronwall’s Lemma. We obtain a bound on
$\int_{-R}^{R}|z_{\varepsilon}(t,x)|dx$ which does not depend on $R$ (albeit
it depends on $\varepsilon$). By sending $R\to+\infty$ we establish (3.3).
Step 3: we remove the assumptions $u_{0}\in\mathrm{Lip}(\mathbb{R})$ and
$\eta\in C^{2}(]-\infty,0[)$, $\eta^{\prime\prime}\in L^{1}(]-\infty,0[)$. We
fix $u_{0}$ and $\eta$ as in the statement of Theorem 1.1 and term
$u_{\varepsilon}$ the solution of the Cauchy problem (1.1) and
$w_{\varepsilon}$ the corresponding convolution terms. Next, we construct
suitable sequences $\\{u_{0n}\\}$ and $\\{\eta_{n}\\}$ satisfying the
assumptions of Theorem 1.1, the further conditions
$u_{0n}\in\mathrm{Lip}(\mathbb{R})$, $\eta_{n}\in C^{2}(]-\infty,0[)$,
$\eta^{\prime\prime}_{n}\in L^{1}(]-\infty,0[)$ and such that

(3.11)
$u_{0n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}u_{0}\;\text{weakly${}^{\ast}$
in
$L^{\infty}(\mathbb{R})$},\quad\mathrm{TotVar}\,u_{0n}\leq\mathrm{TotVar}\,u_{0},\quad\lim_{n\to+\infty}\|u_{n0}-u_{n}\|_{L^{1}(\mathbb{R})}=0$

and that

(3.12) $\eta_{n}\to\eta\;\text{strongly in
$L^{1}(\mathbb{R})$},\quad\\{\eta_{n}(0)\\}\;\text{is uniformly bounded}.$

We can for instance define $u_{0n}$ by convolving $u_{0}$ with a suitable
family of convolution kernels $\\{\rho_{n}\\}$ and then use (2.8) to obtain
the last condition in (3.11).

We term $u_{\varepsilon n}$ the sequence of solutions of the Cauchy problem
(1.1) and $w_{\varepsilon n}$ the corresponding sequence of convolution terms.
By recalling (2.1) and by combining (2.5) with (3.12) and the Ascoli-Arzelà
theorem we get that

$u_{\varepsilon
n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}v_{\varepsilon}\quad\text{weakly${}^{\ast}$
in $L^{\infty}(\mathbb{R}_{-}\times\mathbb{R})$},\qquad w_{\varepsilon n}\to
p_{\varepsilon}\quad\text{in $C^{0}(K)$, for every
$K\subseteq\mathbb{R}_{-}\times\mathbb{R}$ compact}$

for suitable functions $v_{\varepsilon}\in
L^{\infty}(\mathbb{R}_{-}\times\mathbb{R})$ and
$p_{\varepsilon}\in\mathrm{Lip}(\mathbb{R}_{-}\times\mathbb{R})$. Owing to
(3.12) we can pass to the limit in the equality $w_{\varepsilon
n}(t,x)=\int_{\mathbb{R}_{-}}\eta_{n}(\xi)u_{\varepsilon
n}(t,x-\varepsilon\xi)d\xi$ and arrive at
$p_{\varepsilon}(t,x)=\int_{\mathbb{R}_{-}}\eta(\xi)v_{\varepsilon}(t,x-\varepsilon\xi)d\xi$.
Also, by passing to the limit in the distributional formulation we infer that
$v_{\varepsilon}$ is a bounded distributional solution of the Cauchy problem
(1.1) and by uniqueness this implies $v_{\varepsilon}=u_{\varepsilon}$,
$p_{\varepsilon}=w_{\varepsilon}$. By applying Step 1 to the sequence
$w_{\varepsilon n}$, recalling the lower semicontinuity of the total variation
and then passing to the limit we arrive at

(3.13)
$\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)\leq\liminf_{n\to+\infty}\mathrm{TotVar}\,w_{\varepsilon
n}(0,\cdot).$

By combining (3.11) with (3.12) and (2.3) we get

$\lim_{n\to+\infty}\mathrm{TotVar}\,w_{\varepsilon
n}(0,\cdot)=\mathrm{TotVar}\,w_{\varepsilon}(0,\cdot)$

and by plugging the above equality into (3.13) we get (1.9).

##  4\. Proof of Theorem 1.2

We need the following lemma.

######  Lemma 4.1.

Let $\eta$ satisfy (1.5) and assume that $\\{v_{k}\\}$ is a pre-compact
sequence in $L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$ such that
$\|v_{k}\|_{L^{\infty}}\leq\Lambda$ for some $\Lambda>0$ and for every $k$.
Set

$F_{k\varepsilon}(t,x):=\int_{x}^{+\infty}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{\varepsilon}\right)|v_{k}(t,y)-v_{k}(t,x)|dy;$

then the family $\\{F_{k\varepsilon}\\}$ converges to $0$ as $\varepsilon\to
0^{+}$ in $L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$, uniformly in
$k$. In other words, for every $r,T,M>0$, there is $\tilde{\varepsilon}>0$,
depending on $r,T$ and $M$ only, such that if
$\varepsilon\leq\tilde{\varepsilon}$, then

$\int_{0}^{T}\int_{-M}^{M}|F_{k\varepsilon}(t,x)|dxdt\leq r\quad\text{for
every $k$}.$

###### Proof.

We proceed according to the following steps.
Step 1: we rely on the Fréchet-Kolmogorov theorem and infer that the sequence
$\\{v_{k}\\}$ is equicontinuous in $L^{1}_{\mathrm{loc}}$. In particular, for
every $T>0$, $L>0$ and $\nu>0$ there is $\tilde{\tau}(T,L,\nu)>0$ such that,
if $|\tau|<\tilde{\tau}(T,L,\nu)$, then

(4.1)
$\int_{0}^{T}\int_{-L+\tau}^{L-\tau}|v_{k}(t,x+\tau)-v_{k}(t,x)|dxdt\leq\nu\quad\text{for
every $k$.}$

Step 2: we now fix $\delta>0$, choose $R>0$ in such a way that
$\int_{-\infty}^{-R}\eta(z)dz<\delta$ and point out that

(4.2)
$\begin{split}\int_{0}^{T}\int_{-M}^{M}&|F_{k\varepsilon}(t,x)|dxdt\stackrel{{\scriptstyle\xi=\frac{x-y}{\varepsilon}}}{{=}}\int_{0}^{T}\int_{-M}^{M}\int_{\mathbb{R}_{-}}\eta(\xi)|v_{k}(t,x-\varepsilon\xi)-v_{k}(t,x)|d\xi
dxdt\\\
&\stackrel{{\scriptstyle\text{Fubini}}}{{=}}\int_{-R}^{0}\\!\\!\eta(\xi)\int_{0}^{T}\\!\\!\int_{-M}^{M}\\!\\!|v_{k}(t,x-\varepsilon\xi)-v_{k}(t,x)|dxdtd\xi\\\
&\qquad+\int_{-\infty}^{-R}\\!\\!\eta(\xi)\int_{0}^{T}\\!\\!\int_{-M}^{M}\\!\\!|v_{k}(t,x-\varepsilon\xi)-v_{k}(t,x)|dxdtd\xi\stackrel{{\scriptstyle\eqref{e:equic2}}}{{\leq}}\nu+4\delta
TM\underbrace{\|v_{k}\|_{L^{\infty}}}_{\leq\Lambda}\end{split}$

provided $L=M+1$ and $\varepsilon R<\min\\{\tilde{\tau}(T,L,\nu),1\\}$. By the
arbitrariness of $\nu$ and $\delta$ this concludes the proof of Lemma 4.1. ∎

We are now ready to provide the actual proof of Theorem 1.2.

###### Proof of Theorem 1.2.

We fix an entropy-entropy flux pair $(\alpha,\beta)$ for (1.2), namely
$\alpha,\beta:\mathbb{R}\to\mathbb{R}$ are two $C^{2}$ functions satisfying
$\alpha^{\prime\prime}\geq 0$,
$\beta^{\prime}(u)=\alpha^{\prime}(u)[V(u)+uV^{\prime}(u)]$. Also, we fix
$\phi\in C^{\infty}_{c}(\mathbb{R}_{+}\times\mathbb{R})$ satisfying $\phi\geq
0$. By the arbitrariness of $(\alpha,\beta)$ and $\phi$, the proof of Theorem
1.2 boils down to the proof of the inequality

(4.3)
$D_{\alpha}(\phi):=\iint_{\mathbb{R}_{+}\times\mathbb{R}}\big{[}\alpha(u)\partial_{t}\phi+\beta(u)\partial_{x}\phi\big{]}dxdt+\int_{\mathbb{R}}\alpha(u_{0}(x))\phi(0,x)dx\geq
0.$

Note that in particular (4.3) implies that $u$ is a distributional solution of
(1.2). Indeed, by choosing $(\alpha(u),\beta(u))=(u,uV(u))$ and
$(\alpha(u),\beta(u))=(-u,-uV(u))$ we get the inequalities
$\partial_{t}u+\partial_{x}[uV]\leq 0$ and $\partial_{t}u+\partial_{x}[uV]\geq
0$, respectively, which are satisfied in the sense of distributions, and from
this we infer (1.2).

