#  Large-time asymptotic stability of Riemann shocks
of scalar balance laws

V. Duchêne Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
<EMAIL_ADDRESS>L.M. Rodrigues Univ Rennes & IUF, CNRS, IRMAR
- UMR 6625, F-35000 Rennes, France<EMAIL_ADDRESS>Research of LMR was partially supported by the city of Rennes.

###### Abstract

We prove the large-time asymptotic orbital stability of strictly entropic
Riemann shock solutions of first order scalar hyperbolic balance laws, under
piecewise regular perturbations provided that the source term is dissipative
about endstates of the shock. Moreover the convergence towards a shifted
reference state is exponential with a rate predicted by the linearized
equations about constant endstates.

Keywords: Riemann shocks; asymptotic stability; scalar balance laws.

2010 MSC: 35B35, 35L02, 35L67, 35B40, 35L03, 37L15.

## Introduction

In the present contribution, we study the large-time asymptotic behavior of
solutions to first order scalar hyperbolic balance laws, that is, in the one-
dimensional case111While most of this work is restricted to dimension one, we
also discuss in Section 3 how ideas therein extend to multidimensional
settings., of the form

$\partial_{t}u+\partial_{x}\big{(}f(u)\big{)}=g(u),\qquad
u:\mathbb{R}^{+}\times\mathbb{R}\to\mathbb{R}\,,$ (1)

in a neighborhood of strictly entropy-admissible Riemann shocks, that is,
about strictly entropy-admissible traveling waves with profiles piecewise
constant and exhibiting a single discontinuity.

Equations such as (1) are prototypes for dynamics where only convective and
reaction effects are relevant, and, as such, are ubiquitous in applications,
at least as first-order approximations in some particular regimes. In
particular, when $f(u)=c(u)\,u$ and $g(u)=r(u)\,u$, it describes the evolution
of a density $u$ of point particles moving with speed $c$ and reacting at rate
$r$.

In comparison with the purely conservative case encoded by homogeneous
conservation laws (i.e. with $g\equiv 0$), the (local) well-posedness of the
standard initial-value (Cauchy) problem for (1) is not significantly altered
by the addition of sufficiently smooth (say locally Lipschitz) reaction terms
$g$. In particular the theory of Kružkov [Kru70] applies and there exists a
unique bounded local-in-time entropy weak solution for any bounded initial
data. However in contrast the large-time asymptotic behavior of the solutions
is expected to be deeply impacted by the presence of the source term, even
when it does not lead to reaction blow-up, for instance when $g\in
W^{1,\infty}(\mathbb{R})$ or $g$ is dissipative at infinity. This expectation
is consistent with the simple observations that the purely reactive case (with
$f\equiv 0$) assigns a distinguished role to stable zeros of $g$ — that is,
the $u_{\star}$ such that $g(u_{\star})=0$ and $g^{\prime}(u_{\star})<0$ — and
that related growth and decay mechanisms are generically exponentially fast,
hence much stronger than the algebraic decay involved in the purely
conservative large-time dynamics. That the reaction term plays a dominant role
— at least near equilibria — is also supported by the fact that the linearized
operator about zeros of $g$, that is

$L=-f^{\prime}(u_{\star})\partial_{x}+g^{\prime}(u_{\star}),$

on, say, $BUC^{0}(\mathbb{R})$ with domain $BUC^{1}(\mathbb{R})$ when
$f^{\prime}(u_{\star})\neq 0$, $BUC^{0}(\mathbb{R})$ otherwise, is closed
densely defined with spectrum $g^{\prime}(u_{\star})+i\mathbb{R}$ if
$f^{\prime}(u_{\star})\neq 0$, $\\{g^{\prime}(u_{\star})\\}$ otherwise. In
other words, the spectral stability of zeros of $g$ as equilibria of (1)
agrees with their stability as equilibria of the purely reactive equation.
Incidentally note that we have used notation $BUC^{k}$ to denote the set of
${\mathcal{C}}^{k}$ functions whose derivatives up to order $k$ are bounded
and uniformly continuous.

Another good grasp at new large-time phenomena (compared to the conservative
case) is already obtained from the analysis of the structure of relative
equilibria, namely in the present case of traveling waves. Since the presence
of a source term discards self-similarity, these are the most natural
candidates to serve as asymptotic profiles or building blocks of a large-time
description. Traveling waves of (1) are given as
$u(t,x)={\underline{U}}(x-\sigma t)$ with wavespeed $\sigma$ and waveprofile
${\underline{U}}$ solving

$(f({\underline{U}})-\sigma\,{\underline{U}})^{\prime}=g({\underline{U}})\,.$

One striking novelty in the non-homogeneous setting is the existence of
traveling wave solutions with non-trivial profiles, whereas in the
conservative case only piecewise constant profiles are available and the only
spatially periodic entropy-admissible profiles are constant. The most obvious
ones are obtained by picking two consecutive non degenerate zeros $u_{-}$ and
$u_{+}$ of $g$, a speed $\sigma\notin f^{\prime}([u_{-},u_{+}])$ and solving
${\underline{U}}^{\prime}=g({\underline{U}})/(f^{\prime}({\underline{U}})-\sigma)$
between these two zeros. Yet, in this configuration one of the two endstates
is spectrally unstable and the corresponding front inherits this instability.
More interesting waves are obtained if one allows the presence of a sonic, or
characteristic, point in the profile, that is, a point where
$f^{\prime}({\underline{U}})-\sigma$ vanishes. Necessarily then the wavespeed
$\sigma$ must be equal to the sound speed $f^{\prime}(u_{\star})$ at a zero
$u_{\star}$ of $g$. In the non degenerate bistable case when $u_{-}$,
$u_{\star}$ and $u_{+}$ are three consecutive zeros of $g$ with
$g^{\prime}(u_{-})<0$, $g^{\prime}(u_{+})<0$ and $g^{\prime}(u_{\star})>0$ and
$\sigma=f^{\prime}(u_{\star})$, $f^{\prime\prime}(u_{\star})\neq 0$, one
indeed derives spectrally stable waves in this way, that are fronts connecting
$u_{-}$ and $u_{+}$ through $u_{\star}$. As a consequence of the foregoing
discussion, note that the presence of a nondegenerate source term selects a
discrete set of constant solutions, but also a discrete set of wavespeeds for
stable fronts. Beyond (discontinuous or smooth) fronts and constant solutions,
the equation may also support spatially periodic traveling waves. These are
however necessarily discontinuous and, as a consequence of Lax’s admissibility
condition, each of their smooth part must also contain a sonic point (see
[JNR+18] for details, on a closely related system case).

Under rather natural assumptions on $f$ and $g$ — including the strict
convexity of $f$ and the dissipativity at infinity of $g$ —, it has been
proved that starting from an $L^{\infty}$ initial data that is either
spatially periodic or is constant near $-\infty$ and near $\infty$, the large-
time dynamics is indeed well captured in $L^{\infty}$ topology by piecing
together traveling waves (constants, fronts or periodic waves). In the
periodic setting [FH93, Lyb94, Sin95, Sin97a], every solution approaches
asymptotically either a periodic (necessarily discontinuous) traveling wave,
or a constant equilibrium. Moreover, periodic traveling waves are actually
unstable and the rate of convergence is exponential in the latter case whereas
it may be arbitrarily slow in the former case, even when restricting to
solutions that are initially close and do converge to a periodic traveling
wave. Starting from data with essentially compact support [Sin96, MS97], the
large-time asymptotics may a priori involve several blocks of different kinds
(constants, fronts or periodics). Yet the scenario generating periodic blocks
is also non generic and unstable. Note that at the level of regularity
considered there the strict convexity assumption on $f$ plays a key role as it
impacts the structure of possible discontinuities. The few contributions
relaxing the convexity assumption add severe restrictions on $g$ or on the
initial data, for instance linearity of $g$ in [Lyb92], Riemann initial data
in [Sin97b, Mas00] and monotonicity of the initial data in [Mas98].

At a technical level, one key ingredient in the proofs of the aforementioned
series of investigations are generalized characteristics of Dafermos [Daf77].
They provide a formulation of the equation that is well suited to comparison
principles thus to asymptotics in $L^{\infty}$ topology. Our goal here is in a
neighborhood of one stable traveling wave (of a specific kind) to complete the
picture with a description in stronger topologies assuming more regularity but
less localization on initial data. By doing so we expect to contribute to put
on a par the stability theory for (1) with the one successfully derived over
the years for parabolic systems (see for instance [KP13] for the stability of
constants, fronts and solitary waves, and [JNRZ14] for periodic waves). In
particular, we derive our asymptotics under spectral stability assumptions
that are sharp up to the exclusion of limit cases. Among the difficulties to
overcome in carrying out such a general program are the absence of
regularization effects sufficiently strong to rely on a Duhamel formula based
on a linearization about the reference wave and the presence of
discontinuities and/or of sonic points in the profiles themselves that alter
even the nature of the underlying spectral problems.

Whereas in a companion paper [DR] we do study waves exhibiting sonic points,
we restrict here as announced to the stability of Riemann shocks, that is, to
waves given by ${\underline{u}}(t,x)={\underline{U}}(x-(\psi_{0}+\sigma t))$
with initial shock position $\psi_{0}\in\mathbb{R}$, speed
$\sigma\in\mathbb{R}$ and wave profile ${\underline{U}}$ such that

${\underline{U}}(x)=\begin{cases}{\underline{u}}_{-}&\text{ if }x<0\\\
{\underline{u}}_{+}&\text{ if }x>0\end{cases}$

where $({\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{2}$,
${\underline{u}}_{+}\neq{\underline{u}}_{-}$. The function ${\underline{u}}$
is indeed an entropy-admissible solution provided that

$g({\underline{u}}_{+})=0\,,\qquad g({\underline{u}}_{-})=0\,,\qquad
f({\underline{u}}_{+})-f({\underline{u}}_{-})=\sigma({\underline{u}}_{+}-{\underline{u}}_{-})\,,$

and Oleinik’s condition holds

$\begin{cases}\qquad\qquad\qquad\sigma\,\geq\,f^{\prime}({\underline{u}}_{+})\,,&\\\\[5.0pt]
\frac{f(\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+})-f({\underline{u}}_{-})}{\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+}-{\underline{u}}_{-}}\geq\frac{f(\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+})-f({\underline{u}}_{+})}{\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+}-{\underline{u}}_{+}}&\qquad\textrm{for
any }\ \tau\in(0,1)\,,\\\\[5.0pt] \qquad\qquad\qquad
f^{\prime}({\underline{u}}_{-})\,\geq\,\sigma\,.\end{cases}$ (2)

One may readily check that

$g^{\prime}({\underline{u}}_{+})\leq 0\quad\text{ and }\quad
g^{\prime}({\underline{u}}_{-})\leq 0$ (3)

are necessary to exclude spectral instability of ${\underline{u}}$. We prove
asymptotic orbital stability in $W^{1,\infty}$ topology, with sharp
exponential decay rates and asymptotic phase, under $BUC^{1}$ perturbations
possibly jointly with perturbations on the position and the strength of the
discontinuity jump when (2) and (3) hold with strict inequalities. Likewise,
we also provide stability results under $BUC^{k}$ perturbations for any $k\geq
1$. We stress that at this stage no convexity assumption is needed. Yet, our
approach may also be extended to cases when perturbations are only piecewise
$BUC^{1}$ with a finite number222Yet for exposition purposes, we only provide
details about the case where this number is at most one. of discontinuities of
shock-type, and then we do assume that
$f^{\prime\prime}({\underline{u}}_{-})\neq 0$ and
$f^{\prime\prime}({\underline{u}}_{+})\neq 0$ (or only half of it if shock-
type discontinuities are only introduced on one side of the reference
discontinuity).

One important point contrasting with the purely conservative case is that near
${\underline{u}}$ the positions of discontinuities arising from piecewise
smooth perturbations with smooth parts sufficiently small in $BUC^{1}$ may be
predicted at leading order from the linearized dynamics. This may be intuited
by analogy from the consideration of solutions near ${\underline{u}}\equiv 0$
to

$\partial_{t}u+\partial_{x}\left(\alpha\frac{u^{2}}{2}\right)=-\beta\,u$

with $\alpha\in\mathbb{R}$, $\beta\geq 0$. On the latter basic explicit
example, by studying $\partial_{x}u$ along characteristics, one readily checks
that the existence of a classical solution and the persistence of regularity
holds globally forward-in-time if and only if
$\alpha\,\partial_{x}(u(0,\cdot))\geq-\beta$. Hence when $\beta>0$ and
$\alpha\neq 0$, shock formation may be prevented by assuming asymmetric
initial smallness on the derivative of the initial data. Incidentally note
that this asymmetry is fundamental in [Mas98]. This also hints at a
classification of discontinuities in initial data between shock-like
discontinuities across which $f^{\prime}$ decreases and rarefaction-like
discontinuities across which $f^{\prime}$ increases. The latter are removable
by a density argument in the sense that the generated dynamics may be
approximated by the one arising from a family of initial data where the
discontinuity is absent. In particular, provided results are proved under
sharp asymmetric smallness conditions, there is no loss in generality in
assuming that any discontinuity is of shock-type.

Though we hope that similar analyses could be carried out in some
system333During the finalization of the present contribution we have been
informed that a system case has been analyzed in [YZ18] with distinct but not
disjoint techniques. Parts of the arguments used in [YZ18] actually originate
in private communications of the second author of the present contribution to
the second author of [YZ18]. cases, we use here crucially the scalar structure
to analyze the evolution of the piecewise regularity in the following way.
First we extend each smooth part of the initial data to a function on
$\mathbb{R}$, that is either close to ${\underline{u}}_{-}$ or close to
${\underline{u}}_{+}$. Then we propagate each of the extended initial data and
achieve suitable estimates on the corresponding dynamics near stable constant
states. At last we use the evolved extensions to determine the evolution of
shock locations by solving the corresponding Rankine-Hugoniot conditions and
glue them along the shock curves to obtain the solution for the original
discontinuous initial data. In particular along the way in order to carry out
the second step we prove a $BUC^{1}$ asymptotic stability result for constant
solutions ${\underline{u}}$, that is, constant functions with value a zero
${\underline{u}}$ of $g$, such that $g^{\prime}({\underline{u}})<0$. Though in
principle the foregoing result could be proved —yet much less readily than
$L^{\infty}$ asymptotics— with classical characteristics and comparison
principles444Similar results could also be obtained by energy estimates
provided one relaxes the essentially sharp $BUC^{1}$ framework to the
$L^{2}$-based $H^{2}$ space. (along the lines in [Li94, Chapter 4]), we choose
to use tools as close as possible to those in the classical stability theory
[KP13, JNRZ14], relying on resolvent estimates and semigroup theory. However,
as mentioned hereinabove, since regularization effects are too weak, it is not
sufficient to consider the linearized dynamics. Instead we prove that spectral
assumptions yield decay estimates for all nearby —time and space dependent—
linear dynamics, hence actually use the evolution system (see [Paz83, Chapter
5]) rather than semigroup framework.

In the rest of the present paper, we first study the asymptotic stability of
constant states under regular perturbations in $BUC^{1}(\mathbb{R})$, as
stated in Section 1.1 and proved in Sections 1.2 and 1.3. In Section 1.4 we
extend our analysis to the case where a constant state is perturbed by a
(small) shock. Then we turn to our main concern, the asymptotic stability of
(large) Riemann shocks, under perturbations that are either regular (Section
2.1) or piecewise regular with a small shock (Section 2.2). Next, in Section 3
we investigate extensions to multidimensional settings. At last, in Section 4,
we provide some further insights on limitations and extensions of the present
analysis.

