# Nonlinear orbital stability of stationary shock profiles for the Lax-Wendroff scheme

Jean-François Coulombel, Grégory Faye

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Jean-François Coulombel, Grégory Faye. Nonlinear orbital stability of stationary shock profiles for the Lax-Wendroff scheme. 2024. hal-04790143

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Distributed under a Creative Commons Attribution 4.0 International License
Nonlinear orbital stability of stationary shock profiles
for the Lax-Wendroff scheme

Jean-François COULOMBEL & Grégory FAYE<fnref n="1" />

November 19, 2024

<footnote n="1">¹Institut de Mathématiques de Toulouse - UMR 5219, Université de Toulouse ; CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France. Research of J.-F. C. was supported by ANR project HEAD under grant agreement ANR-24-CE40-3260. G.F. acknowledges support from Labex CIMI under grant agreement ANR-11-LABX-0040, from ANR project Indyana under grant agreement ANR-21-CE40-0008 and from ANR project HEAD under grant agreement ANR-24-CE40-3260. Emails: jean-francois.coulombel@math.univ-toulouse.fr, gregory.faye@math.univ-toulouse.fr</footnote>
Abstract

In this article we study the spectral, linear and nonlinear stability of stationary shock profile solutions to the Lax-Wendroff scheme for hyperbolic conservation laws. We first clarify the spectral stability of such solutions depending on the convexity of the flux for the underlying conservation law. The main contribution of this article is a detailed study of the so-called Green's function for the linearized numerical scheme. As evidenced on numerical simulations, the Green's function exhibits a highly oscillating behavior ahead of the leading wave before this wave reaches the shock location. One of our main results gives a quantitative description of this behavior. Because of the existence of a one-parameter family of stationary shock profiles, the linearized numerical scheme admits the eigenvalue 1 that is embedded in its continuous spectrum, which gives rise to several contributions in the Green's function. Our detailed analysis of the Green's function describes these contributions by means of a so-called activation function. For large times, the activation function describes how the mass of the initial condition accumulates along the eigenvector associated with the eigenvalue 1 of the linearized numerical scheme. We can then obtain sharp decay estimates for the linearized numerical scheme in polynomially weighted spaces, which in turn yield a nonlinear orbital stability result for spectrally stable stationary shock profiles. This nonlinear result is obtained despite the lack of uniform $\ell^1$ estimates for the Green's function of the linearized numerical scheme, the lack of such estimates being linked with the dispersive nature of the numerical scheme. This dispersive feature is in sharp contrast with previous results on the orbital stability of traveling waves or discrete shock profiles for parabolic perturbations of conservation laws.
**Mathematics Subject Classification 2020:** 65M06, 65M12, 47B35, 35L65, 35L67.

**Keywords:** hyperbolic conservation laws, shock waves, difference approximations, stability, Lax-Wendroff scheme.

For the purpose of open access, this work is distributed under a Creative Commons Attribution — 4.0
International licence: https://creativecommons.org/licenses/by/4.0/.

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<table><tr><td>A</td><td>The exact and approximate Green's functions for the Cauchy problem</td><td>101</td></tr><tr><td>A.1</td><td>The exact and approximate Green's functions. Main results</td><td>102</td></tr><tr><td>A.2</td><td>The uniform bound</td><td>107</td></tr><tr><td>A.3</td><td>The fast decaying side</td><td>108</td></tr><tr><td>A.4</td><td>The oscillating side</td><td>112</td></tr><tr><td>A.5</td><td>Proof of Theorem A.3</td><td>123</td></tr><tr><td>A.6</td><td>Consequences</td><td>126</td></tr><tr><td>A.7</td><td>Higher order estimates</td><td>128</td></tr></table>
# Notation

Throughout this article, we let $\mathbb{N}^*$ denote the set of positive integers $\{1, 2, 3, \dots\}$ and $\mathbb{N}$ denote the set of integers (including 0). We let $\mathbb{R}$ denote the set of real numbers, $\mathbb{R}^+ := [0, +\infty)$ the set of nonnegative real numbers and $\mathbb{R}^{*+} := (0, +\infty)$ the set of positive numbers. We also let $\mathbb{C}$ denote the set of complex numbers. We shall use the notation:

$$ \begin{aligned} \mathcal{U} &:= \{\zeta \in \mathbb{C} \mid |\zeta| > 1\}, & \mathbb{D} &:= \{\zeta \in \mathbb{C} \mid |\zeta| < 1\}, & \mathbb{S}^1 &:= \{\zeta \in \mathbb{C} \mid |\zeta| = 1\}, \\ \overline{\mathcal{U}} &:= \mathcal{U} \cup \mathbb{S}^1, & \overline{\mathbb{D}} &:= \mathbb{D} \cup \mathbb{S}^1. \end{aligned} $$

If $w$ is a complex number, the notation $B_r(w)$ stands for the (round) open ball in $\mathbb{C}$ centered at $w$ and with radius $r > 0$, that is $B_r(w) := \{z \in \mathbb{C} \mid |z - w| < r\}$. We shall also use “square balls” in the complex plane. Namely, for any $r > 0$, the notation $\mathbf{B}_r(w)$ stands for the open set:

$$ \mathbf{B}_r(w) := \{z \in \mathbb{C} \mid \max (|\mathrm{Re}(z - w)|, |\mathrm{Im}(z - w)|) < r\}. $$

The notation $\overline{B}_r(w)$, resp. $\overline{\mathbf{B}}_r(w)$, refers to the closure of $B_r(w)$, resp. $\mathbf{B}_r(w)$, in $\mathbb{C}$. For any positive number $r$, we let $r\mathbb{S}^1$ denote the circle in $\mathbb{C}$ centered at the origin and with radius $r$. We also let $\mathcal{M}_{n,k}(\mathbb{C})$ denote the set of $n \times k$ matrices with complex entries. If $n = k$, we simply write $\mathcal{M}_n(\mathbb{C})$.

For $q \in [1, +\infty)$, we let $\ell^q(\mathbb{Z}; \mathbb{C})$ denote the space of complex valued sequences $\mathbf{v} = (v_j)_{j \in \mathbb{Z}}$ indexed by $\mathbb{Z}$ and such that the quantity:

$$ \sum_{j \in \mathbb{Z}} |v_j|^q $$

is finite. The $1/q$-th power of this quantity defines a norm with which $\ell^q(\mathbb{Z}; \mathbb{C})$ becomes a Banach space. This norm is denoted $\| \cdot \|_{\ell^q}$. For $q = +\infty$, we let $\ell^\infty(\mathbb{Z}; \mathbb{C})$ denote the space of bounded complex valued sequences indexed by $\mathbb{Z}$. This space is equipped with the norm:

$$ \sup_{j \in \mathbb{Z}} |v_j|, $$

which we shall refer to as $\| \cdot \|_{\ell^\infty}$. When equipped with this norm, the space $\ell^\infty(\mathbb{Z}; \mathbb{C})$ is a Banach algebra (the product between two sequences being here understood in the pointwise sense $(v w)_j := v_j w_j$ for each $j \in \mathbb{Z}$). Sequences will be denoted with bold letters, while their evaluation at a given integer will be denoted with standard letters.

We let $C$, resp. $c$, denote some (large, resp. small) positive constants that may vary throughout the text (sometimes within the same line). A typical example is the inequality:

$$ \forall x \in \mathbb{R}^+, \quad x \exp(-cx) \le C \exp(-cx), $$
where, of course, the constant $c$ on the right-hand side of the inequality is not the same as in the left-hand side. The dependence of the constants on the various involved parameters is made precise throughout the article.

In order to avoid overloading some expressions, we sometimes write:

$$ \sum_{m=0}^{x} $$

for a sum that runs over the integer $m$ from $m = 0$ to $m = x$, even when $x$ is a positive real number that is not an integer. In that case, it is understood that the sum runs up to the largest number that is less than $x$. This will allow us to avoid using the integer part of several quantities.
# Chapter 1

# Introduction

This article is devoted to a detailed stability analysis of stationary shock profiles for the Lax-Wendroff scheme. The Lax-Wendroff scheme is a finite difference approximation of hyperbolic conservation laws that is formally second order accurate (at least for smooth solutions), see [19, 21]. As any second order accurate numerical scheme, the Lax-Wendroff scheme gives rise to spurious oscillations once the solution to the conservation law develops discontinuities, that is, once shock waves have appeared. This is evidenced for instance in the numerical simulation reported in Figure 2.4 below. This undesirable numerical feature is usually corrected by introducing flux limiters or other more involved numerical treatments (essentially non-oscillatory schemes, weighted non-oscillatory schemes and so on). Our goal here is to show that despite the formation of oscillatory wave trains in the numerical computation of shock waves, the Lax-Wendroff scheme gives rise to *stable* stationary shock profiles for *convex* or *concave* fluxes, and for even other situations. The whole point is to clarify the meaning of “stable” in the previous sentence.

Following a long line of research, we aim at studying this stability problem by first showing a *spectral* stability result, then turning this spectral stability result into a *linear* stability result by proving sharp decay estimates for the linearized numerical scheme. Obtaining these sharp linear decay estimates is the cornerstone of our work. The final step of the analysis is to use the linear decay estimates in order to obtain an *orbital* stability result for the nonlinear dynamics. The approach is definitely not new. For the Lax-Wendroff scheme, it was followed by Smyrlis [28] who dealt with the exact same problem as the one we are looking at here. However, the functional framework that we use is much larger than the one in [28] where the introduction of *exponential* weights stabilizes the linearized operator. Actually, the introduction of exponential weights gives rise to a spectral gap for some carefully chosen parameters, thus bypassing a detailed stability analysis that would take the flux properties into account. We consider here a larger and, probably, more natural functional framework in which the spectrum of the linearized operator depends on the flux of the conservation law.

The overall strategy is the same as the one followed in the contributions [14, 6, 7]. We start from a given discrete shock profile, which is quite easy for the Lax-Wendroff scheme since this numerical scheme exhibits *exact*, piecewise constant, stationary discrete shock profiles that are the mere projections on the numerical grid of the continuous step shock. We linearize the Lax-Wendroff scheme around this stationary solution and try to locate the spectrum of the linearized evolution operator. Enlarging the functional framework has some major impacts, the first of which being that we have to take into account the (neutral) eigenvalue 1 for the linearized numerical scheme. This eigenvalue arises by “translation invariance”, as explained in [28] (see also [27] for a general overview on discrete shock profiles). Translation invariance
means here that there exists a one-parameter smooth family of stationary discrete shock profiles<fnref n="1" />. The
rest of the spectrum is located by analyzing the so-called Evans function associated with the shock profile.
The Evans function plays the role of a characteristic polynomial for the linearized operator. The general
construction of the Evans function is detailed in [27], see also [14, 6]. In our case, the construction is
far easier because our reference discrete shock is piecewise constant so the Evans function reduces to
what could be referred to as a Lopatinskii determinant (by analogy with the stability analysis of shock
waves [2]). Our problem is similar to the stability analysis of the discrete shock profiles for the Godunov
scheme [5] in which the shock profiles are also piecewise constant. For scalar equations, as we consider
here, the expression of the Lopatinskii determinant is simple enough so that we can analyze the location
of its zeroes in the case where the flux of the conservation law is either convex or concave. Namely, we
shall show that for convex or concave fluxes, the Lopatinskii determinant has only 1 as a (simple) zero
in the region $\mathcal{W}$ of the complex plane. This corresponds, in the terminology of [27] to a *spectrally stable*
situation. Our analysis encompasses a particular (symmetric) case that was considered in the seminal
work [15].

Once we have located the spectrum of the operator, we can construct the so-called spatial Green's
function, that is the fundamental solution of the resolvent equation. For spectrally stable configurations,
we show that the spatial Green's function has a meromorphic extension near 1 and it can also be holomor-
phically extended near any other point of the unit circle. This is a key step towards the decomposition
and the proof of sharp decay bounds for the Green's function of the linearized evolution operator. In the
parabolic case, the Green's function satisfies uniform (in time) $\ell^1$ estimates (in space) and it also satisfies
decaying (in time) $\ell^\infty$ estimates (in space), just like the heat kernel. For the Lax-Wendroff scheme, there
is unfortunately no hope to obtain such favorable uniform bounds. Indeed, it is already known that when
applied to the transport equation, the Green's function of the Lax-Wendroff scheme does not enjoy uni-
form $\ell^1$ estimates. This failure of $\ell^1$ stability, that is linked to the *dispersive* behavior of the Lax-Wendroff
scheme, has been identified in a general context by Thomée [29] for convolution operators. The analysis
of $\ell^1$ instability was later refined in [16, 17] in order to make the instability growth rates percise. Rather
than following [16, 17] for the Lax-Wendroff scheme, we shall build here on the recent work [9] by one
of the authors. The analysis in [9] gives a precise description of the Green's function for this numerical
scheme in the context of the Cauchy problem for the transport equation<fnref n="2" />. We extend here the analysis of
[9] to the context of the shock profile stability analysis in which Fourier analysis is no longer available due
to spatial variations. We therefore substitute the so-called spatial dynamics approach in place of Fourier
analysis and use the inverse Laplace transform to obtain a representation formula for the Green's function
of the linearized operator. The main difficulty that we shall face arises from the singularity of the spatial
Green's function at the eigenvalue 1 (that is imbedded in the continuous spectrum). This pole gives rise
to a leading contribution in the Green's function which, following [7], we refer to as an *activation function*.
This function can be thought of as a primitive of the Green's function for the Cauchy problem. Detailed
bounds on this activation function are given in Appendix A at the end of this article. In particular, a
lengthy -though crucial- argument here is to obtain a uniform bound for the activation function. This is
rather trivial in the parabolic setting thanks to uniform $\ell^1$ estimates for the Green's function but such
uniform $\ell^1$ estimates are known to fail in the dispersive setting. Our uniform bound for the activation
function is reminiscent of the main result in [13] where such a uniform bound is also the cornerstone of

<footnote n="1">¹For continuous problems, such as parabolic perturbations of conservation laws, the one-parameter family of profiles is obtained by simply translating in space a given profile, but the construction of shock profiles is unfortunately more complicated for discrete problems.</footnote>

<footnote n="2">²The quantitative estimates in [9] are more accurate in many regimes than the ones in [16, 17] and this is crucial for several arguments that we use below.</footnote>
the argument.

Once we have an accurate description of the Green’s function with sharp bounds for each term in its decomposition, the nonlinear orbital stability result follows more or less by using standard tools as in [1]. Since our Green’s function does not satisfy uniform $\ell^1$ estimates and exhibits a dispersive instability$^3$, we can not complete an orbital stability for small $\ell^1$ perturbations. We need to work in *polynomially weighted* $\ell^p$ spaces and carefully play with the weights in the definition of the norms in order to recover some integrability in time, which is a crucial feature of bootstrap arguments. This is less favorable than the situation for semi-discrete shocks that is dealt with in [1] where the time translation invariance also allows to use a larger functional framework. The fully discrete situation does not give as many tools to deal with the nonlinear argument.

The plan of this article is the following. In Chapter 2 we introduce the numerical scheme, we give the expression of the reference shock profile that we consider and state our main results. For the sake of clarity, we have split the main results in four theorems, the first two being devoted to *spectral stability*, the third one being devoted to *linear stability*, and the fourth and main one being devoted to *nonlinear orbital stability*. We also report on some numerical simulations that illustrate these results. Chapter 3 is devoted to the spectral analysis. The linear decay estimates are proved in Chapter 4 with the help of several key estimates that are given in Appendix A. Eventually, the nonlinear analysis is detailed in Chapter 5.

<footnote>$^3$The $\ell^\infty$ decay of the Green's function for the Lax-Wendroff scheme is also slower than for the parabolic situation that is considered in [1].</footnote>
# Chapter 2

# The Lax-Wendroff scheme. Notation and main results

## 2.1 The Lax-Wendroff scheme and its stationary shock profiles

We consider in this article a scalar conservation law of the form:

$$ \partial_t u + \partial_x f(u) = 0, \quad x \in \mathbb{R}, \qquad (2.1) $$

with a smooth flux $f \in C^\infty(\mathbb{R}; \mathbb{R})$. Since the solutions to the Cauchy problem for (2.1) generically develop singularities in finite time [12, 26], we directly consider piecewise smooth solutions to (2.1), which we assume to be stationary for simplicity:

$$ u(t,x) = \begin{cases} u_{\ell}, & x < 0, \\ u_{r}, & x > 0. \end{cases} \qquad (2.2) $$

For (2.2) to be a weak solution to (2.1), the so-called Rankine-Hugoniot relation must hold:

$$ f(u_{\ell}) = f(u_{r}), \qquad (2.3) $$

and we also enforce the so-called entropy criterion [26]:

$$ f'(u_r) < 0 < f'(u_\ell). \qquad (2.4) $$

The Lax entropy inequalities (2.4) imply that the characteristics stemming from either side of the shock wave (that is located here at $x = 0$) enter the shock in positive time, as depicted in Figure 2.1.

