# An informal account of recent results on initial-boundary value problems for
systems of conservation laws

Laura V. Spinolo  L.V.S. CNR-IMATI, via Ferrata 5, I-27100 Pavia, Italy.
<EMAIL_ADDRESS>,  Fabio Ancona  F.A. Dipartimento di Matematica “Tullio
Levi-Civita”, Università degli Studi di Padova, Via Trieste 63, 35131 Padua,
Italy<EMAIL_ADDRESS>and  Andrea Marson  A.M. Dipartimento di
Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste
63, 35131 Padua, Italy<EMAIL_ADDRESS>
Abstract. This note aims at providing a rather informal and hopefully
accessible overview of the fairly long and technical work [4]. In that paper,
the authors established new global-in-time existence results for admissible
solutions of nonlinear systems of conservation laws defined in domains with
boundaries. The main novelty in [4] is that the solution is constructed by
taking into account the underlying viscous mechanism, which is relevant
because, in the case of initial-boundary value problems, different viscous
approximations yield in general different limits. This note will frame the
analysis of [4] in the relevant context, compare the main result with the
previous existing literature, and touch upon the most innnovative technical
points of the proof.

Keywords: systems of conservation laws, initial-boundary value problems,
hyperbolic systems, wave front-tracking, boundary characteristic case, mixed
hyperbolic-parabolic systems, boundary layers

MSC (2010): 35L65, 35B30

##  1\. Introduction

The goal of this note is to provide an handwaving overview of some recent
progress on the analysis of the initial-boundary value problem for nonlinear
systems of conservation laws in the form

(1.1) $\mathbf{g}(\mathbf{v})_{t}+\mathbf{f}(\mathbf{v})_{x}=\mathbf{0}.$

In the above expression, the unknown $\mathbf{v}$ depends on the variables
$(t,x)$, with $x$ one-dimensional, and attains values in $\mathbb{R}^{N}$. The
functions $\mathbf{g},\,\mathbf{f}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ are smooth
and satisfy suitable assumptions that we touch upon in the following. The
exposition in this note is mainly based on the recent paper [4], where (1.1)
is coupled with the viscous approximation

(1.2)
$\mathbf{g}(\mathbf{v}^{\varepsilon})_{t}+\mathbf{f}(\mathbf{v}^{\varepsilon})_{x}=\varepsilon\Big{(}\mathbf{D}(\mathbf{v}^{\varepsilon})\mathbf{v}^{\varepsilon}_{x}\Big{)}_{x}.$

In the above expression, $\mathbf{D}$ is a positive semi-definite $N\times N$
matrix depending on the physical model under consideration. We discuss in the
following the precise assumptions we impose on $\mathbf{D}$, for the time
being we only mention that we rely on the analysis in the fundamental works by
Kawashima and Shizuta, see [30]. In particular, our assumptions apply in the
case of the compressible Navier-Stokes (or Navier-Stokes-Fourier) equations
and the viscous MHD equations, which are the most natural choices for
approximating the Euler and inviscid MHD equations, respectively. From the
analytical standpoint, a very relevant feature of these cases (and of most, if
not all, the physically relevant cases) is that the matrix $\mathbf{D}$ is
singular, so that (1.2) is a _mixed hyperbolic-parabolic_ system. This
accounts for severe technical challenges that we touch upon in the following.

Coupling (1.1) with the underlying viscous mechanism is especially important
in the case of initial-boundary value problems because, as we will discuss in
the following, different choices of $\mathbf{D}$ in (1.2) yield in general
different solutions of (1.1) in the vanishing $\varepsilon$ limit. The main
result of [4], which is also Theorem 3.1 below, establishes global-in-time
existence of admissible solutions of (1.1) consistent with the underlying
viscous mechanism (1.2).

The outline of this note is as follows. In §2 we discuss the main motivation
for the analysis in [4] by providing some background information and focusing
in particular on the viscous approximation of initial-boundary value problems.
In §3 we state the main result of [4], extensively comment on it, and provide
a very high level overview of the proof. In §4 we compare the main result of
[4] with the existing literature, and finally in §5 we explain what we feel
are, from the technical standpoint, the most interesting points of the proof
of Theorem 3.1.

##  2\. Background information

###  2.1. Nonlinear systems of conservation laws in one space variable, and
their viscous approximation

The archetype of (1.1) are the celebrated compressible Euler equations
modeling the dynamics of an ideal and compressible fluid. In this case, $N=3$
and the component of the unknown $\mathbf{v}$ represent the fluid density,
velocity, and internal energy. Other famous examples include the inviscid
Magneto-Hydro-Dynamic (MHD) equations describing the propagation of plane
waves in an electrically charged, compressible and ideal fluid.

Note however that the present note, despite using the Euler and MHD equation
as a guiding thread for the exposition, aims at discussing results that apply
to _general_ systems of conservation laws. In this framework and in the case
of the Cauchy problem, Glimm [24] established existence of global-in-time,
admissible solutions of (1.1) by relying on a suitable random choice
approximation method now named after him. Uniqueness results were obtained in
a series of works by Bressan _et al._ , see [10] and also the very recent
contributions [13, 14]. A fundamental tool in the proof of the above
uniqueness results is the analysis of the so-called _wave front-tracking
scheme_ , a suitable approximation algorithm through piecewise constant
functions that we briefly discuss in §5.2 below. Note furthermore, and more
importantly, that this well-posedness theory requires that the total variation
of the initial data is sufficiently small, with the smallness threshold
depending on the specific system under consideration. Well-posedness results
for data of large total variation are only available for very specific
systems, like those in the so-called _Temple class_ [40], or in special
perturbative regimes, see [31]. This is a somehow unavoidable drawback of the
theory since there are explicit examples [28] showing the if the total
variation of the initial datum is finite, but large, then the admissible
solution can blow-up in finite time. The blow-up system in [28] is not
physical (i.e., it does not admit strictly convex entropies), but the recent
contributions [11, 12] provide strong indications of a possible finite time
blow-up of the total variation in the case of the so-called $p$-system of
isentropic gas dynamics in Lagrangian coordinates, arguably one of the
simplest systems with a clear physical meaning.