To establish (4.3) we set
$\eta_{\varepsilon}(\xi):=\frac{1}{\varepsilon}\eta\left(\frac{\xi}{\varepsilon}\right)$,
choose a sequence $\\{\varepsilon_{k}\\}$ and set

(4.4)
$D^{\varepsilon_{k}}_{\alpha}(\phi):=\iint_{\mathbb{R}_{+}\times\mathbb{R}}\big{[}\alpha(w_{\varepsilon_{k}})\partial_{t}\phi+\beta(w_{\varepsilon_{k}})\partial_{x}\phi\big{]}dxdt+\int_{\mathbb{R}}\alpha(w_{\varepsilon_{k}}(0,x))\phi(0,x)dx.$

We recall that by assumption $w_{\varepsilon_{k}}\to w$ in
$L^{1}(]0,T[\times\mathbb{R})$ and that
$w_{\varepsilon_{k}}(0,x)=u_{0}\ast\eta_{\varepsilon_{k}}\to u_{0}$ in
$L^{1}(\mathbb{R})$. This implies that
$\lim_{k\to+\infty}D^{\varepsilon_{k}}_{\alpha}(\phi)=D_{\alpha}(\phi)$ and
hence that establishing (4.3) amounts to show that

(4.5) $\lim_{k\to+\infty}D^{\varepsilon_{k}}_{\alpha}(\phi)\geq 0.$

To establish (4.5) we proceed according to the following steps.
Step 1: we write $D^{\varepsilon_{k}}_{\alpha}(\phi)$ in a more convenient
form. To this end we consider the equation at the first line of (1.1),
convolve it with $\eta_{\varepsilon_{k}}$ and then multiply the result times
$\alpha^{\prime}(w_{\varepsilon_{k}})$. We arrive at

$\partial_{t}\alpha(w_{\varepsilon_{k}})+\alpha^{\prime}(w_{\varepsilon_{k}})\partial_{x}\big{[}[V(w_{\varepsilon_{k}})u_{\varepsilon_{k}}]\ast\eta_{\varepsilon_{k}}\big{]}=0.$

Next, we multiply the above equation times $\phi$ and integrate over
$\mathbb{R}_{+}\times\mathbb{R}$. Owing to the integration by parts formula
this yields

(4.6)
$\iint_{\mathbb{R}_{+}\times\mathbb{R}}\alpha(w_{\varepsilon_{k}})\partial_{t}\phi+[(V(w_{\varepsilon_{k}})u_{\varepsilon_{k}})\ast\eta_{\varepsilon_{k}}]\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})\phi]dxdt+\int_{\mathbb{R}}\alpha(w_{\varepsilon_{k}}(0,x))\phi(0,x)dx=0.$

We now point that, owing to the equality
$\beta^{\prime}(u)=\alpha^{\prime}(u)[V(u)+uV^{\prime}(u)]$, we have

(4.7)
$\begin{split}\iint_{\mathbb{R}_{+}\times\mathbb{R}}\beta(w_{\varepsilon_{k}})\partial_{x}\phi\
dxdt&=-\iint_{\mathbb{R}_{+}\times\mathbb{R}}\beta^{\prime}(w_{\varepsilon_{k}})\partial_{x}w_{\varepsilon_{k}}\phi\
dxdt\\\
&=-\iint_{\mathbb{R}_{+}\times\mathbb{R}}\alpha^{\prime}(w_{\varepsilon_{k}})[V(w_{\varepsilon_{k}})+w_{\varepsilon_{k}}V^{\prime}(w_{\varepsilon_{k}})]\partial_{x}w_{\varepsilon_{k}}\phi\
dxdt\\\
&=-\iint_{\mathbb{R}_{+}\times\mathbb{R}}\alpha^{\prime}(w_{\varepsilon_{k}})\partial_{x}[V(w_{\varepsilon_{k}})w_{\varepsilon_{k}}]\phi\
dxdt=\iint_{\mathbb{R}_{+}\times\mathbb{R}}V(w_{\varepsilon_{k}})w_{\varepsilon_{k}}\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})\phi]dxdt.\end{split}$

By comparing (4.4) with (4.6) and using (4.7) we then get

(4.8)
$\begin{split}D^{\varepsilon_{k}}_{\alpha}(\phi)&\stackrel{{\scriptstyle\eqref{e:alphaweek},\eqref{e:pino}}}{{=}}\iint_{\mathbb{R}_{+}\times\mathbb{R}}\underbrace{\big{[}V(w_{\varepsilon_{k}})w_{\varepsilon_{k}}-[V(w_{\varepsilon_{k}})u_{\varepsilon_{k}}]\ast\eta_{\varepsilon_{k}}\big{]}}_{S_{\varepsilon}(t,x)}\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})\phi]dxdt\\\
&=\iint_{\mathbb{R}_{+}\times\mathbb{R}}S_{\varepsilon}(t,x)\alpha^{\prime}(w_{\varepsilon_{k}})\partial_{x}\phi
dxdt+\underbrace{\iint_{\mathbb{R}_{+}\times\mathbb{R}}S_{\varepsilon}(t,x)\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})]\phi
dxdt}_{T_{1}^{\varepsilon_{k}}}.\end{split}$

Note that

(4.9)
$\begin{split}T_{1}^{\varepsilon_{k}}&=\iint_{\mathbb{R}_{+}\times\mathbb{R}}\int_{x}^{+\infty}\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})](t,x)\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)u_{\varepsilon_{k}}(t,y)\Big{[}V(w_{\varepsilon_{k}}(t,x))-V(w_{\varepsilon_{k}}(t,y))\Big{]}dydxdt\\\
=&\iint_{\mathbb{R}_{+}\times\mathbb{R}}u_{\varepsilon_{k}}(t,y)\omega_{\varepsilon_{k}}(t,y)dydt,\end{split}$

provided

$\begin{split}\omega_{\varepsilon_{k}}(t,y):=&~{}\int_{-\infty}^{y}\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})](t,x)\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)\Big{[}V(w_{\varepsilon_{k}}(t,x))-V(w_{\varepsilon_{k}}(t,y))\Big{]}dx\\\
=&~{}\int_{-\infty}^{y}\underbrace{\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})](t,x)V(w_{\varepsilon_{k}}(t,x))}_{\partial_{x}I(w_{\varepsilon_{k}})}\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)dx\\\
&~{}-V(w_{\varepsilon_{k}}(t,y))\int_{-\infty}^{y}\partial_{x}[\alpha^{\prime}(w_{\varepsilon_{k}})](t,x)\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)dx.\end{split}$

In the previous expression, we have chosen the $C^{1}$ function $I$ in such a
way that $I^{\prime}(u)=\alpha^{\prime\prime}(u)V(u)$. By applying the
integration by parts formula we then get

(4.10)
$\begin{split}\omega_{\varepsilon_{k}}(t,y)=&I(w_{\varepsilon_{k}}(t,y))\phi(t,y)\frac{\eta(0)}{\varepsilon_{k}}-\int_{-\infty}^{y}I(w_{\varepsilon_{k}}(t,x))\partial_{x}\left[\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)\right]dx\\\
&-V(w_{\varepsilon_{k}}(t,y))\left[\alpha^{\prime}(w_{\varepsilon_{k}}(t,y))\phi(t,y)\frac{\eta(0)}{\varepsilon_{k}}-\int_{-\infty}^{y}\alpha^{\prime}(w_{\varepsilon_{k}}(t,x))\partial_{x}\left[\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)\right]dx\right]\\\
=&~{}\int_{-\infty}^{y}\big{[}I(w_{\varepsilon_{k}}(t,y))-I(w_{\varepsilon_{k}}(t,x))\big{]}\partial_{x}\left[\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)\right]dx\\\
&~{}-V(w_{\varepsilon_{k}}(t,y))\int_{-\infty}^{y}\big{[}\alpha^{\prime}(w_{\varepsilon_{k}}(t,y))-\alpha^{\prime}(w_{\varepsilon_{k}}(t,x))\big{]}\partial_{x}\left[\phi(t,x)\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)\right]dx\\\
&=G^{1}_{\varepsilon_{k}}(t,y)+G^{2}_{\varepsilon_{k}}(t,y)+P_{\varepsilon_{k}}(t,y),\phantom{\int}\end{split}$

provided

(4.11) $\displaystyle
G^{1}_{\varepsilon_{k}}(t,y):=\int_{-\infty}^{y}\big{[}I(w_{\varepsilon_{k}}(t,y))-I(w_{\varepsilon_{k}}(t,x))\big{]}\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)\partial_{x}\phi(t,x)dx$
$\displaystyle
G^{2}_{\varepsilon_{k}}(t,y)=-V(w_{\varepsilon_{k}}(t,y))\int_{-\infty}^{y}\big{[}\alpha^{\prime}(w_{\varepsilon_{k}}(t,y))-\alpha^{\prime}(w_{\varepsilon_{k}}(t,x))\big{]}\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)\partial_{x}\phi(t,x)dx$

and

(4.12)
$\begin{split}P_{\varepsilon_{k}}(t,y):&=\int_{-\infty}^{y}H(w_{\varepsilon}(t,x),w_{\varepsilon}(t,y))\phi(t,x)\partial_{x}\left[\frac{1}{\varepsilon}\eta\left(\frac{x-y}{\varepsilon}\right)\right]dx\\\
&=\frac{1}{\varepsilon^{2}}\int_{-\infty}^{y}H(w_{\varepsilon}(t,x),w_{\varepsilon}(t,y))\phi(t,x)\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)dx,\end{split}$

where

(4.13) $H(a,b):=I(b)-I(a)-V(b)(\alpha^{\prime}(b)-\alpha^{\prime}(a)).$

By plugging (4.11) and (4.12) into (4.10) and then recalling (4.8) and (4.9)
we then arrive at

(4.14)
$D^{\varepsilon_{k}}_{\alpha}(\phi)=\iint_{\mathbb{R}_{+}\times\mathbb{R}}S_{\varepsilon}(t,x)\alpha^{\prime}(w_{\varepsilon_{k}})\partial_{x}\phi
dxdt+\iint_{\mathbb{R}_{+}\times\mathbb{R}}u_{\varepsilon_{k}}(t,y)[G^{1}_{\varepsilon_{k}}(t,y)+G^{2}_{\varepsilon_{k}}(t,y)+P_{\varepsilon_{k}}(t,y)]dydt$

Step 2: we establish (4.5).
Step 2A: we show that

(4.15)
$D^{\varepsilon_{k}}_{\alpha}(\phi)\geq\underbrace{\iint_{\mathbb{R}_{+}\times\mathbb{R}}S_{\varepsilon}(t,x)\alpha^{\prime}(w_{\varepsilon_{k}})\partial_{x}\phi
dxdt}_{T^{\varepsilon_{k}}_{2}}+\underbrace{\iint_{\mathbb{R}_{+}\times\mathbb{R}}u_{\varepsilon_{k}}(t,y)G^{1}_{\varepsilon_{k}}(t,y)dydt}_{T^{\varepsilon_{k}}_{3}}+\underbrace{\iint_{\mathbb{R}_{+}\times\mathbb{R}}u_{\varepsilon_{k}}(t,y)G^{2}_{\varepsilon_{k}}(t,y)]dydt}_{T^{\varepsilon_{k}}_{4}}.$

Owing to (4.14) and since $u_{\varepsilon_{k}}\geq 0$ it suffices to show that
$P_{\varepsilon_{k}}\geq 0$. To this end we recall (4.12) and (4.13) and that
$I^{\prime}(u)=\alpha^{\prime\prime}(u)V(u)$. We point out that

$\frac{\partial H}{\partial
a}(u,b)=-I^{\prime}(u)+V(b)\alpha^{\prime\prime}(u)=\alpha^{\prime\prime}(u)[V(b)-V(u)]$

and by recalling that $\alpha^{\prime\prime}\geq 0$ and that $V^{\prime}\leq
0$ we conclude that the last expression is non-positive if $u\leq b$ and non-
negative if $u\geq b$. This implies that $u=b$ is a minimum for $H(u,b)$ and
since $H(b,b)=0$ we obtain the inequality $H(a,b)\geq 0$ for every
$(a,b)\in\mathbb{R}^{2}.$ By plugging this information into (4.12) and
recalling that $\phi\geq 0$ and that $\eta^{\prime}\geq 0$ by (1.5), we
conclude that $P_{\varepsilon_{k}}\geq 0$.
Step 2B: to establish (4.5) it suffices to show that the right-hand side of
(4.15) vanishes in the $k\to+\infty$ limit. To this end we recall the explicit
expression (4.8) of $S_{\varepsilon_{k}}$ and point out that