Our goal here is to understand the influence of a high order discretization procedure on the stability of the shock (2.2) and more specifically whether dispersion in a numerical scheme may rule out linear or nonlinear stability. Let us recall that a rather complete existence and stability theory for monotone schemes has been developed in [18] but monotone schemes are at most first order accurate [19, 21]. We therefore pursue the analysis of [28] and try to develop a rather complete stability analysis of discrete shock profiles for the Lax-Wendroff scheme. This is a model situation for a high order finite difference scheme and we hope that some of our arguments below may prove useful for other discretization procedures of conservation laws.

We therefore introduce a space step $\Delta x > 0$ and a time step $\Delta t > 0$ that are always chosen such that the ratio $\lambda := \Delta t/\Delta x$ is kept constant. For any couple of integers $n \in \mathbb{N}$ and $j \in \mathbb{Z}$, the solution to (2.1)
<figure><img src="image_4.png" /><figcaption>Figure 2.1: The characteristics on either side of the shock.</figcaption></figure>

is approximated, on the time-space cell $[n \Delta t, (n+1) \Delta t) \times [j \Delta x, (j+1) \Delta x)$ by a constant value $u_j^n$, that is iteratively defined with respect to $n$ according to the formula:

$$ \forall (n, j) \in \mathbb{N} \times \mathbb{Z}, \quad u_j^{n+1} = u_j^n - \lambda (\mathcal{F}_\lambda(u_j^n, u_{j+1}^n) - \mathcal{F}_\lambda(u_{j-1}^n, u_j^n)), \quad (2.5) $$

with a numerical flux $\mathcal{F}_\lambda$ that is defined by:

$$ \forall (u, v) \in \mathbb{R}^2, \quad \mathcal{F}_\lambda(u, v) := \frac{1}{2} (f(u) + f(v)) - \frac{\lambda}{2} f'\left(\frac{u+v}{2}\right) (f(v) - f(u)). \quad (2.6) $$

The numerical scheme (2.5)-(2.6), that is referred to as the Lax-Wendroff scheme and dates back to [20], is a formally second order accurate approximation procedure for (2.1), see [19, 21]. The initial condition $(u_j^0)_{j \in \mathbb{Z}}$ for (2.5) will be chosen as a small perturbation of a discretized version of the shock wave (2.2).

A very specific feature of the numerical scheme (2.5)-(2.6) is the fact that it captures *exactly* the stationary piecewise constant solutions to (2.1). Namely, if we consider the weak solution (2.2) to (2.1), then the following sequence:

$$ \bar{u}_j^n := \begin{cases} u_\ell, & j \le 0, \\ u_r, & j \ge 1, \end{cases} \quad n \in \mathbb{N}, \quad (2.7) $$

defines a stationary solution to (2.5)-(2.6) since it does not depend on $n$ and satisfies (we omit the index $n$ in $\bar{u}_j$ from now on):

$$ \forall j \in \mathbb{Z}, \quad \mathcal{F}_\lambda(\bar{u}_j, \bar{u}_{j+1}) = \mathcal{F}_\lambda(\bar{u}_{j-1}, \bar{u}_j) = f(u_\ell) = f(u_r), $$

the final equality coming from the Rankine-Hugoniot relation (2.3). The aim of this article is to study the stability of the stationary solution $\bar{u} = (\bar{u}_j)_{j \in \mathbb{Z}}$ defined in (2.7) with respect to the dynamics of (2.5)-(2.6). Some numerical experiments are reported below in Section 2.3. Let us recall that the piecewise constant discrete shock (2.7) is not the only discrete shock for the Lax-Wendroff scheme (2.5) associated with (2.2). As explained in [28], a stationary discrete shock profile for (2.2) is a real valued sequence $\mathbf{u} = (u_j)_{j \in \mathbb{Z}}$ such that:

$$ \forall j \in \mathbb{Z}, \quad \mathcal{F}_\lambda(u_j, u_{j+1}) = \mathcal{F}_\lambda(u_{j-1}, u_j), $$
and

$$ \lim_{j \to -\infty} u_j = u_\ell, \quad \lim_{j \to +\infty} u_j = u_r. $$

The sequence in (2.7) is a particular case of a stationary discrete shock profile but there are many other ones. We refer to [27] and references therein for a thorough description of the existence theory (both in the scalar and system cases), and to [28] for the specific case of the Lax-Wendroff scheme (2.5)-(2.6) in the scalar case. We recall the following result that is one of the achievements in [28]:

**Theorem 2.1** (Smyrlis [28]). Let the shock (2.2) satisfy the Rankine-Hugoniot condition (2.3) and the entropy condition (2.4). Let $\lambda$ satisfy the so-called CFL condition:

$$ \max(\lambda f'(u_\ell), \lambda |f'(u_r)|) < 1. \quad (2.8) $$

Then there exist $\underline{\theta} > 0$ and a one-parameter family of stationary discrete shock profiles $\mathbf{v}^\theta = (v_j^\theta)_{j \in \mathbb{Z}}$, $\theta \in (-\underline{\theta}, \underline{\theta})$, that satisfies the following properties:

(i) $\mathbf{v}^0 = \bar{\mathbf{u}}$ is the piecewise constant discrete shock (2.7),

(ii) for every $j \in \mathbb{Z}$, the map $\theta \mapsto v_j^\theta$ is $\mathcal{C}^\infty$ on the interval $(-\underline{\theta}, \underline{\theta})$,

(iii) there exist $\delta > 0$ and $C > 0$ such that for any $\theta \in (-\underline{\theta}, \underline{\theta})$, the discrete shock profile $\mathbf{v}^\theta$ converges towards its end states exponentially fast at rate $\delta$, namely:

$$ \forall j \in \mathbb{N}, \quad |v_j^\theta - u_r| + |v_{-j}^\theta - u_\ell| \le C e^{-\delta j}, $$

and furthermore

$$ \forall j \in \mathbb{Z}, \quad |v_j^\theta - v_j^0| \le C |\theta| e^{-\delta |j|}, $$

(iv) for every $\theta$ in the interval $(-\underline{\theta}, \underline{\theta})$, the “excess mass” of $\mathbf{v}^\theta$ equals $\theta$, namely<fnref n="1" />:

$$ \sum_{j \in \mathbb{Z}} v_j^\theta - v_j^0 = \theta. $$

Since the numerical scheme (2.5)-(2.6) is conservative, it preserves mass. More precisely, given an initial sequence $\mathbf{h}$ in $\ell^1(\mathbb{Z}; \mathbb{R})$, we want to consider the dynamics of (2.5)-(2.6) for the initial condition $\mathbf{v}^0 + \mathbf{h}$ (see the numerical experiments in Section 2.3). Let $(u_j^n)_{j \in \mathbb{Z}, n \in \mathbb{N}}$ denote the corresponding solution to (2.5)-(2.6). By mass conservation, we have:

$$ \forall n \in \mathbb{N}, \quad \sum_{j \in \mathbb{Z}} u_j^n - v_j^0 = \sum_{j \in \mathbb{Z}} u_j^0 - v_j^0 = \sum_{j \in \mathbb{Z}} h_j. $$

In particular, if we can prove that $(u_j^n)_{j \in \mathbb{Z}, n \in \mathbb{N}}$ converges, as $n$ tends to $+\infty$, towards a stationary discrete shock profile, then this limit can only be $\mathbf{v}^\theta$ where $\theta$ denotes the mass of the initial perturbation $\mathbf{h}$. In other words, we rather intend to show that the whole curve $\{\mathbf{v}^\theta, \theta \in (-\underline{\theta}, \underline{\theta})\}$ of stationary discrete shock profiles is *orbitally stable*.

To study the stability of (2.7), or rather the orbital stability of the curve $\{\mathbf{v}^\theta, \theta \in (-\underline{\theta}, \underline{\theta})\}$, we follow a common approach that is based on first studying the spectral stability of (2.7) and then on proving

<footnote n="1">¹The series is indeed convergent because stationary shock profiles converge exponentially fast towards their end states.</footnote>
linear and nonlinear decay estimates. We therefore introduce the linearized numerical scheme that is obtained by linearizing (2.5)-(2.6) around the constant solution $\mathbf{v}^0$ that is given by (2.7). For future use, we introduce the notation:

$$ \alpha_{\ell,r} := \lambda f'(u_{\ell,r}), \quad \alpha_m := \lambda f'\left(\frac{u_{\ell} + u_r}{2}\right). \qquad (2.9) $$

The linearization of (2.5)-(2.6) around the discrete shock profile (2.7) leads to the iteration:

$$ \mathbf{w}^{n+1} = \mathcal{L} \mathbf{w}^n, \quad \mathbf{w}^n = (w_j^n)_{j \in \mathbb{Z}}, \qquad (2.10) $$

and the bounded linear operator $\mathcal{L} : \ell^q(\mathbb{Z}; \mathbb{C}) \to \ell^q(\mathbb{Z}; \mathbb{C})$ is defined by:

$$ (\mathcal{L} \mathbf{w})_j := \begin{cases} w_j - \frac{\alpha_r}{2} (w_{j+1} - w_{j-1}) + \frac{\alpha_r^2}{2} (w_{j+1} - 2w_j + w_{j-1}), & j \ge 2, \\ w_j - \frac{\alpha_\ell}{2} (w_{j+1} - w_{j-1}) + \frac{\alpha_\ell^2}{2} (w_{j+1} - 2w_j + w_{j-1}), & j \le -1, \\ w_0 - \frac{1}{2} (\alpha_r w_1 - \alpha_\ell w_{-1}) + \frac{1}{2} (\alpha_r \alpha_m w_1 - \alpha_\ell (\alpha_\ell + \alpha_m) w_0 + \alpha_\ell^2 w_{-1}), & j = 0, \\ w_1 - \frac{1}{2} (\alpha_r w_2 - \alpha_\ell w_0) + \frac{1}{2} (\alpha_r^2 w_2 - \alpha_r (\alpha_r + \alpha_m) w_1 + \alpha_\ell \alpha_m w_0), & j = 1. \end{cases} \qquad (2.11) $$

Let us observe that the values $\alpha_\ell = -\alpha_r$ and $\alpha_m = 0$ correspond to the particular “symmetric” case studied in [15]. This case corresponds for instance to an even function $f$ with respect to the mid-point $(u_\ell + u_r)/2$. Actually, the spectral stability result of [15] is, to some extent, the starting point for our analysis in Chapter 3 below.

## 2.2 Main results

Our first main result is a spectral stability result for the stationary discrete shock (2.7) when the flux $f$ is either convex or concave.

**Theorem 2.2** (Spectral stability for convex or concave fluxes). Let the flux $f$ in (2.1) be either convex or concave, and let the weak solution (2.2) satisfy the Rankine-Hugoniot relation (2.3) and the entropy inequalities (2.4). Then under the condition<fnref n="2" />:

$$ \max(\alpha_{\ell}, |\alpha_{r}|) < 1, \qquad (2.12) $$

the operator $\mathcal{L}$ has no spectrum in $\overline{\mathcal{U}} \setminus \{1\}$. Moreover, 1 is an eigenvalue of $\mathcal{L}$ in $\ell^q(\mathbb{Z}; \mathbb{C})$ for any $q \in [1, +\infty]$.

Actually, it will turn out in Chapter 3 that in the case of a convex or concave flux $f$, the operator $\mathcal{L}$ will have no spectrum in the exterior of the curve<fnref n="3" />:

$$ \left\{ 1 - 2\alpha^2 \sin^2 \frac{\xi}{2} + \mathbf{i}\alpha \sin \xi \mid \xi \in \mathbb{R} \right\}, \quad \alpha := \max(\alpha_{\ell}, |\alpha_r|). \qquad (2.13) $$

<footnote n="2">²This condition is a mere restriction on the ratio $\lambda = \Delta t/\Delta x$. We recall that $\alpha_\ell$ and $\alpha_r$ are defined in (2.9). The condition (2.12) is the exact same one as in the existence theorem for stationary discrete shock profiles, see (2.8).</footnote>

<footnote n="3">³This curve is actually an ellipse that is centered at $1 - \alpha^2$.</footnote>
The exterior of this curve will always be denoted $\mathcal{O}$ in Chapter 3. It can be parametrized as:

$$ \mathcal{O} = \{z = x + iy \in \mathbb{C} | (x - 1 + \alpha^2)^2 + \alpha^2 y^2 > \alpha^4\}, $$

see for instance Figure 3.1 in Chapter 3 below where $\mathcal{O}$ corresponds to the complement of the grey shaded area.

We wish to encompass slightly more general situations than the sole case of convex (or concave) fluxes. Indeed, our goal is to show that spectral stability of a stationary discrete shock is a sufficient condition for linear and orbital nonlinear stability. In practice, this requires having at our disposal a convenient tool that locates the spectrum of $\mathcal{L}$ just as the characteristic polynomial does so for a matrix. Constructing such a tool, which we shall refer to as a Lopatinskii determinant (in analogy with the shock wave theory for systems of conservation laws, see e.g. [2]), is the purpose of part of Chapter 3 below. Examples of numerical calculations of such Lopatinskii determinants can be found in [4] in the context of numerical boundary conditions for transport equations but the analysis is entirely similar here.

Going beyond the case of a convex or concave flux requires making a stability assumption that is given as Assumption 1 in Chapter 3 below. This stability assumption is meant to exclude the possibility of having some spectrum of $\mathcal{L}$ in $\overline{\mathcal{U}} \setminus \{1\}$. Our hope is that Assumption 1 can be “easily” verified (or proved not to hold) in specific situations. The generalization of Theorem 2.2 is the following result.

**Theorem 2.3.** Let the weak solution (2.2) satisfy the Rankine-Hugoniot relation (2.3) and the entropy inequalities (2.4). Then under the condition (2.12) on the parameter $\lambda$, we can define a function $\Delta$ that is holomorphic on a neighborhood of $\overline{\mathcal{U}}$ (see (3.3) in Chapter 3). Then 1 is an eigenvalue of $\mathcal{L}$ in $\ell^q(\mathbb{Z}; \mathbb{C})$ for any $q \in [1, +\infty]$. Moreover, under Assumption 1 below on the location of the zeroes of $\Delta$, the operator $\mathcal{L}$ has no spectrum in $\overline{\mathcal{U}} \setminus \{1\}$.

Once we know that spectral stability holds, we expect that the favorable localization of the spectrum of $\mathcal{L}$ will imply some decay estimates for the semigroup of operators $\{\mathcal{L}^n | n \in \mathbb{N}\}$. We gather here these estimates but, to some extent, our main result is rather Theorem 4.1 that will be stated in Chapter 4 below. Theorem 4.1 gives accurate bounds for the Green's function of the operator $\mathcal{L}$ so that, with classical convolution estimates, we can infer estimates on the norm of $\mathcal{L}^n$ in some polynomially weighted spaces. These spaces are defined as follows. We recall that $\ell^q(\mathbb{Z}; \mathbb{C})$ denotes the space of complex valued sequences that are indexed by $\mathbb{Z}$ and such that their $q$-th power is integrable (bounded sequences if $q = +\infty$). The norm is denoted $\| \cdot \|_{\ell^q}$. Given a real number $\gamma \ge 0$, we then introduce the polynomially weighted $\ell^q$ space:

$$ \ell_{\gamma}^{q}(\mathbb{Z}; \mathbb{C}) := \{ \mathbf{h} \in \ell^{q}(\mathbb{Z}; \mathbb{C}) \mid ((1 + |j|^{\gamma}) h_j)_{j \in \mathbb{Z}} \in \ell^{q}(\mathbb{Z}; \mathbb{C}) \}. \quad (2.14) $$

For $\mathbf{h} \in \ell_{\gamma}^q$, we denote:

$$ \|\mathbf{h}\|_{\ell_{\gamma}^q} := \left\| \left( (1 + |j|^{\gamma}) h_j \right)_{j \in \mathbb{Z}} \right\|_{\ell^q}, $$

the norm of $\mathbf{h}$, so that $\ell_{\gamma}^q(\mathbb{Z}; \mathbb{C})$ is endowed with a Banach space structure. Our main decay estimates for the operator $\mathcal{L}$ read as follows.

**Theorem 2.4.** Let the weak solution (2.2) satisfy the Rankine-Hugoniot relation (2.3) and the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the condition (2.12) and let Assumption 1 on the zeroes
of $\Delta$ be satisfied. Then for any real numbers $\gamma_2 \ge \gamma_1 \ge 0$, there exists a constant $C$ such that, for any $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{C})$ that satisfies:

$$ \sum_{j \in \mathbb{Z}} h_j = 0, $$

there holds:

$$ \begin{aligned} \forall n \ge 1, \quad & \|\mathcal{L}^n \mathbf{h}\|_{\ell_{\gamma_1}^1} \le \frac{C}{n^{\gamma_2-\gamma_1-1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \\ & \|\mathcal{L}^n \mathbf{h}\|_{\ell_{\gamma_1}^\infty} \le \frac{C}{n^{\gamma_2-\gamma_1+\min(1/3, \gamma_1)}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. \end{aligned} $$

The first estimate in Theorem 2.4 is consistent with all known results on the behavior of the Lax-Wendroff scheme for the transport equation. Indeed, if one chooses $\gamma_2 = \gamma_1 = 0$, the first estimate in Theorem 2.4 corresponds to the well-known (weak) instability of the Lax-Wendroff scheme in either the $\ell^1$ or $\ell^\infty$ norm. The growth $n^{1/8}$ is known to be sharp, see e.g. [16, 17, 9]. Since we wish to obtain a nonlinear orbital stability result by means of a bootstrap argument, some decay should be gained one way or another, and the introduction of the polynomial weight is a way to compensate for the weak instability of the numerical scheme (and to gain enough time integrability as well). If we compare with the more standard Cauchy problem on $\mathbb{Z}$ for the transport equation, the main new difficulty that we are facing here is the eigenvalue 1 for the operator $\mathcal{L}$, this eigenvalue being imbedded into the continuous spectrum. Nevertheless, our decay estimates in Theorem 2.4 (and some complementary estimates that are detailed in Chapter 4) are strong enough to yield the following orbital nonlinear stability result for the family of stationary discrete shock profiles exhibited in Theorem 2.1.