As mentioned before, in this note we will recover the inviscid system (1.1) as
the limit of the underlying viscous mechanism (1.2). Indeed, the viscous
system (1.2) formally boils down to (1.1) as the parameter $\varepsilon$
vanishes. However, despite considerable efforts and advances, the analytical
understanding of the vanishing viscosity limit is still badly incomplete and a
general convergence proof is missing. In the _artificial viscosity_ case, that
is when the matrix $\mathbf{D}$ is the identity, the right-hand side of (1.2)
boils down to $\varepsilon\mathbf{v}^{\varepsilon}_{xx}$ and a very complete
result was established by Bianchini and Bressan in [7]. In the _physical case_
the situation is much less understood and fewer results are available. Among
them, the remarkable work by G.Q. Chen and Perepelitsa [17] applies to system
of compressible and barotropic gas dynamics, that is to the $2\times 2$
Navier-Stokes equations in Eulerian coordinates, and relies on the compensated
compactness techniques discussed in the pivotal works by DiPerna [19, 20]. We
also refer to the very recent contribution by G. Chen, Kang and Vasseur [15],
where the authors manage to recover small $BV$ solutions of the system of
compressible and barotropic inviscid gas dynamics as strong limits of
solutions of the physical viscous approximation, that is the $2\times 2$
Navier-Stokes equations.

###  2.2. On the viscous approximation of nonlinear systems of conservation
laws in domains with boundaries

In the case of initial-boundary value problem, the analysis of the vanishing
viscosity limit of (1.2) is further complicated by the presence of boundary
layer phenomena, that account for a loss of boundary condition and for a sharp
transient behavior of the solution of the viscous system close to the domain
boundary. Nevertheless, partial convergence results have been obtained in some
works, see for instance [3, 16, 23, 26, 29, 35, 39]. See also [37, 38] for a
general introduction to initial-boundary value problems for conservation laws.

The signature feature of the viscous approximation of the initial-boundary
value problems, and a striking difference with the Cauchy problem case, is
that the vanishing $\varepsilon$ limit of (1.2) in general _depends_ on
$\mathbf{D}$ and changes as $\mathbf{D}$ changes, see [23]. Remarkably, this
happens even in the most elementary linear case: consider the linear viscous
system

(2.1)
$\mathbf{v}^{\varepsilon}_{t}+\mathbf{A}\mathbf{v}^{\varepsilon}_{x}=\varepsilon\mathbf{D}\mathbf{v}^{\varepsilon}_{xx},$

where $\mathbf{A}$ and $\mathbf{D}$ are (constant) $N\times N$ matrices. To
highlight the heart of the matter and avoid some technicalities, let us assume
that $\mathbf{A}$ is symmetric and invertible and that $\mathbf{D}$ is
symmetric and positive definite. We assume that (2.1) is defined on the domain
$x>0$, $t\geq 0$ and we couple (2.1) with the Riemann-type data

(2.2)
$\mathbf{v}^{\varepsilon}(0,x)=\mathbf{v}_{0},\qquad\mathbf{v}^{\varepsilon}(t,0)=\mathbf{v}_{b},$

where $\mathbf{v}_{0}$ and $\mathbf{v}_{b}$ are given states in
$\mathbb{R}^{N}$. In this case, fairly standard energy-type estimates ensure
that $\mathbf{v}^{\varepsilon}$ converges weakly in
$L^{2}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$ to a weak solution
$\mathbf{v}$ of the linear advection equation

(2.3) $\mathbf{v}_{t}+\mathbf{A}\mathbf{v}_{x}=0$

which has the following structure: it is piecewise constant, with contact
discontinuities occurring along $p$ half-lines departing from the origin. Here
$p$ denotes the number of _positive_ eigenvalues of $\mathbf{A}$. The solution
$\mathbf{v}$ attains the boundary datum $\mathbf{\bar{v}}:=\mathbf{v}(t,0)$,
which does not depend on $t$ since $\mathbf{v}$ is self-similar. Most
importantly, the relation between $\mathbf{\bar{v}}$ and the boundary datum
$\mathbf{v}_{b}$ imposed on the viscous approximation is the following: there
is a so-called _boundary layer_ $\mathbf{w}:[0,+\infty[\to\mathbb{R}^{N}$ such
that

(2.4)
$\left\\{\begin{array}[]{ll}\mathbf{A}\mathbf{w}^{\prime}=\mathbf{D}\mathbf{w}^{\prime\prime}\\\
\mathbf{w}(0)=\mathbf{v}_{b},\quad\lim_{y\to+\infty}\mathbf{w}(y)=\mathbf{\bar{v}}.\end{array}\right.$

Note that the equation at the first line of the above system dictates that the
boundary layer $\mathbf{w}$ is, up to a change of variables, a steady solution
of the viscous equation (2.1). Fairly classical results on linear systems
imply that the boundary value problem (2.4) admits a solution if and only if
$\mathbf{v}_{b}-\mathbf{\bar{v}}$ belongs to the so-called _stable subspace_ ,
that is to the generalized eigenspace of $\mathbf{D}^{-1}\mathbf{A}$
associated to eigenvalues with negative real part. Since this eigenspace
obviously _depends_ on $\mathbf{D}$, so does the boundary value
$\mathbf{\bar{v}}$ and hence the limit solution $\mathbf{v}$.