##  1 Asymptotic stability of constant states

###  1.1 Asymptotic stability under shockless perturbations

In this section, first we show the asymptotic stability of constant states
with respect to regular perturbations under the natural spectral condition.

######  Proposition 1.1.

Let $g\in{\mathcal{C}}^{2}(\mathbb{R})$ and ${\underline{u}}\in\mathbb{R}$ be
such that

$g({\underline{u}})=0\qquad\textrm{and}\qquad
g^{\prime}({\underline{u}})<0\,.$ (4)

Then for any $C_{0}>1$, there exists $\epsilon>0$ such that for any
$f\in{\mathcal{C}}^{2}(\mathbb{R})$, for any $v_{0}\in BUC^{1}(\mathbb{R})$
satisfying

$\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\epsilon\,,$

the unique maximal classical solution to (1),
$u\in{\mathcal{C}}^{0}([0,T_{*}(v_{0}));BUC^{1}(\mathbb{R}))\cap{\mathcal{C}}^{1}([0,T_{*}(v_{0}));BUC^{0}(\mathbb{R}))$
with $T_{*}(v_{0})\in(0,\infty]$, generated by the the initial data
$u\left|{}_{{}_{t=0}}\right.={\underline{u}}+v_{0}$ satisfies for any ${0\leq
t<T_{*}(v_{0})}$

$\big{\lVert}u-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}$

and if moreover $\partial_{x}v_{0}\in L^{1}(\mathbb{R})$

$\big{\lVert}\partial_{x}u(t,\cdot)\big{\rVert}_{L^{1}(\mathbb{R})}\leq\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{1}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

The foregoing proposition is a _conditional_ asymptotic stability result.
Proximity is guaranteed only as long as the solution persists as a classical
solution. A strong sign that the result tells nothing about persistence of
regularity is that the required smallness is independent of $f$ and does not
involve derivatives of $v_{0}$. This should be contrasted with the explicit
example discussed in the introduction.

In a framework involving a smallness condition with more regularity, one may
prove

######  Proposition 1.2.

Let $f,\,g\in{\mathcal{C}}^{2}(\mathbb{R})$ and ${\underline{u}}\in\mathbb{R}$
be such that

$g({\underline{u}})=0\qquad\textrm{and}\qquad
g^{\prime}({\underline{u}})<0\,.$

Then for any $C_{0}>1$, there exists $\epsilon>0$ such that for any $v_{0}\in
BUC^{1}(\mathbb{R})$ satisfying

$\big{\lVert}v_{0}\big{\rVert}_{W^{1,\infty}(\mathbb{R})}\leq\epsilon\,,$

the initial data $u\left|{}_{{}_{t=0}}\right.={\underline{u}}+v_{0}$ generates
a global unique classical solution to (1), ${u\in
BUC^{1}(\mathbb{R}^{+}\times\mathbb{R})}$, and it satisfies for any $t\geq 0$

$\displaystyle\big{\lVert}u(t,\cdot)-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\
;$
$\displaystyle\big{\lVert}\partial_{x}u(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

Assuming local convexity/concavity, one may relax part of the foregoing
smallness condition

######  Proposition 1.3.

Let $f,\,g\in{\mathcal{C}}^{2}(\mathbb{R})$ and ${\underline{u}}\in\mathbb{R}$
be such that

$g({\underline{u}})=0\,,\qquad
g^{\prime}({\underline{u}})<0\qquad\textrm{and}\qquad
f^{\prime\prime}({\underline{u}})\neq 0\,.$

Then for any $C_{0}>1$, there exists $\epsilon>0$ such that for any $v_{0}\in
BUC^{1}(\mathbb{R})$ satisfying

$\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\epsilon\qquad\textrm{and}\qquad\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}\,\leq\epsilon\,,$

the initial data $u\left|{}_{{}_{t=0}}\right.={\underline{u}}+v_{0}$ generates
a global unique classical solution to (1), ${u\in
BUC^{1}(\mathbb{R}^{+}\times\mathbb{R})}$, and it satisfies for any $t\geq 0$

$\displaystyle\big{\lVert}u(t,\cdot)-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}u(t,\cdot))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\big{\lVert}\partial_{x}u(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

By a classical approximation/compactness argument one then deduces

######  Corollary 1.4.

Let $f,\,g\in{\mathcal{C}}^{2}(\mathbb{R})$ and ${\underline{u}}\in\mathbb{R}$
be such that

$g({\underline{u}})=0\,,\qquad
g^{\prime}({\underline{u}})<0\qquad\textrm{and}\qquad
f^{\prime\prime}({\underline{u}})\neq 0\,.$

Then for any $C_{0}>1$, there exists $\epsilon>0$ such that for any $v_{0}\in
BV_{loc}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ such that
$(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}v_{0})_{-}\in
L^{\infty}(\mathbb{R})$ and

$\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\epsilon\qquad\textrm{and}\qquad\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}\,\leq\epsilon\,,$

the initial data $u\left|{}_{{}_{t=0}}\right.={\underline{u}}+v_{0}$ generates
a global unique entropy solution to (1) and it satisfies for a.e. $t\geq 0$

$\displaystyle\big{\lVert}u(t,\cdot)-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}u(t,\cdot))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$

and if moreover $v_{0}\in BV(\mathbb{R})$

$\big{\lVert}u(t,\cdot)\big{\rVert}_{TV(\mathbb{R})}\,\leq\,\big{\lVert}v_{0}\big{\rVert}_{TV(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$

while if $\partial_{x}v_{0}\in L^{\infty}(\mathbb{R})$

$\big{\lVert}\partial_{x}u(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}\,\leq\,\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

To enlighten the content of Corollary 1.4, we stress that it allows
discontinuous initial data generating small rarefaction waves but not shocks.
This does not mean that a similar result cannot hold when small shocks are
present but simply that in general, as the explicit example of the
introduction shows, they cannot be obtained by a limiting process building on
global classical solutions. This is consistent with expectations drawn from
general theory, see for instance [Bre00, Chapter 9, Problem 6].

######  Remark 1.5.

An examination of proofs shows that one may relax everywhere the assumption
that $g\in{\mathcal{C}}^{2}$. It is sufficient that $g\in{\mathcal{C}}^{1}$
and that the modulus of continuity

$\omega(r)\,=\,\max_{|u-{\underline{u}}|\leq
r}\|g^{\prime}(u)-g^{\prime}({\underline{u}})\|$

is such that $r\mapsto\omega(r)/r$ is locally integrable. This includes the
case when $g\in{\mathcal{C}}^{1,\alpha}$, $\alpha>0$. Indeed the key property
is that for any positive $C$ and $\theta$

$\int_{0}^{\infty}\omega(C\,\varepsilon\,e^{-\theta\,t})\,\operatorname{d}t\,=\,\frac{1}{\theta}\int_{0}^{C\,\varepsilon}\,\frac{\omega(r)}{r}\,\operatorname{d}r\
\stackrel{{\scriptstyle\varepsilon\to 0^{+}}}{{\longrightarrow}}\ 0\,.$

The exponential decay in time also holds for higher order derivatives without
further restriction on sizes of perturbations.

######  Proposition 1.6.

Under the assumptions of either Proposition 1.2 or Proposition 1.3, if one
assumes additionally that $f\in{\mathcal{C}}^{k+1}(\mathbb{R})$,
$g\in{\mathcal{C}}^{k}(\mathbb{R})$ with $k\in\mathbb{N}$, $k\geq 2$ then
there exists $C_{k}>0$, depending on $f$, $g$ and $k$ but not on the initial
data $v_{0}$, such that if $v_{0}\in BUC^{k}(\mathbb{R})$ additionally to
constraints in either Proposition 1.2 or Proposition 1.3, then the global
unique classical solution to (1) emerging from the initial data
${\underline{u}}+v_{0}$ satisfies $u\in
BUC^{k}(\mathbb{R}^{+}\times\mathbb{R})$ and for any $t\geq 0$

$\big{\lVert}\partial_{x}^{k}u(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}\partial_{x}^{k}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}e^{C_{k}\,\|v_{0}\|_{W^{1,\infty}}\,(1+\|v_{0}\|_{W^{1,\infty}}^{k-1})}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

The local well-posedness theory for (1) at the various levels of regularity
considered here is standard. Note in particular that in the foregoing
statements without any further constraint uniqueness holds also on any finite
time interval. Though we shall not repeat it henceforth this remark applies
equally well to all our uniqueness claims. Thus the main upshots of
Propositions 1.2 and 1.3, Corollary 1.4 and Proposition 1.6 are global
existence of classical solutions and exponential decay in time. For the
classical well-posedness theory for scalar balance laws, due to Kružkov, the
reader is referred to [Kru70] and555Unfortunately, as most of textbooks, for
expository reasons [Bre00, Chapter 6] restricts to conservation laws. Yet for
local-in-time issues, such as well-posedness, changes needed to extend from
conservation laws to balance laws are immaterial. [Bre00, Chapter 6].

For our purposes it is expedient to introduce $v\stackrel{{\scriptstyle\rm
def}}{{=}}u-{\underline{u}}$ and as long as classical solutions are concerned
work with the following quasilinear form of (1)

$\partial_{t}v+f^{\prime}({\underline{u}}+v)\partial_{x}v-g^{\prime}({\underline{u}})v=g({\underline{u}}+v)-g({\underline{u}})-g^{\prime}({\underline{u}})v.$
(5)

Note in particular that in the above formulation one cannot allow any
“regularity loss" due to a linearization of the transport term. Bearing this
in mind, prior to the consideration of a mild formulation of (5) we analyze
linear equations of the form

$\partial_{t}v+a\partial_{x}v-bv=r$ (6)

where $a$ is close to $f^{\prime}({\underline{u}})$ and $b$ close to
$g^{\prime}({\underline{u}})$ in a suitable sense. Let us anticipate that to
deal with the mild formulation of (5) and prove Propositions 1.1 and 1.2 we
could stick to the case where $b=g^{\prime}({\underline{u}})$. We shall use
the extra flexibility in the choice of $b$ only when tracking asymmetric
regularity involved in Proposition 1.3 and Corollary 1.4.

As a preliminary let us discuss the linearized equation

$\partial_{t}v+f^{\prime}({\underline{u}})\partial_{x}v-g^{\prime}({\underline{u}})v=0\,.$

A notion of solution may be obtained through the classical semigroup
formalism. For instance one may consider
$L=-f^{\prime}({\underline{u}})\partial_{x}+g^{\prime}({\underline{u}})$ on
either $L^{p}(\mathbb{R})$, $1\leq p<\infty$, or $BUC^{0}(\mathbb{R})$ with
domain $W^{1,p}(\mathbb{R})$ or $BUC^{1}(\mathbb{R})$, if
$f^{\prime}({\underline{u}})\neq 0$ and $L^{p}(\mathbb{R})$ or
$BUC^{0}(\mathbb{R})$, otherwise. The operator $L$ is then closed densely-
defined with spectrum $g^{\prime}({\underline{u}})+i\,\mathbb{R}$ if
$f^{\prime}({\underline{u}})\neq 0$, $\\{g^{\prime}({\underline{u}})\\}$
otherwise. In particular, $g^{\prime}({\underline{u}})>0$ would yield spectral
instability whereas as follows from the analysis below suitable resolvent
estimates show that $g^{\prime}({\underline{u}})<0$ provides linear asymptotic
stability with exponential rates. We refer the reader to [Paz83, N96] for
background on semigroups and their large-time behaviors.

It is already apparent here that though this does not alter significantly the
stability properties, the vanishing of transport term impacts dramatically the
regularity structure of the spectral problem. As long as we restrict to
classical solutions near a constant steady state this is immaterial since
going to a uniformly moving frame may remove possible vanishings. This would
however not be possible near the continuous stable traveling fronts described
in the introduction. In general the presence of an essential characteristic
point is a serious cause of trouble, and the reader is referred to [JNR+18,
DR] for an example of its impact on spectral problems.

As a consequence it is convenient to change coordinate frame. Explicitly for
any $\sigma\in\mathbb{R}$, by introducing ${\widetilde{v}}$ through
${\widetilde{v}}(t,x)=v(t,x+\sigma t)$ one replaces (6) with

$\partial_{t}{\widetilde{v}}+({\widetilde{a}}-\sigma)\partial_{x}{\widetilde{v}}-{\widetilde{b}}{\widetilde{v}}={\widetilde{r}}\,,$

with $({\widetilde{a}},{\widetilde{b}},{\widetilde{r}})$ defined by
$({\widetilde{a}},{\widetilde{b}},{\widetilde{r}})(t,x)=(a,b,r)(t,x+\sigma
t)$. Implicitly some of our assumptions on $a$ will build on the fact that one
may choose $\sigma$ so that ${\widetilde{a}}-\sigma$ is bounded away from
zero.

###  1.2 Linear equations

To consider (6) with time-dependent $a$ and $b$, we may either rely on or
mimic the available abstract theory for evolution systems, as described in
[Paz83, Chapter 5]. In any case the needed elementary block is the solution of
problems where $a$ and $b$ are independent of time.

As a consequence we first consider this case. With this restriction we are
back to the semigroup framework that may be analyzed directly by resolvent
estimates. In the present section we always assume that $a\,,\ b\in
BUC^{0}(\mathbb{R})$ with $a$ bounded away from zero. For such $a$, $b$,
$L_{a,\,b}=-a\partial_{x}+b$ is elliptic666Or, in a more standard terminology,
$i\,L_{a,\,b}$ is elliptic., and is a closed, densely-defined operator on
either $L^{p}(\mathbb{R})$ with domain $W^{1,p}(\mathbb{R})$, $1\leq
p<\infty$, or on $BUC^{0}(\mathbb{R})$ with domain $BUC^{1}(\mathbb{R})$. The
key basic estimate is

######  Lemma 1.7.

Assume $a,\,b\in BUC^{0}(\mathbb{R})$ with $a$ bounded away from zero.
(i). Then for any $\lambda\in\mathbb{C}$ such that

$\Re(\lambda)>\sup_{\mathbb{R}}b(\cdot)\,,$

for any $F\in BUC^{0}(\mathbb{R})$, there exists a unique
$\check{v}(\,\cdot\,;\lambda)\in BUC^{1}(\mathbb{R})$ such that

$(\lambda-L_{a,\,b})\,\check{v}(\,\cdot\,;\lambda)\,=\,F$

and moreover

$\big{\lVert}\check{v}(\,\cdot\,;\lambda)\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\frac{1}{\Re\lambda-\sup_{\mathbb{R}}b(\cdot)}\big{\lVert}F\big{\rVert}_{L^{\infty}(\mathbb{R})}\,.$

If $b$ is constant and $F\in W^{1,1}(\mathbb{R})$, then
$\check{v}(\,\cdot\,;\lambda)\in W^{1,1}(\mathbb{R})$ and

$\big{\lVert}\partial_{x}\check{v}(\,\cdot\,;\lambda)\big{\rVert}_{L^{1}(\mathbb{R})}\leq\frac{1}{\Re\lambda-b}\big{\lVert}\partial_{x}F\big{\rVert}_{L^{1}(\mathbb{R})}\,.$

Moreover if $\lambda\in\mathbb{R}$,
$\lambda\in(\sup_{\mathbb{R}}b(\cdot),\infty)$ and $F\geq 0$ then
$\check{v}(\,\cdot\,;\lambda)\geq 0$.
(ii). Assume moreover that

$a\in BUC^{1}(\mathbb{R})\,,\qquad b\textrm{ is
constant}\,,\qquad\Re(\lambda)>b-\inf_{\mathbb{R}}a^{\prime}(\cdot)\,,$

and $F\in W^{1,\infty}(\mathbb{R})$. Then

$\big{\lVert}\partial_{x}\check{v}(\,\cdot\,;\lambda)\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\frac{1}{\Re\lambda-b+\inf_{\mathbb{R}}a^{\prime}(\cdot)}\big{\lVert}\partial_{x}F\big{\rVert}_{L^{\infty}(\mathbb{R})}\,.$

###### Proof.

Let us begin with the uniqueness part. If $(\lambda-
L_{a,\,b})\,\check{v}(\,\cdot\,;\lambda)=0$ then actually

$\check{v}(\,x\,;\lambda)\,=\,e^{\int_{0}^{x}\frac{b(z)-\lambda}{a(z)}\,\operatorname{d}z}\,\check{v}_{0}$

for some constant $\check{v}_{0}\in\mathbb{C}$. Then if $a$ is positive and
bounded away from zero and $\Re(\lambda)>\sup_{\mathbb{R}}b(\cdot)$, the
boundedness near $x=-\infty$ implies $\check{v}_{0}=0$ since $\lvert
e^{\int_{0}^{x}\frac{b(z)-\lambda}{a(z)}\,\operatorname{d}z}\rvert\geq
e^{|x|\,\frac{\Re(\lambda)-\sup_{\mathbb{R}}b(\cdot)}{\|a\|_{L^{\infty}}}}$
when $x<0$. Likewise if $a$ is negative and bounded away from zero and
$\Re(\lambda)>\sup_{\mathbb{R}}b(\cdot)$, boundedness near $x=\infty$ yields
$\check{v}_{0}=0$.