**Theorem 2.5.** Let the weak solution (2.2) satisfy the Rankine-Hugoniot relation (2.3) and the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the condition (2.12) and let Assumption 1 on the zeroes of $\Delta$ be satisfied. Let now $\beta, \sigma \in \mathbb{R}^+$ satisfy $\beta + \sigma \ge \frac{5}{12}$ and $0 \le \sigma < \beta + \frac{1}{8}$. We define the constant:

$$ \gamma := \sigma + \beta + \frac{1}{8}. \qquad (2.15) $$

Then there exist some positive real numbers $C_0, \epsilon > 0$ such that for any sequence $\mathbf{h} \in \ell_{\gamma}^1(\mathbb{Z}; \mathbb{R})$ satisfying
$$ \|\mathbf{h}\|_{\ell_{\gamma}^1} < \epsilon, $$

then one has

$$ \theta := \sum_{j \in \mathbb{Z}} h_j \in (-\underline{\theta}, \underline{\theta}), $$

and the solution $(\mathbf{u}^n)_{n \in \mathbb{N}}$ of the Lax-Wendroff scheme (2.5)-(2.6) with the initial condition $\mathbf{u}^0 := \bar{\mathbf{u}} + \mathbf{h}$ is well-defined. Furthermore, if we introduce the sequence $(\mathbf{p}^n)_{n \in \mathbb{N}}$ defined as

$$ \forall n \in \mathbb{N}, \quad \mathbf{p}^n := \mathbf{u}^n - \mathbf{v}^\theta, $$

then for all $n \in \mathbb{N}$ one has $\mathbf{p}^n \in \ell_{\beta}^1(\mathbb{Z}; \mathbb{R})$ together with the estimates:

$$ \forall n \in \mathbb{N}, \quad \|\mathbf{p}^n\|_{\ell_{\beta}^1} \le \frac{C_0}{(1+n)^\sigma} \|\mathbf{h}\|_{\ell_{\gamma}^1}, \quad \text{and} \quad \|\mathbf{p}^n\|_{\ell_{\beta}^\infty} \le \frac{C_0}{(1+n)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}, $$

so that $\mathbf{u}^n$ tends to $\mathbf{v}^\theta$ in $\ell_{\beta}^\infty(\mathbb{Z}; \mathbb{R})$ (and also in $\ell_{\beta}^1(\mathbb{Z}; \mathbb{R})$ if $\sigma$ is positive).
<figure><img src="image_5.png" /><figcaption>Figure 2.2: Discrete shock profiles for the Lax-Wendroff scheme applied to the Burgers equation. Left: the reference shock (2.2). Middle: a discrete shock profile with same end states but negative mass difference ($\theta < 0$ in Theorem 2.1). Right: a discrete shock profile with same end states but positive mass difference ($\theta > 0$ in Theorem 2.1).</figcaption></figure>

## 2.3 Numerical experiments

We first present some numerical computations of stationary shock profiles. We consider for simplicity the Burgers equation, which corresponds to the convex flux function $f(u) = u^2/2$. By Theorem 2.2 above, every stationary discrete shock of the form (2.7) with $u_\ell > 0$ and $u_r = -u_\ell$ is spectrally and therefore nonlinearly orbitally stable (in the sense of Theorem (2.5)) if the parameter $\lambda$ is chosen to satisfy the CFL stability condition. Figure 2.2 displays three possible stationary discrete shock profiles for the Lax-Wendroff scheme with the same end states $u_\ell = 1/2$, $u_r = -1/2$. The Rankine-Hugoniot conditions (2.3) and the Lax entropy condition (2.4) are satisfied. The CFL parameter $\lambda$ is chosen to be $1/2$ so that the CFL condition (2.12) is also satisfied.

As follows from Theorem 2.1, discrete shock profiles can be parametrized, at least for small enough mass perturbations, by their mass difference with respect to the reference discrete shock profile (2.7). The first graph on the left of Figure 2.2 corresponds to the reference shock (2.2) with end states $u_\ell = 1/2$, $u_r = -1/2$. The value of that discrete shock at $j = 0$ is highlighted in red. The middle and right graphs in Figure 2.2 correspond to stationary discrete shock profiles as given by Theorem 2.1, one being with negative mass difference (middle graph), and the other one being with positive mass difference (on the right of Figure 2.2).

It turns out that the family $\{v^\theta\}$ of stationary discrete shock profiles given in Theorem 2.1 can be parametrized globally for the Burgers equation. This was already mentioned in [28] and we report here on the numerical computation of the whole family. As a first observation, we remark that the translation of the shock profile (2.7), namely:

$$ \begin{cases} u_{\ell}, & j \le 1, \\ u_{r}, & j \ge 2, \end{cases} \qquad (2.16) $$

is also a stationary discrete shock profile for (2.5) with mass difference $u_\ell - u_r = 1$ with respect to (2.7) (the only difference with (2.7) is in the cell labeled with the index $j = 1$). In other words, if we can parametrize the family $v^\theta$ of Theorem 2.1 for $\theta \in [0, 1]$ with $v^0$ being equal to the stationary discrete shock profile (2.7) and $v^1$ being equal to the translated discrete shock profile (2.16), then by repeated translations -either to the left or to the right- one can parametrize a global family $\{v^\theta | \theta \in \mathbb{R}\}$ where $\theta$ still refers to the mass perturbation. This is illustrated in Figure 2.3 where we plot the evaluation at $j = 0$ and $j = 1$ of the family of stationary discrete shock profiles with $\theta \in [-2, 2]$. For $\theta = 1$, both $v_0^1$ and $v_1^1$ equal $1/2$ in agreement with (2.16). For $\theta = -1$, the translation of the reference discrete shock is to the left. Both curves that are plotted in Figure 2.3 are translates one of the other.
<figure><img src="image_6.png" /><figcaption>Figure 2.3: The evaluation at $j = 0$ and $j = 1$ of the family of stationary discrete shock profiles $v^\theta$, $\theta \in \mathbb{R}$.</figcaption></figure>

We now report on some computations that illustrate the dynamics of the numerical scheme (2.5). In Figure 2.4, we give several plots corresponding to the evolution of a zero mass perturbation of the reference discrete shock (2.7). By appealing to Theorem 2.5, we expect that the numerical solution converges asymptotically towards (2.7) and this is what we observe. The convergence is illustrated in Figure 2.4 where two waves, one emanating from the right and one from the left of the shock, hit the shock one after the other, giving first rise to a translation of the initial shock to the right (due to positive mass excess) and then going back to the reference discrete shock (2.7). Another view of this computation is given in Figure 2.5 where the plot is in the $(j, n)$ plane. The translation of the shock to the right after the first wave hit the shock is more visible.

As a conclusion, we illustrate the behavior of the Lax-Wendroff scheme (2.5) for a positive mass initial perturbation, see Figures 2.6 and 2.7. The asymptotic state is a stationary discrete shock that corresponds to a non-integer value of $\theta > 0$, thus displaying the typical oscillations associated with the Lax-Wendroff scheme. Despite these oscillations, these solutions are spectrally and nonlinearly stable.
<figure><img src="image_7.png" /><figcaption>Figure 2.4: Evolution of a perturbation with zero mass of the reference discrete shock (2.2). First line (from left to right): the initial condition, the solution at $n = 40$, the solution at $n = 150$. Second line (from left to right): the solution at $n = 400$, the solution at $n = 700$, the solution at $n = +\infty$ (convergence towards the reference shock (2.2)).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 2.5: Evolution of a perturbation with zero mass of the reference discrete shock (2.2).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 2.6: Evolution of a perturbation with positive mass of the reference discrete shock (2.2). First line (from left to right): the initial condition, the solution at $n = 100$, the solution at $n = 200$. Second line (from left to right): the solution at $n = 300$, the solution at $n = 400$ (convergence towards a discrete shock profile).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 2.7: Evolution of a perturbation with positive mass of the reference discrete shock (2.2).</figcaption></figure>
# Chapter 3

## Spectral stability

Localizing the spectrum of the linearized operator $\mathcal{L}$ proceeds in several steps. In the first Section below, we analyze the eigenvalue problem<fnref n="1" /> $\mathcal{L} \mathbf{v} = z \mathbf{v}$ for $z$ in the exterior $\mathcal{O}$ of the curve (2.13). We show that the existence of a nonzero eigensequence $\mathbf{v} \in \ell^q(\mathbb{Z}; \mathbb{C})$ is equivalent to a scalar equation $\Delta(z) = 0$, where the so-called Lopatinskii determinant $\Delta$ is a holomorphic function on $\mathcal{O}$, and, actually, even on a larger region of the complex plane. This result is independent of the considered space $\ell^q(\mathbb{Z}; \mathbb{C})$, $1 \le q \le +\infty$, and the Lopatinskii determinant $\Delta$ does not depend on $q$. Since the discrete shock profile $\bar{u}$ in (2.7) is piecewise constant and the numerical scheme (2.5) only involves a three point stencil, the Lopatinskii determinant $\Delta$ is explicitly computable<fnref n="2" />. We analyze the zeroes of the function $\Delta$ in the case of a convex (or concave) flux $f$ in (2.1) which seems to be new, up to our knowledge. The symmetric case $\alpha_r = -\alpha_\ell$, $\alpha_m = 0$ is dealt with in [15]. We shall also show in Section 3.4 that for a non-convex flux, the Lopatinskii determinant $\Delta$ can have zeroes in the unstable region $\mathcal{U} = \{z \in \mathbb{C} \mid |z| > 1\}$ or that it can have a double root at 1. In Section 3.2, we use our knowledge on $\Delta$ to compute the so-called spatial Green's function, that is, the solution to the resolvent problem:

$$ (z \operatorname{Id} - \mathcal{L}) \mathcal{G}^{j_0}(z) = \delta_{j_0}, $$

where $\delta_{j_0}$ denotes the discrete Dirac mass located at the index $j_0 \in \mathbb{Z}$:

$$ \forall j \in \mathbb{Z}, \quad (\delta_{j_0})_j := \begin{cases} 1, & \text{if } j = j_0, \\ 0, & \text{otherwise.} \end{cases} $$

The construction of the spatial Green's function and the estimates we find on it will directly show that the operator $\mathcal{L}$ has no spectrum outside the curve (2.13) as long as $f$ is convex or concave, as claimed in Theorem 2.2. Theorem 2.3, that considers more general flux functions $f$, will follow from similar arguments. The spatial Green's function is our starting point for the analysis of the large time behavior of the linearized numerical scheme (2.10).

<footnote n="1">¹Since the only discrete shock profile that will appear in this Chapter is the piecewise constant one in (2.7), we feel free to use the notation $\mathbf{v}$ for generic sequences. No possible confusion can be made with the family of discrete shock profiles $\mathbf{v}^\theta$ that will not be mentioned in this Chapter.</footnote>

<footnote n="2">²The situation is opposite to the case of the Lax-Friedrichs scheme, for instance, where the discrete shock profiles depend on the spatial variable and the localization of the spectrum of the linearized operator involves an Evans function that is not accessible analytically, see [14, 6] or [27]. Shock profiles such as (2.2) for the Lax-Wendroff scheme rather look like the stationary shock profiles for the Godunov scheme that are studied in [5].</footnote>
## 3.1 The Lopatinskii determinant

We consider a complex number $z$ in the exterior $\mathcal{O}$ of the curve (2.13) and we shall be first looking for solutions $\mathbf{v} = (v_j)_{j \in \mathbb{Z}} \in \ell^q(\mathbb{Z}; \mathbb{C})$ to the eigenvalue problem $\mathcal{L} \mathbf{v} = z \mathbf{v}$. Our goal is to reduce the existence of a non-trivial solution $\mathbf{v}$ to an equation of the form $\Delta(z) = 0$ where the holomorphic function $\Delta$ plays the role of a characteristic polynomial for the operator $\mathcal{L}$. Specifying the relation $(\mathcal{L} \mathbf{v})_j = z v_j$ to those indices $j \le -1$ or $j \ge 2$ (see (2.11) for the definition of the operator $\mathcal{L}$), we are led to solving the dispersion relations:

$$ \frac{\alpha_{\ell}(\alpha_{\ell}-1)}{2} \kappa^2 + (1 - \alpha_{\ell}^2 - z) \kappa + \frac{\alpha_{\ell}(\alpha_{\ell}+1)}{2} = 0, \qquad (3.1a) $$

$$ \frac{\alpha_r(\alpha_r - 1)}{2} \kappa^2 + (1 - \alpha_r^2 - z) \kappa + \frac{\alpha_r(\alpha_r + 1)}{2} = 0. \qquad (3.1b) $$

The behavior of the roots of the above dispersion relations (3.1) is encoded in the following result.

**Lemma 3.1.** Let the conditions (2.4) and (2.12) be satisfied, and let $z \in \mathbb{C}$ belong to the exterior $\mathcal{O}$ of the curve (2.13). Then the dispersion relation (3.1a) has one solution $\kappa_{\ell}(z) \in \mathcal{U}$ and one solution $\kappa_{\ell}^{u}(z) \in \mathbb{D} \setminus \{0\}$. Both functions depend holomorphically on $z$ over $\mathcal{O}$, and they can be holomorphically extended to the set:

$$ \mathbb{C} \setminus \left\{ 1 - \alpha_{\ell}^2 + i t \alpha_{\ell} \sqrt{1 - \alpha_{\ell}^2} \mid t \in [-1, 1] \right\}, $$

on which they satisfy $\kappa_{\ell}(z) \neq \kappa_{\ell}^{u}(z)$ and $\kappa_{\ell}(z) \kappa_{\ell}^{u}(z) \neq 0$.

Furthermore, assuming still that $z \in \mathbb{C}$ belongs to the exterior $\mathcal{O}$ of the curve (2.13), the dispersion relation (3.1b) has one solution $\kappa_r(z) \in \mathbb{D} \setminus \{0\}$ and one solution $\kappa_r^{u}(z) \in \mathcal{U}$. Both functions depend holomorphically on $z$ over $\mathcal{O}$, and they can be holomorphically extended to the set:

$$ \mathbb{C} \setminus \left\{ 1 - \alpha_r^2 + i t \alpha_r \sqrt{1 - \alpha_r^2} \mid t \in [-1, 1] \right\}, $$

on which they satisfy $\kappa_r(z) \neq \kappa_r^{u}(z)$ and $\kappa_r(z) \kappa_r^{u}(z) \neq 0$.