The above elementary argument describes the very basic mechanism yielding the
dependence of the vanishing viscosity approximation on the underlying viscous
system. Note that it is a mechanism, basically due to boundary layer
phenomena, that acts in the initial-boundary case only, and indeed in the case
of the Cauchy problem it can be shown that, at least in suitable small total
variation regimes, the limit of the vanishing viscosity (1.2), if any, does
not depend on $\mathbf{D}$, see [6]. As a side remark, the dependence on the
underlying viscous mechanism of solutions of conservation laws defined on
domains with boundary has also very relevant consequences from the numerical
viewpoint. Very loosely speaking, this is due to the fact that most numerical
schemes for conservation laws contain the so-called _numerical viscosity_. In
standard numerical schemes, the numerical viscosity is modeled upon the
_artificial viscosity_ , namely the viscosity mechanism obtained by choosing
as viscosity matrix $\mathbf{D}$ the identity. As a consequence, standard
numerical scheme provide an approximation of the inviscid limit of the
artificial viscosity, which, in the case of initial-boundary value problems,
differs from the physically relevant solution, which is the limit of the
_physical_ viscosity. To fix this issue, one can introduce new numerical
schemes where the numerical viscosity is modeled upon the physical viscosity,
see [33] for related numerical experiments.

##  3\. Main result

In this section we state the main result of [4], which is Theorem 3.1 below.
More precisely, in §3.1 we provide the formulation of the boundary condition
assigned on (1.1), a non trivial issue in view of the discussion in §2.2. In
§3.2 we provide the precise statement of Theorem 3.1, and make some comments.
Finally, in §3.3 we discuss the proof of Theorem 3.1 at a fairly high level,
and refer to §5 for more technical comments.

###  3.1. Formulation of the initial-boundary value problem

Going back to the analytic study of the initial-boundary value problem for
(1.1), from the discussion in §2.2 and in particular from the considerations
on the linear case (2.3) we infer that the key point to understand the
dependence of the viscous limit on the viscosity matrix $\mathbf{D}$ is to
unveil the relation between the boundary datum imposed on the _viscous_ system
(1.2) and the trace of the solution of the inviscid conservation law (1.1).
Towards this end, we first of all point out that, owing to the singularity of
the matrix $\mathbf{D}$, the initial-boundary value problem for (1.2) is in
general overdetermined if one imposes a _full_ boundary condition like
$\mathbf{v}^{\varepsilon}(t,0)=\mathbf{v}_{b}(t)$. To see this, let us
consider a specific example: the first line of both the Navier-Stokes and the
viscous MHD equation written in Eulerian coordinates expresses mass
conservation and reads

$\partial_{t}\rho+\partial_{x}[\rho u]=0,$

where $\rho$ and $u$ denote the fluid density and velocity, respectively. If
$u<0$ at the boundary then assigning both the boundary and the initial
condition on $\rho$ would yield an overdetermined problem. Going back to the
general case, under fairly resonable assumptions on the mixed hyperbolic-
parabolic system one can introduce a slightly involved formulation of the
boundary condition to restore well-posedness, namely one couples (1.2) with
the initial and boundary conditions

(3.1)
$\mathbf{v}^{\varepsilon}(0,\cdot)=\mathbf{v}_{0},\qquad\boldsymbol{\widetilde{\beta}}(\mathbf{v}^{\varepsilon}(\cdot,0),\mathbf{v}_{b})=\mathbf{0}_{N}.$

The precise definition of the function
$\boldsymbol{\widetilde{\beta}}:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}^{N}$
is provided in [4, §2.2], but the very basic idea underpinning the
construction of $\boldsymbol{\widetilde{\beta}}$ is imposing a full boundary
condition on the _parabolic_ component of $\mathbf{v}^{\varepsilon}$, and a
boundary condition along the characteristic fields of the _hyperbolic_
component entering the domain. In particular, going back to specific example
of the Navier-Stokes equations, prescribing
$\boldsymbol{\widetilde{\beta}}(\mathbf{v}^{\varepsilon}(\cdot,0),\mathbf{v}_{b})=\mathbf{0}_{N}$
means that we _always_ assign the values of the fluid velocity $u$ and of the
internal energy at the boundary, and that if $u>0$ at the domain boundary we
also assign the value of the fluid density, otherwise we do not.

With the above notation in place, we can now discuss the boundary condition we
impose on the inviscid system (1.1). We proceed step by step and at first we
introduce an additional assumption that considerably simplifies the matter. We
term the domain boundary $x=0$ _not characteristic_ if all the eigenvalues of
the jacobian matrix $\mathbf{D}\mathbf{f}(\mathbf{u})$ are bounded away from
$0$, for every $\mathbf{u}\in\mathbb{R}^{N}$. In other words, in the non
characteristic boundary case we can easily discriminate between the
characteristic field of (1.1) _entering_ and _leaving_ the domain at the
domain boundary. In this framework, we impose the initial and boundary
condition on (1.1) by requiring that

(3.2) $\mathbf{v}(0,\cdot)=\mathbf{v}_{0}\;\;\text{a.e. on
$\mathbb{R}_{+}$},\qquad\mathbf{v}(\cdot,0)\sim_{\mathbf{D}}\mathbf{v}_{b}\;\;\text{a.e.
on $\mathbb{R}_{+}$},$

where the relation $\sim_{\mathbf{D}}$ is defined as follows: given
$\mathbf{\bar{v}},\mathbf{v}_{b}\in\mathbb{R}^{N}$ we say that
$\mathbf{\bar{v}}\sim_{\mathbf{D}}\mathbf{v}_{b}$ if there is a boundary layer
$\mathbf{w}:\mathbb{R}_{+}\to\mathbb{R}^{N}$ such that

(3.3)
$\left\\{\begin{array}[]{ll}\mathbf{D}(\mathbf{w})\mathbf{w}^{\prime}=\mathbf{f}(\mathbf{w})-\mathbf{f}(\mathbf{\bar{v}})\\\
\boldsymbol{\widetilde{\beta}}(\mathbf{w}(0),\mathbf{v}_{b})=\mathbf{0}_{N},\quad\lim_{y\to+\infty}\mathbf{w}(y)=\mathbf{\bar{v}}.\end{array}\right.$

Keeping in mind that
$\boldsymbol{\widetilde{\beta}}(\mathbf{w}(0),\mathbf{v}_{b})=\mathbf{0}_{N}$
is the way we assign the boundary condition on the viscous system (1.2), we
conclude that (3.3) is, up to space integration, the exact nonlinear analogous
of (2.4).