(4.16)
$\begin{split}S_{\varepsilon_{k}}(t,x)=&~{}\int_{x}^{+\infty}\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{\varepsilon_{k}}\right)\Big{[}w_{\varepsilon_{k}}(t,x)V(w_{\varepsilon_{k}}(t,x))-u_{\varepsilon_{k}}(t,y)V(w_{\varepsilon_{k}}(t,y))\Big{]}dy\\\
=&~{}\int_{x}^{+\infty}\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{\varepsilon_{k}}\right)u_{\varepsilon_{k}}(t,y)\Big{[}V(w_{\varepsilon_{k}}(t,x))-V(w_{\varepsilon_{k}}(t,y))\Big{]}dy.\end{split}$

Owing to (2.1), $\|u_{\varepsilon_{k}}(t,y)\|_{L^{\infty}}\leq 1$; by applying
Lemma 4.1 with $v_{k}:=V(w_{\varepsilon_{k}})$ we get that
$S_{\varepsilon_{k}}$ converges to $0$ in
$L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$. Since $\phi$ is
compactly supported, this implies that
$\lim_{k\to+\infty}T^{\varepsilon_{k}}_{2}=0$. We now show that
$\lim_{k\to+\infty}T^{\varepsilon_{k}}_{3}=0$. To this end, we point out that

$\begin{split}|T^{\varepsilon_{k}}_{3}|&\stackrel{{\scriptstyle\eqref{e:G},\eqref{e:perdopo2}}}{{\leq}}\iint_{\mathbb{R}_{+}\times\mathbb{R}}u_{\varepsilon_{k}}(t,y)\int_{-\infty}^{y}\big{|}I(w_{\varepsilon_{k}}(t,y))-I(w_{\varepsilon_{k}}(t,x))\big{|}\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)|\partial_{x}\phi(t,x)|dxdydt\\\
&\stackrel{{\scriptstyle\text{Fubini}}}{{=}}\iint_{\mathbb{R}_{+}\times\mathbb{R}}|\partial_{x}\phi(t,x)|\int_{x}^{+\infty}\underbrace{u_{\varepsilon_{k}}(t,y)}_{\leq
1\;\text{by~{}\eqref{e:maxpu}}}\big{|}I(w_{\varepsilon_{k}}(t,y))-I(w_{\varepsilon_{k}}(t,x))\big{|}\frac{1}{\varepsilon_{k}}\eta\left(\frac{x-y}{{\varepsilon_{k}}}\right)dydxdt.\end{split}$

Since $\phi$ is compactly supported, by applying Lemma 4.1 with
$v_{k}:=I(w_{\varepsilon_{k}})$ we conclude that $T^{\varepsilon_{k}}_{3}$
vanishes in the $\varepsilon\to 0^{+}$ limit. By relying on an analogous
argument we show that $\lim_{k\to+\infty}T^{\varepsilon_{k}}_{4}=0$ and owing
to (4.15) this concludes the proof of (4.5). ∎

##  5\. Proof of Theorem 1.3

###  5.1. Convergence proof

Let $u$ be the entropy admissible solution of (1.2), we now establish (1.11).
Step 1: we show that the family $\\{w_{\varepsilon}\\}$ is pre-compact in the
strong topology of $L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$. To
this end, we point out that, for every $T>0$, we have

(5.1)
$\int_{0}^{T}\int_{\mathbb{R}}|\partial_{x}w_{\varepsilon}|(t,x)dxdt\stackrel{{\scriptstyle\eqref{e:tv}}}{{\leq}}T\mathrm{TotVar}w_{\varepsilon}(0,\cdot)\stackrel{{\scriptstyle\eqref{e:tv0}}}{{\leq}}T\,\mathrm{TotVar}\
u_{0}$

and

$\begin{split}\int_{0}^{T}\int_{\mathbb{R}}&\left|\frac{1}{\varepsilon^{2}}\int_{x}^{+\infty}\eta^{\prime}\left(\frac{x-y}{\varepsilon}\right)[V(w_{\varepsilon}(t,x))-V(w_{\varepsilon}(t,y))]u_{\varepsilon}(t,y)dy\right|dxdt\\\
&\stackrel{{\scriptstyle\xi=\frac{x-y}{\varepsilon}}}{{=}}\frac{1}{\varepsilon}\int_{0}^{T}\int_{\mathbb{R}}\left|\int_{\mathbb{R}_{-}}\eta^{\prime}(\xi)[V(w_{\varepsilon}(t,x-\varepsilon\xi))-V(w_{\varepsilon}(t,y))]u_{\varepsilon}(t,x-\varepsilon\xi)d\xi\right|dxdt\\\
&\leq\frac{1}{\varepsilon}\int_{0}^{T}\int_{\mathbb{R}^{-}}|\eta^{\prime}(\xi)|\int_{\mathbb{R}}|V(w_{\varepsilon}(t,x-\varepsilon\xi))-V(w_{\varepsilon}(t,y))||u_{\varepsilon}(t,x-\varepsilon\xi)|dxd\xi
dt\\\
&\stackrel{{\scriptstyle\eqref{e:maxpu},\eqref{e:tvchar}}}{{\leq}}T\sup_{t\in]0,T[}\mathrm{TotVar}[V\circ
w_{\varepsilon}(t,\cdot)]\int_{\mathbb{R}^{-}}|\eta^{\prime}(\xi)\xi|d\xi\\\
&\stackrel{{\scriptstyle\eqref{e:maxpw}}}{{\leq}}T\,\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\sup_{t\in]0,T[}\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)\int_{\mathbb{R}^{-}}|\eta^{\prime}(\xi)\xi|d\xi\\\
&\stackrel{{\scriptstyle\eqref{e:tv},\eqref{e:tv0}}}{{\leq}}T\,\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\mathrm{TotVar}\,u_{0}\int_{\mathbb{R}^{-}}|\eta^{\prime}(\xi)\xi|d\xi\stackrel{{\scriptstyle\eqref{e:etader}}}{{\leq}}T\,\sup_{w\in]0,1[}|V^{\prime}(w)|\mathrm{TotVar}\,u_{0}.\end{split}$

Owing to (3.1) and recalling (5.1), this implies that
$\int_{0}^{T}\int_{\mathbb{R}}|\partial_{t}w_{\varepsilon}|dxdt$ is also
bounded uniformly in $\varepsilon$. We recall (2.2) and by applying the Helly-
Kolmogorov-Fréchet compactness theorem we eventually get the desired pre-
compactness result.
Step 2: fix a sequence $\varepsilon_{k}\to 0^{+}$, then owing to Step 1 and up
to subsequences $w_{\varepsilon_{k}}\to u$ in
$L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$ for some function $u\in
L^{\infty}(\mathbb{R}_{+}\times\mathbb{R})$. Owing to Theorem 1.2, $u$ is the
entropy admissible solution of (1.2) and by the uniqueness of such solution
this yields the first convergence result in (1.11).
Step 3: let $u$ be as in Step 2, we now show that
$u_{\varepsilon}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}u$ weakly∗ in
$L^{\infty}(\mathbb{R}^{+}\times\mathbb{R})$. Owing to (2.1), the family
$\\{u_{\varepsilon}\\}$ is pre-compact in
$L^{\infty}(\mathbb{R}^{+}\times\mathbb{R})$ endowed with the weak∗ topology.
To conclude, it suffices to show that any accumulation point $v$ satisfies
$v=u$. To this end, we recall that, for any $\varphi\in
C^{\infty}_{c}(\mathbb{R}_{+}\times\mathbb{R})$ we have the identity

$\iint_{\mathbb{R}_{+}\times\mathbb{R}}\varphi
w_{\varepsilon}dxdt=\iint_{\mathbb{R}_{+}\times\mathbb{R}}\varphi[u_{\varepsilon}\ast\eta_{\varepsilon}]dxdt=\iint_{\mathbb{R}_{+}\times\mathbb{R}}[\varphi\ast\check{\eta}_{\varepsilon}]u_{\varepsilon}dxdt\quad\text{provided
$\check{\eta}_{\varepsilon}(x):=\eta_{\varepsilon}(-x)$}.$

By passing to the limit in the above inequality and using Step 2 and the
arbitrariness of $\varphi$ we then arrive at $v=u$.

###  5.2. Proof of the convergence rate (1.12)

We first introduce the so-called Kružkov’s entropy-entropy flux pairs

(5.2)
$\alpha_{c}(u):=|u-c|,\qquad\beta_{c}(u)=\mathrm{sign}(u-c)[V(u)u-V(c)c],\qquad
c\in\mathbb{R}$

and establish a preliminary result.

######  Proposition 5.1.

Assume that $u_{0}$, $V$ and $\eta$ satisfy (1.3) (1.4) and (1.5),
respectively, and let $w_{\varepsilon}$ be as in (1.6), where
$u_{\varepsilon}$ is the solution of the Cauchy problem (1.1). If
$\eta(\xi)\xi\in L^{1}(\mathbb{R})$ there is a constant $K>0$ which only
depends on $V$ and $\eta$ and satisfies the following properties. Fix
$\varepsilon>0$, then there is a function $E_{\varepsilon}\in
L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$ such that

(5.3)
$D^{\varepsilon}_{c}(\phi):=\iint_{\mathbb{R}_{+}\times\mathbb{R}}\big{[}\alpha_{c}(w_{\varepsilon})\partial_{t}\phi+\beta_{c}(w_{\varepsilon})\partial_{x}\phi\big{]}dxdt\geq-\iint_{\mathbb{R}_{+}\times\mathbb{R}}E_{\varepsilon}|\partial_{x}\phi|dxdt\text{
}$

for every $c\in\mathbb{R}$ and every test function $\phi\in
C^{\infty}_{c}(]0,+\infty[\times\mathbb{R}),\;\phi\geq 0$; also,

(5.4) $\int_{\mathbb{R}}|E_{\varepsilon}(t,x)|dx\leq
K\varepsilon\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)\quad\text{for a.e.
$t\in\mathbb{R}_{+}$}.$

###### Proof.

We fix a test function $\phi\in
C^{\infty}_{c}(]0,+\infty[\times\mathbb{R}),\;\phi\geq 0$ and proceed
according to the following steps.
Step 1: we fix a (classical) entropy-entropy flux pair, i.e. $\alpha,\beta\in
C^{2}(\mathbb{R})$ satisfy $\alpha^{\prime\prime}\geq 0$ and
$\beta^{\prime}(u)=\alpha^{\prime}(u)[uV^{\prime}(u)+V(u)]$. We recall (4.4)
and (4.15) and conclude that