Let us quickly observe that, for $\beta \in [-1, 1]$, the compact domain $D_{\beta}$ that is delimited by the ellipse:

$$ \left\{ 1 - 2 \beta^2 \sin^2 \frac{\xi}{2} + i \beta \sin \xi \mid \xi \in \mathbb{R} \right\} = \left\{ 1 - \beta^2 + \beta^2 \cos \xi + i \beta \sin \xi \mid \xi \in \mathbb{R} \right\}, $$

satisfies $D_{\beta_1} \subset D_{\beta_2}$ as long as $|\beta_1| \le |\beta_2|$. This is the reason why the exterior $\mathcal{O}$ of the curve (2.13) does not contain any element of the two curves:

$$ \left\{ 1 - 2 \alpha_{\ell}^2 \sin^2 \frac{\xi}{2} + i \alpha_{\ell} \sin \xi \mid \xi \in \mathbb{R} \right\} \quad \text{and} \quad \left\{ 1 - 2 \alpha_r^2 \sin^2 \frac{\xi}{2} + i \alpha_r \sin \xi \mid \xi \in \mathbb{R} \right\}. $$

We also observe that the segment:

$$ \left\{ 1 - \alpha_{\ell}^2 + i t \alpha_{\ell} \sqrt{1 - \alpha_{\ell}^2} \mid t \in [-1, 1] \right\}, $$

is located in the closed ball of $\mathbb{C}$ centered at 0 and with radius $\sqrt{1 - \alpha_{\ell}^2}$. Moreover, it is located within the curve (2.13). Same for the analogous segment associated with the “right” state $u_r$ rather than with
$u_\ell$ (with obvious modifications). In what follows, we shall mainly be interested in the fact that some quantities can be holomorphically extended through the unit circle $S^1$. The exterior $\mathcal{O}$ of the curve (2.13) contains some elements of $\mathbb{D}$ far from the point 1 where it is tangent to $S^1$, and this is the reason why, sometimes, we need to consider the set:

$$ \mathcal{O} \cup \{ \zeta \in \mathbb{C} \mid |\zeta| > \max(\sqrt{1 - \alpha_{\ell}^2}, \sqrt{1 - \alpha_r^2}) \}. $$

<figure><img src="image_11.png" /><figcaption>Figure 3.1: Locating the spectrum of the operator $\mathcal{L}$. In blue: the unit circle. In black: the curve (2.13). The region $\mathcal{O}$ is the complement of the grey shaded area. In red: the segment $[1 - \alpha_{\ell}^2 - i\alpha_{\ell}\sqrt{1 - \alpha_{\ell}^2}, 1 - \alpha_{\ell}^2 + i\alpha_{\ell}\sqrt{1 - \alpha_{\ell}^2}]$ outside of which one can holomorphically extend $\kappa_{\ell}$ and $\kappa_{\ell}^u$. The chosen parameter is $\alpha_{\ell} = \sqrt{3}/2$ with $\alpha_{\ell} \ge |\alpha_r|$ so that $\alpha = \alpha_{\ell}$.</figcaption></figure>

**Proof of Lemma 3.1.** We shall only give the proof of Lemma 3.1 for the case of Equation (3.1a), the case of Equation (3.1b) being entirely similar. We begin with some preliminary observations. First of all, the curve (2.13) is a closed curve that is enclosed within the closed unit disk $\bar{\mathbb{D}}$, and that encompasses a strictly convex region. It is actually an ellipse that is centered at $1 - \alpha^2$ with axis of half-length $\alpha^2$ and $\alpha$. This ellipse is tangent to the unit circle $S^1$ from within at 1, see Figure 3.1 for an illustration. As we have pointed out above, thanks to our choice for $\alpha$, the curve (2.13) encompasses both curves:

$$ \{1 - 2\alpha_{\ell,r}^2 \sin^2 \frac{\xi}{2} - i\alpha_{\ell,r} \sin \xi \mid \xi \in \mathbb{R}\}. $$

Let us also note that the exterior $\mathcal{O}$ of the curve (2.13) is a connected set and that, when $z$ belongs to $\mathcal{O}$, Equation (3.1a) has no root $\kappa \in S^1$, for otherwise we would have:

$$ z = 1 - 2\alpha_{\ell}^2 \sin^2 \frac{\theta}{2} - i\alpha_{\ell} \sin \theta $$
for some $\theta \in \mathbb{R}$, and this fact would imply that $z$ belongs either to the interior of the curve (2.13) or to its boundary (depending whether $\alpha = \alpha_\ell$ or $\alpha = \alpha_r$), which is precluded here by our assumption $z \in \mathcal{O}$. Consequently, the number of roots of Equation (3.1a) in $\mathcal{U}$, resp. in $\mathbb{D}$, does not depend on $z \in \mathcal{O}$, and these two numbers (whose sum is 2) are determined by letting $z$ tend to infinity. In that case, one root of Equation (3.1a) tends to zero and the other one tends to infinity, which gives the first half of Lemma 3.1.

Once the functions $\kappa_\ell$ and $\kappa_\ell^u$ have been defined on the exterior $\mathcal{O}$ of the curve (2.13), it only remains to determine how they can be holomorphically extended. The product $\kappa_\ell(z) \kappa_\ell^u(z)$ equals $-(1+\alpha_\ell)/(1-\alpha_\ell)$. It is then rather easy to observe that the two roots of (3.1a) have same modulus if and only if $z$ belongs to the segment:

$$ \{1 - \alpha_{\ell}^{2} + \mathbf{i}t \alpha_{\ell} \sqrt{1 - \alpha_{\ell}^{2}} \mid t \in [-1, 1]\}, $$

and that segment is located within the interior of the curve (2.13). If $\alpha = \alpha_\ell$, the segment is actually part of one axis of the ellipse. Away from that segment, we can therefore always extend $\kappa_\ell(z)$ as the root of largest modulus to (3.1a), and since that root is necessarily simple$^3$, it depends locally holomorphically on $z$. The proof of Lemma 3.1 is thus complete. $\square$

**Remark 3.1.** We observe from (3.1) that in the case $\alpha_r = -\alpha_\ell$, there holds $\kappa_r(z) = 1/\kappa_\ell(z)$ for any $z$ in the exterior $\mathcal{O}$ of the curve (2.13).

Specifying the relation $(\mathcal{L} \mathbf{v})_j = z v_j$ to the indices $j \ge 2$, we obtain:

$$ \forall j \ge 2, \quad \frac{\alpha_r (\alpha_r - 1)}{2} v_{j+1} + (1 - \alpha_r^2 - z) v_j + \frac{\alpha_r (\alpha_r + 1)}{2} v_{j-1} = 0, $$

and because the sequence $\mathbf{v} = (v_j)_{j \in \mathbb{Z}}$ belongs to $\ell^q(\mathbb{Z}; \mathbb{C})$, with $1 \le q \le +\infty$, we obtain, thanks to Lemma 3.1, the expression:

$$ \forall j \ge 1, \quad v_j = v_1 \kappa_r(z)^{j-1}. $$

Since $\kappa_r(z)$ belongs to the unit disk $\mathbb{D}$, the sequence $(\kappa_r(z)^{j-1})_{j \ge 1}$ has exponential decay and therefore belongs to any of the spaces $\ell^p(\mathbb{Z}; \mathbb{C})$, $1 \le p \le \infty$. Specifying now to the indices $j \le -1$, we obtain in a similar way:

$$ \forall j \le 0, \quad v_j = v_0 \kappa_\ell(z)^j. $$

There now remains to determine whether we can find a nonzero pair $(v_0, v_1) \in \mathbb{C}^2$ such that, with the sequence $\mathbf{v} = (v_j)_{j \in \mathbb{Z}} \in \ell^q(\mathbb{Z}; \mathbb{C})$ defined by:

$$ v_j := \begin{cases} v_1 \kappa_r(z)^{j-1}, & \text{if } j \ge 1, \\ v_0 \kappa_\ell(z)^j, & \text{if } j \le 0, \end{cases} $$

then there holds $(\mathcal{L} \mathbf{v})_0 = z v_0$ and $(\mathcal{L} \mathbf{v})_1 = z v_1$, which would provide us with a nonzero solution to the eigenvalue problem $\mathcal{L} \mathbf{v} = z \mathbf{v}$. Substituting the values of $v_2$ and $v_{-1}$ and collecting the terms, we find that the eigenvalue problem $\mathcal{L} \mathbf{v} = z \mathbf{v}$ amounts to solving the $2 \times 2$ linear system:

$$ \begin{bmatrix} \frac{\alpha_{\ell}}{2} (\alpha_{\ell} - \alpha_{m} + (1 - \alpha_{\ell}) \kappa_{\ell}(z)) & -\frac{\alpha_{r}}{2} (1 - \alpha_{m}) \\ \frac{\alpha_{\ell}}{2} (1 + \alpha_{m}) & \frac{\alpha_{r}}{2} \left( \alpha_{r} - \alpha_{m} - \frac{1 + \alpha_{r}}{\kappa_{r}(z)} \right) \end{bmatrix} \begin{bmatrix} v_0 \\ v_1 \end{bmatrix} = 0. \quad (3.2) $$

<footnote>$^3$The endpoints of the segment correspond precisely to the two values of $z$ at which (3.1a) has a double root, the so-called “glancing” points. These two glancing points lie here in the unit disk $\mathbb{D}$ since the Lax-Wendroff scheme is dissipative.</footnote>
Here we have used the relations (3.1a) and (3.1b) in order to simplify some coefficients of the linear system (3.2) and also used the fact that $\kappa_r(z)$ is nonzero. Up to the harmless nonzero factor $\alpha_\ell \alpha_r/4$, we are led to study the so-called Lopatinskii determinant $\Delta$ associated with the linear system (3.2). Its expression is explicitly given here by:

$$ \Delta(z) := 1 - \alpha_m^2 + (\alpha_\ell - \alpha_m + (1 - \alpha_\ell) \kappa_\ell(z)) \left( \alpha_r - \alpha_m - \frac{1 + \alpha_r}{\kappa_r(z)} \right). \quad (3.3) $$

If $\Delta(z) = 0$, then we can find a nonzero solution to (3.2), which means that we can find a nonzero solution to the eigenvalue problem $\mathcal{L} \mathbf{v} = z \mathbf{v}$. The converse is also true. We have therefore reduced, for $z \in \mathcal{O}$, the eigenvalue problem for the operator $\mathcal{L}$ to determining the zeroes of the function $\Delta$.

**Remark 3.2.** In the symmetric case $\alpha_r = -\alpha_\ell$ (and therefore $\kappa_r(z) = 1/\kappa_\ell(z)$), that is considered in [15], the expression (3.3) of the Lopatinskii determinant reduces to:

$$ \Delta(z) = (1 - \alpha_\ell)(1 - \kappa_\ell(z))(1 + \alpha_\ell + (1 - \alpha_\ell)\kappa_\ell(z)), $$

independently of the value of $\alpha_m$. For $z \in \mathcal{O}$, we have $\kappa_\ell(z) \in \mathcal{U}$ so $\kappa_\ell(z)$ can not equal 1. Hence, for $z \in \mathcal{O}$, we have $\Delta(z) = 0$ if and only if:

$$ \kappa_{\ell}(z) = - \frac{1 + \alpha_{\ell}}{1 - \alpha_{\ell}}. $$

Plugging this value in (3.1a), we obtain $z = 1 \notin \mathcal{O}$, meaning that $\Delta$ does not vanish on $\mathcal{O}$. This spectral stability result is independent of the convexity or concavity properties of the flux $f$.

We explain below how to analyze the general case of a convex (or concave) flux $f$. We summarize the properties of the Lopatinskii determinant $\Delta$ in the following result. It is important to observe that the first part of Lemma 3.2, namely the holomorphy properties of $\Delta$ is independent of $f$.

**Lemma 3.2.** Let the weak solution (2.2) satisfy the Rankine-Hugoniot relation (2.3) and the entropy inequalities (2.4). Then under the condition (2.12) on the parameter $\lambda$, the Lopatinskii determinant $\Delta$ defined in (3.3) is holomorphic on the open set:

$$ \mathcal{O} \cup \{ \zeta \in \mathbb{C} \mid |\zeta| > \max(\sqrt{1 - \alpha_{\ell}^2}, \sqrt{1 - \alpha_r^2}) \}, $$

and $\Delta(1) = 0$. Furthermore, if the flux $f$ in (2.1) is either convex or concave, then $\Delta$ satisfies:

* $\Delta'(1) \neq 0$,

* $\Delta$ does not vanish on $\mathcal{O}$.

The fact that $\Delta$ vanishes at 1 is associated with the presence of an eigenvalue at 1 for the operator $\mathcal{L}$. This is discussed in more details below (we refer to [27] for a general discussion on this fact). In the terminology of [27, Definition 4.1], if the flux $f$ is either convex or concave, the stationary discrete shock profile (2.2) is thus spectrally stable since the non-vanishing of $\Delta$ on $\mathcal{O}$ will imply that the spectrum of $\mathcal{L}$ is located within $\mathbb{D} \cup \{1\}$ (see the final argument below in Section 3.3 after our construction of the spatial Green's function).
Proof of Lemma 3.2. The holomorphy of $\Delta$ follows from that of $\kappa_\ell$ and $\kappa_r^{-1}$ as given in Lemma 3.1. Indeed, $\kappa_\ell$ is holomorphic on:

$$ \mathbb{C} \setminus \{1 - \alpha_{\ell}^{2} + i t \alpha_{\ell} \sqrt{1 - \alpha_{\ell}^{2}} \mid t \in [-1, 1]\}, $$

and the segment that has to be excluded is contained in the closed ball $\overline{B_{\sqrt{1-\alpha_{\ell}^2}(0)}}$ and within the spectral curve (2.13). Arguing similarly with $\kappa_r^{-1}$, one sees that $\Delta$ is holomorphic on $\mathcal{O} \cup \{\zeta \in \mathbb{C} \mid |\zeta| > \max(\sqrt{1-\alpha_{\ell}^2}, \sqrt{1-\alpha_r^2})\}$.

For the behavior of $\Delta$ at 1, we have:

$$ \kappa_{\ell}(1) = - \frac{1 + \alpha_{\ell}}{1 - \alpha_{\ell}} \in \mathcal{W}, \quad \kappa_{r}(1) = - \frac{1 + \alpha_{r}}{1 - \alpha_{r}} \in \mathbb{D}, $$

and we thus compute $\Delta(1) = 0$ (plug the above values at $z = 1$ in (3.3)). This proves the first part of Lemma 3.2, which holds independently of the convexity or concavity of $f$.

Differentiating $\Delta$ with respect to $z$, we also get:

$$ \Delta'(1) = (1 - \alpha_{\ell})(1 - \alpha_{m})\kappa'_{\ell}(1) - (1 + \alpha_{m})\frac{(1 - \alpha_{r})^2}{1 + \alpha_{r}}\kappa'_{r}(1), $$

and the derivatives $\kappa'_\ell(1)$, $\kappa'_r(1)$ are obtained by differentiating (3.1a) and (3.1b) and evaluating at $z = 1$:

$$ \kappa'_{\ell}(1) = - \frac{1 + \alpha_{\ell}}{\alpha_{\ell}(1 - \alpha_{\ell})}, \quad \kappa'_{r}(1) = - \frac{1 + \alpha_{r}}{\alpha_{r}(1 - \alpha_{r})}. $$

We end up with the expression:

$$ \Delta'(1) = - \frac{(1 + \alpha_{\ell})(1 - \alpha_{m})}{\alpha_{\ell}} + \frac{(1 - \alpha_{r})(1 + \alpha_{m})}{\alpha_{r}}. \qquad (3.4) $$

Let us from now on assume that the flux $f$ is either convex or concave. The crucial consequence is that we have $\alpha_m \in [\alpha_r, \alpha_\ell]$, and we recall the entropy inequalities (2.4) as well as the stability restriction (2.12). This implies that $\alpha_m$ belongs to the open interval $(-1, 1)$ and therefore $\Delta'(1)$ is the sum of two negative quantities. It therefore does not vanish.

It remains to study the other possible zeroes of $\Delta$ and specifically to show that $\Delta$ does not vanish on $\mathcal{O}$. We are first going to expand the expression of $\Delta$ in (3.3). Because of the form of the dispersion relations (3.1a) and (3.1b), we first write:

$$ \kappa_{\ell}(z) = \frac{z - 1 + \alpha_{\ell}^{2} + W_{\ell}(z)}{\alpha_{\ell}(\alpha_{\ell} - 1)}, \quad W_{\ell}(z)^{2} = (z - 1 + \alpha_{\ell}^{2})^{2} + \alpha_{\ell}^{2}(1 - \alpha_{\ell}^{2}), $$

$$ \kappa_r(z) = \frac{z - 1 + \alpha_r^2 + W_r(z)}{\alpha_r(\alpha_r - 1)}, \quad W_r(z)^2 = (z - 1 + \alpha_r^2)^2 + \alpha_r^2(1 - \alpha_r^2), $$

where we do not mind at this stage which square root should be picked for $W_\ell(z)$ and $W_r(z)$. Plugging these expressions for $\kappa_\ell(z)$ and $\kappa_r(z)$ in (3.3), we obtain the expression:

$$ \alpha_{\ell} \alpha_{r} \Delta(z) = Z^2 + \alpha_{m}(\alpha_{\ell} + \alpha_{r}) Z + \alpha_{\ell} \alpha_{r} + (Z + \alpha_{m} \alpha_{r}) W_{\ell} - (Z + \alpha_{m} \alpha_{\ell}) W_{r} - W_{\ell} W_{r}, \quad (3.5) $$
where here and from now on, we use the notation $Z := z - 1$ and we rather consider $W_\ell$ and $W_r$ as functions of $Z$.