We now move towards the most general case and take into account the
possibility that an eigenvalue of the jacobian matrix
$\mathbf{D}\mathbf{f}\mathbf{(}\mathbf{u})$ attains the value $0$: this is
usually refered to as the _boundary characteristic case_. Note that for
instance the boundary $x=0$ is characteristic for the compressible Euler
equations written in Eulerian coordinates if the fluid velocity either
vanishes, or is close (in modulus) to the sound speed. The boundary $x=0$ is
always characteristic for the compressible Euler equations written in
Lagrangian coordinates, as the second eigenvalue vanishes identically.

Compared to the treatment of the non-characteristic case, the analysis of the
boundary characteristic case involves additional severe technical challenges
that we touch upon in the following. For the time being, we mention that in
the boundary characteristic case we have to take into account the possibility
of a shock or contact discontinuity that sits exactly at the domain boundary.
With the above preliminary considerations in place, we can now introduce the
general definition of the relation $\sim_{\mathbf{D}}$. For simplicity, from
now on we focus on the case where the characteristic field associated to the
vanishing eigenvalue is either genuinely nonlinear or linearly degenerate: in
the following, we term this field _boundary characteristic field_. Note that
in many physically relevant systems, like the compressible Euler equations
written in both Eulerian and Lagrangian coordinates, every vector field (and
henceforth, also the boundary characteristic field, if any) is either
genuinely nonlinear or linearly degenerate.

######  Definition 3.1.

Given system (1.2) and $\mathbf{\bar{v}}$, $\mathbf{v}_{b}\in\mathbb{R}^{N}$,
we say that “$\mathbf{\bar{v}}\sim_{\mathbf{D}}\mathbf{v}_{b}$” if there is
$\mathbf{\underline{v}}\in\mathbb{R}^{N}$ such that the following conditions
are both satisfied:

  * i)

$\mathbf{f}(\mathbf{\bar{v}})=\mathbf{f}(\mathbf{\underline{v}})$; if the
boundary characteristic field is genuinely nonlinear, we also require that the
$0$-speed shock between $\mathbf{\bar{v}}$ (on the right) and
$\mathbf{\underline{v}}$ (on the left) is Lax admissible111if the boundary
characteristic field is linearly degenerate, then the discontinuity is a
contact discontinuity and the Lax admissibility condition is automatically
satisfied;

  * ii)

there is a so-called “boundary layer”
$\mathbf{w}:\mathbb{R}_{+}\to\mathbb{R}^{N}$ such that

(3.4)
$\left\\{\begin{array}[]{ll}\mathbf{D}(\mathbf{w})\mathbf{w}^{\prime}=\mathbf{f}(\mathbf{w})-\mathbf{f}(\mathbf{\underline{v}})\\\
\boldsymbol{\widetilde{\beta}}(\mathbf{w}(0),\mathbf{v}_{b})=\mathbf{0},\quad\lim_{y\to+\infty}\mathbf{w}(y)=\mathbf{\underline{v}}.\end{array}\right.$

Note that, if we apply Definition 3.1 in the non characteristic boundary case,
if $\mathbf{\bar{v}}$ is confined in a small enough neighborhood of
$\mathbf{\underline{v}}$ the identity
$\mathbf{f}(\mathbf{\bar{v}})=\mathbf{f}(\mathbf{\underline{v}})$ implies by
the Local Invertibility Theorem that
$\mathbf{\bar{v}}=\mathbf{\underline{v}}$, and hence (3.4) boils down to
(3.3).

###  3.2. Global-in-time existence of admissible solutions

With Definition 3.1 in place, we can now state the main result of [4].

######  Theorem 3.1.

Assume that $\mathbf{g}$, $\mathbf{f}$, $\mathbf{D}$ satisfy Hypotheses
$1,\cdots,5$ in [4, §2], and fix $\mathbf{v}^{\ast}\in\mathbb{R}^{N}$; then
there is a constant $\delta^{\ast}>0$ only depending on the functions
$\mathbf{g}$, $\mathbf{f}$, $\mathbf{D}$ in system (1.2) such that the
following holds. If $\mathbf{v}_{0},\mathbf{v}_{b}\in BV(\mathbb{R}_{+})$
satisfy

(3.5) $\mathrm{TotVar}\ \mathbf{v}_{0}+\mathrm{TotVar}\
\mathbf{v}_{b}+|\mathbf{v}_{0}(0^{+})-\mathbf{v}_{b}(0^{+})|\leq\delta^{\ast},\quad|\mathbf{v}_{0}(0^{+})-\mathbf{v}^{\ast}|\leq\delta^{\ast}$

then there is a global-in-time, Lax admissible distributional solution
$\mathbf{v}\in BV_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R}_{+})$ of (1.1)
satisfying the initial and boundary conditions (3.2) in the sense of
Definition 3.1. Also, if system (1.1) admits a convex entropy then the
solution we construct is entropy admissible.