(5.5)
$D^{\varepsilon}_{\alpha}(\phi)\geq-\max_{w\in[0,1]}|\alpha^{\prime}(w)|\iint_{\mathbb{R}_{+}\times\mathbb{R}}|S_{\varepsilon}(t,x)||\partial_{x}\phi|dxdt+\iint_{\mathbb{R}_{+}\times\mathbb{R}}u_{\varepsilon_{k}}(t,y)[G^{1}_{\varepsilon}(t,y)+G^{2}_{\varepsilon}(t,y)]dydt,$

where $S_{\varepsilon}$, $G^{1}_{\varepsilon}$, $G^{2}_{\varepsilon}$ are as
in (4.16) and (4.11), respectively. We now set
$L(u):=I(u)-\alpha^{\prime}(u)V(u)$ and, recalling the equality
$I^{\prime}(u)=\alpha^{\prime\prime}(u)V(u)$, conclude that
$L^{\prime}(u)=-\alpha^{\prime}(u)V^{\prime}(u)$. We then point out that

$\begin{split}(G^{1}_{\varepsilon}+G^{2}_{\varepsilon})(t,y)=&~{}\int_{-\infty}^{y}\big{[}L(w_{\varepsilon}(t,y))-L(w_{\varepsilon}(t,x))\big{]}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{{\varepsilon}}\right)\partial_{x}\phi(t,x)dx\\\
&+\int_{-\infty}^{y}\alpha^{\prime}(w_{\varepsilon}(t,x))\big{[}V(w_{\varepsilon}(t,y))-V(w_{\varepsilon}(t,x))\big{]}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{{\varepsilon}}\right)\partial_{x}\phi(t,x)dx,\end{split}$

which owing to Fubini’s theorem and by recalling (2.1) yields

(5.6)
$\begin{split}&\iint_{\mathbb{R}_{+}\times\mathbb{R}}(G^{1}_{\varepsilon}+G^{2}_{\varepsilon})(t,y)u_{\varepsilon}(t,y)dydt\\\
\geq&-\iint_{\mathbb{R}_{+}\times\mathbb{R}}|\partial_{x}\phi|(t,x)\int_{x}^{+\infty}\big{|}L(w_{\varepsilon}(t,y))-L(w_{\varepsilon}(t,x))\big{|}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{{\varepsilon}}\right)dydxdt\\\
&-\max_{w\in[0,1]}|\alpha^{\prime}(w)|\iint_{\mathbb{R}_{+}\times\mathbb{R}}|\partial_{x}\phi|(t,x)\int_{x}^{+\infty}\big{|}V(w_{\varepsilon}(t,y))-V(w_{\varepsilon}(t,x))\big{|}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{{\varepsilon}}\right)dydxdt\\\
&\stackrel{{\scriptstyle
L^{\prime}=-\alpha^{\prime}V^{\prime}}}{{\geq}}\\!\\!-2\max_{w\in[0,1]}|\alpha^{\prime}(w)|\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\iint_{\mathbb{R}_{+}\times\mathbb{R}}|\partial_{x}\phi|(t,x)\int_{x}^{+\infty}\\!\\!\big{|}w_{\varepsilon}(t,y)-w_{\varepsilon}(t,x)\big{|}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{{\varepsilon}}\right)dydxdt.\end{split}$

Step 2: we fix $c\in\mathbb{R}$ and consider the Kružkov’s entropy-entropy
flux pairs defined in (5.2). We construct a sequence $(\alpha_{n},\beta_{n})$
of $C^{2}$ entropy-entropy flux pairs such that $\alpha_{n}\to\alpha_{c}$ and
$\beta_{n}\to\beta_{c}$ as $n\to+\infty$ in $C^{0}(\mathbb{R})$ and
$|\alpha^{\prime}_{n}|\leq 1$ for every $n$. By comparing (4.4) and (5.3) and
recalling that $\phi(0,\cdot)\equiv 0$ we get that
$D^{\varepsilon}_{\alpha_{n}}(\phi)\to D^{\varepsilon}_{c}(\phi)$ as
$n\to+\infty$. By passing to the limit in (5.5) and using (5.6) we arrive at
the inequality

$\begin{split}D^{\varepsilon}_{c}(\phi)&\geq-\int_{0}^{T}\\!\\!\int_{\mathbb{R}}\\!\\!|S_{\varepsilon}(t,x)||\partial_{x}\phi|dxdt\\\
&-\iint_{\mathbb{R}_{+}\times\mathbb{R}}|\partial_{x}\phi|(t,x)\underbrace{2\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\int_{x}^{+\infty}\\!\\!\big{|}w_{\varepsilon}(t,y)-w_{\varepsilon}(t,x)\big{|}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{{\varepsilon}}\right)dy}_{A_{\varepsilon}(t,x)}dxdt.\end{split}$

We now set $E_{\varepsilon}:=S_{\varepsilon}+A_{\varepsilon}$ and by recalling
(4.16) point out that

$\begin{split}\int_{\mathbb{R}}|S_{\varepsilon}(t,x)|dx&=\int_{\mathbb{R}}\int_{x}^{+\infty}\underbrace{|u_{\varepsilon}(t,y)|}_{\leq
1\ \text{by
\eqref{e:maxpu}}}\big{|}V(w_{\varepsilon}(t,y))-V(w_{\varepsilon}(t,x))\big{|}\frac{1}{\varepsilon}\eta\left(\frac{x-y}{{\varepsilon}}\right)dydx\\\
&\stackrel{{\scriptstyle\xi=\frac{x-y}{\varepsilon}}}{{\leq}}\int_{\mathbb{R}}\int_{\mathbb{R}_{-}}\big{|}V(w_{\varepsilon}(t,x-\varepsilon\xi))-V(w_{\varepsilon}(t,x))\big{|}\eta(\xi)d\xi
dx\\\
&\stackrel{{\scriptstyle\text{Fubini}}}{{=}}\int_{\mathbb{R}_{-}}\eta(\xi)\int_{\mathbb{R}}\big{|}V(w_{\varepsilon}(t,x-\varepsilon\xi))-V(w_{\varepsilon}(t,x))\big{|}dxd\xi\\\
&\leq\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\int_{\mathbb{R}_{-}}\eta(\xi)\int_{\mathbb{R}}\big{|}w_{\varepsilon}(t,x-\varepsilon\xi)-w_{\varepsilon}(t,x)\big{|}dxd\xi\\\
&\stackrel{{\scriptstyle\eqref{e:tvchar}}}{{\leq}}\varepsilon\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\mathrm{TotVar}[w_{\varepsilon}](t,\cdot)\underbrace{\int_{\mathbb{R}_{-}}\eta(\xi)|\xi|d\xi}_{C_{\eta}}{\leq}C_{\eta}\varepsilon\operatorname*{ess\,sup}_{w\in]0,1[}|V^{\prime}(w)|\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot).\end{split}$

By relying on the same computations we control the integral of
$|A_{\varepsilon}|$ and eventually arrive at (5.4). ∎

We now establish the proof of (1.12). We follow an argument due to Kuznetsov
[19], which in turn relies on the doubling-of-variables technique by Kružkov
[18]. We detail the argument for the sake of completeness. First, we apply
Proposition 5.1 and recall that $u$ is an entropy admissibile solution of
(1.2); we conclude that for every test function $\phi\in
C^{\infty}(({]0,+\infty[}\times\mathbb{R})^{2})$, $\phi\geq 0$ we have

(5.7)
$\begin{split}&\iint_{\mathbb{R}_{+}\times\mathbb{R}}\big{[}|w_{\varepsilon}(t^{\prime},x^{\prime})-u(t,x)|\partial_{t^{\prime}}\phi+q(w_{\varepsilon}(t^{\prime},x^{\prime}),u(t,x))\partial_{x^{\prime}}\phi\big{]}dx^{\prime}dt^{\prime}\geq-\iint_{\mathbb{R}_{+}\times\mathbb{R}}E(t^{\prime},x^{\prime})|\partial_{x^{\prime}}\phi|dt^{\prime}dx^{\prime}\\\
&\iint_{\mathbb{R}_{+}\times\mathbb{R}}\big{[}|w_{\varepsilon}(t^{\prime},x^{\prime})-u(t,x)|\partial_{t}\phi+q(w_{\varepsilon}(t^{\prime},x^{\prime}),u(t,x))\partial_{x}\phi\big{]}dxdt\geq~{}0,\end{split}$

provided $q(a,b)=\mathrm{sign}(a-b)\big{(}aV(a)-bV(b)\big{)}$. We now choose
the test function $\phi$ by setting

(5.8)
$\phi(t,x,t^{\prime},x^{\prime})=\psi\left(\frac{t+t^{\prime}}{2}\right)\chi\left(\frac{x+x^{\prime}}{2}\right)\gamma_{\nu_{1}}(t-t^{\prime})\gamma_{\nu_{2}}(x-x^{\prime}),$

where $\psi\in C^{\infty}_{c}(]0,+\infty[)$, $\chi\in
C^{\infty}_{c}(\mathbb{R})$ satisfy $\psi,\chi\geq 0$. Also,
$\nu_{1},\nu_{2}>0$ are two parameters and we have used the notation
$\gamma_{\nu_{i}}(x):=\nu_{i}^{-1}\gamma(\nu_{i}^{-1}x)$, where $\gamma$ is a
standard convolution kernel satisfying

$\gamma\in C^{\infty}_{c}(]-1,1[),\quad\gamma\geq
0,\quad\int_{\mathbb{R}}\gamma=1.$

We plug (5.8) into (5.7), integrate the first equation with respect to $dxdt$
and the second equation with respect to $dx^{\prime}dt^{\prime}$ and add the
resulting equations; we get

(5.9)
$\iiiint_{(\mathbb{R}_{+}\times\mathbb{R})^{2}}\\!\\!\\!\big{[}|w_{\varepsilon}-u|\psi^{\prime}\chi+q(w_{\varepsilon},u)\psi\chi^{\prime}\big{]}\gamma_{\nu_{1}}\gamma_{\nu_{2}}dx^{\prime}dt^{\prime}dxdt\geq-\iiiint_{(\mathbb{R}_{+}\times\mathbb{R})^{2}}\\!\\!\\!E^{\varepsilon}(t^{\prime},x^{\prime})|\partial_{x^{\prime}}\phi|dx^{\prime}dt^{\prime}dxdt,$

where we used the equalities
$\partial_{t}\phi+\partial_{t^{\prime}}\phi=\psi^{\prime}\chi\
\gamma_{\nu_{1}}\gamma_{\nu_{2}}$ and
$\partial_{x}\phi+\partial_{x^{\prime}}\phi=\psi\chi^{\prime}\
\gamma_{\nu_{1}}\gamma_{\nu_{2}}$. In the previous expression and in the
following if not otherwise specified the functions $\chi$ and $\chi^{\prime}$
are evaluated at $(x+x^{\prime})/2$ and the functions $\gamma_{\nu_{1}}$ and
$\gamma_{\nu_{2}}$ at $t-t^{\prime}$ and $x-x^{\prime}$, respectively. To
control the right-hand side of (5.9), we first of all point out that