Let us now assume that $z \in \mathcal{O}$ is a point where the Lopatinskii determinant $\Delta$ vanishes. Then (3.5) provides us with an expression for the product $W_\ell W_r$ which we can raise to the square (this is the reason why we do not care at this stage about which square root should be preferred). Namely, we introduce the polynomials:

$$Q(Z) := Z^2 + \alpha_m(\alpha_\ell + \alpha_r)Z + \alpha_\ell \alpha_r, \quad P_{\ell,r}(Z) := Z + \alpha_m \alpha_{\ell,r},$$

and we see from (3.5) that if $\Delta$ vanishes at $z \in \mathcal{O}$, then $Z = z - 1$ satisfies:

$$W_\ell W_r = Q(Z) + P_r(Z) W_\ell - P_\ell(Z) W_r.$$

Squaring both left and right-hand sides, and then substituting the expression:

$$P_r(Z) W_\ell - P_\ell(Z) W_r = W_\ell W_r - Q(Z),$$

we obtain the relation:

$$2(Q - P_\ell P_r) W_\ell W_r = W_\ell^2 W_r^2 + Q^2 + P_r^2 W_\ell^2 + P_\ell^2 W_r^2.$$

Expanding each side with respect to $Z$ and simplifying by the nonzero factor $\alpha_\ell \alpha_r$, we obtain:

$$(1 - \alpha_m^2) W_\ell W_r = \{1 + \alpha_m^2 + 2 \alpha_\ell \alpha_r - 2 \alpha_m (\alpha_\ell + \alpha_r)\} Z^2 + \alpha_\ell \alpha_r (1 - \alpha_m^2) (2Z + 1).$$

Raising one last time to the square and collecting the various terms, we end up with the following polynomial equation for $Z$:

$$Z^2 (4 \gamma Z^2 + 2 \beta Z + \beta) = 0, \qquad (3.6)$$

with the parameters $\beta$ and $\gamma$ being defined by:

$$\beta := (1 - \alpha_m^2) \{ (\alpha_\ell - \alpha_r)^2 - (\alpha_m(\alpha_\ell + \alpha_r) - 2 \alpha_\ell \alpha_r)^2 \}, \qquad (3.7)$$

$$\gamma := (\alpha_m - \alpha_r)(\alpha_\ell - \alpha_m)(1 + \alpha_\ell \alpha_r - \alpha_m(\alpha_\ell + \alpha_r)). \qquad (3.8)$$

Let us summarize where we are at this stage. We have shown that if $z \in \mathcal{O}$ is a point where the Lopatinskii determinant $\Delta$ vanishes, then $Z = z - 1$ is a root of (3.6), with coefficients $\beta, \gamma$ given by (3.7)-(3.8). Since the function:

$$\alpha_m \in [\alpha_r, \alpha_\ell] \mapsto (\alpha_\ell - \alpha_r)^2 - (\alpha_m(\alpha_\ell + \alpha_r) - 2 \alpha_\ell \alpha_r)^2$$

is concave, its minimum on the segment $[\alpha_r, \alpha_\ell]$ is attained either at $\alpha_\ell$ or $\alpha_r$ (that is, at one of the endpoints) and this minimum is therefore necessarily positive. In other words, we have $\beta > 0$ for all relevant values of $\alpha_m$ (let us recall that $\alpha_m \in [\alpha_r, \alpha_\ell]$ follows from the convexity or concavity of $f$).

In the cases $\alpha_m = \alpha_r$ or $\alpha_m = \alpha_\ell$, the coefficient $\gamma$ vanishes. In these two extreme cases, it is clear that all the roots of (3.6) are real. If $\alpha_m$ belongs to the open interval $(\alpha_r, \alpha_\ell)$, then $\gamma$ is also positive, and we need to compute the discriminant of the second order factor in (3.6) in order to determine the location of its two roots. Since $\beta$ is positive, the discriminant has the same sign as the quantity$^4$:

$$\beta - 4\gamma = \left( (\alpha_\ell + \alpha_r)(1 + \alpha_m^2) - 2(1 + \alpha_\ell \alpha_r)\alpha_m \right)^2 \geq 0.$$

<footnote>$^4$The key final argument is that the quantity $\beta - 4\gamma$ can be explicitly factorized !</footnote>
This means that for $\alpha_m \in [\alpha_r, \alpha_l]$, all the roots of (3.6) are real, which means that the only possible roots of $\Delta$ in $\mathcal{O}$ are real.

To complete the proof of Lemma 3.2, we thus restrict to real values of the parameter $z \in \mathcal{O}$, that is we consider either $z \in (-\infty, 1-2\alpha^2)$ or $z \in (1, +\infty)$ (see Figure 3.1 for visualizing the set $\mathcal{O} \cap \mathbb{R}$). Let us first consider the case $z > 1$, that is $Z = z - 1 > 0$ for which we have, with the standard determination of the square root$^5$:

$$ \kappa_{\ell}(z) = \frac{Z + \alpha_{\ell}^2 + \sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2}}{\alpha_{\ell}(\alpha_{\ell} - 1)}, \quad \kappa_r(z) = \frac{Z + \alpha_r^2 - \sqrt{Z^2 + 2 \alpha_r^2 Z + \alpha_r^2}}{\alpha_r(\alpha_r - 1)}. $$

We then compute, for $z > 1$, the expression:

$$ \alpha_{\ell} \alpha_{r} \Delta(z) = Z^2 + \alpha_{m}(\alpha_{\ell} + \alpha_{r}) Z + \alpha_{\ell} \alpha_{r} + (Z + \alpha_{m} \alpha_{r}) \sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2} \\ + (Z + \alpha_{m} \alpha_{\ell}) \sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2} + \sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2} \sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2}, $$

where we recall that $\alpha_\ell$, $\alpha_m$ and $\alpha_r$ satisfy:

$$ -1 < \alpha_r \le \alpha_m \le \alpha_\ell < 1, \quad \alpha_r < 0 < \alpha_\ell. $$

The function $\Delta$ is real valued and smooth (that is, analytic) on $[1, +\infty)$. It vanishes at 1, and its derivative is given, for $z \ge 1$, by:

$$ \begin{aligned} \alpha_{\ell} \alpha_{r} \Delta'(z) = & 2Z + \alpha_{m}(\alpha_{\ell} + \alpha_{r}) + \sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2} + \sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2} \\ & + (Z + \alpha_{m} \alpha_{r}) \frac{Z + \alpha_{\ell}^2}{\sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2}} + (Z + \alpha_{m} \alpha_{\ell}) \frac{Z + \alpha_{r}^2}{\sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2}} \quad (3.9) \\ & + (Z + \alpha_{\ell}^2) \frac{\sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2}}{\sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2}} + (Z + \alpha_{r}^2) \frac{\sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2}}{\sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2}}. \end{aligned} $$

For $z \ge 1$, we have $Z \ge 0$, and this implies:

$$ \frac{Z + \alpha_{\ell}^2}{\sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2}} \ge \alpha_{\ell}, \quad \frac{Z + \alpha_{r}^2}{\sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2}} \ge |\alpha_{r}|, $$

which gives (deriving lower bounds for the above three blue terms in (3.9)), for $z \ge 1$:

$$ \begin{aligned} \alpha_{\ell} \alpha_{r} \Delta'(z) \ge D(\alpha_{m}, Z) := & \alpha_{m}(\alpha_{\ell} + \alpha_{r}) + 2 \alpha_{\ell} |\alpha_{r}| + \sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2} + \sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2} \\ & + (Z + \alpha_{m} \alpha_{r}) \frac{Z + \alpha_{\ell}^2}{\sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2}} + (Z + \alpha_{m} \alpha_{\ell}) \frac{Z + \alpha_{r}^2}{\sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2}}. \end{aligned} \quad (3.10) $$

<footnote>$^5$Since $Z$ is now restrained to real values only, we shall only take square roots of positive real numbers.</footnote>
We now consider the quantity $D(\alpha_m, Z)$ that is defined in (3.10) and try to show that it does not vanish for $\alpha_m \in [\alpha_r, \alpha_\ell]$ and $Z > 0$. This is shown by deriving a positive lower bound. Indeed, the quantity $D(\alpha_m, Z)$ for $\alpha_\ell \alpha_r \Delta'(z)$ is an affine function with respect to $\alpha_m \in [\alpha_r, \alpha_\ell]$. Its value is therefore not smaller than its values at the endpoints of the interval $[\alpha_r, \alpha_\ell]$. For $\alpha_m = \alpha_r$, we compute:

$$
\begin{align*}
D(\alpha_r, Z) &= \alpha_\ell |\alpha_r| + \alpha_r^2 + \sqrt{Z^2 + 2\alpha_\ell^2 Z + \alpha_\ell^2} + \sqrt{Z^2 + 2\alpha_r^2 Z + \alpha_r^2} \\
&\quad + (Z + \alpha_r^2) \frac{Z + \alpha_\ell^2}{\sqrt{Z^2 + 2\alpha_\ell^2 Z + \alpha_\ell^2}} + (Z - \alpha_\ell |\alpha_r|) \frac{Z + \alpha_r^2}{\sqrt{Z^2 + 2\alpha_r^2 Z + \alpha_r^2}} \\
&\geq \alpha_\ell |\alpha_r| \left( 1 - \underbrace{\frac{Z + \alpha_r^2}{\sqrt{Z^2 + 2\alpha_r^2 Z + \alpha_r^2}}}_{\geq 0} \right) + \sqrt{Z^2 + 2\alpha_\ell^2 Z + \alpha_\ell^2} + \sqrt{Z^2 + 2\alpha_r^2 Z + \alpha_r^2}.
\end{align*}
$$

We thus obtain the (far from optimal but nevertheless sufficient !) uniform lower bound:

$$
\forall Z \ge 0, \quad D(\alpha_r, Z) \ge \alpha_\ell + |\alpha_r| > 0,
$$

and similar arguments lead to the analogous estimate:

$$
\forall Z \ge 0, \quad D(\alpha_\ell, Z) \ge \alpha_\ell + |\alpha_r| > 0.
$$

For $\alpha_m \in [\alpha_r, \alpha_\ell]$, we have thus obtained the lower bound:

$$
\forall z \ge 1, \quad \alpha_\ell \alpha_r \Delta'(z) \ge \alpha_\ell + |\alpha_r| > 0,
$$

which implies that $\underline{\Delta}$ does not vanish on the open interval $(1, +\infty)$. (Let us recall that $\underline{\Delta}$ vanishes at 1.)

It remains to examine the case $z \in (-\infty, 1 - 2\alpha^2)$, for which we now have $Z \le -2\alpha^2$ and we get the expressions:

$$
\kappa_{\ell}(z) = \frac{Z + \alpha_{\ell}^{2} - \sqrt{Z^{2} + 2 \alpha_{\ell}^{2} Z + \alpha_{\ell}^{2}}}{\alpha_{\ell} (\alpha_{\ell} - 1)}, \quad \kappa_{r}(z) = \frac{Z + \alpha_{r}^{2} + \sqrt{Z^{2} + 2 \alpha_{r}^{2} Z + \alpha_{r}^{2}}}{\alpha_{r} (\alpha_{r} - 1)},
$$

which yields:

$$
\alpha_{\ell} \alpha_{r} \underline{\Delta}(z) = Z^2 + \alpha_{m}(\alpha_{\ell} + \alpha_{r}) Z + \alpha_{\ell} \alpha_{r} - (Z + \alpha_{m} \alpha_{r}) \sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2} \\
\qquad - (Z + \alpha_{m} \alpha_{\ell}) \sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2} + \sqrt{Z^2 + 2 \alpha_{\ell}^2 Z + \alpha_{\ell}^2} \sqrt{Z^2 + 2 \alpha_{r}^2 Z + \alpha_{r}^2}.
$$

Once again, the right-hand side of the latter equality, which we denote $\mathbb{D}(\alpha_m, Z)$, is an affine function with respect to $\alpha_m$ and we are going to derive a lower bound for either $\alpha_m = \alpha_r$ or $\alpha_m = \alpha_\ell$, which will give a lower bound for any value of $\alpha_m \in [\alpha_r, \alpha_\ell]$. For $\alpha_m = \alpha_r$ and $Z \le -2\alpha^2$, we have:

$$
\begin{align*}
\mathbb{D}(\alpha_r, Z) &= Z^2 + \alpha_r^2 Z + \alpha_\ell \alpha_r Z \\
&\quad + (|Z| - \alpha_r^2) \sqrt{Z^2 + 2\alpha_\ell^2 Z + \alpha_\ell^2} + (|Z| + \alpha_\ell |\alpha_r|) \sqrt{Z^2 + 2\alpha_r^2 Z + \alpha_r^2} \\
&\quad + \sqrt{Z^2 + 2\alpha_\ell^2 Z + \alpha_\ell^2} \sqrt{Z^2 + 2\alpha_r^2 Z + \alpha_r^2} - \alpha_\ell |\alpha_r|.
\end{align*}
$$
Recalling that $\alpha$ stands for the maximum of $|\alpha_r|$ and $\alpha_\ell$, we find that both functions:

$$Z \mapsto Z^2 + 2\alpha_{\ell}^2 Z + \alpha_{\ell}^2, \quad Z \mapsto Z^2 + 2\alpha_{r}^2 Z + \alpha_{r}^2$$

are decreasing on $(-\infty, -2\alpha^2]$, and we thus derive the lower bounds:

$$\forall Z \le -2\alpha^2, \quad Z^2 + 2\alpha_{\ell,r}^2 Z + \alpha_{\ell,r}^2 \ge \alpha_{\ell,r}^2.$$

Using these lower bounds in the expression of $\mathcal{D}(\alpha_r, Z)$ leads to:

$$\forall Z \le -2\alpha^2, \quad \begin{aligned} \mathcal{D}(\alpha_r, Z) &\ge Z^2 + \alpha_r^2 Z + \alpha_\ell |\alpha_r| |Z| \\ &\quad + \sqrt{Z^2 + 2\alpha_\ell^2 Z + \alpha_\ell^2} \sqrt{Z^2 + 2\alpha_r^2 Z + \alpha_r^2} - \alpha_\ell |\alpha_r| \\ &\ge Z^2 + \alpha_r^2 Z \ge 2\alpha^2 (2\alpha^2 - \alpha_r^2) \ge 2\alpha^4 > 0. \end{aligned}$$

Similar arguments lead to:

$$\forall Z \le -2\alpha^2, \quad \mathcal{D}(\alpha_\ell, Z) \ge 2\alpha^4 > 0,$$

and we have therefore proved that $\Delta(z)$ does not vanish on the interval $z \in (-\infty, 1 - 2\alpha^2]$. In other words, the Lopatinskii determinant $\Delta$ does not vanish on $\mathcal{O}$. The conclusion of Lemma 3.2 follows. $\square$

Lemma 3.2 has an immediate consequence regarding the solvability of the eigenvalue problem. The proof is a straightforward application of all above results on the Lopatinskii determinant $\Delta$.

**Corollary 3.1.** Let the flux $f$ in (2.1) be either convex or concave. Let the weak solution (2.2) satisfy the Rankine-Hugoniot relation (2.3) and the entropy inequalities (2.4). Then, under the CFL condition (2.12), for any $z \in \mathcal{O}$ and $1 \le q \le +\infty$, the only solution $v \in l^q(\mathbb{Z}; \mathbb{C})$ to the eigenvalue problem $\mathcal{L}v = zv$ is the zero sequence.

## 3.2 The spatial Green's function

In this section, we intend to construct the so-called Green's function, which amounts to inverting the operator $zId - \mathcal{L}$. We have already seen that the Lopatinskii determinant $\Delta$ plays a crucial role in the location of the eigenvalues of the operator $\mathcal{L}$. We are going to show below that the condition $\Delta(z) \ne 0$ is actually necessary and sufficient for a complex number $z \in \overline{\mathcal{U}} \setminus \{1\}$ to lie in the resolvent set of $\mathcal{L}$. We do not wish any longer to assume that the flux $f$ is either convex or concave. We rather wish to deal more generally with spectrally stable configurations. We shall therefore substitute Assumption 1 below in place of the convexity (or concavity) assumption on $f$. In particular, there is now no obvious reason for the parameter $\alpha_m$ in (2.9) to belong to the interval $[\alpha_r, \alpha_\ell]$. We make from now on the following assumption.

**Assumption 1.** The Lopatinskii determinant $\Delta$ in (3.3) associated with the discrete shock (2.7) satisfies:

* $\Delta'(1) \ne 0$,

* $\Delta$ does not vanish on $\overline{\mathcal{U}} \setminus \{1\}$.

As we have seen in the proof of Lemma 3.2, the holomorphy of $\Delta$ on an open set that contains $\overline{\mathcal{U}}$ always holds as long as the CFL condition (2.12) is satisfied (the CFL condition (2.12) allows us to define the modes $\kappa_{r,\ell}$ by Lemma 3.1, and therefore the function $\Delta$). The property $\Delta(1) = 0$ also holds independently of the convexity properties of $f$. Moreover, Lemma 3.2 shows that Assumption 1 is satisfied whenever the flux $f$ is either convex or concave. There is no real difficulty to generalize the analysis of the previous Section in order to consider a slightly larger context than the sole case of a convex or concave flux $f$. We can indeed extend Corollary 3.1 and obtain the following result.
**Corollary 3.2.** Let the weak solution (2.2) satisfy the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. Then for any $z \in \overline{\mathcal{U}} \setminus \{1\}$ and $1 \le q \le +\infty$, the only solution $v \in \ell^q(\mathbb{Z}; \mathbb{C})$ to the eigenvalue problem $\mathcal{L}v = zv$ is the zero sequence.

For $z$ in the resolvent set of the operator $\mathcal{L}$, we denote by $\mathcal{G}^{j_0}(z) = (\mathcal{G}_j^{j_0}(z))_{j \in \mathbb{Z}} \in \ell^q(\mathbb{Z}; \mathbb{C})$ the solution to the resolvent problem:

$$ (z \operatorname{Id} - \mathcal{L}) \mathcal{G}^{j_0}(z) = \delta_{j_0}, \qquad (3.11) $$

where $\delta_{j_0} = (\delta_{j_0}(j))_{j \in \mathbb{Z}}$ stands for the discrete Dirac mass located at the index $j_0 \in \mathbb{Z}$. The following result gives an explicit expression for the spatial Green's function $\mathcal{G}^{j_0}(z)$ for $z \in \overline{\mathcal{U}} \setminus \{1\}$. This makes use of the fact that the Lopatinskii determinant does not vanish (which is the reason for Assumption 1). Corollary 3.2 above shows that the solution to (3.11) is necessarily unique for $z \in \overline{\mathcal{U}} \setminus \{1\}$.