In the statement of the above result, $\mathrm{TotVar}\ \mathbf{v}$ and $BV$
denote the total variation of the function $\mathbf{v}$ and the space of
bounded total variation functions, respectively. Also, $\mathbf{v}_{0}(0^{+})$
and $\mathbf{v}_{b}(0^{+})$ denotes the right limit of $\mathbf{v}_{0}$ and
$\mathbf{v}_{b}$ at $0$, respectively. Note furthermore that, as a function of
total variation, the function $\mathbf{v}$ has a well-defined trace at $x=0$.

We refer to §4 below for a more detailed comparison between Theorem 3.1 and
previous related results. Here we just point out that the main novelty of
Theorem 3.1 is that, in the boundary characteristic case, it provides what is
to the best of our knowledge the first global existence result for solutions
of nonlinear systems of conservation laws (1.1) consistent with the underlying
viscous mechanism (1.2). This consistency is encoded in the boundary condition
of Definition 3.1. As we discussed before, enclosing information on the
underlying viscous mechanism in the definition of solution is fundamental,
because different viscous approximation yield different inviscid limits in the
case of initial-boundary value problems.

Rather than providing the exact statement of the hypotheses of Theorem 3.1
concerning $\mathbf{g}$, $\mathbf{f}$, $\mathbf{D}$, which would require the
introduction of some heavy notation, we now informally discuss them. First of
all, we stress that Hypotheses $1,\cdots,5$ in [4, §2] are all satisfied by
our archetipycal physical examples, that is by the compressible Euler and MHD
equations, written in both Eulerian and Lagrangian coordinates. More in
detail, Hypothesis 1 states that system (1.2) is, up to a change of the
dependent variables, in the _normal form_ , in the Kawashima-Shizuta sense
[30], that is it allows for a specific block decomposition. Hypothesis 2
states that system (1.2) satisfies the so-called _Kawashima-Shizuta condition_
, which heuristically speaking is a coupling condition that rules out the
possibility of singling out in (1.2) a purely hyperbolic component. Hypothesis
3 is a fairly standard strict hyperbolicity assumption. Hypothesis 4 is also
fairly common in the conservation laws analysis framework and dictates that
every characteristic field of (1.1) is either linearly degenerate or genuinely
nonlinear. Finally, Hypothesis 5 requires that system (1.2) satisfies some
slightly technical conditions introduced in the previous works [8, 9] whose
meaning is, loosely speaking, to ensure that the boundary layers system (3.4)
can be written in a tractable form, a nontrivial requirement given the
singularity of the viscosity matrix $\mathbf{D}$.

Concerning the small total variation assumption (3.5), this is obviously
highly restrictive, but it is basically a necessary price to pay if the goal
is to deal with fairly general systems, since as mentioned before there are by
now very strong indications that, if the smallness assumptions fails, then
finite time total variation blow-up may occur even in the case of relativey
simple examples like the $p$-system, see [12].

As a last comment, we point out that Theorem 3.1 is a global existence result
only: we are fairly confident one could establish uniqueness results by
relying on the Standard Riemann Semigroup approach _à la_ Bressan, see also
the related work [2]. However, we decided to postpone the uniqueness proof to
the future given that [4] is already fairly long and technical.

###  3.3. A very non technical overview of the proof of Theorem 3.1

We now briefly and informally comment on the proof of Theorem 3.1, and we
refer to §5 below for some more detailed technical remarks concerning the main
novelties of our construction. In a nutshell, the proof of Theorem 3.1 relies
on the introduction of a new wave front-tracking algorithm, and the most
delicate points of our analysis stem from the fact that we deal with the
boundary characteristic case. At a very high level, the presence of a possibly
vanishing boundary characteristic field implies that one cannot make a clear
distinction between waves that are entering and leaving the domain $x>0$ at
the domain boundary. In particular, one should in principle take into account
the possibility of waves of the boundary characteristic family that are
bounced back and forth at the boundary, a behavior that could lead to the
finite time blow-up of the total variation and to the breakdown of the wave
front-tracking algorithm. At a more technical level, our analysis relies on a
detailed description of the structure of the boundary layers $\mathbf{w}$
solving (3.4) that was provided in the previous works [8, 9]. If the boundary
is characteristic, the boundary layers may have a component lying on a
suitable center manifold: by slightly perturbing this component one can obtain
waves (rarefaction waves, shocks or contact discontinuities) of the boundary
characteristic family that travel with very small albeit positive speed and
hence enter the domain. This possibility severely complicates our analysis,
and requires the introduction of several new ideas to obtain useful estimates
on the total variation increase at times where a boundary characteristic wave
hits the boundary, estimates that in the wave front-tracking jargon are called
_interaction estimates_. We come back to this point in §5.

Note that relying on wave front-tracking techniques often accounts for fairly
long and technical proofs, but also carries several advantages and paves the
way for further developments. In particular, the introduction of a suitable
wave front-tracking algorithm is pivotal to the proof of the uniqueness
results for (1.1) due to Bressan and collaborators, see [10]. Also, it is
known that for instance a $BV$ solution $\mathbf{v}$ of the conservation law
(1.1) recovered as the limit of a wave front-tracking algorithm enjoys better
regularity properties than a generic function of bounded variation, see for
instance the analysis in [10, Ch. 10].

##  4\. Comparison with previous results

###  4.1. The scalar case

To start our little overview of global-in-time existence results for solutions
of (1.1) defined on domains with boundary we briefly touch up the scalar case
$N=1$. In this case, and for conservation laws in several space dimensions,
the paper [5] by Bardos, Le Roux and Nedelec extends the analysis of the
milestone work of Kružkov to the case of domains with boundaries. In
particular, it establishes convergence of the vanishing viscosity
approximation and uniqueness of (a suitable notion of) entropy admissible
solution.