(5.10)
$|\partial_{x^{\prime}}\phi(t,x,t^{\prime},x^{\prime})|\leq\frac{1}{2}\|\psi\chi^{\prime}\|_{C^{0}}\gamma_{\nu_{1}}\gamma_{\nu_{2}}+\|\psi\chi\|_{C^{0}}\gamma_{\nu_{1}}\frac{1}{\nu^{2}_{2}}\left|\gamma^{\prime}\left(\frac{x-x^{\prime}}{\nu_{2}}\right)\right|.$

Next, we choose $t_{1}$ and $t_{2}$ such that $\mathrm{supp}\
\psi\subseteq]t_{1},t_{2}[$ and arrive at

(5.11)
$\begin{split}\iiiint_{(\mathbb{R}_{+}\times\mathbb{R})^{2}}&E^{\varepsilon}(t^{\prime},x^{\prime})|\partial_{x^{\prime}}\phi|dxdtdx^{\prime}dt^{\prime}\leq~{}\int_{t_{1}-\nu_{1}/2}^{t_{2}+\nu_{1}/2}\int_{\mathbb{R}}E^{\varepsilon}(t^{\prime},x^{\prime})\left(\iint_{\mathbb{R}_{+}\times\mathbb{R}}|\partial_{x^{\prime}}\phi|dxdt\right)dx^{\prime}dt^{\prime}\\\
\stackrel{{\scriptstyle\eqref{e:ddue}}}{{\leq}}&~{}\int_{t_{1}-\nu_{1}/2}^{t_{2}+\nu_{1}/2}\int_{\mathbb{R}}E^{\varepsilon}(t^{\prime},x^{\prime})\left[\frac{1}{2}\|\psi\chi^{\prime}\|_{C^{0}}+\|\psi\chi\|_{C^{0}}\frac{\|\gamma^{\prime}\|_{L^{1}}}{\nu_{2}}\right]dx^{\prime}dt^{\prime}\\\
\stackrel{{\scriptstyle\eqref{e:inte}}}{{\leq}}&K\varepsilon[t_{2}-t_{1}+\nu_{1}]\left[\frac{1}{2}\|\psi\chi^{\prime}\|_{C^{0}}+\|\psi\chi\|_{C^{0}}\frac{\|\gamma^{\prime}\|_{L^{1}}}{\nu_{2}}\right]\operatorname*{ess\,sup}_{t\in\mathbb{R}_{+}}\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)\\\
\stackrel{{\scriptstyle\eqref{e:tv},\eqref{e:tv0}}}{{\leq}}&K\varepsilon[t_{2}-t_{1}+\nu_{1}]\left[\frac{1}{2}\|\psi\chi^{\prime}\|_{C^{0}}+\|\psi\chi\|_{C^{0}}\frac{\|\gamma^{\prime}\|_{L^{1}}}{\nu_{2}}\right]\mathrm{TotVar}\,u_{0}.\end{split}$

We now let $\nu_{1}\to 0^{+}$ and then consider a sequence $\psi_{n}$ such
that

$\psi_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathbf{1}_{]t_{1},t_{2}[}\;\text{weakly${}^{\ast}$
in $L^{\infty}(\mathbb{R}_{+})$},\quad\|\psi_{n}\|_{C^{0}}\leq 1$

and by taking the $n\to+\infty$ limit in both (5.9) and (5.11) and recalling
that $w_{\varepsilon},u\in
C^{0}(\mathbb{R}_{+},L^{1}_{\mathrm{loc}}(\mathbb{R}))$ we arrive at

(5.12)
$\begin{split}\iint_{\mathbb{R}^{2}}&|w_{\varepsilon}(t_{2},x^{\prime})-u(t_{2},x)|\chi\gamma_{\nu_{2}}dxdx^{\prime}-\iint_{\mathbb{R}^{2}}|w_{\varepsilon}(t_{1},x^{\prime})-u(t_{1},x)|\chi\gamma_{\nu_{2}}dxdx^{\prime}\\\
&\leq\int_{t_{1}}^{t_{2}}\iint_{\mathbb{R}^{2}}q(w_{\varepsilon},u)\chi^{\prime}\gamma_{\nu_{2}}dxdx^{\prime}dt+K\varepsilon[t_{2}-t_{1}]\left[\frac{1}{2}\|\chi^{\prime}\|_{C^{0}}+\|\chi\|_{C^{0}}\frac{\|\gamma^{\prime}\|_{L^{1}}}{\nu_{2}}\right]\mathrm{TotVar}\,u_{0}.\end{split}$

Assume $\|\chi\|_{C^{0}}\leq 1$; then we have

(5.13)
$\begin{split}\iint_{\mathbb{R}^{2}}&|w_{\varepsilon}(t_{2},x^{\prime})-w_{\varepsilon}(t_{2},x)|\chi\gamma_{\nu_{2}}dxdx^{\prime}\stackrel{{\scriptstyle\xi=\frac{x-x^{\prime}}{\nu_{2}}}}{{\leq}}\int_{\mathbb{R}}\gamma(\xi)\int_{\mathbb{R}}|w_{\varepsilon}(t_{2},x-\nu_{2}\xi)-w_{\varepsilon}(t_{2},x)|dxd\xi\\\
&\stackrel{{\scriptstyle\eqref{e:tvchar}}}{{\leq}}C_{\gamma}\nu_{2}\mathrm{TotVar}\,w_{\varepsilon}(t_{2},\cdot)\stackrel{{\scriptstyle\eqref{e:tv},\eqref{e:tv0}}}{{\leq}}C_{\gamma}\nu_{2}\mathrm{TotVar}\,u_{0}\end{split}$

and analogously

(5.14)
$\begin{split}\iint_{\mathbb{R}^{2}}&|w_{\varepsilon}(t_{1},x^{\prime})-w_{\varepsilon}(t_{1},x)|\chi\gamma_{\nu_{2}}dxdx^{\prime}\leq
C_{\gamma}\nu_{2}\mathrm{TotVar}\,u_{0}.\end{split}$

We now point out that

(5.15)
$\begin{split}\int_{\mathbb{R}}&|w_{\varepsilon}(t_{2},x)-u(t_{2},x)|\int_{\mathbb{R}}\chi\gamma_{\nu_{2}}dx^{\prime}dx=\iint_{\mathbb{R}^{2}}|w_{\varepsilon}(t_{2},x)-u(t_{2},x)|\chi\gamma_{\nu_{2}}dxdx^{\prime}\\\
&\leq\iint_{\mathbb{R}^{2}}|w_{\varepsilon}(t_{2},x)-w_{\varepsilon}(t_{2},x^{\prime})|\chi\gamma_{\nu_{2}}dxdx^{\prime}+\iint_{\mathbb{R}^{2}}|w_{\varepsilon}(t_{2},x^{\prime})-u(t_{2},x)|\chi\gamma_{\nu_{2}}dxdx^{\prime}\\\
\end{split}$

and that

(5.16)
$\begin{split}\iint_{\mathbb{R}^{2}}|w_{\varepsilon}(t_{1},x^{\prime})-u(t_{1},x)|\chi\gamma_{\nu_{2}}&dxdx^{\prime}\leq\iint_{\mathbb{R}^{2}}|w_{\varepsilon}(t_{1},x^{\prime})-w_{\varepsilon}(t_{1},x)|\chi\gamma_{\nu_{2}}dxdx^{\prime}\\\
&+\int_{\mathbb{R}}|w_{\varepsilon}(t_{1},x)-u(t_{1},x)|\int_{\mathbb{R}}\chi\gamma_{\nu_{2}}dx^{\prime}dx.\end{split}$

By plugging the above inequalities into (5.12) and recalling (5.13), (5.14)
and the inequality $\chi\leq 1$ we arrive at

$\begin{split}\int_{\mathbb{R}}&|w_{\varepsilon}(t_{2},x)-u(t_{2},x)|\int_{\mathbb{R}}\chi\gamma_{\nu_{2}}dx^{\prime}dx\leq
2C_{\gamma}\nu_{2}\mathrm{TotVar}\,u_{0}\\\
&+\int_{\mathbb{R}}|w_{\varepsilon}(t_{1},x)-u(t_{1},x)|\int_{\mathbb{R}}\chi\gamma_{\nu_{2}}dx^{\prime}dx{+}\int_{t_{1}}^{t_{2}}\iint_{\mathbb{R}^{2}}q(w_{\varepsilon},u)\chi^{\prime}\gamma_{\nu_{2}}dxdx^{\prime}dt\\\
&+K\varepsilon[t_{2}-t_{1}]\left[\frac{1}{2}\|\chi^{\prime}\|_{C^{0}}+\frac{\|\gamma^{\prime}\|_{L^{1}}}{\nu_{2}}\right]\mathrm{TotVar}\,u_{0}\end{split}$

and by using the inequality

(5.17)
$\begin{split}\int_{\mathbb{R}}|w_{\varepsilon}(0,x)-u_{0}(x)|\int_{\mathbb{R}}&\chi\gamma_{\nu_{2}}dx^{\prime}dx\leq\int_{\mathbb{R}}|w_{\varepsilon}(0,x)-u_{0}(x)|dx\\\
&\stackrel{{\scriptstyle\eqref{e:w}}}{{=}}\|u_{0}\ast\eta_{\varepsilon}-u_{0}\|_{L^{1}}\stackrel{{\scriptstyle\eqref{e:tvl1}}}{{\leq}}C_{\eta}\varepsilon\mathrm{TotVar}\,u_{0}\end{split}$

and letting $t_{1}\to 0^{+}$ we get

(5.18)
$\begin{split}\int_{\mathbb{R}}&|w_{\varepsilon}(t_{2},x)-u(t_{2},x)|\int_{\mathbb{R}}\chi\gamma_{\nu_{2}}dx^{\prime}dx\leq
2C_{\gamma}\nu_{2}\mathrm{TotVar}\,u_{0}\\\
&+C_{\eta}\varepsilon\mathrm{TotVar}\,u_{0}+\int_{0}^{t_{2}}\iint_{\mathbb{R}^{2}}q(w_{\varepsilon},u)\chi^{\prime}\gamma_{\nu_{2}}dxdx^{\prime}dt+K\varepsilon
t_{2}\left[\frac{1}{2}\|\chi^{\prime}\|_{C^{0}}+\frac{\|\gamma^{\prime}\|_{L^{1}}}{\nu_{2}}\right]\mathrm{TotVar}\,u_{0}.\end{split}$