**Proposition 3.1.** Let the weak solution (2.2) satisfy the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. Then for any $z \in \overline{\mathcal{U}} \setminus \{1\}$ and for any $j_0 \in \mathbb{Z}$, there exists a unique solution $\mathcal{G}^{j_0}(z) \in \ell^q(\mathbb{Z}; \mathbb{C})$ to the equation (3.11).

For $j_0 \ge 1$, this sequence $\mathcal{G}^{j_0}(z) = (\mathcal{G}_j^{j_0}(z))_{j \in \mathbb{Z}}$ is explicitly given by:

$$ \mathcal{G}_j^{j_0}(z) = \begin{cases} -\frac{2(1-\alpha_m)}{\alpha_\ell \Delta(z)} \kappa_r^u(z)^{1-j_0} \kappa_\ell(z)^j, & \text{if } j \le 0, \\ -\frac{2(\alpha_\ell - \alpha_m + (1-\alpha_\ell)\kappa_\ell(z))}{\alpha_r \Delta(z)} \kappa_r^u(z)^{1-j_0} \kappa_r(z)^{j-1} \\ \qquad +\frac{2(\kappa_r^u(z)^{1-j_0} \kappa_r(z)^{j-1} - \kappa_r^u(z)^{j-j_0})}{\alpha_r(1-\alpha_r)(\kappa_r^u(z) - \kappa_r(z))}, & \text{if } 1 \le j \le j_0, \\ -\frac{2(\alpha_\ell - \alpha_m + (1-\alpha_\ell)\kappa_\ell(z))}{\alpha_r \Delta(z)} \kappa_r^u(z)^{1-j_0} \kappa_r(z)^{j-1} \\ \qquad +\frac{2(\kappa_r^u(z)^{1-j_0} \kappa_r(z)^{j-1} - \kappa_r(z)^{j-j_0})}{\alpha_r(1-\alpha_r)(\kappa_r^u(z) - \kappa_r(z))}, & \text{if } j > j_0, \end{cases} \qquad (3.12) $$
and for $j_0 \le 0$, $\mathcal{G}^{j_0}(z)$ is explicitly given by:

$$ \mathcal{G}_j^{j_0}(z) = \left\{ \begin{array}{ll} \frac{2(1+\alpha_m)}{\alpha_r \Delta(z)} \kappa_\ell^u(z)^{-j_0} \kappa_r(z)^{j-1}, & \text{if } j \ge 1, \\ \\ -\frac{2(\alpha_r - \alpha_m - (1+\alpha_r)\kappa_r(z)^{-1})}{\alpha_\ell \Delta(z)} \kappa_\ell^u(z)^{-j_0} \kappa_\ell(z)^j & \\ \\ +\frac{2(\kappa_\ell^u(z)^{-j_0} \kappa_\ell(z)^j - \kappa_\ell^u(z)^{j-j_0})}{\alpha_\ell(1-\alpha_\ell)(\kappa_\ell(z) - \kappa_\ell^u(z))}, & \text{if } j_0 \le j \le 0, \\ \\ -\frac{2(\alpha_r - \alpha_m - (1+\alpha_r)\kappa_r(z)^{-1})}{\alpha_\ell \Delta(z)} \kappa_\ell^u(z)^{-j_0} \kappa_\ell(z)^j & \\ \\ +\frac{2(\kappa_\ell^u(z)^{-j_0} \kappa_\ell(z)^j - \kappa_\ell(z)^{j-j_0})}{\alpha_\ell(1-\alpha_\ell)(\kappa_\ell(z) - \kappa_\ell^u(z))}, & \text{if } j < j_0 \le 0. \end{array} \right. \quad (3.13) $$

The result of Proposition 3.1 is independent of the space $\ell^q(\mathbb{Z}, \mathbb{C})$ that is considered since for any $j_0 \in \mathbb{Z}$ and $z \in \mathcal{W} \setminus \{1\}$, the sequence $\mathcal{G}^{j_0}(z)$ belongs to the intersection of all spaces $\ell^p(\mathbb{Z}, \mathbb{C})$, $p \in [1, +\infty]$ (it has exponential decay at infinity with respect to $j$).

*Proof of Proposition 3.1.* We shall give the proof of Proposition 3.1 in the case $\alpha_m \ne 1$. This assumption is used below to rewrite the second order scalar recurrence relation (3.11) as a first order recurrence relation for a vector $V_{j+1}$ in terms of $V_j$. In the case $\alpha_m = 1$, which is left to the interested reader, one should rather write a first order recurrence relation for a vector $V_{j-1}$ in terms of $V_j$, that is, going backwards.

Let us, for a moment, consider the slightly more general problem where we let $\mathbf{h} = (h_j)_{j \in \mathbb{Z}}$ be a given sequence in $\ell^q(\mathbb{Z}, \mathbb{C})$ and take $z \in \mathcal{W} \setminus \{1\}$. We then wish to construct a solution $\mathbf{v}(z) \in \ell^q(\mathbb{Z}; \mathbb{C})$ to the resolvent equation:

$$ (z \operatorname{Id} - \mathcal{L}) \mathbf{v}(z) = \mathbf{h}. $$

To do so, we first introduce the augmented vectors

$$ \forall j \in \mathbb{Z}, \quad V_j(z) := \begin{pmatrix} v_{j-1}(z) \\ v_j(z) \end{pmatrix} \in \mathbb{C}^2, \quad \text{and} \quad H_j := \begin{pmatrix} 0 \\ h_j \end{pmatrix} \in \mathbb{C}^2. $$

Using the definition of $\mathcal{L}$, we obtain that

$$ \left\{ \begin{array}{ll} V_{j+1}(z) = \mathbb{M}_r(z)V_j(z) + \mathbb{A}_r H_j, & j \ge 2, \\ V_{j+1}(z) = \mathbb{M}_\ell(z)V_j(z) + \mathbb{A}_\ell H_j, & j \le -1, \\ V_2(z) = \mathbb{M}_{2,1}(z)V_1(z) + \mathbb{A}_r H_1, \\ V_1(z) = \mathbb{M}_{1,0}(z)V_0(z) + \mathbb{A}_{r,m} H_0, \end{array} \right. \quad (3.14) $$

where the above matrices in (3.14) are defined as

$$ \mathbb{M}_k(z) := \begin{pmatrix} 0 & 1 \\ \frac{1+\alpha_k}{1-\alpha_k} & \frac{2(1-\alpha_k^2-z)}{\alpha_k(1-\alpha_k)} \end{pmatrix}, \quad \mathbb{A}_k := \begin{pmatrix} 0 & 0 \\ 0 & \frac{2}{\alpha_k(1-\alpha_k)} \end{pmatrix}, \quad k \in \{r, \ell\}, $$
and<fnref n="6" />

$$
\begin{align*}
\mathbb{M}_{2,1}(z) &:= \begin{pmatrix} 0 & 1 \\ \frac{\alpha_{\ell}(1+\alpha_m)}{\alpha_r(1-\alpha_r)} & \frac{2-\alpha_r(\alpha_r+\alpha_m)-2z}{\alpha_r(1-\alpha_r)} \end{pmatrix}, \\
\mathbb{M}_{1,0}(z) &:= \begin{pmatrix} 0 & 1 \\ \frac{\alpha_{\ell}(1+\alpha_{\ell})}{\alpha_r(1-\alpha_m)} & \frac{2-\alpha_{\ell}(\alpha_{\ell}+\alpha_m)-2z}{\alpha_r(1-\alpha_m)} \end{pmatrix}, \\
\mathbb{A}_{r,m} &:= \begin{pmatrix} 0 & 0 \\ 0 & \frac{2}{\alpha_r(1-\alpha_m)} \end{pmatrix}.
\end{align*}
$$

Keeping the notation of Lemma 3.1, we denote, for $z \in \overline{\mathcal{W}} \setminus \{1\}$, by $\kappa_r(z) \in \mathbb{D}$ and $\kappa_r^u(z) \in \mathcal{W}$ the two eigenvalues of $\mathbb{M}_r(z)$ (these are the roots of the dispersion relation (3.1b)), while we denote by $\kappa_\ell(z) \in \mathcal{W}$ and $\kappa_\ell^u(z) \in \mathbb{D}$ the two eigenvalues of $\mathbb{M}_\ell(z)$ (these are the roots of the dispersion relation (3.1a)). Upon denoting $\Pi_{r,\ell}^{s,u}(z)$ the corresponding stable/unstable spectral projections together with $\mathbb{E}_{r,\ell}^{s,u}(z)$ the associated eigenspaces, we have that<fnref n="7" />:

$$ \mathbb{C}^2 = \mathbb{E}_k^s(z) \oplus \mathbb{E}_k^u(z), \quad k \in \{r, \ell\}, $$

with

$$ \mathbb{E}_k^s(z) := \operatorname{Span} \begin{pmatrix} 1 \\ \kappa_k(z) \end{pmatrix}, \quad \mathbb{E}_k^u(z) := \operatorname{Span} \begin{pmatrix} 1 \\ \kappa_k^u(z) \end{pmatrix}, \quad k \in \{r, \ell\}. $$

Integrating the stable and unstable parts in (3.14), we have that

$$ \forall j \ge 2, \quad \left\{ \begin{aligned} \Pi_r^u(z)V_j(z) &= -\sum_{p=0}^{+\infty} \kappa_r^u(z)^{-1-p} \Pi_r^u(z) \mathbb{A}_r H_{j+p}, \\ \Pi_r^s(z)V_j(z) &= \kappa_r(z)^{j-2} \Pi_r^s(z)V_2(z) + \sum_{p=2}^{j-1} \kappa_r(z)^{j-p-1} \Pi_r^s(z) \mathbb{A}_r H_p, \end{aligned} \right. \qquad (3.15) $$

from which we already deduce that

$$ \Pi_r^u(z)V_2(z) = -\sum_{p=0}^{+\infty} \kappa_r^u(z)^{-1-p} \Pi_r^u(z) \mathbb{A}_r H_{2+p}. $$

On the other hand, we get

$$ \forall j \le 0, \quad \left\{ \begin{aligned} \Pi_{\ell}^{u}(z)V_{j}(z) &= \sum_{p=0}^{+\infty} \kappa_{\ell}^{u}(z)^{p} \Pi_{\ell}^{u}(z) \mathbb{A}_{\ell} H_{j-p-1}, \\ \Pi_{\ell}^{s}(z)V_{j}(z) &= \kappa_{\ell}(z)^{j} \Pi_{\ell}^{s}(z)V_{0}(z) - \sum_{p=j}^{-1} \kappa_{\ell}(z)^{p} \Pi_{\ell}^{s}(z) \mathbb{A}_{\ell} H_{j-p-1}, \end{aligned} \right. \qquad (3.16) $$

<footnote n="6">⁶The definition of $\mathbb{M}_{1,0}(z)$ and of $\mathbb{A}_{r,m}$ uses the assumption $\alpha_m \neq 1$.</footnote>

<footnote n="7">⁷It should be kept in mind that the stable eigenvalue for the “right” state is $\kappa_r(z) \in \mathbb{D}$ since it corresponds to the dynamics for $j \ge 1$, while the stable eigenvalue for the “left” state is $\kappa_\ell(z) \in \mathcal{W}$ since it corresponds to the dynamics for $j \le 0$.</footnote>
such that

$$ \Pi_{\ell}^{u}(z)V_{0}(z) = \sum_{p=0}^{+\infty} \kappa_{\ell}^{u}(z)^{p} \Pi_{\ell}^{u}(z)A_{\ell}H_{-p-1}. $$

We notice that both vectors $\Pi_r^s(z)V_2(z)$ and $\Pi_\ell^s(z)V_0(z)$ still need to be determined, and if we are able to do so, then we shall have a solution to (3.14) at our disposal. First, we use the remaining two equations in (3.14) to obtain that

$$ V_2(z) = \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)V_0(z) + \mathbb{M}_{2,1}(z)A_{r,m}H_0 + A_rH_1, $$

which we write instead as:

$$ \Pi_r^s(z)V_2(z) - \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_\ell^s(z)V_0(z) = \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_\ell^u(z)V_0(z) - \Pi_r^u(z)V_2(z) + \mathbb{M}_{2,1}(z)A_{r,m}H_0 + A_rH_1. $$

Upon writing

$$ \Pi_r^s(z)V_2(z) = \underbrace{\chi_2(z)}_{\in\mathbb{C}} E_r^s(z), \quad E_r^s(z) := \begin{pmatrix} 1 \\ \kappa_r(z) \end{pmatrix}, $$

and

$$ \Pi_\ell^s(z)V_0(z) = \underbrace{\chi_0(z)}_{\in\mathbb{C}} E_\ell^s(z), \quad E_\ell^s(z) := \begin{pmatrix} 1 \\ \kappa_\ell(z) \end{pmatrix}, $$

with the vector $(\chi_2(z), \chi_0(z)) \in \mathbb{C}^2$ still to be determined, we have that

$$ \Pi_r^s(z)V_2(z) - \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_\ell^s(z)V_0(z) = \underbrace{(E_r^s(z) - \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)E_\ell^s(z))}_{:= \mathbb{B}(z) \in \mathcal{M}_2(\mathbb{C})} \begin{pmatrix} \chi_2(z) \\ \chi_0(z) \end{pmatrix}. \quad (3.17) $$

We thus obtain the last two relations:

$$ \chi_2(z) = e_1^t \mathbb{B}(z)^{-1} (\mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_\ell^u(z)V_0(z) - \Pi_r^u(z)V_2(z) + \mathbb{M}_{2,1}(z)A_{r,m}H_0 + A_rH_1), $$

$$ \chi_0(z) = e_2^t \mathbb{B}(z)^{-1} (\mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_\ell^u(z)V_0(z) - \Pi_r^u(z)V_2(z) + \mathbb{M}_{2,1}(z)A_{r,m}H_0 + A_rH_1), $$

with $(e_1, e_2)$ the canonical basis of $\mathbb{C}^2$, at least as long as the matrix $\mathbb{B}(z) \in \mathcal{M}_2(\mathbb{C})$ in (3.17) is invertible. The invertibility property of the matrix $\mathbb{B}(z)$ is summarized in the following result which is a mere consequence of the analysis in the previous Section and of Assumption 1.

**Lemma 3.3.** Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. Then, for any $z \in \mathcal{W} \setminus \{1\}$, the matrix $\mathbb{B}(z)$ defined in (3.17) is invertible and its determinant $\Delta(z)$ satisfies:

$$ \forall z \in \overline{\mathcal{W}} \setminus \{1\}, \quad \Delta(z) = -\frac{\alpha_\ell \kappa_\ell(z)}{\alpha_r(1-\alpha_r)(1-\alpha_m)} \Delta(z), $$
with $\Delta(z)$ the Lopatinskii determinant given in (3.3). Moreover, the matrix $\mathbb{B}(z)$ is given for $z \in \overline{\mathcal{U}} \setminus \{1\}$ by:

$$ \mathbb{B}(z) = \begin{pmatrix} 1 & b_1(z) \\ \kappa_r(z) & b_2(z) \end{pmatrix}, $$

$$ \text{with} \quad b_1(z) = - \frac{\alpha_\ell \kappa_\ell(z) (\alpha_\ell - \alpha_m + (1 - \alpha_\ell) \kappa_\ell(z))}{\alpha_r (1 - \alpha_m)}, $$

$$ b_2(z) = - \frac{\alpha_\ell \kappa_\ell(z) (\alpha_r (1 - \alpha_m^2) - (2(z-1) + \alpha_r(\alpha_r + \alpha_m))(\alpha_\ell - \alpha_m + (1 - \alpha_\ell) \kappa_\ell(z)))}{\alpha_r^2 (1 - \alpha_r) (1 - \alpha_m)}. $$

We omit the proof of Lemma 3.3, which is a mere algebra exercise that uses the definition of the matrices $M_{2,1}(z)$, $M_{1,0}(z)$ and the dispersion relations (3.1). The above methodology shows how to construct a solution $\mathbf{v}(z)$ to the resolvent equation for any source term $\mathbf{h}$ and $z \in \overline{\mathcal{U}} \setminus \{1\}$. From Corollary 3.2, we know that the solution to the resolvent equation is necessarily unique in $\ell^q(\mathbb{Z}; \mathbb{C})$ for any $z \in \overline{\mathcal{U}} \setminus \{1\}$ since the eigenvalue problem does not have any nontrivial solution$^8$. We are now going to specify the above calculations to the case $\mathbf{h} = \delta_{j_0}$ (the discrete Dirac mass located at $j_0 \in \mathbb{Z}$) and therefore derive the expression of the spatial Green's function $G^{j_0}(z)$. We split the calculations according to the location of $j_0$ with respect to the discontinuity in the discrete shock $\bar{\mathbf{u}}$.