From now on for definiteness we assume that $a$ is positive and bounded away
from zero. Note that there is no loss of generality since one may go from this
case to the opposite one by reversing $x$ into $-x$.

Let $F\in BUC^{0}(\mathbb{R})$. One readily checks when
$\Re(\lambda)>\sup_{\mathbb{R}}b(\cdot)$ that

$\check{v}(x;\lambda)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{-\infty}^{x}e^{\int_{y}^{x}\frac{b(z)-\lambda}{a(z)}\,\operatorname{d}z}\frac{F(y)}{a(y)}\,\operatorname{d}y$

defines $\check{v}(\,\cdot\,;\lambda)\in BUC^{1}(\mathbb{R})$ and that

$\displaystyle\lvert\check{v}(x;\lambda)\rvert$
$\displaystyle\leq\frac{\big{\lVert}F\big{\rVert}_{L^{\infty}(\mathbb{R})}}{\Re\lambda-\sup_{\mathbb{R}}b(\cdot)}\int_{-\infty}^{x}e^{\int_{y}^{x}\frac{b(z)-\Re\lambda}{a(z)}\,\operatorname{d}z}\frac{\Re\lambda-b(y)}{a(y)}\,\operatorname{d}y=\frac{\big{\lVert}F\big{\rVert}_{L^{\infty}(\mathbb{R})}}{\Re\lambda-\sup_{\mathbb{R}}b(\cdot)}\,,$
$\displaystyle\partial_{x}\check{v}(x;\lambda)$
$\displaystyle=\frac{F(x)}{a(x)}+\int_{-\infty}^{x}\frac{b(x)-\lambda}{a(x)}e^{\int_{y}^{x}\frac{b(z)-\lambda}{a(z)}\,\operatorname{d}z}\frac{F(y)}{a(y)}\,\operatorname{d}y\,.$

It is also straightforward to check that if moreover $\lambda\in\mathbb{R}$
and $F\geq 0$ then $\check{v}(\cdot;\lambda)\geq 0$. When moreover $b$ is
constant and $\partial_{x}F\in L^{1}(\mathbb{R})+L^{\infty}(\mathbb{R})$, the
latter expression may be integrated by parts into

$\displaystyle\partial_{x}\check{v}(x;\lambda)$
$\displaystyle=\int_{-\infty}^{x}e^{\int_{y}^{x}\frac{b-\lambda}{a(z)}\,\operatorname{d}z}\frac{\partial_{y}F(y)}{a(x)}\,\operatorname{d}y$
$\displaystyle=\int_{-\infty}^{x}e^{\int_{y}^{x}\frac{b-\lambda-a^{\prime}(z)}{a(z)}\,\operatorname{d}z}\frac{\partial_{y}F(y)}{a(y)}\,\operatorname{d}y\,.$

The latter expression may be used to obtain the
$\dot{W}^{1,\infty}\to\dot{W}^{1,\infty}$ bound as we have derived the
$L^{\infty}\to L^{\infty}$ bound. Concerning the former expression it may be
integrated in $x$ to deduce the $\dot{W}^{1,1}\to\dot{W}^{1,1}$ bound since
when $b$ is constant for any $y$

$\int_{y}^{\infty}e^{\int_{y}^{x}\frac{b-\Re(\lambda)}{a(z)}\,\operatorname{d}z}\frac{\operatorname{d}x}{a(x)}\,=\,\frac{1}{\Re(\lambda)-b}\,.$

∎

With the above frozen-time resolvent estimates, for general coefficients $a,b$
one may first change frame to ensure that $a$ is bounded away from zero then
apply general theorems on evolution systems. See for instance [Paz83, Chapter
5, Theorem 3.1] with $X=BUC^{0}(\mathbb{R})$ and $Y=BUC^{1}(\mathbb{R})$, and
apply [Paz83, Chapter 5, Theorem 2.3] to reduce the verification of assumption
$(H_{2})$ there to the case where $b$ is constant.

######  Proposition 1.8.

Let $T\in(0,\infty]$, $a\in{\mathcal{C}}^{0}([0,T),BUC^{1}(\mathbb{R}))$,
$b\in{\mathcal{C}}^{0}([0,T),BUC^{0}(\mathbb{R}))$. Then the family of
operators ${\mathcal{L}}_{t}=L_{a(t,\cdot),\,b(t,\cdot)}$ generates an
evolution system ${\mathcal{S}}_{a,b}$ on $BUC^{0}(\mathbb{R})$ such that for
any $v_{0}\in BUC^{0}(\mathbb{R})$, any $0\leq s\leq t<T$

$\big{\lVert}{\mathcal{S}}_{a,b}(s,t)\,v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq
e^{\int_{s}^{t}\sup_{\mathbb{R}}b(\tau,\cdot)\,\operatorname{d}\tau}\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\,,$

and $v_{0}\geq 0$ implies ${\mathcal{S}}_{a,b}(s,t)\,v_{0}\geq 0$. If moreover
$b$ is constant, then $v_{0}\in BUC^{0}(\mathbb{R})\cap W^{1,1}(\mathbb{R})$
yields for any $0\leq s\leq t<T$

$\big{\lVert}\partial_{x}{\mathcal{S}}_{a,b}(s,t)\,v_{0}\big{\rVert}_{L^{1}(\mathbb{R})}\leq
e^{(t-s)\,b}\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{1}(\mathbb{R})}\,,$

and from $v_{0}\in W^{1,\infty}(\mathbb{R})$ stems for any $0\leq s\leq t<T$,

$\big{\lVert}\partial_{x}{\mathcal{S}}_{a,b}(s,t)\,v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq
e^{(t-s)\,b-\int_{s}^{t}\inf_{\mathbb{R}}\partial_{x}a(\tau,\cdot)\,\operatorname{d}\tau}\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\,.$

###  1.3 The shockless nonlinear problem

In this section we complete the proofs of results from Section 1.1.

###### Proof of Proposition 1.1..

Let $\epsilon\in(0,1]$. Pick a classical solution $u={\underline{u}}+v$
starting from ${\underline{u}}+v_{0}$ such that
$\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\epsilon$. Then if
$u$ exists (as a classical solution) on $[0,t_{0})$, for any $0\leq t<t_{0}$

$v(t,\cdot)\,=\,{\mathcal{S}}_{f^{\prime}({\underline{u}}+v),\,g^{\prime}({\underline{u}})}v_{0}+\int_{0}^{t}{\mathcal{S}}_{f^{\prime}({\underline{u}}+v),\,g^{\prime}({\underline{u}})}(s,t)\big{(}g({\underline{u}}+v)-g({\underline{u}})-g^{\prime}({\underline{u}})v\big{)}(s,\cdot)\,\operatorname{d}s\,.$

Therefore if moreover for any $t\in[0,t_{0})$,
$\big{\lVert}v(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq
2\epsilon\,e^{g^{\prime}({\underline{u}})\,t}$, then for any $t\in[0,t_{0})$

$e^{-g^{\prime}({\underline{u}})\,t}\big{\lVert}v(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}+2\epsilon\,C_{g}\,\int_{0}^{t}e^{g^{\prime}({\underline{u}})\,s}\,\left(e^{-g^{\prime}({\underline{u}})\,s}\big{\lVert}v(s,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}\right)\operatorname{d}s$

where
$C_{g}=\tfrac{1}{2}\big{\lVert}g^{\prime\prime}\big{\rVert}_{L^{\infty}([{\underline{u}}-2\epsilon,{\underline{u}}+2\epsilon])}$,
so that for any $t\in[0,t_{0})$,

$\big{\lVert}v(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\,e^{g^{\prime}({\underline{u}})\,t}\,e^{2\epsilon\,C_{g}\int_{0}^{t}e^{g^{\prime}({\underline{u}})\,s}\operatorname{d}s}\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}\,e^{g^{\prime}({\underline{u}})\,t}\,e^{\epsilon\,\frac{2C_{g}}{|g^{\prime}({\underline{u}})|}}\,.$
(7)

If $\epsilon$ is small enough to ensure that
$\exp(\epsilon\,\frac{2C_{g}}{|g^{\prime}({\underline{u}})|})<2$ then a
continuity argument yields that estimate (7) holds as long as $u$ persists as
a classical solution. Since
$\exp(\epsilon\,\frac{2C_{g}}{|g^{\prime}({\underline{u}})|})$ may be brought
arbitrarily close to $1$ by choosing $\epsilon$ small, this proves the
$L^{\infty}$ part of Proposition 1.1. With this bound in hands we deduce even
more directly that if moreover $\partial_{x}v_{0}\in L^{1}$ then

$\big{\lVert}\partial_{x}v(t,\cdot)\big{\rVert}_{L^{1}(\mathbb{R})}\leq\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{1}(\mathbb{R})}\,e^{g^{\prime}({\underline{u}})\,t}\,e^{\epsilon\,\frac{2C_{0}\,C_{g}}{|g^{\prime}({\underline{u}})|}}\,.$

This achieves the proof by taking $\epsilon$ even smaller if needed. ∎

The proof of Proposition 1.2 being completely similar, we omit it.

###### Proof of Proposition 1.3..

First we fix $\epsilon\in(0,1]$ sufficient small to satisfy conclusions of
Proposition 1.1 and to ensure that on
$[{\underline{u}}-C_{0}\epsilon,{\underline{u}}+C_{0}\epsilon]$,
$f^{\prime\prime}$ is of the sign of $f^{\prime\prime}({\underline{u}})$. To
proceed we use that if $u={\underline{u}}+v$ persists as a classical solution
on $[0,t_{0})$ then for $t\in[0,t_{0})$

$\partial_{x}v(t,\cdot)={\mathcal{S}}_{f^{\prime}({\underline{u}}+v),\,g^{\prime}({\underline{u}}+v)-f^{\prime\prime}({\underline{u}}+v)\partial_{x}v}(0,t)\,\partial_{x}v_{0}$

thus by linearity and preservation of non negativity

$(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\partial_{x}v(t,\cdot))_{-}\leq{\mathcal{S}}_{f^{\prime}({\underline{u}}+v),\,g^{\prime}({\underline{u}}+v)-f^{\prime\prime}({\underline{u}}+v)\partial_{x}v}(0,t)\,(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\partial_{x}v_{0})_{-}\,.$

Therefore if moreover for any $t\in[0,t_{0})$,
$\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\partial_{x}v(t,\cdot))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq
2\epsilon\,e^{g^{\prime}({\underline{u}})\,t}$, then for any $t\in[0,t_{0})$

$\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\partial_{x}v(t,\cdot))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}e^{g^{\prime}({\underline{u}})\,t}e^{\epsilon\frac{2C_{f}+C_{0}\,C_{g}}{|g^{\prime}({\underline{u}})|}}$

with
$C_{f}=\big{\lVert}f^{\prime\prime}\big{\rVert}_{L^{\infty}([{\underline{u}}-C_{0}\epsilon,{\underline{u}}+C_{0}\epsilon])}$
and
$C_{g}=\big{\lVert}g^{\prime\prime}\big{\rVert}_{L^{\infty}([{\underline{u}}-C_{0}\epsilon,{\underline{u}}+C_{0}\epsilon])}$.
By choosing $\epsilon$ sufficiently small so that
$e^{\epsilon\frac{2C_{f}+C_{0}\,C_{g}}{|g^{\prime}({\underline{u}})|}}\leq\min(\\{2,C_{0}\\})$,
one deduces that if $u={\underline{u}}+v$ persists as a classical solution on
$[0,t_{0})$ then for $t\in[0,t_{0})$

$\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\partial_{x}v(t,\cdot))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

One concludes by noticing that this implies that if $u={\underline{u}}+v$
persists as a classical solution on $[0,t_{0})$ then for $t\in[0,t_{0})$

$\big{\lVert}\partial_{x}v(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$

which rules out finite-time blow-up. ∎

###### Proof of Proposition 1.6..

Propagation of regularity being classical, we focus on the decay estimate.
Note that since we already know that $v$ is small in $L^{\infty}$

$\displaystyle\partial_{x}^{k}v(t,\cdot)$
$\displaystyle={\mathcal{S}}_{f^{\prime}({\underline{u}}+v),\,g^{\prime}({\underline{u}})}(0,t)\,\partial_{x}^{k}v_{0}$
$\displaystyle+\int_{0}^{t}{\mathcal{S}}_{f^{\prime}({\underline{u}}+v),\,g^{\prime}({\underline{u}})}(s,t)\Big{(}c_{0}(v(s,\cdot))\,v(s,\cdot)\,\partial_{x}^{k}v(s,\cdot)+\sum_{\begin{subarray}{c}2\leq
m\leq|\alpha|\\\ \alpha\in(\mathbb{N}^{*})^{m},\
|\alpha|\in\\{k,k+1\\}\end{subarray}}c_{\alpha}(v(s,\cdot))\,\prod_{i=1}^{m}\partial_{x}^{\alpha_{i}}v(s,\cdot)\Big{)}\operatorname{d}s$

with $c_{0}$, $c_{\alpha}$ bounded. Note moreover that for any $1\leq\ell\leq
k$, for some $C\geq 0$ and any function $w$

$\big{\lVert}\partial_{x}^{\ell}w\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq
C\,\min(\\{\big{\lVert}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{\frac{k-\ell}{k}}\big{\lVert}\partial_{x}^{k}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{\frac{\ell}{k}},\
\big{\lVert}\partial_{x}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{\frac{k-\ell}{k-1}}\big{\lVert}\partial_{x}^{k}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{\frac{\ell-1}{k-1}}\\})$

so that for any $2\leq m\leq k+1$, $\alpha\in(\mathbb{N}^{*})^{m}$,
$|\alpha|\in\\{k,k+1\\}$, there exists $C^{\prime}$ and $C^{\prime\prime}$
such that for any $w$

$\displaystyle\big{\lVert}\prod_{i=1}^{m}\partial_{x}^{\alpha_{i}}w\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq
C^{\prime}\,\min(\\{\big{\lVert}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{m-\frac{|\alpha|}{k}}\big{\lVert}\partial_{x}^{k}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{\frac{|\alpha|}{k}},\
\big{\lVert}\partial_{x}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{m-\frac{|\alpha|-m}{k-1}}\big{\lVert}\partial_{x}^{k}w\big{\rVert}_{L^{\infty}(\mathbb{R})}^{\frac{|\alpha|-m}{k-1}}\\})$
$\displaystyle\leq
C^{\prime\prime}\,\big{\lVert}w\big{\rVert}_{W^{1,\infty}}^{m-1}\,\big{\lVert}\partial_{x}^{k}w\big{\rVert}_{L^{\infty}(\mathbb{R})}\,.$

The proof is then concluded by first invoking the bounds of either Proposition
1.2 or Proposition 1.3 jointly with those of Proposition 1.8 then applying the
Grönwall lemma. ∎

###### Proof of Corollary 1.4..

An initial data as in Corollary 1.4 may be approximated through cut-off with
sufficiently slow cut-off functions and convolution with positive kernels by
initial data satisfying constraints of Proposition 1.3. Bounds of Propositions
1.1 and 1.3, jointly with equation (1), are then sufficient to extract a
subsequence converging pointwise and uniformly bounded. With the latter one
may take limits in weak formulations encoding the notion of entropy solution,
hence proving the existence of an entropy solution starting from the
prescribed initial data and satisfying claimed bounds. We refer the reader to
[Bre00, Section 6.2] for details on the latter compactness arguments. ∎

###  1.4 Perturbation by small shocks

In this section we extend Proposition 1.3 to the case where the perturbation
contains a shock.