Assume for a moment we have shown that

(5.19) $\int_{\mathbb{R}}|w_{\varepsilon}(t,x)-u(t,x)|dx<+\infty\;\text{for
every $t\in\mathbb{R}_{+}$};$

then we can consider a sequence of test functions $\\{\chi_{n}\\}$ such that
$\chi_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}1$ weakly∗ in
$L^{\infty}(\mathbb{R})$, $0\leq\chi_{n}\leq 1$ and
$\|\chi^{\prime}_{n}\|_{C^{0}}\to 0^{+}$ as $n\to+\infty$. By passing to the
$n\to+\infty$ limit in (5.18) we arrive at

$\begin{split}\int_{\mathbb{R}}&|w_{\varepsilon}(t_{2},x)-u(t_{2},x)|dx\leq
2C_{\gamma}\nu_{2}\mathrm{TotVar}\,u_{0}+C_{\eta}\varepsilon\mathrm{TotVar}\,u_{0}+K\varepsilon
t_{2}\frac{\|\gamma^{\prime}\|_{L^{1}}}{\nu_{2}}\mathrm{TotVar}\,u_{0}\end{split}$

and by choosing $\nu_{2}=\sqrt{\varepsilon t_{2}}$ and by relying on the
arbitrariness of $t_{2}$ we get (1.12). We are thus left to establish (5.19):
to this end, we recall that $q(a,b)=\mathrm{sign}(a-b)\big{(}aV(a)-bV(b))$,
which implies that
$|q(w_{\varepsilon},u)|\leq\operatorname*{ess\,sup}_{w\in]0,1[}\\{|V(w)|+|V^{\prime}(w)|\\}|w_{\varepsilon}(t,x)-u(t,x^{\prime})|$.
We can then choose a test function111Note that, technically speaking, the only
compactly supported function satisfying the inequality
$|\chi^{\prime}|\leq\chi$ is the constant $0$. However, one can use an easy
approximation argument and show that the inequality (5.18) holds for instance
for any function in the Schwartz space $\mathcal{S}(\mathbb{R})$. $\chi$ such
that $|\chi^{\prime}|\leq\chi$ and conclude that

$\int_{0}^{t_{2}}\iint_{\mathbb{R}^{2}}q(w_{\varepsilon},u)\chi^{\prime}\gamma_{\nu_{2}}dxdx^{\prime}dt\leq\operatorname*{ess\,sup}_{w\in]0,1[}\\{|V(w)|+|V^{\prime}(w)|\\}\int_{0}^{t_{2}}\iint_{\mathbb{R}^{2}}|w_{\varepsilon}(t,x)-u(t,x^{\prime})|\chi\gamma_{\nu_{2}}dxdx^{\prime}dt.$

By controlling the above term with the same argument as in (5.16), applying
Grönwall’s lemma and recalling (5.17) we establish a bound on the left-hand
side of (5.18) independent of $\chi$, and this eventually implies (5.19) by
the arbitrariness of $t_{2}$.

##  6\. Proof of Theorem 1.4

In §6.1 we provide an overview of the construction of the counter-example, and
we describe the basic ideas involved. Next, in §6.2, §6.3 and §6.4 we detail
the construction. Throughout this section we assume $V(w)=1-w$,
$\eta=\mathbbm{1}_{]-1,0[}$, which implies that (1.6) and (2.3) boil down to

(6.1)
$w_{\varepsilon}(t,x):=\frac{1}{\varepsilon}\int_{x}^{x+\varepsilon}u(t,y)dy\quad\text{and}\quad\frac{\partial
w_{\varepsilon}}{\partial
x}(t,x)=\frac{u_{\varepsilon}(t,x+\varepsilon)-u_{\varepsilon}(t,x)}{\varepsilon},$

respectively. Note that, since both $V$ and $\eta$ are now fixed, the
construction of the counter-example boils down to the construction of the
initial datum $u_{0}$.

###  6.1. Outline

Our construction is reminiscent of the one in [11], in particular we assume
that $u_{0}(x)=1$ for a.e. $x>0$, which implies that $u_{\varepsilon}(t,x)=1$
for a.e. $(t,x)$ such that $x>0$. Also, as in [11] the total variation
increase mechanism is triggered by a countable number of “building blocks”
located on the negative real axis. The building block we use here, however, is
different from the one in [11], and capturing the total variation increase
mechanism requires a finer analysis. We now provide an handwaving description
of the main ideas involved, and we refer to the following paragraphs for the
rigorous argument.

The basic building block is represented in Figure 1 and consists of two
rectangles of height $h$ and length $\ell$, located at distance $2\ell$ and
$6\ell$ from the origin, respectively.

Figure 1. In red the building block $v_{h\ell}$ triggering the total variation
increase for $\varepsilon=4\ell$

In other words,

(6.2) $v_{h\ell}(x):=\left\\{\begin{array}[]{ll}0&x<-7\ell\\\
h&-7\ell<x<-6\ell\\\ 0&-6\ell<x<-3\ell\\\ h&-3\ell<x<-2\ell\\\ 0&x>-2\ell\\\
\end{array}\right.\qquad\ell\in]0,1[,\;h\in]0,1[.$

We now fix $\varepsilon>0$, set
$u_{0}(x)=v_{h\ell}(x)+\mathbbm{1}_{]0,+\infty[}(x)$ and consider the Cauchy
problem (1.1); let $w_{\varepsilon}$ be as in (1.6), then it is fairly easy to
see that

(6.3)
$\lim_{x\to-\infty}w_{\varepsilon}(t,x)=0,\quad\lim_{x\to+\infty}w_{\varepsilon}(t,x)=1\quad\text{for
every $t\geq 0$ and every $\varepsilon>0$.}$

Assume that $\varepsilon=4\ell$; by using formula (6.1) we realize (see
equation (6.13)) that $w_{\varepsilon}(0,\cdot)$ is a monotone non-decreasing
function and hence $\mathrm{TotVar}\;w_{\varepsilon}(0,\cdot)=1$ by (6.3).
Assume for a moment that we have shown that, for some time $t>0$, we have
$\partial_{x}w_{\varepsilon}(t,\cdot)<0$ on some interval, then by using (6.3)
we infer $\mathrm{TotVar}\;w_{\varepsilon}(t,\cdot)~{}>~{}1$, that is we have
an increase of the total variation.

To establish the inequality $\partial_{x}w_{\varepsilon}(t,\cdot)<0$ we rely
on (6.1) and on the method of characteristics. More precisely, in the
following we term $X_{\varepsilon}(t,s,\xi)$ the solution of the Cauchy
problem

(6.4)
$\left\\{\begin{array}[]{ll}\displaystyle{\frac{dX_{\varepsilon}}{dt}=1-w_{\varepsilon}(t,X_{\varepsilon})}\\\
\displaystyle{X_{\varepsilon}(s,s,\xi)=\xi,\phantom{\int}}\end{array}\right.$

in other words $X_{\varepsilon}(\cdot,s,\xi)$ is the characteristic line that
attains the value $\xi$ at time $t=s$. Note that the equation at the first
line of (1.1) implies that, if $u_{0}(x)=0$ then
$u_{\varepsilon}(t,X_{\varepsilon}(t,0,x))=0$ for every $t$, that is
$u_{\varepsilon}$ vanishes along the whole characteristic starting at $x$.
Also, if $u_{0}(y)>0$ then $u_{\varepsilon}(t,X_{\varepsilon}(t,0,y))>0$ for
every $t$. In particular this yields that if $x\in]-2\ell,0[$ and
$y\in]-7\ell,-6\ell[$ then $u_{\varepsilon}(t,X_{\varepsilon}(t,0,x))=0$ and
$u_{\varepsilon}(t,X_{\varepsilon}(t,0,y))>0$ for every $t$, respectively. The
key observation is now that, since one can show that
$w_{\varepsilon}(0,-6\ell)<w_{\varepsilon}(0,-2\ell)$ then the initial speed
of the characteristic $X_{\varepsilon}(\cdot,0,-6\ell)$ is _greater_ than the
initial speed of the characteristic $X_{\varepsilon}(\cdot,0,-2\ell)$. Let us
now consider a point $y$ just at the left of $-6\ell$, then $u_{0}(y)=h>0$ and
hence $u_{\varepsilon}(t,X_{\varepsilon}(t,0,y))>0$ for every $t$. On the
other hand, let us consider the point $X_{\varepsilon}(t,0,y)+\varepsilon$ and
recall that $\varepsilon=4\ell$: since the characteristic line
$X_{\varepsilon}(\cdot,0,-2\ell)$ is slower than $X_{\varepsilon}(\cdot,0,y)$,
then the value $X_{\varepsilon}(t,0,y)+\varepsilon$ overtakes
$X_{\varepsilon}(\cdot,0,-2\ell)$, and this in turn implies that
$u_{\varepsilon}(t,X_{\varepsilon}(t,0,y)+\varepsilon)=0$. Summing up, we have
$u_{\varepsilon}(t,X_{\varepsilon}(t,0,y))>0$ and
$u_{\varepsilon}(t,X_{\varepsilon}(t,0,y)+\varepsilon)=0$, and owing to (6.1)
this yields $\partial_{x}w_{\varepsilon}(t,X_{\varepsilon}(t,0,y))<0$.

To complete the construction we fix a sequence $\varepsilon_{n}$, construct a
corresponding sequence of building blocks $v_{h_{n}\ell_{n}}$ with
$\varepsilon_{n}=4\ell_{n}$, and juxtapose them on the negative real axis, in
other words

(6.5)
$u_{0}(x):=\sum_{n=1}^{\infty}v_{h_{n}\ell_{n}}(x)+\mathbbm{1}_{]0,+\infty[}(x),\quad\ell_{n}:=\varepsilon_{n}/4.$

The main obstruction we have now to tackle is that the previous considerations
show that the building block $v_{h_{n}\ell_{n}}$ triggers a total variation
increase at scale $\varepsilon_{n}=4\ell_{n}$, but these considerations fail
when $\varepsilon_{n}\neq 4\ell_{n}$. Hence, at a given scale
$\varepsilon_{n}$ it may in principle happen that the total variation
_increase_ given by building block $v_{h_{n}\varepsilon_{n}/4}$ is smaller
than the total variation _decrease_ triggered by the other building blocks in
$u_{0}$, and that the overall effect is that the total variation decreases. To
rule out this possibility, we have to rely on a finer analysis and make the
previous qualitative considerations quantitative.

###  6.2. Preliminary results

We recall (6.4). Note that, by the equation at the first line of (1.1), we
have

(6.6)
$\int_{X_{\varepsilon}(t,s,\xi_{1})}^{X_{\varepsilon}(t,s,\xi_{2})}u_{\varepsilon}(t,y)dy=\int_{\xi_{1}}^{\xi_{2}}u_{\varepsilon}(s,y)dy\quad\text{for
every $s,t\in\mathbb{R}$, $\xi_{1}<\xi_{2}$},$

that is the mass between two characteristic lines is always conserved. We now
recall some basic properties established in [11].