Let us recall that we denote by $G^{j_0}(z) = (G_j^{j_0}(z))_{j \in \mathbb{Z}} \in \ell^q(\mathbb{Z}, \mathbb{C})$ the solution to the resolvent equation:

$$ (z \operatorname{Id} - \mathcal{L}) G^{j_0}(z) = \delta_{j_0}. $$

This solution exists and is unique for any $z \in \overline{\mathcal{U}} \setminus \{1\}$. We also introduce the corresponding augmented vectors

$$ \forall j \in \mathbb{Z}, \quad G_j^{j_0}(z) := \begin{pmatrix} G_{j-1}^{j_0}(z) \\ G_j^{j_0}(z) \end{pmatrix} \in \mathbb{C}^2, \quad \text{and} \quad H_j^{j_0} := \begin{pmatrix} 0 \\ \delta_{j_0}(j) \end{pmatrix} \in \mathbb{C}^2. $$

**Case I: $j_0 \ge 2$.** We first recall that for $j \ge 2$ we have

$$ \Pi_r^u(z) G_j^{j_0}(z) = - \sum_{p=0}^{+\infty} \kappa_r^u(z)^{-1-p} \Pi_r^u(z) A_r H_{j+p}^{j_0}, $$

which yields two cases:

* if $2 \le j \le j_0$ then

$$ \Pi_r^u(z) G_j^{j_0}(z) = -\kappa_r^u(z)^{-1+j-j_0} \Pi_r^u(z) A_r e_2; $$

* if $j > j_0$ then

$$ \Pi_r^u(z) G_j^{j_0}(z) = 0. $$

Next, for $j \le 0$, we readily get that

$$ \Pi_{\ell}^{u}(z) G_{j}^{j_{0}}(z) = \sum_{p=0}^{+\infty} \kappa_{\ell}^{u}(z)^{p} \Pi_{\ell}^{u}(z) A_{\ell} H_{j-p-1}^{j_{0}} = 0. $$

<footnote>$^8$Let us recall that the existence of a nontrivial solution to the eigenvalue problem $\mathcal{L} \mathbf{v} = z \mathbf{v}$ is equivalent to $\Delta(z) = 0$, which is precluded for $z \in \overline{\mathcal{U}} \setminus \{1\}$ by Assumption 1.</footnote>
Using the above results, we deduce that

$$
\begin{align*}
\mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_{\ell}^{u}(z)G_{0}^{j0}(z) - \Pi_{r}^{u}(z)G_{2}^{j0}(z) + \mathbb{M}_{2,1}(z)\mathbb{A}_{r,m}H_{0}^{j0} + \mathbb{A}_{r}H_{1}^{j0} &= -\Pi_{r}^{u}(z)G_{2}^{j0}(z) \\
&= \kappa_{r}^{u}(z)^{1-j0} \Pi_{r}^{u}(z)\mathbb{A}_{r}e_{2},
\end{align*}
$$

which implies that (the matrix $\mathbb{B}(z)$ is invertible for $z \in \overline{\mathcal{W}} \setminus \{1\}$ by Lemma 3.3):

$$
\Pi_r^s(z) G_2^{j0}(z) = \kappa_r^u(z)^{1-j0} [e_1^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2] E_r^s(z),
$$

$$
\Pi_{\ell}^{s}(z) G_{0}^{j0}(z) = \kappa_{r}^{u}(z)^{1-j0} \left[ e_{2}^{t} \mathbb{B}(z)^{-1} \Pi_{r}^{u}(z) \mathbb{A}_{r} e_{2} \right] E_{\ell}^{s}(z).
$$

Now, for $j \ge 2$, we observe that

$$
\Pi_r^s(z) G_j^{j0}(z) = \kappa_r(z)^{j-2} \Pi_r^s(z) G_2^{j0}(z) + \sum_{p=2}^{j-1} \kappa_r(z)^{j-p-1} \Pi_r^s(z) \mathbb{A}_r H_p^{j0},
$$

which yields two cases:

• if $2 \le j \le j_0$ then

$$
\Pi_r^s(z) G_j^{j0}(z) = \kappa_r(z)^{j-2} \kappa_r^u(z)^{1-j0} [e_1^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2] E_r^s(z);
$$

• if $j > j_0$ then

$$
\Pi_r^s(z) G_j^{j0}(z) = \kappa_r(z)^{j-2} \kappa_r^u(z)^{1-j0} \left[ e_1^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2 \right] E_r^s(z) + \kappa_r(z)^{j-j_0-1} \Pi_r^s(z) \mathbb{A}_r e_2.
$$

Finally, for $j \le 0$, we have

$$
\begin{align*}
\Pi_{\ell}^{s}(z) G_{j}^{j0}(z) &= \kappa_{\ell}(z)^{j} \Pi_{\ell}^{s}(z) G_{0}^{j0}(z) - \sum_{p=j}^{-1} \kappa_{\ell}(z)^{p} \Pi_{\ell}^{s}(z) \mathbb{A}_{\ell} H_{j-p-1}^{j0} \\
&= \kappa_{\ell}(z)^{j} \Pi_{\ell}^{s}(z) G_{0}^{j0}(z) = \kappa_{\ell}(z)^{j} \kappa_{r}^{u}(z)^{1-j0} \left[ e_{2}^{t} \mathbb{B}(z)^{-1} \Pi_{r}^{u}(z) \mathbb{A}_{r} e_{2} \right] E_{\ell}^{s}(z).
\end{align*}
$$

As a consequence, summarizing the above results, we have obtained for $j_0 \ge 2$ that

$$
G_j^{j0}(z) = \begin{cases}
\kappa_r(z)^{j-2} \kappa_r^u(z)^{1-j0} [e_1^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2] E_r^s(z) + \kappa_r(z)^{j-j_0-1} \Pi_r^s(z) \mathbb{A}_r E_2, & j > j_0, \\
\kappa_r(z)^{j-2} \kappa_r^u(z)^{1-j0} [e_1^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2] E_r^s(z) - \kappa_r^u(z)^{j-j_0-1} \Pi_r^u(z) \mathbb{A}_r E_2, & 2 \le j \le j_0, \\
\kappa_r^u(z)^{1-j0} [e_2^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2] \mathbb{M}_{1,0}(z) E_\ell^s(z), & j = 1, \\
\kappa_\ell(z)^j \kappa_r^u(z)^{1-j0} [e_2^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2] E_\ell^s(z), & j \le 0.
\end{cases}
$$

To recover the expression for the spatial Green's function $G_j^{j0}(z)$, one just notes that

$$
G_j^{j0}(z) = e_2^t G_j^{j0}(z),
$$

meaning that $G_j^{j0}(z) \in \mathbb{C}$ is the second coordinate in the vector $G_j^{j0}(z) \in \mathbb{C}^2$. Further computations give

$$
e_2^t \Pi_r^s(z) \mathbb{A}_r e_2 = \frac{2 \kappa_r(z)}{\alpha_r (1 - \alpha_r) (\kappa_r(z) - \kappa_r^u(z))}, \quad e_2^t \Pi_r^u(z) \mathbb{A}_r e_2 = \frac{2 \kappa_r^u(z)}{\alpha_r (1 - \alpha_r) (\kappa_r^u(z) - \kappa_r(z))},
$$
and

$$e_1^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2 = \frac{2(b_2(z) - b_1(z)\kappa_r^u(z))}{\alpha_r(1 - \alpha_r)\Delta(z)(\kappa_r^u(z) - \kappa_r(z))}, \quad e_2^t \mathbb{B}(z)^{-1} \Pi_r^u(z) \mathbb{A}_r e_2 = \frac{2}{\alpha_r(1 - \alpha_r)\Delta(z)}.$$

This yields for $j_0 \ge 2$ that

$$G_j^{j_0}(z) = \left\{ \begin{array}{ll} \kappa_r(z)^{j-1} \kappa_r^u(z)^{1-j_0} \frac{2(b_2(z) - b_1(z)\kappa_r^u(z))}{\alpha_r(1-\alpha_r)\Delta(z)(\kappa_r^u(z) - \kappa_r(z))} & j \ge j_0, \\ +\kappa_r(z)^{j-j_0} \frac{2(b_2(z) - b_1(z)\kappa_r^u(z))}{\alpha_r(1-\alpha_r)(\kappa_r(z) - \kappa_r^u(z))}, & j \ge j_0, \\ \kappa_r(z)^{j-1} \kappa_r^u(z)^{1-j_0} \frac{2(b_2(z) - b_1(z)\kappa_r^u(z))}{\alpha_r(1-\alpha_r)\Delta(z)(\kappa_r^u(z) - \kappa_r(z))} & 2 \le j \le j_0, \\ -\kappa_r^u(z)^{j-j_0} \frac{2(b_2(z) - b_1(z)\kappa_r^u(z))}{\alpha_r(1-\alpha_r)(\kappa_r(z) - \kappa_r^u(z))}, & j = 1, \\ \kappa_r^u(z)^{1-j_0} \frac{2(\alpha_\ell(1+\alpha_\ell) + \kappa_\ell(z)(2-\alpha_\ell(\alpha_\ell+\alpha_m)-2z))}{\alpha_r^2(1-\alpha_r)(1-\alpha_m)\Delta(z)}, & j = 1, \\ \kappa_\ell(z)^{1+j} \kappa_r^u(z)^{1-j_0} \frac{2(\alpha_\ell(1+\alpha_\ell) + \kappa_\ell(z)(2-\alpha_\ell(\alpha_\ell+\alpha_m)-2z))}{\alpha_r(1-\alpha_r)\Delta(z)}, & j \le 0. \end{array} \right.$$

Next, we remark two points (by using Lemma 3.3):

$$\frac{2(\alpha_{\ell}(1 + \alpha_{\ell}) + \kappa_{\ell}(z)(2 - \alpha_{\ell}(\alpha_{\ell} + \alpha_{m}) - 2z))}{\alpha_{r}^{2}(1 - \alpha_{r})(1 - \alpha_{m})\Delta(z)} = -\frac{2b_{1}(z)}{\alpha_{r}(1 - \alpha_{r})\Delta(z)},$$

and

$$\frac{2(b_2(z) - b_1(z)\kappa_r^u(z))}{\alpha_r(1 - \alpha_r)\Delta(z)(\kappa_r^u(z) - \kappa_r(z))} = -\frac{2b_1(z)}{\alpha_r(1 - \alpha_r)\Delta(z)} + \frac{2}{\alpha_r(1 - \alpha_r)(\kappa_r^u(z) - \kappa_r(z))},$$

so that the above expressions for $G_j^{j_0}(z)$ simplify to

$$G_j^{j_0}(z) = \left\{ \begin{array}{ll} -\kappa_r(z)^{j-1} \kappa_r^u(z)^{1-j_0} \frac{2b_1(z)}{\alpha_r(1-\alpha_r)\Delta(z)} + \frac{2(\kappa_r(z)^{j-1} \kappa_r^u(z)^{1-j_0} - \kappa_r(z)^{j-j_0})}{\alpha_r(1-\alpha_r)(\kappa_r^u(z) - \kappa_r(z))}, & j \ge j_0, \\ -\kappa_r(z)^{j-1} \kappa_r^u(z)^{1-j_0} \frac{2b_1(z)}{\alpha_r(1-\alpha_r)\Delta(z)} + \frac{2(\kappa_r(z)^{j-1} \kappa_r^u(z)^{1-j_0} - \kappa_r(z)^{j-j_0})}{\alpha_r(1-\alpha_r)(\kappa_r^u(z) - \kappa_r(z))}, & 1 \le j \le j_0, \\ \kappa_\ell(z)^{1+j} \kappa_r^u(z)^{1-j_0} \frac{2b_1(z)}{\alpha_r(1-\alpha_r)\Delta(z)}, & j \le 0. \end{array} \right.$$

To conclude the analysis of this first case ($j_0 \ge 2$), we recall the expression of the coefficient $b_1(z)$ and the link between the determinant $\Delta(z)$ of $\mathbb{B}(z)$ and the Lopatinskii determinant $\Delta(z)$ (see Lemma 3.3). This gives, for any $j_0 \ge 2$, the expression (3.12) for $G_j^{j_0}(z)$ as given in Proposition 3.1.

**Case II:** $j_0 = 1$. We directly notice that for $j \ge 2$ we have

$$\Pi_r^u(z)G_j^1(z) = -\sum_{p=0}^{+\infty} \kappa_r^u(z)^{-1-p} \Pi_r^u(z) \mathbb{A}_r H_{j+p}^1 = 0,$$

together with

$$\forall j \le 0, \quad \Pi_{\ell}^{u}(z)G_{j}^{1}(z) = \sum_{p=0}^{+\infty} \kappa_{\ell}^{u}(z)^{p} \Pi_{\ell}^{u}(z) \mathbb{A}_{\ell}H_{j-p-1}^{1} = 0.$$
Using the above two results, we deduce that

$$ \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_{\ell}^{u}(z)G_{0}^{1}(z) - \Pi_{r}^{u}(z)G_{2}^{1}(z) + \mathbb{M}_{2,1}(z)\mathbb{A}_{r,m}H_{0}^{1} + \mathbb{A}_{r}H_{1}^{1} = \mathbb{A}_{r}e_{2}, $$

which implies that

$$ \Pi_{r}^{s}(z)G_{2}^{1}(z) = [e_{1}^{t}\mathbb{B}(z)^{-1}\mathbb{A}_{r}e_{2}] E_{r}^{s}(z), $$

$$ \Pi_{\ell}^{s}(z)G_{0}^{1}(z) = [e_{2}^{t}\mathbb{B}(z)^{-1}\mathbb{A}_{r}e_{2}] E_{\ell}^{s}(z). $$

Now, for $j \ge 2$, we get that

$$ \begin{aligned} \Pi_{r}^{s}(z)G_{j}^{1}(z) &= \kappa_{r}(z)^{j-2}\Pi_{r}^{s}(z)G_{2}^{1}(z) + \sum_{p=2}^{j-1} \kappa_{r}(z)^{j-p-1}\Pi_{r}^{s}(z)\mathbb{A}_{r}H_{p}^{1} = \kappa_{r}(z)^{j-2}\Pi_{r}^{s}(z)G_{2}^{1}(z) \\ &= \kappa_{r}(z)^{j-2} [e_{1}^{t}\mathbb{B}(z)^{-1}\mathbb{A}_{r}e_{2}] E_{r}^{s}(z). \end{aligned} $$

Finally, for $j \le 0$, we have

$$ \begin{aligned} \Pi_{\ell}^{s}(z)G_{j}^{1}(z) &= \kappa_{\ell}(z)^{j}\Pi_{\ell}^{s}(z)G_{0}^{1}(z) - \sum_{p=j}^{-1} \kappa_{\ell}(z)^{p}\Pi_{\ell}^{s}(z)\mathbb{A}_{\ell}H_{j-p-1}^{1} = \kappa_{\ell}(z)^{j}\Pi_{\ell}^{s}(z)G_{0}^{1}(z) \\ &= \kappa_{\ell}(z)^{j} [e_{2}^{t}\mathbb{B}(z)^{-1}\mathbb{A}_{\ell}e_{2}] E_{\ell}^{s}(z). \end{aligned} $$

As a consequence, when $j_0 = 1$, the spatial Green's function reads (in vector form):

$$ G_{j}^{1}(z) = \begin{cases} \kappa_{r}(z)^{j-2} [e_{1}^{t}\mathbb{B}(z)^{-1}\mathbb{A}_{r}e_{2}] E_{r}^{s}(z), & j \ge 2, \\ \kappa_{\ell}(z)^{j} [e_{2}^{t}\mathbb{B}(z)^{-1}\mathbb{A}_{r}e_{2}] E_{\ell}^{s}(z), & j \le 0. \end{cases} $$

Retaining only the second coordinate in each vector (or the first for $j = 2$, which gives the expression of $G_1^1(z)$), we obtain the expressions:

$$ G_{j}^{1}(z) = \begin{cases} -\kappa_{r}(z)^{j-1} \frac{2b_{1}(z)}{\alpha_{r}(1-\alpha_{r})\Delta(z)}, & j \ge 1, \\ \kappa_{\ell}(z)^{j+1} \frac{2b_{2}(z)}{\alpha_{r}(1-\alpha_{r})\Delta(z)}, & j \le 0. \end{cases} $$

The expression (3.12) (with $j_0 = 1$) for $G_j^1(z)$ as given in Proposition 3.1 follows from the expression of $b_1(z)$ and of the determinant $\Delta(z)$, see Lemma 3.3.