We provide a description of the solution $u$ as regular on

$\Omega^{\psi}\stackrel{{\scriptstyle\rm
def}}{{=}}\mathbb{R}_{+}\times\mathbb{R}\setminus\\{\,(t,\psi(t))\,|\,t\geq
0\,\\}$

where $\psi$ follows the position of the shock.

######  Remark 1.9.

It may be convenient to think of $u$ as being of the form

$u:(t,x)\mapsto{\widetilde{u}}(t,x-\psi(t))$

with smooth unknowns $\psi:\mathbb{R}_{+}\to\mathbb{R}$ and
${\widetilde{u}}:\mathbb{R}_{+}\times\mathbb{R}^{\star}\to\mathbb{R}$. Though
we shall not use this form explicitly here (partly because it is not
convenient when two shocks are present), it underlies our strategy and
statements. In particular, henceforth $\partial_{x}u$ will not denote the
distributional derivative of $u\in\mathcal{D}^{\prime}(\mathbb{R})$ but its
smooth part

$\partial_{x}u:(t,x)\mapsto\partial_{x}{\widetilde{u}}(t,x-\psi(t))\,.$

Similarly, for $k\in\mathbb{N}^{\star}$ and $\Omega$ an open domain, we denote
$W^{k,\infty}(\Omega)$ (resp. $BUC^{k}(\Omega)$), the space of functions such
that the restrictions to each connected component of $\Omega$,
$\tilde{\Omega}$, belongs to $W^{k,\infty}(\check{\Omega})$ (resp.
$BUC^{k}(\check{\Omega})$). These spaces are endowed with the obvious norms
consistent with this definition.

For such a $u$ to satisfy the equation in distributional sense we require $u$
to satisfy it in a classical sense on $\Omega^{\psi}$ and that also holds the
Rankine-Hugoniot condition, for any $t\geq 0$

$f(u_{r}(t))-f(u_{l}(t))=\psi^{\prime}(t)(u_{r}(t)-u_{l}(t))$

where $u_{l}(t)=\lim_{\delta\searrow 0}u(t,\psi(t)-\delta)$ and
$u_{r}(t)=\lim_{\delta\searrow 0}u(t,\psi(t)+\delta)$. Moreover when
$f^{\prime\prime}({\underline{u}})\neq 0$ then if $u$ is sufficiently close to
${\underline{u}}$, its admissibility as an entropy solution is equivalent to
Lax’s condition [Bre00, Section 4.5]

$f^{\prime}(u_{r}(t))<f^{\prime}(u_{l}(t)),\qquad t\geq 0\,.$

Of course this requires initially $f^{\prime}(u_{r}(0))<f^{\prime}(u_{l}(0))$.
Recall however that discontinuities with
$f^{\prime}(u_{r}(0))>f^{\prime}(u_{l}(0))$ are already covered by Corollary
1.4.

######  Proposition 1.10.

Let $f,\,g\in{\mathcal{C}}^{2}(\mathbb{R})$ and ${\underline{u}}\in\mathbb{R}$
be such that

$g({\underline{u}})=0\,,\qquad
g^{\prime}({\underline{u}})<0\qquad\textrm{and}\qquad
f^{\prime\prime}({\underline{u}})\neq 0\,.$

For any $C_{0}>1$, there exist $\epsilon>0$ and $C>0$ such that for any
$\psi_{0}\in\mathbb{R}$ and $\tilde{v}_{0}\in BUC^{1}(\mathbb{R}^{\star})$
satisfying

$\big{\lVert}\tilde{v}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}\leq\epsilon\quad\text{
and
}\quad\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}{\widetilde{v}}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}\leq\epsilon\,,$
(8)

and

$\lim_{\delta\searrow
0}f^{\prime}({\underline{u}}+\tilde{v}_{0}(\delta))<\lim_{\delta\searrow
0}f^{\prime}({\underline{u}}+\tilde{v}_{0}(-\delta)),$

there exists $\psi\in{\mathcal{C}}^{2}(\mathbb{R}^{+})$ and $u\in
BUC^{1}(\Omega^{\psi})$ with initial data $\psi(0)=\psi_{0}$ and
$u(0,\cdot)=({\underline{u}}+{\widetilde{v}}_{0})(\cdot+\psi_{0})$ such that
$u$ is an entropy solution to (1) and satisfies for any $t\geq 0$

$\displaystyle\big{\lVert}u(t,\cdot-\psi(t))-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R}\setminus\\{\psi(t)\\})}$
$\displaystyle\leq\big{\lVert}\tilde{v}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}u(t,\cdot-\psi(t)))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}\setminus\\{\psi(t)\\})}$
$\displaystyle\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}\tilde{v}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\big{\lVert}\partial_{x}u(t,\cdot-\psi(t))\big{\rVert}_{L^{\infty}(\mathbb{R}\setminus\\{\psi(t)\\})}$
$\displaystyle\leq\big{\lVert}\partial_{x}\tilde{v}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\lvert\psi^{\prime}(t)-f^{\prime}({\underline{u}})\rvert$
$\displaystyle\leq\big{\lVert}\tilde{v}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C\,e^{g^{\prime}({\underline{u}})\,t}\,,$

and moreover there exists $\psi_{\infty}$ such that

$\lvert\psi_{\infty}-\psi_{0}\rvert\,\leq\big{\lVert}\tilde{v}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C\,,$

and for any $t\geq 0$

$\lvert\psi(t)-\psi_{\infty}-t\,f^{\prime}({\underline{u}})\rvert\,\leq\big{\lVert}\tilde{v}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C\,e^{g^{\prime}({\underline{u}})\,t}\,.$

######  Remark 1.11.

The consideration of perturbation by small shocks is partly motivated by the
fact that smooth perturbations, small in $L^{\infty}$ but not in
$W^{1,\infty}$, may indeed form shocks in finite time. Note however that
whereas Proposition 1.1 does follow smooth solutions until shock formation,
Proposition 1.10 cannot be used right after shock formation since it requires
(asymmetric) smallness of the smooth part of the gradient. Indeed Proposition
1.10 is a counterpart to Proposition 1.3 whereas an analog to Proposition 1.1
would be more appropriate near a shock formation. Note however that then the
“smooth" part of solutions would then be controlled only in $W^{1,1}$.

######  Remark 1.12.

Since the problem is known to be globally well-posed in the class of
$L^{\infty}$ entropy solutions, one may rightfully wonder whether the result
could be extended to such a general class. Such an extension would lead us a
way beyond the scope of the present contribution, focused on piece-wise smooth
solutions, and very close to front-tracking algorithms. Without going that
far, let us now give some hints about first steps required to extend our
strategy in this direction. Note first that it is straightforward to extend
Proposition 1.10 to cases when the initial data contains discontinuities
leading to rarefaction waves and an arbitrary number of well-separated shocks.
Going beyond the latter case to allow for interacting shocks seems a more
tedious task but seemingly still achievable with arguments in the spirit of
those expounded in the present contribution. In particular even in the latter
case one expects that no new discontinuity arises and that paths of
discontinuities could be predicted by linearized dynamics. However to relax
constraints on derivatives, one would need to follow the path sketched in
Remark 1.11 or to approximate $L^{\infty}$ initial data by piece-wise smooth
initial data containing only flat or almost flat smooth parts but an arbitrary
large number of shocks. In both cases the prediction of the regularity
structure would be a much harder task.

###### Proof of Proposition 1.10.

To spare notational complexity, we assume henceforth that $\psi_{0}=0$ and
accordingly drop tildes on ${\widetilde{v}}_{0}$. The general case may be
dealt with either by using translation invariance or by propagating notational
changes.

We recall that the proof strategy is the following. Given an initial data
$v_{0}$ satisfying (8), we define two extensions $v_{0,\pm}$, defined on
$\mathbb{R}$, satisfying $v_{0,\pm}=v_{0}$ on $\mathbb{R}^{\pm}$, and
fulfilling the hypotheses of Proposition 1.3 near ${\underline{u}}$. Consider
$u_{\pm}$ the two global unique classical solutions to (1) emerging from the
initial data $u_{\pm}\left|{}_{{}_{t=0}}\right.={\underline{u}}+v_{0,\pm}$.
The solution $u$ is constructed by patching together $u_{+}$ and $u_{-}$ along
the curve $\psi(t)$ defined through the Rankine-Hugoniot condition.

The first step is carried out thanks to the following Lemma.

######  Lemma 1.13.

For any $C_{0}^{(0)}>1$ and any $v_{0}\in BUC^{1}(\mathbb{R}^{\star})$ there
exist $v_{0,\pm}\in BUC^{1}(\mathbb{R})$ satisfying

$v_{0}(x)=\begin{cases}v_{0,+}(x)&\text{ if }x>0,\\\ v_{0,-}(x)&\text{ if
}x<0,\end{cases}$

and

$\displaystyle\big{\lVert}v_{0,\pm}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{0}^{(0)}\,,$
$\displaystyle\big{\lVert}(\partial_{x}v_{0,\pm})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}(\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}\,,$
$\displaystyle\big{\lVert}(-\partial_{x}v_{0,\pm})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}(-\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}\,,$
$\displaystyle\big{\lVert}\partial_{x}v_{0,\pm}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}\,.$

###### Proof.

Since the situation is symmetric we only show how to extend the right part of
$v_{0}$. To do so let us introduce

$v_{0}(0^{+})\stackrel{{\scriptstyle\rm def}}{{=}}\lim_{x\searrow
0}v_{0}(x)\qquad\textrm{and}\qquad\partial_{x}v_{0}(0^{+})\stackrel{{\scriptstyle\rm
def}}{{=}}\lim_{x\searrow 0}\partial_{x}v_{0}(x)$

whose existence is guaranteed by uniform continuity.

We set

$\delta=\frac{2(C_{0}^{(0)}-1)\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{+})}}{\max(\\{1,|\partial_{x}v_{0}(0^{+})|\\})}$

and define

$v_{0,+}(x)=\begin{cases}v_{0}(0^{+})-\frac{\delta}{2}\partial_{x}v_{0}(0^{+})&\text{
if }x\in(-\infty,-\delta]\,,\\\
v_{0}(0^{+})+\big{(}x+\frac{1}{2}\delta^{-1}x^{2}\big{)}\partial_{x}v_{0}(0^{+})&\text{
if }x\in(-\delta,0]\,,\\\ v_{0}(x)&\text{ if }x>0\,.\end{cases}$

One readily checks that $v_{0,+}$ satisfies all prescribed constraints. ∎

We can now proceed with the proof of Proposition 1.10. We denote $C_{0}$ the
prescribed amplifying constant and $\epsilon$ the smallness parameter as in
the statement. First we apply Lemma 1.13 with amplification constant
$C_{0}^{(0)}=\sqrt{C_{0}}$ to receive extensions $v_{0,\pm}$. Then we apply
twice Proposition 1.3, with initial perturbations $v_{0,\pm}$ near
${\underline{u}}$ and prescribed amplification factors
$C_{0}^{\pm}=\sqrt{C_{0}}$. This is licit provided we constrain $\epsilon$ by

$\sqrt{C_{0}}\epsilon\leq\epsilon_{0}$

where $\epsilon_{0}$ encodes the smallness constraint arising from Proposition
1.3. Hence the existence of $u_{\pm}\in
BUC^{1}(\mathbb{R}^{+}\times\mathbb{R})$ global unique classical solutions to
(1) with initial data
$u_{\pm}\left|{}_{{}_{t=0}}\right.={\underline{u}}+v_{0,\pm}$ satisfying for
any $t\geq 0$,

$\displaystyle\big{\lVert}u_{\pm}(t,\cdot)-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}u_{\pm}(t,\cdot))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}))\,\partial_{x}v_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,,$
$\displaystyle\big{\lVert}\partial_{x}u_{\pm}(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq\big{\lVert}\partial_{x}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

We shall construct our solution, $u$, through the following formula:

$u(t,x)=\begin{cases}u_{-}(t,x)&\text{ if }x<\psi(t),\\\ u_{+}(t,x)&\text{ if
}x>\psi(t),\end{cases}$ (9)

where the discontinuity curve, $\psi$, is defined through the Rankine-Hugoniot
condition

$\big{(}u_{+}(t,\psi(t))-u_{-}(t,\psi(t))\big{)}\psi^{\prime}(t)\,=\,f(u_{+}(t,\psi(t)))-f(u_{-}(t,\psi(t)))\,.$

To this aim, we introduce the slope function associated with $f$:

$s_{f}:\,\mathbb{R}\times\mathbb{R}\to\mathbb{R},\quad(a,b)\mapsto\int_{0}^{1}f^{\prime}\big{(}a+\tau(b-a)\big{)}\,\operatorname{d}\tau.$
(10)

We have $s_{f}\in{\mathcal{C}}^{1}(\mathbb{R}\times\mathbb{R})$. In
particular, $(t,x)\mapsto s_{f}(u_{-}(t,x),u_{+}(t,x))\in
BUC^{1}(\mathbb{R}_{+}\times\mathbb{R})$, hence there exists a unique
$\psi\in{\mathcal{C}}^{2}(\mathbb{R}_{+})$ satisfying $\psi(0)=0$ and for any
$t\geq 0$,

$\psi^{\prime}(t)\,=\,s_{f}(u_{-}(t,\psi(t)),u_{+}(t,\psi(t)))\,.$

It follows that $\psi$ satisfies the Rankine-Hugoniot condition as well as the
claimed estimates. Indeed, we have for any $t\geq 0$

$\psi^{\prime}(t)-f^{\prime}({\underline{u}})\,=\,s_{f}(u_{-}(t,\psi(t)),u_{+}(t,\psi(t)))-s_{f}({\underline{u}},{\underline{u}})$

and since $s_{f}$ is a locally Lipschitz function, the bound on
$\psi^{\prime}(t)-f^{\prime}({\underline{u}})$ stems directly from the known
bounds on
$\big{\lVert}u_{\pm}(t,\cdot)-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R})}$.
Now the bound on $\psi^{\prime}-f^{\prime}({\underline{u}})$ may be integrated
to conclude the desired estimate with

$\psi_{\infty}=\int_{0}^{\infty}(\psi^{\prime}(t)-f^{\prime}({\underline{u}}))\,\operatorname{d}t\,.$

To achieve the proof, we need to ensure that lessening $\epsilon$ if
necessary, the constructed weak solution is an entropy solution. Since
$f^{\prime\prime}({\underline{u}})\neq 0$, we can restrict $\epsilon$ so that
$f$ is either strictly concave or strictly convex on
$[{\underline{u}}-C_{0}\epsilon,{\underline{u}}+C_{0}\epsilon]$ and hence $u$
is an entropy solution if and only if Lax’s condition holds, i.e.