######  Lemma 6.1.

Fix $\varepsilon>0$ and assume $u_{0}\in L^{1}_{\mathrm{loc}}(\mathbb{R})$
satisfies $0\leq u_{0}\leq 1$ and $u_{0}(x)=1$ for a.e. $x>0$. Then
$u_{\varepsilon}(t,x)=1$ for every $t\geq 0$ and a.e. $x>0$ and

(6.7) $X_{\varepsilon}(t,0,0)=0\quad\text{for every $t\geq 0$.}$

Also,

(6.8) $\frac{dX_{\varepsilon}}{dt}(t,s,\xi)\geq 0,\quad\text{for every
$\xi\in\mathbb{R}$, $t,s\in\mathbb{R}$}.$

We also have

######  Lemma 6.2.

Under the same assumptions as in Lemma 6.1, assume furthermore that there is
$x_{0}<0$ such that $u_{0}(x)=0$ for a.e. $x<x_{0}$. Then for every
$\varepsilon>0$ we have

(6.9) $w_{\varepsilon}(t,x)=0\;\text{for every $t>0$,
$x<x_{0}-\varepsilon$},\qquad w_{\varepsilon}(t,x)=1\;\text{for every $t>0$,
$x>0$}.$

###### Proof.

Owing to (6.8), $u_{\varepsilon}(t,x)=0$ for a.e. $x<x_{0}$, $t\geq 0$ and
$\varepsilon>0$. This yields the first equation in (6.9). The second equation
follows from the equality $u_{\varepsilon}(t,x)=1$ for every $t\geq 0$ and
a.e. $x>0$. ∎

Note that (6.9) yields (6.3).

###  6.3. Analysis of the basic building block

The following two lemmas deal with the case $\ell=4\varepsilon$.

######  Lemma 6.3.

Set

(6.10) $u_{0}(x)=v_{h\ell}(x)+s(x)+\mathbbm{1}_{]0,+\infty[}(x),$

where $v_{h\ell}$ is the same as in (6.2) and $s\in L^{\infty}(\mathbb{R})$
satisfies

(6.11) $0\leq s\leq 1,\qquad\text{$s(x)=0$ for a.e.
$x\notin]-\delta,0[$},\;\delta<2\ell.$

If $\varepsilon=4\ell$, then $\mathrm{TotVar}\,w_{\varepsilon}(0,\cdot)=1$.

###### Proof.

We recall the formula

(6.12)
$\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)=\lim_{x\to+\infty}w_{\varepsilon}(t,x)-\lim_{x\to-\infty}w_{\varepsilon}(t,x)+2\int_{\mathbb{R}}\left[\frac{\partial
w_{\varepsilon}}{\partial
x}(t,x)\right]^{-}dx\stackrel{{\scriptstyle\eqref{e:limits}}}{{=}}1+2\int_{\mathbb{R}}\left[\frac{\partial
w_{\varepsilon}}{\partial x}(t,x)\right]^{-}dx,$

where $[\cdot]^{-}$ denotes the negative part. Next, we point out that

(6.13) $\frac{\partial w_{\varepsilon}(0,x)}{\partial
x}=\left\\{\begin{array}[]{ll}0&x<-7\ell-\varepsilon\\\
\displaystyle{\frac{h}{\varepsilon}}&-7\ell-\varepsilon<x<-7\ell\\\
0&-7\ell<x<-6\ell\\\
\displaystyle{\frac{u_{0}(x+\varepsilon)}{\varepsilon}}&-6\ell<x<-4\ell\\\
\displaystyle{\frac{1-u_{0}(x)}{\varepsilon}}&-4\ell<x<0\\\
0&x>0,\end{array}\right.$

which yields $\partial w_{\varepsilon}(0,\cdot)/\partial x\geq 0$ and hence
concludes the proof owing to (6.12). ∎

We now quantify the total variation increase triggered by the initial datum in
(6.10).

######  Lemma 6.4.

Under the same assumptions as in Lemma 6.3, we have

(6.14) $\mathrm{TotVar}\,w_{\varepsilon}(t,\cdot)\geq
1+2\left(\frac{2-h}{4}\right)ht+o(t)\quad t\to 0^{+}.$

###### Proof.

We proceed according to the following steps.
Step 1: we focus on $X_{\varepsilon}(t,0,-2\ell)$. Since

$w_{\varepsilon}(0,-2\ell)\stackrel{{\scriptstyle\eqref{e:wwx},\eqref{e:u0cex}}}{{\geq}}\frac{1}{\varepsilon}\int_{0}^{-2\ell+\varepsilon}1\
dy=1-\frac{2\ell}{\varepsilon}\stackrel{{\scriptstyle\eqref{e:ch}}}{{\implies}}\left.\frac{dX_{\varepsilon}(t,0,-2\ell)}{dt}\right|_{t=0}\leq\frac{2\ell}{\varepsilon},$

then

(6.15)
$X_{\varepsilon}(t,0,-2\ell)\leq-2\ell+\frac{2\ell}{\varepsilon}t+o(t)\stackrel{{\scriptstyle\varepsilon=4\ell}}{{=}}-2\ell+\frac{1}{2}t+o(t)\quad
t\to 0^{+}.$

Step 2: we focus on $X_{\varepsilon}(t,0,-6\ell)$. Note that

$w_{\varepsilon}(0,-6\ell)\stackrel{{\scriptstyle\eqref{e:bblock}}}{{=}}\frac{h\ell}{\varepsilon}\implies\left.\frac{dX_{\varepsilon}(t,0,-6\ell)}{dt}\right|_{t=0}=1-\frac{h\ell}{\varepsilon},$

and by recalling that $\varepsilon=4\ell$ this yields

(6.16)
$X_{\varepsilon}(t,0,-6\ell)+\varepsilon=-2\ell+\left(1-\frac{h}{4}\right)t+o(t)\quad
t\to 0^{+}.$

Step 3: by comparing (6.15) and (6.16) we get that
$X_{\varepsilon}(t,0,-6\ell)+\varepsilon>X_{\varepsilon}(t,0,-2\ell)$ for
$t>0$ small enough and this in turn implies that

$X_{\varepsilon}(0,t,X_{\varepsilon}(t,0,-2\ell)-\varepsilon))\in]-7\ell,-6\ell[$

for $t$ sufficiently small. Since for every $y\in]-7\ell,-6\ell[$ we have

$w_{\varepsilon}(0,\cdot)=\frac{h\ell}{\varepsilon}\stackrel{{\scriptstyle\varepsilon=4\ell}}{{=}}\frac{h}{4}\implies\left.\frac{dX_{\varepsilon}(t,0,y)}{dt}\right|_{t=0}=1-\frac{h}{4}\implies
X_{\varepsilon}(t,0,y)=y+\left(1-\frac{h}{4}\right)t+o(t)\quad t\to 0^{+}$

then owing to (6.15) we get

(6.17)
$X_{\varepsilon}(0,t,X_{\varepsilon}(t,0,-2\ell)-\varepsilon)\leq-6\ell+\left(-\frac{1}{2}+\frac{h}{4}\right)t+o(t)=-6\ell+\left(\frac{-2+h}{4}\right)t+o(t)\quad
t\to 0^{+}.$

Step 4: since $2\ell-\delta>0$ by assumption, then
$X_{\varepsilon}(t,0,-2\ell)<X_{\varepsilon}(t,0,-6\ell)+\varepsilon<X_{\varepsilon}(t,0,-\delta)$
for sufficiently small $t>0$. On the other hand, $u_{\varepsilon}(t,x)=0$ for
every $x\in]X_{\varepsilon}(t,0,-2\ell),X_{\varepsilon}(t,0,-\delta)[$.
Step 5: we point out that, for every $\xi\in]-7\ell,-6\ell[$ we have
$u_{\varepsilon}(t,X_{\varepsilon}(t,0,\xi))>0$. On the other hand, if
$X_{\varepsilon}(t,0,\xi)+\varepsilon\in]X_{\varepsilon}(t,0,-2\ell),X_{\varepsilon}(t,0,-\delta)[$,
then by Step 4 we have
$u_{\varepsilon}(t,X_{\varepsilon}(t,0,\xi)+\varepsilon)=0$ and hence

(6.18) $\frac{\partial w_{\varepsilon}}{\partial
x}(t,X_{\varepsilon}(t,0,\xi))=\frac{u_{\varepsilon}(t,X_{\varepsilon}(t,0,\xi)+\varepsilon)-u_{\varepsilon}(t,X_{\varepsilon}(t,0,\xi))}{\varepsilon}=-\frac{u_{\varepsilon}(t,X_{\varepsilon}(t,0,\xi))}{\varepsilon}<0.$

Step 6: by combining the previous steps we have that, if
$\xi\in]X_{\varepsilon}(0,t,X_{\varepsilon}(t,0,-2\ell)-\varepsilon),-6\ell[$,
then (6.18) holds true. This implies that

$\begin{split}\int_{\mathbb{R}}\left[\frac{\partial w_{\varepsilon}}{\partial
x}(t,x)\right]^{-}dx&\geq-\int_{X_{\varepsilon}(t,0,-2\ell)-\varepsilon}^{X_{\varepsilon}(t,0,-6\ell)}\frac{\partial
w_{\varepsilon}}{\partial
x}(t,x)dx\stackrel{{\scriptstyle\eqref{e:lunedi}}}{{=}}\frac{1}{\varepsilon}\int_{X_{\varepsilon}(t,0,-2\ell)-\varepsilon}^{X_{\varepsilon}(t,0,-6\ell)}u_{\varepsilon}(t,y)dy\\\
&\stackrel{{\scriptstyle\eqref{e:cmass}}}{{=}}\frac{1}{\varepsilon}\int_{X_{\varepsilon}(0,t,X_{\varepsilon}(t,0,-2\ell)-\varepsilon)}^{-6\ell}u_{0}(t,y)dy\stackrel{{\scriptstyle\eqref{e:ottobre2}}}{{\geq}}\left(\frac{2-h}{4}\right)ht+o(t)\quad
t\to 0^{+}\end{split}$

and by recalling (6.12) this yields (6.14). ∎

In the following lemma we consider the case
$\ell>\max\\{\varepsilon+\delta,2\varepsilon\\}$.