###  4.2. Global in time existence results _via_ Glimm-type schemes

Moving towards systems, i.e towards the case $N>1$, one of the very first
works dealing with initial-boundary value problems is the paper by Nishida and
Smoller [34]. It establishes global-in-time existence results for the so-
called _piston problem_ , namely the initial-boundary value problem for the
$p$-system of isentropic gas dynamics in Lagrangian coordinates with the
values of the fluid velocity prescribed at the domain boundary. The analysis
in [34] relies on the introduction of a suitable modification of the
approximation algorithm introduced in [24] and by-now known as _the Glimm
scheme_. Also based on the analysis of a Glimm-type algorithm is the paper by
Liu [32] concerning the full compressible Euler equations written in
Lagrangian coordinates. This is a system of $3$ equations where the second
eigenvalue of the jacobian matrix of the flux is identically $0$, which
implies that the boundary is characteristic. Note, however, that the fact that
the characteristic eigenvalue vanishes identically (and not only at some
point) considerably simplifies the treatment of the boundary characteristic
case as it still allows for a clear discrimination between wave entering and
leaving the domain at the boundary, with the waves of the second family
sitting exactly at the domain boundary and tangent to it. In [32], the author
assigns the value of either the fluid velocity or the pressure at the domain
boundary.

The most general contributions that rely on the introduction of Glimm-type
schemes are the PhD thesis of Goodman [25] and the work by Sablé-Tougeron
[36]. In particular, Goodman establishes global-in-time existence of
admissible solutions for general systems in the non-characteristic boundary
case. Sablé-Tougeron deals with either the non-characteristic boundary case or
the boundary characteristic case where the characteristic eigenvalue vanishes
identically. The boundary condition is assigned by imposing that

(4.1) $\mathbf{b}(\mathbf{v}(t,0))=\mathbf{h}(t)\quad\text{a.e.
$t\in\mathbb{R}_{+}$}.$

In the previous expression, when the domain is $x>0$ the boundary datum
$\mathbf{h}$ attains values in $\mathbb{R}^{p}$, where $p$ is the number of
strictly positive eigenvalues of the jacobian matrix of
$\mathbf{D}\mathbf{f}$. The function
$\mathbf{b}:\mathbb{R}^{N}\to\mathbb{R}^{p}$ is given and must sastify
suitable assumptions. Once again, it is worth stressing that the assumption
that the boundary characteristic eigenvalue vanishes identically is
foundamental here: in the general boundary characteristic case where one
eigenvalue can attain both positive and negative eiegenvalues, the number of
boundary conditions to prescribe at the boundary is not determined _a priori_
as it depends on the sign of the characteristic eigenvalue of the solution at
the boundary, and hence one cannot formulate the boundary condition as in
(4.1). It should be also noted that all the above works [25, 32, 34, 36]
require suitable smallness assumptions on the total variation of the data (or
on some related quantities, as for instance in [34]) and in several cases the
analysis extends to equations defined on the strip $0<x<1$. In particular, in
[36] the author introduces an _ad-hoc_ condition to prevent the possible
amplification of reflected waves in the case.

###  4.3. Results _via_ wave front-tracking algorithms

In [1] Amadori established global existence results for admissible solution of
(1.1) defined on the domain $x>0$. The proof requires, as usual, smallness
assumptions on the data and relies on the introduction of a suitable wave
front-tracking algorithm. In [1] the author considers both non characteristic
and characteristic boundaries, and in the latter case there is no constraint
on the characteristic eigenvalue, which can attain both positive and negative
values. In the non characteristic boundary case, the boundary condition is
prescribed as in (4.1), whereas in the boundary characteristic case Amadori
relies on a formulation introduced in a previous work by Dubois and LeFloch
[22] which we now briefly discuss. Given
$\mathbf{v}_{b}:\mathbb{R}_{+}\to\mathbb{R}^{N}$, in [1] one assigns the
boundary condition in the boundary characteristic case by prescribing that

$\mathbf{v}(t,0)\sim_{\ast}\mathbf{v}_{b}(t)\quad\text{a.e. on
$\mathbb{R}_{+}$},$

where the relation $\sim_{\ast}$ is defined as follows. Consider the Riemann
problem obtained by coupling (1.1) with the initial datum

$\mathbf{v}(0,x)=\left\\{\begin{array}[]{ll}\mathbf{u}^{-}&x<0\\\
\mathbf{u}^{+}&x>0\\\ \end{array}\right.$

and set $\mathbf{u}^{-}:=\mathbf{v}_{b}(t)$, $\mathbf{u}^{+}:=\mathbf{v}(t,0)$
(the trace of the solution $\mathbf{v}$). Then
$\mathbf{v}(t,0)\sim_{\ast}\mathbf{v}_{b}(t)$ if the solution of the Riemann
problem only contains waves that are not entering the domain, namely have non-
positive speed. It is worth stressing that this way of assigning the boundary
condition does not involve any information on the underlying viscous mechanism
and in general it is different from the one provided by Definition 3.1. To
appreciate this difference, and henceforth the difference between the
relations $\sim_{\ast}$ and $\sim_{\mathbf{D}}$, it suffices to consider the
linear case (2.1). The boundary condition considered in [1, 22] dictates that,
for a.e. $t\in\mathbb{R}_{+}$, $\mathbf{v}_{b}(t)-\mathbf{v}(t,0)$ belongs to
the generalized eigenspace of the matrix $\mathbf{A}$ associated to non-
positive eigenvalues, whereas prescribing (3.2) we prescribe that, for a.e.
$t\in\mathbb{R}_{+}$, $\mathbf{v}_{b}(t)-\mathbf{v}(t,0)$ belongs to the
generalized eigenspace of the matrix $\mathbf{D}^{-1}\mathbf{A}$ associated to
non-positive eigenvalues. Since in general the eigenspaces of $\mathbf{A}$ and
$\mathbf{D}^{-1}\mathbf{A}$ do not coincide, the two conditions are different.