$f^{\prime}(u_{+}(t,\psi(t)))<f^{\prime}(u_{-}(t,\psi(t))),\qquad t\geq 0\,.$

Since the corresponding inequality holds at time $t=0$ and $f^{\prime}$ is
one-to-one on $[{\underline{u}}-C_{0}\epsilon,{\underline{u}}+C_{0}\epsilon]$,
it is sufficient to prove that

$w(t):=u_{+}(t,\psi(t))-u_{-}(t,\psi(t))\neq 0,\qquad t>0\,.$

Notice

$\displaystyle w^{\prime}(t)$
$\displaystyle=\left(\partial_{t}u_{+}+\psi^{\prime}(t)\partial_{x}u_{+}-\partial_{t}u_{-}-\psi^{\prime}(t)\partial_{x}u_{-}\right)(t,\psi(t))$
$\displaystyle=\Big{(}g(u_{+})-g(u_{-})+\big{(}\psi^{\prime}(t)-f^{\prime}(u_{+})\big{)}\partial_{x}u_{+}-\big{(}\psi^{\prime}(t)-f^{\prime}(u_{-})\big{)}\partial_{x}u_{-}\Big{)}(t,\psi(t))$
$\displaystyle=\Phi(t,w(t))$

with

$\displaystyle\Phi\,:\,(t,z)\mapsto$ $\displaystyle\
s_{g}(u_{+}(t,\psi(t)),u_{-}(t,\psi(t)))\,z$
$\displaystyle+\Big{(}s_{f}(u_{+}(t,\psi(t)),u_{+}(t,\psi(t))-z)-s_{f}(u_{+}(t,\psi(t)),u_{+}(t,\psi(t)))\Big{)}\partial_{x}u_{+}(t,\psi(t))$
$\displaystyle-\Big{(}s_{f}(u_{-}(t,\psi(t))+z,u_{-}(t,\psi(t)))-s_{f}(u_{-}(t,\psi(t)),u_{-}(t,\psi(t)))\Big{)}\partial_{x}u_{-}(t,\psi(t)).$

Since $\Phi$ is ${\mathcal{C}}^{1}$ and ($\forall t\geq 0$, $\Phi(t,0)=0$), an
application of the Cauchy-Lipschitz theorem concludes the proof. ∎

##  2 Asymptotic stability of shocks

###  2.1 Asymptotic stability under shockless perturbations

In this section under natural spectral assumptions we show the asymptotic
stability under regular perturbations of entropy-admissible Riemann shocks of
(1). More precisely, as described in the introduction we consider a uniformly
traveling wave ${\underline{u}}$,

${\underline{u}}(t,x)={\underline{U}}(x-(\psi_{0}+\sigma t))\,,$

with initial shock position $\psi_{0}\in\mathbb{R}$, speed
$\sigma\in\mathbb{R}$ and wave profile ${\underline{U}}$

${\underline{U}}(x)=\begin{cases}{\underline{u}}_{-}&\text{ if }x<0\\\
{\underline{u}}_{+}&\text{ if }x>0\end{cases}$ (11)

where $({\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{2}$,
${\underline{u}}_{+}\neq{\underline{u}}_{-}$. The problem is invariant by
translation and $\psi_{0}$ is arbitrary, whereas speed and profile are assumed
to satisfy conditions enforcing that ${\underline{u}}$ is a stable entropy
solution. To ensure that ${\underline{u}}$ is a weak solution, we require that
$(\sigma,{\underline{u}}_{-},{\underline{u}}_{+})$ satisfies the equilibrium
condition

$g({\underline{u}}_{+})=0\quad\text{ and }\quad g({\underline{u}}_{-})=0\,;$
(12)

and the Rankine-Hugoniot condition

$f({\underline{u}}_{+})-f({\underline{u}}_{-})=\sigma({\underline{u}}_{+}-{\underline{u}}_{-})\,.$
(13)

(Strict) entropy admissibility then amounts to the following Oleinik condition

$\begin{cases}\qquad\qquad\qquad\sigma\,>\,f^{\prime}({\underline{u}}_{+})\,,&\\\\[5.0pt]
\frac{f(\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+})-f({\underline{u}}_{-})}{\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+}-{\underline{u}}_{-}}>\frac{f(\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+})-f({\underline{u}}_{+})}{\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+}-{\underline{u}}_{+}}&\qquad\textrm{for
any }\ \tau\in(0,1)\,,\\\\[5.0pt] \qquad\qquad\qquad
f^{\prime}({\underline{u}}_{-})\,>\,\sigma\,,\end{cases}$ (14)

and the spectral stability is encoded in

$g^{\prime}({\underline{u}}_{+})<0\quad\text{ and }\quad
g^{\prime}({\underline{u}}_{-})<0\,.$ (15)

######  Remark 2.1.

Note that the entropy condition (14) contributes to the stability properties
of the shock, in a more subtle way than (15). To begin, let us point out that
without assuming condition (14) the proof of Theorem 2.2 below provides a weak
solution to (1). Yet in full generality the solution depends then on the
choices of initial extensions. This lack of uniqueness is even captured by a
direct spectral analysis, as hinted at in Section 4, and the corresponding
linearized dynamics is ill-posed. Replacing (14) with the weaker Lax condition

$f^{\prime}({\underline{u}}_{+})<\sigma<f^{\prime}({\underline{u}}_{-})$

restores spectral stability and ensures uniqueness in a suitable class of
piecewise smooth solutions. The uniqueness in the class of entropy-admissible
solutions if Oleinik’s condition is satisfied is provided by the theory due to
Kružkov [Kru70]. The full condition (14) would also be crucial to the
stability properties of the shock if one allowed perturbations breaking the
large shock into a “sum" of smaller subshocks. See [Bre00, Remark 4.7] for a
more detailed discussion and more generally [Bre00, Chapters 4 and 6] for
classical background on entropy solutions.

As in Section 1.4 we shall solve (1) in the class of piecewise regular
functions and adopt conventions introduced there. The main difference is that
now we require as entropy condition, for any $t\geq 0$

$\begin{cases}\qquad\qquad\qquad\psi^{\prime}(t)\,>\,f^{\prime}(u_{r}(t))\,,&\\\\[5.0pt]
\frac{f(\tau\,u_{l}(t)+(1-\tau)\,u_{r}(t))-f(u_{l}(t))}{\tau\,u_{l}(t)+(1-\tau)\,u_{r}(t)-u_{l}(t)}>\frac{f(\tau\,u_{l}(t)+(1-\tau)\,u_{r}(t))-f(u_{r}(t))}{\tau\,u_{l}(t)+(1-\tau)\,u_{r}(t)-u_{r}(t)}&\qquad\textrm{for
any }\ \tau\in(0,1)\,,\\\\[5.0pt] \qquad\qquad\qquad
f^{\prime}(u_{l}(t))\,>\,\psi^{\prime}(t)\,.\end{cases}$ (16)

where $u_{l}(t)=\lim_{\delta\searrow 0}u(t,\psi(t)-\delta)$ and
$u_{r}(t)=\lim_{\delta\searrow 0}u(t,\psi(t)+\delta)$.

######  Theorem 2.2.

Let $f,g\in\mathcal{C}^{2}(\mathbb{R})$ and
$(\sigma,{\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{3}$ satisfying
(12)-(13)-(14)-(15) and

$f^{\prime\prime}({\underline{u}}_{+})\neq 0\quad\text{ and }\quad
f^{\prime\prime}({\underline{u}}_{-})\neq 0\,.$ (17)

For any $C_{0}>1$, there exists $\epsilon>0$ and $C>0$ such that for any
$\psi_{0}\in\mathbb{R}$ and ${\widetilde{v}}_{0}\in
BUC^{1}(\mathbb{R}^{\star})$ satisfying

$\begin{array}[]{rl}\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}&\leq\epsilon\,,\\\
\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{+}))\,\partial_{x}{\widetilde{v}}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{+})}&\leq\epsilon\,,\\\
\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{-}))\,\partial_{x}{\widetilde{v}}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{-})}&\leq\epsilon\,,\end{array}$
(18)

there exists $\psi\in{\mathcal{C}}^{2}(\mathbb{R}^{+})$ with initial data
$\psi(0)=\psi_{0}$ such that the entropy solution to (1), $u$, generated by
the initial data
$u(0,\cdot)=({\underline{U}}+{\widetilde{v}}_{0})(\cdot+\psi_{0})$,
${\underline{U}}$ being as in (11), belongs to $BUC^{1}(\Omega^{\psi})$ and
satisfies for any $t\geq 0$

$\displaystyle\big{\lVert}u(t,\cdot-\psi(t))-{\underline{u}}_{\pm}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{\pm}))\,\partial_{x}u(t,\cdot-\psi(t)))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{\pm}))\,\partial_{x}{\widetilde{v}}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\big{\lVert}\partial_{x}u(t,\cdot-\psi(t))\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}\partial_{x}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\lvert\psi^{\prime}(t)-\sigma\rvert$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}\,C\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,,$

and moreover there exists $\psi_{\infty}$ such that

$\lvert\psi_{\infty}-\psi_{0}\rvert\,\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C\,,$

and for any $t\geq 0$

$\lvert\psi(t)-\psi_{\infty}-t\,\sigma\rvert\,\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}C\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,.$

######  Remark 2.3.

Theorem 2.2 is a direct counterpart to Proposition 1.3. We could also derive
from it an analogous to Corollary 1.4. Likewise as in Proposition 1.2 we could
relax totally or partly hypothesis (17) if (18) is strengthened. This would
lead to four different versions of Theorem 2.2. We could also provide a
counterpart to Proposition 1.1.

Modifications required to prove the foregoing claims are straightforward and
we have chosen to omit them so as to avoid redundancy.

######  Remark 2.4.

Note that expressed in classical stability terminology (see for instance
[Hen81]) we have proved orbital stability with asymptotic phase. We stress
however that the role of phase shifts is here deeper than in the classical
stability analysis of smooth waves since it is not only required to provide
decay of suitable norms in large-time but also to ensure that these norms are
finite locally in time. In particular here there is no freedom, even in finite
time, in the definition of phase shifts. See [JNR+18, Section 4.1] and [DR]
for related (more elaborate) discussions.

###### Proof of Theorem 2.2..

The proof of Theorem 2.2 follows closely the construction given in the proof
of Proposition 1.10. We also assume henceforth that $\psi_{0}=0$, without loss
of generality. Using Lemma 1.13 and Proposition 1.3, we find that for
$\epsilon>0$ sufficiently small and for any $v_{0}\in
BUC^{1}(\mathbb{R}^{\star})$ satisfying (18), there exist $u_{\pm}\in
BUC^{1}(\mathbb{R}^{+}\times\mathbb{R})$ global classical solutions to (1)
with initial data
$u_{\pm}\left|{}_{{}_{t=0}}\right.={\underline{u}}_{\pm}+v_{0,\pm}$ and
satisfying the desired estimates. We can now construct the solution, $u$,
through (9) where $\psi$ is defined by the differential equation

$\psi^{\prime}(t)\,=\,s_{f}(u_{-}(t,\psi(t)),u_{+}(t,\psi(t)))\,,$

where $s_{f}$ is defined in (10), so that the Rankine-Hugoniot condition as
well as the desired bounds on $\psi$ hold since for any $t\geq 0$,

$\psi^{\prime}(t)-\sigma\,=\,s_{f}(u_{-}(t,\psi(t)),u_{+}(t,\psi(t)))-s_{f}({\underline{u}}_{-},{\underline{u}}_{+})\,.$

Then the last estimates on $\psi$ are obtained by integration with

$\psi_{\infty}=\int_{0}^{\infty}(\psi^{\prime}(t)-\sigma)\,\operatorname{d}t\,.$

To achieve the proof of Theorem 2.2 we only need to ensure that by lessening
$\epsilon$ further if necessary formula (9) ensures (16). For this purpose we
consider

$S_{f}:\,\mathbb{R}\times\mathbb{R}\times[0,1]\to\mathbb{R},\qquad(a,b,\tau)\mapsto
s_{f}(a,\tau\,a+(1-\tau)\,b)-s_{f}(b,\tau\,a+(1-\tau)\,b)$

and observe that it is continuous. Since
$\\{{\underline{u}}_{-}\\}\times\\{{\underline{u}}_{+}\\}\times[0,1]$ is
compact and for any $\tau\in[0,1]$,
$S_{f}({\underline{u}}_{-},{\underline{u}}_{+},\tau)>0$, one may ensure that
provided $\epsilon$ is small enough, for any $(a,b)$ such that
$|a-{\underline{u}}_{-}|\leq C_{0}\epsilon$ and $|b-{\underline{u}}_{+}|\leq
C_{0}\epsilon$, and any $\tau\in[0,1]$, $S_{f}(a,b,\tau)>0$. From this stems
(16) for $u$ built from (9), and the proof is complete. ∎

We now prove that the exponential decay of higher derivatives holds provided
we assume the stronger (symmetric) smallness condition on the first
derivative.

######  Proposition 2.5.

Let $k\in\mathbb{N}$, $k\geq 2$, $f\in{\mathcal{C}}^{k+1}(\mathbb{R}),\
g\in{\mathcal{C}}^{k}(\mathbb{R})$ and
$(\sigma,{\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{3}$ satisfying
(12)-(13)-(14)-(15). There exists $\epsilon>0$ and $C_{k}$ such that for any
$\psi_{0}\in\mathbb{R}$ and ${\widetilde{v}}_{0}\in
BUC^{k}(\mathbb{R}^{\star})$ satisfying

$\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}\leq\epsilon\quad\text{
and
}\quad\big{\lVert}\partial_{x}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}\leq\epsilon\,,$
(19)

there exist $\psi\in{\mathcal{C}}^{k+1}(\mathbb{R}^{+})$ and $u\in
BUC^{k}(\Omega^{\psi})$ with initial data $\psi(0)=\psi_{0}$ and
$u(0,\cdot)=({\underline{U}}+{\widetilde{v}}_{0})(\cdot+\psi_{0})$ such that
$u$ is an entropy solution to (1) and satisfies for any $t\geq 0$ and any
$j\in\\{0,\dots,k\\}$

$\displaystyle\big{\lVert}u(t,\cdot-\psi(t))-{\underline{u}}_{\pm}\big{\rVert}_{W^{j,\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{W^{j,\infty}(\mathbb{R}^{\pm})}C_{k}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\lvert\psi^{(j+1)}(t)\rvert$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{W^{j,\infty}(\mathbb{R}^{\star})}\,C_{k}\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,.$

###### Proof.

The result does not follow directly from Proposition 1.6 applied to $u_{\pm}$
defined in the proof of Theorem 2.2, because the initial data provided by
Lemma 1.13 is not sufficiently regular. Here we rather rely on the following
Lemma deduced from a standard extension theorem [Ada75, Theorem 4.26].