######  Lemma 6.5.

Assume $\ell>\max\\{\varepsilon+\delta,2\varepsilon\\}$ and that $u_{0}$ is
given by (6.10) with $s$ satisfying (6.11); then

(6.19) $\int_{-7\ell-\varepsilon}^{-2\ell}\left|\frac{\partial
w_{\varepsilon}}{\partial x}(0,x)\right|dx=4h$

and

(6.20)
$\int_{X_{\varepsilon}(t,0,-7\ell)-\varepsilon}^{X_{\varepsilon}(t,0,-2\ell)}\left|\frac{\partial
w_{\varepsilon}}{\partial x}(t,x)\right|dx\geq 4h+o(t)\quad\text{as $t\to
0^{+}$}.$

###### Proof.

By using the inequality $\ell-\delta>\varepsilon$ we get

$\frac{\partial w_{\varepsilon}}{\partial
x}(0,x)=\frac{u_{0}(x+\varepsilon)-u_{0}(x)}{\varepsilon}=\left\\{\begin{array}[]{ll}h/\varepsilon&-7\ell-\varepsilon<x<-7\ell\\\
0&-7\ell<x<-6\ell-\varepsilon\\\ -h/\varepsilon&-6\ell-\varepsilon<x<-6\ell\\\
0&-6\ell<x<-3\ell-\varepsilon\\\ h/\varepsilon&-3\ell-\varepsilon<x<-3\ell\\\
0&-3\ell<x<-2\ell-\varepsilon\\\ -h/\varepsilon&-2\ell-\varepsilon<x<-2\ell\\\
\end{array}\right.\qquad
w_{\varepsilon}(0,-7\ell-\varepsilon)=0=w_{\varepsilon}(0,-2\ell)$

and this yields (6.19). We are left to establish (6.20). We point out that

$X_{\varepsilon}(t,0,-6\ell)+\varepsilon<X_{\varepsilon}(t,0,-3\ell),\quad
X_{\varepsilon}(t,0,-2\ell)+\varepsilon<X_{\varepsilon}(t,0,-\delta)$

provided $t$ is sufficiently small. The above inequalities imply

$w_{\varepsilon}(t,X_{\varepsilon}(t,0,-6\ell))=0=w_{\varepsilon}(t,X_{\varepsilon}(t,0,-2\ell))$

and, since we also have
$w_{\varepsilon}(t,X_{\varepsilon}(t,0,-7\ell)-\varepsilon)=0=w_{\varepsilon}(t,X_{\varepsilon}(t,0,-3\ell)-\varepsilon)$
for $t$ sufficiently small, this in turn yields

(6.21)
$\begin{split}\int_{X_{\varepsilon}(t,0,-7\ell)-\varepsilon}^{X_{\varepsilon}(t,0,-2\ell)}\left|\frac{\partial
w_{\varepsilon}}{\partial x}(t,x)\right|dx&\geq
2\sup\\{w_{\varepsilon}(t,x),\;x\in]X_{\varepsilon}(t,0,-7\ell)-\varepsilon,X_{\varepsilon}(t,0,-6\ell)[\\}\\\
&\quad+2\sup\\{w_{\varepsilon}(t,x),\;x\in]X_{\varepsilon}(t,0,-3\ell)-\varepsilon,X_{\varepsilon}(t,0,-2\ell)[\\}\\\
&\geq
2w_{\varepsilon}(t,X_{\varepsilon}(t,0,-7\ell)+2w_{\varepsilon}(t,X_{\varepsilon}(t,0,-3\ell)\phantom{\int}\end{split}$

for $t$ sufficiently small. To evaluate
$w_{\varepsilon}(t,X_{\varepsilon}(t,0,-7\ell))$ we point out that, since
$\eta=\mathbbm{1}_{]-1,0[}$ and $V(w)=1-w$, the equation (3.1) for the
material derivative of $w_{\varepsilon}$ boils down to

$\partial_{t}w_{\varepsilon}+[1-w_{\varepsilon}]\partial_{x}w_{\varepsilon}=u_{\varepsilon}(t,x+\varepsilon)\frac{w_{\varepsilon}(t,x+\varepsilon)-w_{\varepsilon}(t,x)}{\varepsilon}.$

Since $\ell>2\varepsilon$, then
$w_{\varepsilon}(0,-7\ell)=h=w_{\varepsilon}(0,-7\ell+\varepsilon)$ and hence
by the above formula

$\left.\frac{dw_{\varepsilon}(t,X_{\varepsilon}(t,0,-7\ell))}{dt}\right|_{t=0}=0\implies
w_{\varepsilon}(t,X_{\varepsilon}(t,0,-7\ell))=h+o(t)\quad t\to 0^{+}.$

By analogous considerations we get
$w_{\varepsilon}(t,X_{\varepsilon}(t,0,-3\ell))=h+o(t)\quad t\to 0^{+}$ and by
plugging these equalities into (6.21) we arrive at (6.20). ∎

###  6.4. Conclusion of the proof of Theorem 1.4

We fix a sequence $\\{\varepsilon_{n}\\}$ satisfying (1.13) and we take the
same intial datum $u_{0}$ as in (6.5), where $\\{h_{n}\\}$ is any sequence
such that $0\leq h_{n}\leq 1$ and the series $\sum_{n=1}^{\infty}h_{n}$
converges. We now show that, for any $n\in\mathbb{N}$, we can find $t_{n}>0$
satisfying (1.14).
Step 1: if $n=1$ we set $s(x):=\sum_{k=2}^{\infty}v_{h_{k}\ell_{k}}(x),$ which
satisfies (6.11) provided $\delta:=8\ell_{2}<2\ell_{1}$. Since
$\ell_{2}=\varepsilon_{2}/4$, $\ell_{1}=\varepsilon_{1}/4$, this inequality
boils down to $2\varepsilon_{2}<\varepsilon_{1}/2$, which is satisfied owing
to (1.13). By combining Lemma 6.3 and Lemma 6.4 we then get that (1.14) holds
true for $n=1$ and some $t_{1}$ sufficiently small.
Step 2: we now fix $n>1$ and evaluate
$\mathrm{TotVar}\,w_{\varepsilon_{n}}(t,\cdot)$ for $t=0$ and for $t\to
0^{+}$. The basic idea to do so is that we separately consider the
contribution to the total variation of each of the first $n-1$ building blocks
and then the contribution of all the remaining ones. It turns out that to each
of the first $n-1$ building blocks we can apply Lemma 6.5 and hence conclude
that for each of them the total variation can decrease at most of $o(t)$. We
can apply Lemma 6.3 and Lemma 6.4 to the remaining blocks and conclude that
their total variation increases of some factor proportional to $t$. By adding
the two contributions, we arrive at (1.14). We now provide the technical
details.
Step 2A: we show that each of the first $n-1$ evolve independently of the rest
of the solution, at least for sufficiently small $t>0$. To this end, we point
out that, for every $k=1,\dots,n-1$,

(6.22)
$8\ell_{k+1}+\varepsilon_{n}=2\varepsilon_{k+1}+\varepsilon_{n}\stackrel{{\scriptstyle\eqref{e:ticino}}}{{<}}\frac{1}{8}\varepsilon_{k}+\frac{1}{16}\varepsilon_{k}=\frac{3}{16}\varepsilon_{k}<\frac{1}{4}\varepsilon_{k}=\ell_{k}<2\ell_{k}.$

This implies that $-7\ell_{k+1}-\varepsilon_{n}>-2\ell_{k}$ and hence that
$X_{\varepsilon_{n}}(t,0,-7\ell_{k+1})-\varepsilon_{n}>X_{\varepsilon_{n}}(t,0,-2\ell_{k})$
provided $t$ is sufficiently small. In particular, for any such $t$ we have

(6.23)
$\mathrm{TotVar}\,w_{\varepsilon_{n}}(t,\cdot)=\sum_{k=1}^{n-1}\int_{X_{\varepsilon_{n}}(t,0,-7\ell_{k})-\varepsilon_{n}}^{X_{\varepsilon_{n}}(t,0,-2\ell_{k})}\left|\frac{\partial
w_{\varepsilon_{n}}}{\partial
x}(t,x)\right|dx+\int_{X_{\varepsilon_{n}}(t,0,-7\ell_{n})-\varepsilon_{n}}^{0}\left|\frac{\partial
w_{\varepsilon_{n}}}{\partial x}(t,x)\right|dx.$

Step 2B: we now fix $k=1,\dots,n-1$ and evaluate the $k$-th term in the sum in
(6.23). We set $s(x):=\sum_{j=k+1}^{\infty}v_{h_{j},\ell_{j}}(x)$ and
$\delta:=8\ell_{k+1}$. Owing to (6.22) and to (1.13) we have
$\ell_{k}>\max\\{2\varepsilon_{n},\varepsilon_{n}+\delta\\}$. By repeating the
same argument as in the proof of Lemma 6.5 we then get

(6.24) $\int_{-7\ell_{k}-\varepsilon_{n}}^{-2\ell_{k}}\left|\frac{\partial
w_{\varepsilon_{n}}}{\partial
x}(0,x)\right|dx=4h_{k},\qquad\int_{X_{\varepsilon_{n}}(t,0,-7\ell_{k})-\varepsilon_{n}}^{X_{\varepsilon_{n}}(t,0,-2\ell_{k})}\left|\frac{\partial
w_{\varepsilon_{n}}}{\partial x}(t,x)\right|dx\geq 4h_{k}+o(t)\quad t\to
0^{+}.$

Step 2C: we control the second term in (6.23). We set
$s(x):=\sum_{j=n+1}^{\infty}v_{h_{j}\ell_{j}}(x)$ and $\delta:=8\ell_{n+1}$
and point out that $\delta<2\ell_{n}$ by (1.13). By applying Lemma 6.3 and
Lemma 6.4 we get

(6.25) $\int_{-7\ell_{n}-\varepsilon_{n}}^{0}\left|\frac{\partial
w_{\varepsilon_{n}}}{\partial
x}(0,x)\right|dx=1\qquad\int_{X_{\varepsilon_{n}}(t,0,-7\ell_{n})-\varepsilon_{n}}^{0}\left|\frac{\partial
w_{\varepsilon_{n}}}{\partial
x}(t,x)\right|dx=1+2\left(\frac{2-h_{n}}{4}\right)h_{n}t+o(t)\quad t\to
0^{+}.$

Step 2D: we plug (6.24) and (6.25) into (6.23) and conclude that
$\mathrm{TotVar}\,w_{\varepsilon_{n}}(0,\cdot)=4\sum_{k=1}^{n-1}h_{k}+1$ and
that

$\mathrm{TotVar}\,w_{\varepsilon_{n}}(t,\cdot)\geq
4\sum_{k=1}^{n-1}h_{k}+1+2\left(\frac{2-h_{n}}{4}\right)h_{n}t+o(t)\quad t\to
0^{+},$

which establishes (1.14).

## Acknowledgments

M.C. and E.M. are supported by the SNF Grant 182565. G.C. is supported by the
ERC StG 676675 FLIRT. L.V.S. is a member of the PRIN 2020 Project 20204NT8W4
and of the GNAMPA group of INDAM.

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