###  4.4. Vanishing viscosity approximation

To conclude this brief overview, we mention that, in the case of the
_artificial viscosity_ where $\mathbf{D}$ is the identity matrix, there are
some results establishing convergence of the vanishing viscosity approximation
(1.2) in domains with boundaries, see [3, 16, 39]. As a byproduct, these
results yield global existence for solutions of the conservation law (1.1). We
stress once more, however, that in the case of initial-boundary value problems
global-in-time convergence proofs are limited to the artificial viscosity
case, and hence in general the limit does not provide the physically relevant
solution. Convergence results for the physical viscosity have been only
obtained for regular data and on small time intervals (typically, before the
classical solution of the conservation law breaks down), see for instance the
works by Gisclon [23] and by Rousset [35]. See also Joseph and LeFloch [29]
for the convergence of a different self-similar viscous approximation in the
case of Riemann-type data.

##  5\. Some technical comments on the proof of Theorem 3.1

In this paragraph we provide some handwaving comment on the main technical
novelties of the construction in [4]. In §5.1 we make some general comment on
the proof of Theorem 3.1 and mention that the most innovative technical
points, which we discuss in §5.3, are a very detailed boundary analysis, and
some new interaction estimates. In §5.2 to frame the interaction estimates in
the right context we briefly and informally overview the wave front-tracking
construction for the Cauchy problem.

###  5.1. General comments

As mentioned in §3.3, the proof of Theorem 3.1 relies on the introduction of a
new wave front-tracking algorithm. At a very high level, the main novelty of
the scheme discussed in [4] is that it incorporates detailed information on
the underlying viscous approximation and in particular on the transient
behavior as $\varepsilon\to 0^{+}$ of (1.2) at the domain boundary: for
instance, we introduce a new Glimm-type interaction functional that involves,
among other things, the strength of the center component of the boundary layer
sitting at the domain boundary. Note that incorporating information on the
viscous mechanism is possible owing to the detailed analysis of the structure
of the boundary layer profiles (that is, of the solutions of (3.4)) that was
done in [8, 9].

At a lower and more detailed level, it is worth highlighting two main
technical novelties introduced in [4]. The first one is a new interaction
estimate that applies when the boundary characteristic field is genuinely
nonlinear and a wave of the characteristic family hits the boundary. We
comment on this estimate in the following, for the time being we point out
that the boundary characteristic field is genuinely nonlinear if, for
instance, system (1.1) are the compressible Euler equations written in
Eulerian coordinates and the modulus of the fluid velocity is close to the
sound speed. The second technical novelty that we want to single out is a
detailed analysis of the behavior of the wave front-tracking approximation at
the domain boundary, see [4, §9]. This analysis is instrumental in showing
that the limit of our wave front-tracking approximation satisfies indeed the
boundary condition (3.2), and exploits fine properties of the wave front-
tracking algorithm that are discussed in [10, Ch.10].

###  5.2. The wave front-tracking algorithm

Prior to discussing the main technical novelties that we touched upon before,
we have to briefly recall some very basic steps in the construction of the
wave front-tracking algorithm, and we refer to the fundamental references [10,
18, 27] for a complete discussion. In a nutshell, any wave front-tracking
scheme provides an approximation of the admissible solution of (1.1) through
piecewise constant functions with a finite number of discontinuity lines, the
so-called wave fronts after which the algorithm is named. In the case of the
Cauchy problem, the first step of the algorithm is constructing a piecewise
constant approximation of the initial datum. At every discontinuity point of
the approximate initial datum, one constructs a piecewise constant solution of
the Riemann problem with data given by the left and right limits of the
approximate initial datum at the discontinuity point. In particular, to obtain
a piecewise constant solution one must approximate rarefaction waves through a
wave fan involving finitely many states separared by discontinuity lines. Once
the approximation of the solution of the Riemann problem at each single
discontinuity is determined, one can construct a local-in-time approximate
solution of the Cauchy problem by juxtaposing the approximate solutions of the
Riemann problems. This solution can be extended in time until two distinct
wave fronts cross each other: to further extend the solution, one
approximately solves the Riemann problem determined at the interaction point,
and in this way defines an approximate solution that is defined up to the
second interaction time, where one approximately solves the corresponding
Riemann problem, and so on. Needless to dwell on the details, two of the main
challenges in constructing global-in-time approximate solutions with a wave
front-tracking algorithm are on the one hand to prevent the formation of
infinitely many wave fronts in finite time, and on the other to control the
total variation growth. To tackle the first challenge, a by-now-standard
technique, which we also use in [4], is the introduction of so-called _non-
physical_ fronts: very loosely speaking, wave fronts that are somehow
negligible are forced to travel with very high speed and to have minimal
interactions with the “physical” fronts, see [10, Ch. 7] for the precise
construction. To tackle the second issue and find a uniform bound on the total
variation of the approximate solution a key point is establishing the so-
called interaction estimates, namely extracting detailed information on the
strength of the waves generated when two wave fronts interact, in terms of the
strength of the interacting waves. As a matter of fact, establishing suitable
interaction estimates is one of the key points of the constrution of the wave
front-tracking algorithm, and the precise form of these estimate determines
the specific form of the Glimm-type functionals used to control the total
variation.