######  Lemma 2.6.

Let $k\in\mathbb{N}$, $k\geq 2$. There exists $C_{k}>0$ such that for any
$v_{0}\in BUC^{k}(\mathbb{R}^{\star})$, there exist
$v_{0,\pm}\in{\mathcal{C}}^{k}(\mathbb{R})$ satisfying

$v_{0}(x)=\begin{cases}v_{0,+}(x)&\text{ if }x>0,\\\ v_{0,-}(x)&\text{ if
}x<0,\end{cases}$

and for any $j\in\mathbb{N}$, $0\leq j\leq k$,

$\big{\lVert}\partial_{x}^{j}v_{0,\pm}\big{\rVert}_{L^{\infty}(\mathbb{R})}\leq\big{\lVert}\partial_{x}^{j}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{k}\,.$
(20)

Replacing Lemma 1.13 and Proposition 1.3 with Lemma 2.6 and Proposition 1.6,
the proof of Proposition 2.5 is then almost identical to the proof of Theorem
2.2. ∎

We can also obtain a counterpart to Proposition 2.5 with the asymmetric
smallness assumption on first-order derivatives.

######  Proposition 2.7.

Let $k\in\mathbb{N}$, $k\geq 2$, $f\in{\mathcal{C}}^{k+1}(\mathbb{R}),\
g\in{\mathcal{C}}^{k}(\mathbb{R})$ and
$(\sigma,{\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{3}$ satisfying
(12)-(13)-(14)-(15) and (17). There exist $\epsilon>0$ and $C_{k}>0$ such that
for any $\psi_{0}\in\mathbb{R}$ and ${\widetilde{v}}_{0}\in
BUC^{k}(\mathbb{R}^{\star})$ satisfying (18), the entropy solution defined in
Theorem 2.2 satisfies $u\in BUC^{k}(\Omega^{\psi})$,
$\psi\in{\mathcal{C}}^{k+1}(\mathbb{R}^{+})$ and for any $t\geq 0$ and any
$j\in\\{1,\dots,k\\}$

$\displaystyle\big{\lVert}\partial_{x}^{j}u(t,\cdot$
$\displaystyle-\psi(t))\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{W^{j,\infty}(\mathbb{R}^{\pm})}C_{k}(1+\big{\lVert}\partial_{x}^{2}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}^{j-1})e^{C_{k}\,\|{\widetilde{v}}_{0}\|_{W^{1,\infty}(\mathbb{R}^{\pm})}\,(1+\|{\widetilde{v}}_{0}\|_{W^{1,\infty}(\mathbb{R}^{\pm})}^{j-1})}\,\,\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\lvert\psi^{(j+1)}(t)\rvert$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{W^{j,\infty}(\mathbb{R}^{\star})}C_{k}(1+\big{\lVert}\partial_{x}^{2}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star})}^{j-1})e^{C_{k}\,\|{\widetilde{v}}_{0}\|_{W^{1,\infty}(\mathbb{R}^{\star})}\,(1+\|{\widetilde{v}}_{0}\|_{W^{1,\infty}(\mathbb{R}^{\star})}^{j-1})}\,\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,.$

###### Proof.

Although we follow the same strategy as in the earlier results, we need to
ensure that the regular extensions ${\widetilde{v}}_{0,\pm}\in
BUC^{k}(\mathbb{R})$ preserve the asymmetric smallness hypothesis (18). To
this aim, we introduce a smooth cut-off function, $\chi$, such that
$\chi(x)=0$ for $|x|\geq 2/3$, $\chi(x)=1$ for $|x|\leq 1/3$ and
$\chi(x)\in[0,1]$ for $x\in\mathbb{R}$, and define

${\widetilde{v}}_{0,+}(x)=\begin{cases}{\widetilde{v}}_{0}(0^{+})+\int_{0}^{-\delta}w_{0,+}(y)\chi(\delta^{-1}y)\operatorname{d}y&\text{
if }x\in(-\infty,-\delta]\,,\\\
{\widetilde{v}}_{0}(0^{+})+\int_{0}^{x}w_{0,+}(y)\chi(\delta^{-1}y)\operatorname{d}y&\text{
if }x\in(-\delta,0]\,,\\\ {\widetilde{v}}_{0}(x)&\text{ if }x>0\,.\end{cases}$

where $w_{0,+}$ is the extension associated with
$\partial_{x}{\widetilde{v}}_{0}$ provided by Lemma 2.6. When choosing

$\delta=\min\left(\left\\{\frac{c_{0}\,\epsilon}{1+\big{\lVert}\partial_{x}{\widetilde{v}}_{0}\big{\rVert}_{W^{1,\infty}(\mathbb{R}^{+})}},1\right\\}\right),$

with $c_{0}>0$ sufficiently small and defining symmetrically
${\widetilde{v}}_{0,-}$, we derive the following Lemma.

######  Lemma 2.8.

Let $k\in\mathbb{N}$, $k\geq 2$, $C_{0}>1$ and $\epsilon>0$. There exists
$C_{k}>0$ such that for any ${\widetilde{v}}_{0}\in
BUC^{k}(\mathbb{R}^{\star})$ satisfying (18), there exist
${\widetilde{v}}_{0,\pm}\in BUC^{k}(\mathbb{R})$ satisfying

${\widetilde{v}}_{0}(x)=\begin{cases}{\widetilde{v}}_{0,+}(x)&\text{ if
}x>0,\\\ {\widetilde{v}}_{0,-}(x)&\text{ if }x<0,\end{cases}$

and the estimates

$\begin{array}[]{rl}\big{\lVert}{\widetilde{v}}_{0,\pm}\big{\rVert}_{L^{\infty}(\mathbb{R})}&\leq\min(\\{C_{0}\epsilon,\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{W^{1,\infty}(\mathbb{R}^{\star})}C_{k}\\})\,,\\\
\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{+}))\,\partial_{x}{\widetilde{v}}_{0,+})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}&\leq
C_{0}\epsilon\,,\\\
\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{-}))\,\partial_{x}{\widetilde{v}}_{0,-})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R})}&\leq
C_{0}\epsilon\,,\end{array}$

and for any $j\in\mathbb{N}$, $1\leq j\leq k$,

$\big{\lVert}{\widetilde{v}}_{0,\pm}\big{\rVert}_{W^{j,\infty}(\mathbb{R})}\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{W^{j,\infty}(\mathbb{R}^{\pm})}(1+\big{\lVert}\partial_{x}{\widetilde{v}}_{0}\big{\rVert}_{W^{1,\infty}(\mathbb{R}^{\pm})}^{j-1})C_{k}\,.$

We can now follow the proof of Theorem 2.2, replacing Lemma 1.13 and
Proposition 1.3 with Lemma 2.8 and Proposition 1.6. . ∎

######  Remark 2.9.

The non-uniqueness of the intermediate stage of our proofs is particularly
striking here, since even for the same initial data, depending on the level of
regularity we aim at, we build distinct extended solutions. Yet in the end, as
discussed in Remark 2.1, the parts actually used in the final gluing process
are indeed independent of choices in the extension as a consequence of Lax’s
condition, and uniqueness holds by the theory of Kružkov [Kru70].

###  2.2 Perturbation by small shocks

We now elaborate on Proposition 1.10 and Theorem 2.2 and perturb a spectrally
stable strictly entropy-admissible Riemann shock of (1) with a perturbation
containing one shock. For concreteness and concision we assume that the small
shock is located on the left of the large shock, the opposite situation being
deduced by symmetry considerations. Since the Riemann shock is strictly
entropy-admissible, sufficiently small perturbations with a small shock will
produce two paths of discontinuity eventually merging in a single one, the
small shock being essentially absorbed by the large one.

We follow the position of the large shock with $\psi:\
\mathbb{R}\to\mathbb{R}$ and the position of the small shock, as long as it
persists, with $\psi_{s}:\ [0,t^{\star}]\to\mathbb{R}$ where $t^{\star}>0$,
$\psi_{s}(t^{\star})=\psi(t^{\star})$ and, for any $t\in[0,t^{\star})$,
$\psi_{s}(t)<\psi(t)$. In particular we seek for a solution that is a
classical solution on the domain

$\Omega_{\psi,\psi_{s}}\stackrel{{\scriptstyle\rm
def}}{{=}}\mathbb{R}_{+}\times\mathbb{R}\setminus\Big{(}\\{\,(t,\psi_{s}(t))\,|\,t\in[0,t^{\star}]\,\\}\cup\\{\,(t,\psi(t))\,|\,t\geq
0\\}\Big{)}\,$

(see Figure 1).

(a) Original domain (b) Straightened domain

Figure 1: Sketch of the shock paths.

######  Theorem 2.10.

Let $f,g\in\mathcal{C}^{2}(\mathbb{R})$ and
$(\sigma,{\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{3}$ satisfying
(12)-(13)-(14)-(15) and (17). For any $C_{0}>1$, there exists $\epsilon>0$ and
$C>0$ such that for any ${\widetilde{\psi}}_{s,0}<0$ and
$\psi_{0}\in\mathbb{R}$ and any ${\widetilde{v}}_{0}\in
BUC^{1}(\mathbb{R}^{\star}\setminus\\{{\widetilde{\psi}}_{s,0}\\})$ satisfying

$\begin{array}[]{rl}\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star}\setminus\\{{\widetilde{\psi}}_{s,0}\\})}&\leq\epsilon\,,\\\
\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{+}))\,\partial_{x}{\widetilde{v}}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{+})}&\leq\epsilon\,,\\\
\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{-}))\,\partial_{x}{\widetilde{v}}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{-}\setminus\\{{\widetilde{\psi}}_{s,0}\\})}&\leq\epsilon\,,\end{array}$
(21)

there exist a time $t^{\star}\in(0,+\infty)$,

  * •

a ${\mathcal{C}}^{0}$ function $\psi:\,\mathbb{R}_{+}\to\mathbb{R}$ that is
${\mathcal{C}}^{2}$ on $\mathbb{R}_{+}\setminus\\{t^{\star}\\}$ and such that
$\psi(0)=\psi_{0}$,

  * •

a ${\mathcal{C}}^{2}$ function
${\widetilde{\psi}}_{s}:\,[0,t^{\star}]\to\mathbb{R}_{-}$ such that
${\widetilde{\psi}}_{s}$ is negative on $[0,t^{\star})$,
${\widetilde{\psi}}_{s}(0)={\widetilde{\psi}}_{s,0}$ and
${\widetilde{\psi}}_{s}(t^{\star})=0$,

so that, with $\psi_{s}=\psi_{|[0,t^{\star}]}+{\widetilde{\psi}}_{s}$, the
entropy solution to (1), $u$, generated by the initial data
$({\underline{U}}+{\widetilde{v}}_{0})(\cdot+\psi_{0})$ belongs to
$BUC^{1}(\Omega_{\psi,\psi_{s}})$ and satisfies777In the first three
inequalities we sacrifice consistency to the sake of concision and
readilibility and write $\mathbb{R}^{-}$ even when $0\leq t<t^{\star}$ and
notational conventions used elsewhere would require
$\mathbb{R}^{-}\setminus\\{{\widetilde{\psi}}_{s}(t)\\}$ or
$\mathbb{R}^{-}\setminus\\{{\widetilde{\psi}}_{s,0}\\}$. for any $t\geq 0$

$\displaystyle\big{\lVert}u(t,\cdot-\psi(t))-{\underline{u}}_{\pm}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{\pm}))\,\partial_{x}u(t,\cdot-\psi(t)))_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}(\operatorname{sgn}(f^{\prime\prime}({\underline{u}}_{\pm}))\,\partial_{x}{\widetilde{v}}_{0})_{-}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\big{\lVert}\partial_{x}u(t,\cdot-\psi(t))\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}$
$\displaystyle\leq\big{\lVert}\partial_{x}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\pm})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\lvert\psi_{s}^{\prime}(t)-f^{\prime}({\underline{u}}_{-})\rvert$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{-}\setminus\\{{\widetilde{\psi}}_{s,0}\\})}\,C\,e^{g^{\prime}({\underline{u}}_{-})\,t}\,,\qquad
t\leq t^{\star}\,,$ $\displaystyle\lvert\psi^{\prime}(t)-\sigma\rvert$
$\displaystyle\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star}\setminus\\{{\widetilde{\psi}}_{s,0}\\})}\,C\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,,$

and moreover there exists $\psi_{\infty}$ such that

$\lvert\psi_{\infty}-\psi_{0}\rvert\,\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star}\setminus\\{{\widetilde{\psi}}_{s,0}\\})}C\,,$

and for any $t\geq 0$

$\lvert\psi(t)-\psi_{\infty}-t\,\sigma\rvert\,\leq\big{\lVert}{\widetilde{v}}_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star}\setminus\\{{\widetilde{\psi}}_{s,0}\\})}C\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,.$

###### Proof.

Here again we follow the extension/patching strategy used for the previous
results, assume without loss of generalty that $\psi_{0}=0$, and
correspondingly drop some tildes. With a straightforward adaptation of Lemma
1.13 and using Proposition 1.3, we find that for $\epsilon>0$ sufficiently
small and for any $v_{0}\in
BUC^{1}(\mathbb{R}^{\star}\setminus\\{\psi_{s,0}\\})$ satisfying (21), there
exists $u_{l},u_{c},u_{r}\in BUC^{1}(\mathbb{R}^{+}\times\mathbb{R})$ global
classical solutions to (1) with initial data such that

$\begin{cases}u_{l}(0,x)={\underline{u}}_{-}+v_{0}&\text{ if
$x<\psi_{s,0}$}\\\ u_{c}(0,x)={\underline{u}}_{-}+v_{0}&\text{ if
$x\in(\psi_{s,0},0)$}\\\ u_{r}(0,x)={\underline{u}}_{+}+v_{0}&\text{ if
$x>0$}\end{cases}$

and satisfying the desired estimates.