###  5.3. Main technical novelties of [4]

In the case of the initial-boundary value problem (1.1),(3.2), the first step
in our construction is establishing piecewise constant approximations of the
initial datum $\mathbf{v}_{0}$ and of the boundary datum $\mathbf{v}_{b}$. At
every discontinuity point in the approximate boundary datum we approximately
solve the so-called boundary Riemann problem, see [4, §4] for the precise
construction and in particular §4.1 for an heuristic overview of the basic
ideas underpinning the analysis. Without dwelling on the details, what is
worth pointing out here is that our approximate solution of the boundary
Riemann problem relies on the exact solution discussed in [8, 9], and provides
precise information on the boundary layer profiles satisfying (3.4), and in
particular on the size of their component on a suitable center manifold. In
the following, we term this size $\xi_{k}$, with $k$ denoting the index of the
boundary characteristic family222in other words, the $k$-th eigenvalue of
$\mathbf{D}\mathbf{f}(\mathbf{u})$ is the one that can vanish, and as we will
see it will appear in our main interaction estimate.

As in the case of the Cauchy problem, for the initial-boundary value problem
by juxtaposing approximate solutions of Riemann and boundary Riemann problems
we can define a local-in-time solution which can be extended up to the first
interaction time. Note that, in the case of the initial-boundary value
problem, interactions occur at times where either two wave fronts collide, or
a wave front hits the boundary. To handle the first case we basically proceed
as in the case of the Cauchy problem and employ the interactions estimates
established in [10, Ch 7]. The non standard part is in the handling of the
second type of collision. In what follows we focus on the most innovative part
of our contribution and assume that the boundary characteristic field is
genuinely nonlinear. We consider a wave front of the boundary characteristic
family that hits the domain boundary $x=0$ at the time $\tau$, see Figure 1
for a representation.

$t$$\tau$ Figure 1. A wave front of the boundary characteristic family (red)
hitting the boundary at time $\tau$, and the waves entering the domain that
are generated at the interaction.

Our main interaction estimate is equation (6.5) in [4] and states that there
is a constant $C$, only depending on system (1.2) and on the value
$\mathbf{v}^{\ast}$ in the statement of Theorem 3.1, such that the following
holds: if we term $|\Delta V(\tau)|$ the increase of the total variation of
the approximate solution at time $\tau$ we have

(5.1) $|\Delta V(\tau)|\leq C|s_{k}|([\varsigma_{k}]^{-}+|\xi_{k}|).$

In the previous expression, $s_{k}$ denotes the size of the hitting wave, and
$\xi_{k}$ is either the size of the center component of the boundary layer
sitting at the domain boundary before the interaction, or the size of the
$0$-speed Lax admissible shock located at $x=0$ before the interaction. As a
matter of fact, the cases where the boundary layer has a center component and
where there is a Lax shock tangent at the boundary are mutually exclusive
owing to the genuine nonlinearity assumption, see [4, §4.3] for the details of
the analysis. Finally, $[\varsigma_{k}]^{-}$ denotes the negative part of the
quantity $\varsigma_{k}$, which in turn is the shock speed if the hitting
front is a shock wave, and the eigenvalue of the boundary characteristic
family evaluated at the leftmost state if the hitting front is a rarefaction
wave. To the best of our knowledge, (5.1) is the first boundary interaction
estimate involving the speed of the hitting wave, as well as the first one
involving the size of the boundary layer sitting at the domain boundary. From
the technical standpoint, the proof of (5.1) is rather delicate and it is
complicated by the fact that the functions involved in the definition of the
solution of the boundary Riemann problem (and in particular the so-called
_characteristic wave fan curve of admissible states_ defined in [4, §4.3]) are
only Lipschitz continuous and fail to attain $C^{1}$ regularity.

After establishing our interaction estimates, in [4] we introduce a new Glimm-
type functional, which in turn allows us to control the total variation growth
and hence establish strong $L^{1}_{\mathrm{loc}}$ compactness of the wave
front-tracking approximation _via_ the Helly-Kolmogorov Compactness Theorem.
What is then left to prove is that any accumulation point of the wave-front
tracking approximation is a solution of the initial-boundary value problem
(1.1),(3.2). If some parts of the limit analysis are by now fairly standard
and rely on the well-established techniques discussed in [10], showing that
the any accumulation point satisfies the boundary condition in the sense of
Definition 3.1 is a highly nontrivial point whose proof requires an accurate
analysis of the boundary behavior of the approximate solutions and relies on
the powerful techniques discussed in [10, Ch. 10]. We now touch upon some
technical details, and to highlight the heart of the matter we focus on the
case of a genuinely nonlinear boundary characteristic field, which has a
richer and more interesting behavior. Heuristically speaking, there are two
basic mechanisms that could in principle prevent the limit of a sequence of
wave front-tracking approximation from attaining the boundary condition in the
sense of Definition 3.1: on the one hand, in the approximation there might be
a sequence of shocks with vanishing speed that accumulate at the domain
boundary. On the other, in the approximation there might a sequence of
boundary layers with nontrivial center components, that in the limit converge
to a $0$-speed Lax shock. In both cases, and at the price of a fairly
technical analysis, in [4] we show that the limit satisfies Definition 3.1
with a non-trivial $0$-speed shock sitting at the domain boundary. This is
very looosely speaking one of the two main ingredients of the boundary
analysis in [4], the other one being a compactness result for the boundary
trace of the flux functions of the wave front-tracking approximations, see [4,
Lemma 9.2]. This is a nontrivial result since we expect that the trace of the
solutions may instead have an higly oscillatory behavior, see for instance the
related example in [21, §4.3].

## Acknowledgments

LVS wishes to thank Professors Mikhail Feldman, Dehua Wang, Wei Xiang and Tong
Yang for the invitation to contribute to the celebration of Professor Gui-
Qiang Chen’s birthday. All authors are members of the GNAMPA group of INDAM
and of the PRIN Project 20204NT8W4 (PI Stefano Bianchini). FA and LVS are also
members of the PRIN 2022 PNRR Project P2022XJ9SX (PI Roberta Bianchini) and
LVS is a member of the PRIN Project 2022YXWSLR (PI Paolo Antonelli).

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