We may now identify shock locations. Let $\psi_{l}$ and $\psi_{r}$ be defined
by the differential equations

$\psi_{l}^{\prime}(t)\,=\,s_{f}(u_{l}(t,\psi_{l}(t)),u_{c}(t,\psi_{l}(t)))\text{
and
}\psi_{r}^{\prime}(t)\,=\,s_{f}(u_{c}(t,\psi_{r}(t)),u_{r}(t,\psi_{r}(t)))$

with initial data $\psi_{l}(0)=\psi_{l,0}$ and $\psi_{r}(0)=0$. Then we
observe that $\psi_{l},\psi_{r}\in{\mathcal{C}}^{2}(\mathbb{R}^{+})$ and

$\displaystyle\lvert\psi_{l}^{\prime}(t)-f^{\prime}({\underline{u}}_{-})\rvert$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{-}\setminus\\{\psi_{s,0}\\})}\,C\,e^{g^{\prime}({\underline{u}}_{-})\,t}\,,$
$\displaystyle\lvert\psi_{r}^{\prime}(t)-\sigma\rvert$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{\star}\setminus\\{\psi_{s,0}\\})}\,C\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,.$

Since $f^{\prime}({\underline{u}}_{-})>\sigma$ and $\psi_{s,0}<0$ this implies
that the time

$t^{\star}=\operatorname{arg\,min}\ \\{t\in\mathbb{R}^{+}\ |\
\psi_{l}(t)=\psi_{r}(t)\\}$

is postive and finite. At last $\psi_{f}$ is defined by the differential
equation

$\psi_{f}^{\prime}(t)\,=\,s_{f}(u_{l}(t,\psi_{f}(t)),u_{r}(t,\psi_{f}(t)))$

with “initial” data $\psi(t^{\star})=\psi_{r}(t^{\star})$. Note that
$\psi_{f}\in{\mathcal{C}}^{2}(\mathbb{R}^{+})$ and that
$\lvert\psi_{f}^{\prime}(t)-\sigma\rvert$ also decays exponentially with the
same estimate as $\lvert\psi_{r}^{\prime}(t)-\sigma\rvert$. Then we set
$\psi_{s}=(\psi_{l})_{|[0,t^{\star}]}$ and

$\psi:\,\mathbb{R}\to\mathbb{R}\,,\qquad
t\mapsto\begin{cases}\psi_{r}(t)&\text{ if $0\leq t\leq t^{\star}$}\\\
\psi_{f}(t)&\text{ if $t>t^{\star}$}\end{cases}\,.$

Again the last estimates on $\psi$ are obtained by integration with

$\psi_{\infty}=\int_{0}^{\infty}(\psi^{\prime}(t)-\sigma)\,\operatorname{d}t\,.$

We can now construct the solution $u$. For any $t\in[0,t^{\star}]$, we define

$u(t,x)=\begin{cases}u_{l}(t,x)&\text{ if }x<\psi_{s}(t)\\\ u_{c}(t,x)&\text{
if }\psi_{s}(t)<x<\psi(t)\\\ u_{r}(t,x)&\text{ if }x>\psi(t)\end{cases}\,.$

For subsequent times $t\in[t^{\star},+\infty)$, we set

$u(t,x)=\begin{cases}u_{l}(t,x)&\text{ if }x<\psi(t)\\\ u_{r}(t,x)&\text{ if
}x>\psi(t)\end{cases}\,.$

One easily checks that the function $u$ is an entropy solution as soon as
$\epsilon$ is sufficiently small, following the proof of Proposition 1.10
(along the path $\\{(t,\psi_{s}(t))\,|\,0\leq t\leq t^{\star}\\}$) and
Proposition 2.2 (along the path $\\{(t,\psi(t))\,|\,t\geq 0\\}$). ∎

##  3 Transverse stability in the multidimensional framework

In the present section we consider some generalizations of the main results
proved so far to multidimensional settings. In particular we now replace (1)
with

$\partial_{t}u+\operatorname{div}\big{(}f(u)\big{)}=g(u)$ (22)

where the spatial variable $x$ belongs to $\mathbb{R}^{d}$, $d\in\mathbb{N}$,
and $f:\mathbb{R}\to\mathbb{R}^{d}$.

Most of adaptations are rather straightforward and we only sketch main
variations required in the process. We aim not at gaining new insights on the
general stability problem but at demonstrating a certain robustness of the
one-dimensional arguments.

Starting from spatial dimension $2$ the range of possible geometries for
discontinuities becomes too wide to be reasonably covered here, even if one
restricts to a few typical cases. Therefore, in the multidimensional case we
only consider shockless perturbations.

###  3.1 Asymptotic stability of constant states

Propositions 1.2 and 1.6 translate almost verbatim to the multidimensional
case.

######  Proposition 3.1.

Let $f\in{\mathcal{C}}^{2}(\mathbb{R};\mathbb{R}^{d})$,
$g\in{\mathcal{C}}^{2}(\mathbb{R})$ and ${\underline{u}}\in\mathbb{R}$ be such
that

$g({\underline{u}})=0\qquad\textrm{and}\qquad
g^{\prime}({\underline{u}})<0\,.$

Then for any $C_{0}>1$, there exists $\epsilon>0$ such that for any $v_{0}\in
BUC^{1}(\mathbb{R}^{d})$ satisfying

$\big{\lVert}v_{0}\big{\rVert}_{W^{1,\infty}(\mathbb{R}^{d})}\leq\epsilon\,,$

the initial data $u\left|{}_{{}_{t=0}}\right.={\underline{u}}+v_{0}$ generates
a global unique classical solution to (22), $u\in
BUC^{1}(\mathbb{R}^{+}\times\mathbb{R}^{d})$, and it satisfies for any $t\geq
0$

$\displaystyle\big{\lVert}u(t,\cdot)-{\underline{u}}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\
;$ $\displaystyle\big{\lVert}\nabla
u(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})}$
$\displaystyle\leq\big{\lVert}\nabla
v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})}C_{0}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

In the foregoing statement and henceforth, we measure vectors in
$\mathbb{R}^{d}$ with the Euclidean norm.

######  Proposition 3.2.

Under the assumptions of Proposition 3.1, if one assumes additionally that
$f\in{\mathcal{C}}^{k+1}(\mathbb{R};\mathbb{R}^{d})$,
$g\in{\mathcal{C}}^{k}(\mathbb{R})$ with $k\in\mathbb{N}$, $k\geq 2$ then
there exists $C_{k}>0$, depending on $f$, $g$ and $k$ but not on the initial
data $v_{0}$, such that if $v_{0}\in BUC^{k}(\mathbb{R}^{d})$ additionally to
constraints in Proposition 3.1, then the global unique classical solution to
(22) emerging from the initial data ${\underline{u}}+v_{0}$ satisfies $u\in
BUC^{k}(\mathbb{R}^{+}\times\mathbb{R}^{d})$ and for any $\alpha$ multi-index
of size $|\alpha|=k$ and $t\geq 0$

$\big{\lVert}\partial^{\alpha}u(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}\leq\big{\lVert}\partial^{\alpha}v_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}e^{C_{k}\,\|v_{0}\|_{W^{1,\infty}}\,(1+\|v_{0}\|_{W^{1,\infty}}^{k-1})}\,e^{g^{\prime}({\underline{u}})\,t}\,.$

Proofs are also essentially identical. The proof involves now the
consideration of operators $L_{a,\,b}=-a\cdot\nabla+b$, but still with $a$
close to $f^{\prime}({\underline{u}})$ and $b$ close to
$g^{\prime}({\underline{u}})$. Since transport operators are not elliptic when
$d>1$, their domains — the set of $v\in BUC^{0}(\mathbb{R}^{d})$ such that
$L_{a,\,b}v\in BUC^{0}(\mathbb{R}^{d})$ — cannot be identified with a
classical function space. Note however that this does not alter any part of
the argument and that in particular one may still apply [Paz83, Chapter 5,
Theorem 3.1] with $X=BUC^{0}(\mathbb{R}^{d})$ and $Y=BUC^{1}(\mathbb{R}^{d})$
even though $Y$ is now a common core, rather than a common domain.

Key estimates are provided in the following lemma.

######  Lemma 3.3.

Assume $a\in BUC^{1}(\mathbb{R}^{d};\mathbb{R}^{d})$, $b\in
BUC^{0}(\mathbb{R}^{d})$.
(i). Then for any $\lambda\in\mathbb{C}$ such that

$\Re(\lambda)>\sup_{\mathbb{R}^{d}}b(\cdot)\,,$

for any $F\in BUC^{0}(\mathbb{R}^{d})$, there exists a unique
$\check{v}(\,\cdot\,;\lambda)\in BUC^{0}(\mathbb{R}^{d})$ such that

$(\lambda-L_{a,\,b})\,\check{v}(\,\cdot\,;\lambda)\,=\,F$

and moreover

$\big{\lVert}\check{v}(\,\cdot\,;\lambda)\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}\leq\frac{1}{\Re\lambda-\sup_{\mathbb{R}^{d}}b(\cdot)}\big{\lVert}F\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}\,.$

(ii). Assume moreover that

$b\textrm{ is constant}\qquad\textrm{ and
}\qquad\Re(\lambda)>b+\sup_{\mathbb{R}^{d}}\|\operatorname{d}a\|(\cdot)$

then for any $F\in BUC^{1}(\mathbb{R}^{d})$, $\check{v}(\,\cdot\,;\lambda)\in
BUC^{1}(\mathbb{R}^{d})$ and

$\big{\lVert}\nabla\check{v}(\,\cdot\,;\lambda)\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}\leq\frac{1}{\Re\lambda-b-\sup_{\mathbb{R}^{d}}\|\operatorname{d}a\|(\cdot)}\big{\lVert}\nabla
F\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}\,.$

###### Proof.

One may proceed as in the one-dimensional case, with generalized formula

$\check{v}(x;\lambda)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{-\infty}^{0}e^{\int_{s}^{0}(b(X(\sigma;x))-\lambda)\,\operatorname{d}\sigma}F(X(s;x))\,\operatorname{d}s$

where $X(\cdot,x)$ is such that $X(0,x)=x$ and $\forall s\in\mathbb{R}$,
$\partial_{s}X(s;x)=a(X(s;x))$. Since $a$ is Lipschitz, $X(\cdot,x)$ is indeed
globally well-defined. When moreover $b$ is constant one derives from the
latter

$\displaystyle\operatorname{d}_{x}\check{v}(x;\lambda)(h)$
$\displaystyle=\int_{-\infty}^{0}e^{\int_{s}^{0}(b-\lambda)\,\operatorname{d}\sigma}\operatorname{d}F(X(s;x))(\operatorname{d}_{x}X(s;x)(h))\,\operatorname{d}s$

and the claim follows from the observation that for any $s\leq 0$

$\|\operatorname{d}_{x}X(s;x)(h)\|\leq
e^{\int_{s}^{0}\|\operatorname{d}a(X(\sigma;x))\|\,\operatorname{d}\sigma}\|h\|\leq
e^{\int_{s}^{0}\|\operatorname{d}a\|_{L^{\infty}}\,\operatorname{d}\sigma}\|h\|\,.$

∎

###  3.2 Asymptotic stability of Riemann shocks

We now turn to the stability of plane Riemann shocks. For the sake of clarity
and without loss of generality, we fix the direction of propagation of the
reference Riemann shock, split spatial variables accordingly
$x=(\xi,y)\in\mathbb{R}\times\mathbb{R}^{d-1}$, and correspondingly
$f=(f_{\shortparallel},f_{\perp})$.

We consider a plane wave ${\underline{u}}$,

${\underline{u}}(t,x)={\underline{U}}(\xi-(\psi_{0}+\sigma t))\,,$

with $\psi_{0}\in\mathbb{R}$, $\sigma\in\mathbb{R}$ and ${\underline{U}}$ such
that

${\underline{U}}(\xi)=\begin{cases}{\underline{u}}_{-}&\text{ if }\xi<0\\\
{\underline{u}}_{+}&\text{ if }\xi>0\end{cases}$ (23)

where $({\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{2}$,
${\underline{u}}_{+}\neq{\underline{u}}_{-}$ are such that

$g({\underline{u}}_{+})=0\,,\quad g({\underline{u}}_{-})=0\quad\text{ and
}\quad
f_{\shortparallel}({\underline{u}}_{+})-f_{\shortparallel}({\underline{u}}_{-})=\sigma({\underline{u}}_{+}-{\underline{u}}_{-})$
(24)

and we assume that ${\underline{u}}$ is strictly entropy-admissible, that is

$\begin{cases}\qquad\qquad\qquad\sigma\,>\,f_{\shortparallel}^{\prime}({\underline{u}}_{+})\,,&\\\\[5.0pt]
\frac{f_{\shortparallel}(\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+})-f_{\shortparallel}({\underline{u}}_{-})}{\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+}-{\underline{u}}_{-}}>\frac{f_{\shortparallel}(\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+})-f_{\shortparallel}({\underline{u}}_{+})}{\tau\,{\underline{u}}_{-}+(1-\tau)\,{\underline{u}}_{+}-{\underline{u}}_{+}}&\qquad\textrm{for
any }\ \tau\in(0,1)\,,\\\\[5.0pt] \qquad\qquad\qquad
f_{\shortparallel}^{\prime}({\underline{u}}_{-})\,>\,\sigma\,,\end{cases}$
(25)

(though this does not play a strong role in our analysis; see Remark 2.1).
Moreover we require

$g^{\prime}({\underline{u}}_{+})<0\quad\text{ and }\quad
g^{\prime}({\underline{u}}_{-})<0\,.$ (26)

To accurately account for discontinuities, we introduce

$\displaystyle\Omega^{+}_{\psi_{0}}$ $\displaystyle\stackrel{{\scriptstyle\rm
def}}{{=}}\\{\,(\xi,y)\in\mathbb{R}\times\mathbb{R}^{d-1}\,;\quad\xi>\psi_{0}(y)\,\\}$
$\displaystyle\Omega^{-}_{\psi_{0}}$ $\displaystyle\stackrel{{\scriptstyle\rm
def}}{{=}}\\{\,(\xi,y)\in\mathbb{R}\times\mathbb{R}^{d-1}\,;\quad\xi<\psi_{0}(y)\,\\}$
$\displaystyle\Omega_{+}^{\psi}$ $\displaystyle\stackrel{{\scriptstyle\rm
def}}{{=}}\\{\,(t,\xi,y)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}^{d-1}\,;\quad\xi>\psi(t,y)\,\\}$
$\displaystyle\Omega_{-}^{\psi}$ $\displaystyle\stackrel{{\scriptstyle\rm
def}}{{=}}\\{\,(t,\xi,y)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}^{d-1}\,;\quad\xi<\psi(t,y)\,\\}$
$\displaystyle\Omega_{\psi_{0}}$ $\displaystyle\stackrel{{\scriptstyle\rm
def}}{{=}}\Omega^{+}_{\psi_{0}}\cup\Omega^{-}_{\psi_{0}}\,,\qquad\Omega^{\psi}\stackrel{{\scriptstyle\rm
def}}{{=}}\Omega_{+}^{\psi}\cup\Omega_{-}^{\psi}\,.$

######  Theorem 3.4.

Let $f\in\mathcal{C}^{2}(\mathbb{R};\mathbb{R}^{d})$,
$g\in\mathcal{C}^{2}(\mathbb{R})$ and
$(\sigma,{\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{3}$ satisfying
(24)-(25)-(26). For any $C_{0}>1$, there exists $\epsilon>0$ and $C>0$ such
that for any $\psi_{0}\in BUC^{1}(\mathbb{R}^{d-1})$ and $v_{0}\in
BUC^{1}(\Omega_{\psi_{0}})$ satisfying

$\begin{array}[]{rl}\big{\lVert}v_{0}\big{\rVert}_{W^{1,\infty}(\Omega_{\psi_{0}})}&\leq\epsilon\,,\\\
\big{\lVert}\nabla_{y}\psi_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d-1})}&\leq\epsilon\,,\end{array}$

there exists $\psi\in{\mathcal{C}}^{2}(\mathbb{R}^{+}\times\mathbb{R}^{d-1})$
with initial data $\psi(0,\cdot)=\psi_{0}$ such that the entropy solution to
(22), $u$, generated by the initial data

$u(0,\xi,y)={\underline{U}}(\xi-\psi_{0}(y))+v_{0}(\xi,y)\,,$

${\underline{U}}$ being as in (23), belongs to $BUC^{1}(\Omega^{\psi})$ and
satisfies for any $t\geq 0$

$\displaystyle\big{\lVert}u(t,\cdot,\cdot)-{\underline{u}}_{\pm}\big{\rVert}_{L^{\infty}(\Omega^{\pm}_{\psi(t,\cdot)})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\Omega^{\pm}_{\psi_{0}})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\big{\lVert}\nabla
u(t,\cdot,\cdot)\big{\rVert}_{L^{\infty}(\Omega^{\pm}_{\psi(t,\cdot)};\mathbb{R}^{d})}$
$\displaystyle\leq\big{\lVert}\nabla
v_{0}\big{\rVert}_{L^{\infty}(\Omega^{\pm}_{\psi_{0}};\mathbb{R}^{d})}C_{0}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,,$
$\displaystyle\big{\lVert}\nabla_{y}\psi(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R}^{d-1};\mathbb{R}^{d-1})}$
$\displaystyle\leq\big{\lVert}\nabla_{y}\psi_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d-1};\mathbb{R}^{d-1})}\,C_{0}\,\,+\,\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\Omega_{\psi_{0}})}\,C$
$\displaystyle\big{\lVert}\partial_{t}\psi(t,\cdot)+s_{f_{\perp}}({\underline{u}}_{-},{\underline{u}}_{+})\cdot\nabla_{y}\psi(t,\cdot)-\sigma\big{\rVert}_{L^{\infty}(\mathbb{R}^{d-1})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\Omega_{\psi_{0}})}\,C\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,,$

where $s_{f_{\perp}}$ is as defined as in (10), and moreover there exists
$\psi_{\infty}$ such that

$\big{\lVert}\psi_{\infty}-\psi_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d-1})}\,\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\Omega_{\psi_{0}})}C\,,$

and for any $t\geq 0$

$\big{\lVert}\psi(t,\cdot)-\psi_{\infty}(\cdot-t\,s_{f_{\perp}}({\underline{u}}_{-},{\underline{u}}_{+}))-t\,\sigma\big{\rVert}_{L^{\infty}(\mathbb{R}^{d-1})}\,\leq\big{\lVert}v_{0}\big{\rVert}_{L^{\infty}(\Omega_{\psi_{0}})}C\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,.$

###### Proof.

The proof follows the same scheme as the one of the proof of Theorem 2.2. To
start with, even though one cannot reduce the proof to the case when
$\psi_{0}=0$, the Lipschitz assumption on $\psi_{0}$ is sufficient to extend
both smooth parts of $v_{0}$ so as to apply Proposition 3.1 to both extensions
and receive functions $u_{-}$ and $u_{+}$. Once the shock position is
determined — through the equation $\xi=\psi(t,y)$ —, that is, once $\psi$ is
known, the sought solution is obtained by gluing $u_{-}$ and $u_{+}$.

Once again the evolution of $\psi$ is determined from the Rankine-Hugoniot
conditions. They read

$\partial_{t}\psi(t,\cdot)+s_{f_{\perp}}(u_{-}(t,\psi(t,\cdot),\cdot),u_{+}(t,\psi(t,\cdot),\cdot))\cdot\nabla_{y}\psi(t,\cdot)\,=\,s_{f_{\shortparallel}}(u_{-}(t,\psi(t,\cdot),\cdot),u_{+}(t,\psi(t,\cdot),\cdot))\,.$

At the level of regularity considered here, the local-well posedness of the
foregoing PDE is classical and we may again focus on proving bounds. Uniform
bounds on $\psi(t,\cdot)-\sigma t$ and $\nabla_{y}\psi$, hence also the global
well-posedness, is derived by arguing as in the proof of Proposition 3.1.
Indeed, Lemma 3.3 yields the existence of and bounds for the evolution system
generated by the time-dependent linear operator
$L_{a(t,\cdot),\,0}=-a(t,\cdot)\cdot\nabla$, where
$a\in{\mathcal{C}}^{0}([0,T),BUC^{1}(\mathbb{R}^{d-1}))$. We may then apply
Duhamel’s formula on the equation

$\partial_{t}(\psi(t,\cdot)-\sigma
t)+s_{f_{\perp}}(u_{-}(t,\psi(t,\cdot),\cdot),u_{+}(t,\psi(t,\cdot),\cdot))\cdot\nabla_{y}(\psi(t,\cdot)-\sigma
t)\\\
\,=\,s_{f_{\shortparallel}}(u_{-}(t,\psi(t,\cdot),\cdot),u_{+}(t,\psi(t,\cdot),\cdot))-s_{f_{\shortparallel}}({\underline{u}}_{-},{\underline{u}}_{+})\,,$

and use that $u_{-}-{\underline{u}}_{-}$ and $u_{+}-{\underline{u}}_{+}$ decay
exponentially in time in $W^{1,\infty}$. Then the decay bounds follow from
substracting
$\left(s_{f_{\perp}}(u_{-}(t,\psi(t,\cdot),\cdot),u_{+}(t,\psi(t,\cdot),\cdot))-s_{f_{\perp}}({\underline{u}}_{-},{\underline{u}}_{+})\right)\cdot\nabla_{y}\psi(t,\cdot)$
to each side of the above equation and using again the exponential decay in
time in $L^{\infty}$ of $u_{-}-{\underline{u}}_{-}$ and
$u_{+}-{\underline{u}}_{+}$, with

$\displaystyle\psi_{\infty}(y)$ $\displaystyle\,\stackrel{{\scriptstyle\rm
def}}{{=}}\,\psi_{0}(y)+\int_{0}^{\infty}\frac{\operatorname{d}}{\operatorname{d}t}\left(\psi(\cdot,y+s_{f_{\perp}}({\underline{u}}_{-},{\underline{u}}_{+})\,\cdot)-\sigma\,\cdot\right)(t)\,\operatorname{d}t$
$\displaystyle\,=\,\psi_{0}(y)+\int_{0}^{\infty}\left(\partial_{t}\psi+s_{f_{\perp}}({\underline{u}}_{-},{\underline{u}}_{+})\cdot\nabla_{y}\psi\right)(t,y+s_{f_{\perp}}({\underline{u}}_{-},{\underline{u}}_{+})t)-\sigma\,\operatorname{d}t\,.$

Finally, the $L^{\infty}$ bounds on $u_{\pm}$ allow to guarantee, lessening
$\epsilon$ if necessary, that the strict entropy-admissibility of the shock
propagates for positive times, and the uniqueness of entropy solutions is
provided by the theory of Kružkov [Kru70]. ∎

The proof of Theorem 2.5 also carries over to the multidimensional case. Note
in particular that the Lipschitz condition on $\psi_{0}$ is already sufficient
to extend smooth parts of $v_{0}$ while maintaining higher-order regularity.

######  Proposition 3.5.

Let $k\in\mathbb{N}$, $k\geq 2$,
$f\in{\mathcal{C}}^{k+1}(\mathbb{R};\mathbb{R}^{d}),\
g\in{\mathcal{C}}^{k}(\mathbb{R})$ and
$(\sigma,{\underline{u}}_{-},{\underline{u}}_{+})\in\mathbb{R}^{3}$ satisfying
(24)-(26) and defining a strictly entropic plane wave. There exists
$\epsilon>0$ and $C_{k}$ such that for any $\psi_{0}\in
BUC^{1}(\mathbb{R}^{d})$ and $v_{0}\in BUC^{k}(\Omega_{\psi_{0}})$ satisfying

$\big{\lVert}v_{0}\big{\rVert}_{W^{1,\infty}(\Omega_{\psi_{0}})}\leq\epsilon\quad\text{
and
}\quad\big{\lVert}\nabla_{y}\psi_{0}\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}\leq\epsilon\,,$

there exist $\psi\in{\mathcal{C}}^{2}(\mathbb{R}^{+})$ and $u\in
BUC^{k}(\Omega^{\psi})$ with initial data $\psi(0,\cdot)=\psi_{0}$ and
$u(0,\cdot,\cdot)$ given by

$u(0,\xi,y)={\underline{U}}(\xi-\psi_{0}(y))+v_{0}(\xi,y)\,,$

${\underline{U}}$ being as in (23), such that $u$ is an entropy solution to
(22) and satisfies for any $t\geq 0$ and any $j\in\\{0,\dots,k\\}$

$\displaystyle\big{\lVert}u(t,\cdot,\cdot)-{\underline{u}}_{\pm}\big{\rVert}_{W^{j,\infty}(\Omega_{\psi(t,\cdot)}^{\pm})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{W^{j,\infty}(\Omega_{\psi_{0}}^{\pm})}C_{k}\,e^{g^{\prime}({\underline{u}}_{\pm})\,t}\,.$

If moreover $\psi_{0}\in BUC^{k+1}(\mathbb{R}^{d})$, then $\psi\in
BUC^{k+1}(\mathbb{R}\times\mathbb{R}^{d})$ and for any $t\geq 0$ and any
$j\in\\{1,\dots,k\\}$

$\displaystyle\big{\lVert}(\partial_{t}+s_{f_{\perp}}({\underline{u}}_{-},{\underline{u}}_{+})\cdot\nabla_{y})^{j+1}\psi(t,\cdot)\big{\rVert}_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\leq\big{\lVert}v_{0}\big{\rVert}_{W^{j,\infty}(\Omega_{\psi_{0}})}\,C_{k}\,e^{\max(\\{g^{\prime}({\underline{u}}_{+}),g^{\prime}({\underline{u}}_{-})\\})\,t}\,.$

##  4 Further comments

Finally we conclude our contribution by replacing the present analysis in the
more general framework of stability of discontinuous solutions of hyperbolic
systems of balance laws. In particular, we comment on which feature of Riemann
shocks of scalar laws is involved in each of the derived properties.

To begin with, note that this is from the scalar character of the equations
that stems the fact that the analysis of piecewise smooth solutions may be
reduced to on one hand the analysis of smooth solutions and on the other hand
of how to glue them according to the Rankine-Hugoniot conditions. The strategy
may also be used in multidimensional cases (as in Section 3) or with wave
profiles that are not piecewise constant (as in [DR]). The argument is
crucially used to allow for perturbations adding discontinuities. However, as
long as perturbations do not change the regularity structure, up to some loss
in optimality one may replace the argument with a direct stability analysis
whose starting point is expounded below. In the scalar case, the latter
restriction amounts essentially to requiring smoothness on perturbations,
whereas in the system case it should also include compatibility conditions at
discontinuities, for instance met by perturbations supported far away from the
original discontinuities888We recall that during the finalization of the
present contribution we have been informed that a system case has been
analyzed in [YZ18] and we refer to it for such an example of analysis..
Unfortunately for the moment the consideration of discontinuous perturbations
in a system case still lies beyond reach of both frameworks.

To give an idea of what a direct stability analysis for Riemann shocks would
look like, we now write the corresponding spectral stability problem (in the
one-dimensional case). The question is whether one may solve uniquely in
$(\check{v}(\,\cdot\,;\lambda),\check{\psi}(\lambda))$ and continuously with
respect to any $(F,\varphi)$ the system

$\displaystyle(\lambda$
$\displaystyle+(f^{\prime}({\underline{u}}_{+})-\sigma)\partial_{x}-g^{\prime}({\underline{u}}_{+}))\,\check{v}(\,\cdot\,;\lambda)\,=\,F\qquad\textrm{on
}\mathbb{R}_{+}^{\star}\,,$ $\displaystyle(\lambda$
$\displaystyle+(f^{\prime}({\underline{u}}_{-})-\sigma)\partial_{x}-g^{\prime}({\underline{u}}_{-}))\,\check{v}(\,\cdot\,;\lambda)\,=\,F\qquad\textrm{on
}\mathbb{R}_{-}^{\star}\,,$ $\displaystyle\lambda\check{\psi}(\lambda)$
$\displaystyle-\,\left(\frac{f^{\prime}({\underline{u}}_{+})-\sigma}{{\underline{u}}_{+}-{\underline{u}}_{-}}\check{v}(0^{+};\lambda)-\frac{f^{\prime}({\underline{u}}_{-})-\sigma}{{\underline{u}}_{+}-{\underline{u}}_{-}}\check{v}(0^{-};\lambda)\right)\,=\,\varphi\,.$

An inspection of the proof of Lemma 1.7 shows that since
$f^{\prime}({\underline{u}}_{+})<\sigma$, when
$\Re(\lambda)>g^{\prime}({\underline{u}}_{+})$, $\check{v}(\,\cdot\,;\lambda)$
is uniquely and continuously determined on $\mathbb{R}_{+}^{\star}$ by the
first equation, and obtained there by solving from $+\infty$. Likewise, when
$\Re(\lambda)>g^{\prime}({\underline{u}}_{-})$, $\check{v}(\,\cdot\,;\lambda)$
is uniquely and continuously determined on $\mathbb{R}_{-}^{\star}$ by the
second equation, and obtained there by solving from $-\infty$. Thus when
$\Re(\lambda)>\max(\\{g^{\prime}({\underline{u}}_{-});g^{\prime}({\underline{u}}_{+})\\})$,
the problem is uniquely solved if and only if $\lambda\neq 0$. A closer
examination reveals that it follows from
$f^{\prime}({\underline{u}}_{+})<\sigma<f^{\prime}({\underline{u}}_{-})$ that
the spectrum is

$\left\\{\ \lambda\ ;\
\Re(\lambda)\leq\max(\\{g^{\prime}({\underline{u}}_{-});g^{\prime}({\underline{u}}_{+})\\})\
\right\\}\cup\\{0\\}$

and that when
$\max(\\{g^{\prime}({\underline{u}}_{-});g^{\prime}({\underline{u}}_{+})\\})<0$,
$0$ has multiplicity $1$ (in the sense provided by resolvent singularities).
The consideration of the Riemann shocks may leave the reader with the
deceptive impression that in scalar balance laws the stability of traveling
waves is determined by smooth parts of the profiles whereas shocks play a
passive role, encoding invariance by space translation. This special feature
is however due to the fact that $g({\underline{U}})$ is continuous across
discontinuities. In contrast, in [DR] we provide examples of some cases when
positions of discontinuities are exponentially stable and of some others when
they bring exponential instabilities even when the smooth parts are point-wise
dissipative (in a weighted sense that yields equivalent norms).

In the foregoing paragraph and all through the text, not much care was
required in the sense in which spectral properties were considered. Indeed,
any reasonable functional framework is associated with the same spectrum. This
is also reflected at the nonlinear level in the fact that stability is also
obtained in arbitrarily high regularity. This is easy to see at the spectral
level when profiles are piecewise constant, but it is more fundamentally
related to the ellipticity of underlying operators, hence here to the fact
that $f^{\prime}({\underline{U}})-\sigma$ is bounded away from zero. When this
condition is relaxed (to include sonic points), one may still deduce, at any
level of regularity, nonlinear stability from spectral stability. Yet spectral
properties do depend on functional choices, and specifically on the degree of
regularity under consideration. In particular in [DR] we exhibit an
instability mechanism that is seen in the topology of $BUC^{k}$, $k\geq 1$,
but not in $BUC^{0}$, which demonstrates that not all key properties in scalar
balance laws are captured by $L^{\infty}$-based estimates.

The present paper opens at least two clear close follow-up : the stability of
more general traveling waves of scalar balance laws, and the stability of
Riemann shocks in general systems. The former is studied in details in the
companion paper [DR] where breakthrough obtained here are combined with an
elucidation of the effects of the presence of sonic points in wave profiles so
as to derive an essentially complete picture for balance laws. The latter is
left for future work.

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