**Case III:** $j_0 = 0$. We now feel free to shorten some details of the computations since many steps in Case III are similar to those in Case II ($j_0 = 1$). Since the Dirac mass is located at $j_0 = 0$, we have $\Pi_r^u(z)G_j^0(z) = 0$ for $j \ge 2$ and $\Pi_\ell^u(z)G_j^0(z) = 0$ for $j \le 0$. Using these two facts, we deduce that

$$ \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_{\ell}^{u}(z)G_{0}^{0}(z) - \Pi_{r}^{u}(z)G_{2}^{0}(z) + \mathbb{M}_{2,1}(z)\mathbb{A}_{r,m}H_{0}^{0} + \mathbb{A}_{r}H_{1}^{0} = \mathbb{M}_{2,1}(z)\mathbb{A}_{r,m}e_{2}, $$

which implies that

$$ \Pi_{r}^{s}(z)G_{2}^{0}(z) = [e_{1}^{t}\mathbb{B}(z)^{-1}\mathbb{M}_{2,1}(z)\mathbb{A}_{r,m}e_{2}] E_{r}^{s}(z), $$

$$ \Pi_{\ell}^{s}(z)G_{0}^{0}(z) = [e_{2}^{t}\mathbb{B}(z)^{-1}\mathbb{M}_{2,1}(z)\mathbb{A}_{r,m}e_{2}] E_{\ell}^{s}(z). $$
Now, for $j \ge 2$, we get that

$$ \Pi_r^s(z) G_j^0(z) = \kappa_r(z)^{j-2} [e_1^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{A}_{r,m} e_2] E_r^s(z). $$

and for $j \le 0$, we have

$$ \Pi_{\ell}^{s}(z) G_{j}^{0}(z) = \kappa_{\ell}(z)^{j} [e_{2}^{t} \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{A}_{r,m} e_{2}] E_{\ell}^{s}(z). $$

At this stage, the spatial Green's function (in vector form) for $j_0 = 0$ reads:

$$ G_j^0(z) = \begin{cases} \kappa_r(z)^{j-2} [e_1^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{A}_{r,m} e_2] E_r^s(z), & j \ge 2, \\ \kappa_\ell(z)^j [e_2^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{A}_{r,m} e_2] E_\ell^s(z), & j \le 0. \end{cases} $$

We compute the expressions:

$$ e_1^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{A}_{r,m} e_2 = \frac{2}{\alpha_r(1-\alpha_m)\Delta(z)} \left( b_2(z) - b_1(z) \frac{2(1-z) - \alpha_r(\alpha_r + \alpha_m)}{\alpha_r(1-\alpha_r)} \right), $$

$$ e_2^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{A}_{r,m} e_2 = \frac{2}{\alpha_r(1-\alpha_m)\Delta(z)} \left( -\kappa_r(z) + \frac{2(1-z) - \alpha_r(\alpha_r + \alpha_m)}{\alpha_r(1-\alpha_r)} \right). $$

Looking at either the first or second coordinate of the vector $G_j^0(z)$ for $j \ge 2$, we obtain the expression of $G_j^0(z)$ for any $j \ge 1$. Looking then at the second coordinate of $G_j^0(z)$ for $j \le 0$, we obtain the expression of $G_j^0(z)$ for any $j \le 0$. In the end, we obtain that for $j_0 = 0$, the Green's function reads:

$$ G_j^0(z) = \begin{cases} \kappa_r(z)^{j-1} \frac{2}{\alpha_r(1-\alpha_m)\Delta(z)} \left( b_2(z) - b_1(z) \frac{2(1-z) - \alpha_r(\alpha_r + \alpha_m)}{\alpha_r(1-\alpha_r)} \right), & j \ge 1, \\ \kappa_\ell(z)^{j+1} \frac{2}{\alpha_r(1-\alpha_m)\Delta(z)} \left( -\kappa_r(z) + \frac{2(1-z) - \alpha_r(\alpha_r + \alpha_m)}{\alpha_r(1-\alpha_r)} \right), & j \le 0. \end{cases} $$

The expression of $G_j^0(z)$ for $j \ge 1$ is then simplified one last time by using the expressions of $b_1(z)$ and $b_2(z)$ in Lemma 3.3 and by using the relation between $\Delta(z)$ and $\Delta(z)$, while the expression of $G_j^0(z)$ for $j \le 0$ is simplified by using the dispersion relation (3.1b) as well as the relation between $\Delta(z)$ and $\Delta(z)$. We are then led to the expression (3.13) (with $j_0 = 0$) of $G_j^0(z)$ given in Proposition 3.1.

**Case IV:** $j_0 \le -1$. We directly notice that we have $\Pi_r^u(z)G_j^{j_0}(z) = 0$ for $j \ge 2$, while for $j \le 0$ the expression

$$ \Pi_{\ell}^{u}(z) G_{j}^{j_{0}}(z) = \sum_{p=0}^{+\infty} \kappa_{\ell}^{u}(z)^{p} \Pi_{\ell}^{u}(z) \mathbb{A}_{\ell} H_{j-p-1}^{j_{0}}, $$

gives two cases:

* if $j_0 < j \le 0$ then

$$ \Pi_{\ell}^{u}(z) G_{j}^{j_{0}}(z) = \kappa_{\ell}^{u}(z)^{j-j_{0}-1} \Pi_{\ell}^{u}(z) \mathbb{A}_{\ell} e_{2}; $$

* if $j \le j_0$ then

$$ \Pi_{\ell}^{u}(z) G_{j}^{j_{0}}(z) = 0. $$
Now, using the above results, we deduce that

$$
\begin{align*}
& \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_{\ell}^{u}(z)G_{0}^{j0}(z) - \Pi_{r}^{u}(z)G_{2}^{j0}(z) + \mathbb{M}_{2,1}(z)\mathbb{A}_{r,m}H_{0}^{j0} + \mathbb{A}_{r}H_{1}^{j0} \\
& = \mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_{\ell}^{u}(z)G_{0}^{j0}(z) = \kappa_{\ell}^{u}(z)^{-j0-1}\mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z)\Pi_{\ell}^{u}(z)\mathbb{A}_{\ell}e_{2},
\end{align*}
$$

which implies that

$$
\Pi_r^s(z) G_2^{j0}(z) = \kappa_\ell^u(z)^{-j_0-1} [e_1^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_\ell^u(z) \mathbb{A}_\ell e_2] E_r^s(z),
$$

$$
\Pi_{\ell}^{s}(z) G_{0}^{j0}(z) = \kappa_{\ell}^{u}(z)^{-j_{0}-1} \left[ e_{2}^{t} \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_{\ell}^{u}(z) \mathbb{A}_{\ell} e_{2} \right] E_{\ell}^{s}(z).
$$

Now, for $j \ge 2$, we observe that

$$
\begin{align*}
\Pi_r^s(z) G_j^{j0}(z) &= \kappa_r(z)^{j-2} \Pi_r^s(z) G_2^{j0}(z) \\
&= \kappa_r(z)^{j-2} \kappa_\ell^u(z)^{-j_0-1} \left[ e_1^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_\ell^u(z) \mathbb{A}_\ell e_2 \right] E_r^s(z).
\end{align*}
$$

Finally, for $j \le 0$, we have

$$
\Pi_{\ell}^{s}(z) G_{j}^{j0}(z) = \kappa_{\ell}(z)^{j} \Pi_{\ell}^{s}(z) G_{0}^{j0}(z) - \sum_{p=j}^{-1} \kappa_{\ell}(z)^{p} \Pi_{\ell}^{s}(z) \mathbb{A}_{\ell} H_{j-p-1}^{j0},
$$

which yields two cases:

• if $j \le j_0$ then

$$
\Pi_{\ell}^{s}(z) G_{j}^{j0}(z) = \kappa_{\ell}(z)^{j} \kappa_{\ell}^{u}(z)^{-j_{0}-1} \left[ e_{2}^{t} \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_{\ell}^{u}(z) \mathbb{A}_{\ell} e_{2} \right] E_{\ell}^{s}(z) \\
- \kappa_{\ell}(z)^{j-j_{0}-1} \Pi_{\ell}^{s}(z) \mathbb{A}_{\ell} e_{2};
$$

• if $j_0 < j \le -1$ then

$$
\Pi_{\ell}^{s}(z) G_{j}^{j0}(z) = \kappa_{\ell}(z)^{j} \kappa_{\ell}^{u}(z)^{-j_{0}-1} \left[ e_{2}^{t} \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_{\ell}^{u}(z) \mathbb{A}_{\ell} e_{2} \right] E_{\ell}^{s}(z).
$$

As a consequence, summarizing the above results, we have obtained for $j_0 \le -1$ that

$$
G_j^{j0}(z) = \begin{cases}
\kappa_r(z)^{j-2} \kappa_\ell^u(z)^{-j_0-1} [e_1^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_\ell^u(z) \mathbb{A}_\ell e_2] E_r^s(z), & j \ge 2, \\
\kappa_\ell(z)^j \kappa_\ell^u(z)^{-j_0-1} [e_2^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_\ell^u(z) \mathbb{A}_\ell e_2] E_\ell^s(z) \\
\qquad + \kappa_\ell^u(z)^{j-j_0-1} \Pi_\ell^u(z) \mathbb{A}_\ell e_2, & j_0 < j \le 0, \\
\kappa_\ell(z)^j \kappa_\ell^u(z)^{-j_0-1} [e_2^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_\ell^u(z) \mathbb{A}_\ell e_2] E_\ell^s(z) \\
\qquad - \kappa_\ell(z)^{j-j_0-1} \Pi_\ell^s(z) \mathbb{A}_\ell e_2, & j \le j_0.
\end{cases} \quad (3.18)
$$

For later use, we introduce the notation:

$$
\mathbb{M}_{2,1}(z)\mathbb{M}_{1,0}(z) = \begin{pmatrix} m_1(z) & m_2(z) \\ m_3(z) & m_4(z) \end{pmatrix},
$$

We then compute:

$$
e_2^t \Pi_\ell^s(z) \mathbb{A}_\ell e_2 = \frac{2 \kappa_\ell(z)}{\alpha_\ell(1 - \alpha_\ell)(\kappa_\ell(z) - \kappa_\ell^u(z))}, \quad e_2^t \Pi_\ell^u(z) \mathbb{A}_\ell e_2 = \frac{2 \kappa_\ell^u(z)}{\alpha_\ell(1 - \alpha_\ell)(\kappa_\ell^u(z) - \kappa_\ell(z))},
$$
as well as

$$e_1^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_\ell^u(z) \mathbb{A}_\ell e_2 = \frac{2 \left( b_2(z) (m_1(z) + \kappa_\ell^u(z) m_2(z)) - b_1(z) (m_3(z) + \kappa_\ell^u(z) m_4(z)) \right)}{\alpha_\ell (1 - \alpha_\ell) \Delta(z) (\kappa_\ell^u(z) - \kappa_\ell(z))},$$

$$e_2^t \mathbb{B}(z)^{-1} \mathbb{M}_{2,1}(z) \mathbb{M}_{1,0}(z) \Pi_\ell^u(z) \mathbb{A}_\ell e_2 = \frac{2 \left( -\kappa_r(z) (m_1(z) + \kappa_\ell^u(z) m_2(z)) + m_3(z) + \kappa_\ell^u(z) m_4(z) \right)}{\alpha_\ell (1 - \alpha_\ell) \Delta(z) (\kappa_\ell^u(z) - \kappa_\ell(z))}.$$

Looking at either the first or second coordinate of the vector $G_j^{j0}(z)$ for $j \ge 2$, we obtain the expression of $G_j^{j0}(z)$ for any index $j \ge 1$:

$$G_j^{j0}(z) = \kappa_r(z)^{j-1} \kappa_\ell^u(z)^{-j_0-1} \frac{2(b_2(z)(m_1(z) + \kappa_\ell^u(z)m_2(z)) - b_1(z)(m_3(z) + \kappa_\ell^u(z)m_4(z)))}{\alpha_\ell(1-\alpha_\ell)\Delta(z)(\kappa_\ell^u(z) - \kappa_\ell(z))}.$$

Because of the definition of the matrix $\mathbb{B}(z)$, we have the relations:

$$m_1(z) + \kappa_\ell^u(z) m_2(z) = -b_1(z) + (\kappa_\ell^u(z) - \kappa_\ell(z)) m_2(z),$$

$$m_3(z) + \kappa_\ell^u(z) m_4(z) = -b_2(z) + (\kappa_\ell^u(z) - \kappa_\ell(z)) m_4(z),$$

from which we deduce:

$$\forall j \ge 1, \quad G_j^{j0}(z) = \kappa_r(z)^{j-1} \kappa_\ell^u(z)^{-j_0-1} \frac{2(b_2(z)m_2(z) - b_1(z)m_4(z))}{\alpha_\ell(1-\alpha_\ell)\Delta(z)},$$

and we now use the relations:

$$b_1(z) = -m_1(z) - \kappa_\ell(z)m_2(z), \quad b_2(z) = -m_3(z) - \kappa_\ell(z)m_4(z), \qquad (3.19)$$

to further simplify the expression of $G_j^{j0}(z)$, $j \ge 1$, into:

$$\begin{aligned} G_j^{j0}(z) &= \kappa_r(z)^{j-1} \kappa_\ell^u(z)^{-j_0-1} \frac{2(m_1(z)m_4(z) - m_2(z)m_3(z))}{\alpha_\ell(1-\alpha_\ell)\Delta(z)} \\ &= \kappa_r(z)^{j-1} \kappa_\ell^u(z)^{-j_0-1} \frac{2 \det \mathbb{M}_{2,1}(z) \det \mathbb{M}_{1,0}(z)}{\alpha_\ell(1-\alpha_\ell)\Delta(z)} \\ &= \kappa_r(z)^{j-1} \kappa_\ell^u(z)^{-j_0-1} \frac{2 \alpha_\ell(1+\alpha_\ell)(1+\alpha_m)}{\alpha_r^2(1-\alpha_r)(1-\alpha_m)(1-\alpha_\ell)\Delta(z)}. \end{aligned}$$

To obtain the expression (3.13) of $G_j^{j0}(z)$ for $j_0 \le -1$ and $j \ge 1$, it only remains to use the relation $\kappa_\ell(z) \kappa_\ell^u(z) = -(1+\alpha_\ell)/(1-\alpha_\ell)$ (see (3.1a)), and the link between the determinant $\Delta(z)$ of $\mathbb{B}(z)$ and $\Delta(z)$ (Lemma 3.3).

The only remaining task is to derive the expression (3.13) of $G_j^{j0}(z)$ for $j_0 \le -1$ and $j \le 0$. We go back to (3.18) and retain the second coordinate of the vector $G_j^{j0}(z)$ for $j \le 0$. We obtain:

$$G_j^{j0}(z) = \begin{cases} \kappa_\ell(z)^{j+1} \kappa_\ell^u(z)^{-j_0-1} \frac{2(-\kappa_r(z)(m_1(z) + \kappa_\ell^u(z)m_2(z)) + m_3(z) + \kappa_\ell^u(z)m_4(z))}{\alpha_\ell(1-\alpha_\ell)\Delta(z)(\kappa_\ell^u(z) - \kappa_\ell(z))} & j_0 < j \le 0, \\ + \frac{2 \kappa_\ell^u(z)^{j-j_0}}{\alpha_\ell(1-\alpha_\ell)(\kappa_\ell^u(z) - \kappa_\ell(z))}, & j_0 < j \le 0, \\ \kappa_\ell(z)^{j+1} \kappa_\ell^u(z)^{-j_0-1} \frac{2(-\kappa_r(z)(m_1(z) + \kappa_\ell^u(z)m_2(z)) + m_3(z) + \kappa_\ell^u(z)m_4(z))}{\alpha_\ell(1-\alpha_\ell)\Delta(z)(\kappa_\ell^u(z) - \kappa_\ell(z))} & j_0 < j \le 0, \\ + \frac{2 \kappa_\ell(z)^{j-j_0}}{\alpha_\ell(1-\alpha_\ell)(\kappa_\ell^u(z) - \kappa_\ell(z))}, & j \le j_0. \end{cases}$$
Recalling the relations (3.19), we get:

$$-\kappa_r(z)(m_1(z) + \kappa_{\ell}^u(z)m_2(z)) + m_3(z) + \kappa_{\ell}^u(z)m_4(z) = (\kappa_{\ell}^u(z) - \kappa_{\ell}(z))(m_4(z) - \kappa_r(z)m_2(z)) - \Delta(z),$$

and this simplifies the expression of $\mathcal{G}_j^{j_0}(z)$ into:

$$\mathcal{G}_j^{j_0}(z) = \begin{cases} \kappa_{\ell}(z)^{j+1} \kappa_{\ell}^{u}(z)^{-j_0-1} \frac{2(m_4(z) - \kappa_r(z)m_2(z))}{\alpha_{\ell}(1-\alpha_{\ell})\Delta(z)} & j_0 < j \le 0, \\ + \frac{2(\kappa_{\ell}^{u}(z)^{j-j_0} - \kappa_{\ell}(z)^{j+1} \kappa_{\ell}^{u}(z)^{-j_0-1})}{\alpha_{\ell}(1-\alpha_{\ell})(\kappa_{\ell}^{u}(z) - \kappa_{\ell}(z))}, & j_0 < j \le 0, \\ \kappa_{\ell}(z)^{j+1} \kappa_{\ell}^{u}(z)^{-j_0-1} \frac{2(m_4(z) - \kappa_r(z)m_2(z))}{\alpha_{\ell}(1-\alpha_{\ell})\Delta(z)} & j_0 < j \le 0, \\ + \frac{2(\kappa_{\ell}(z)^{j-j_0} - \kappa_{\ell}(z)^{j+1} \kappa_{\ell}^{u}(z)^{-j_0-1})}{\alpha_{\ell}(1-\alpha_{\ell})(\kappa_{\ell}^{u}(z) - \kappa_{\ell}(z))}, & j \le j_0. \end{cases}$$

It turns out that we can slightly modify the above expression as follows. Using the expressions:

$$\begin{aligned} \Delta(z) &= b_2(z) - \kappa_r(z)b_1(z) = \kappa_r(z)m_1(z) - m_3(z) + \kappa_{\ell}(z)(\kappa_r(z)m_2(z) - m_4(z)), \\ m_1(z) &= \frac{\alpha_{\ell}(1 + \alpha_{\ell})}{\alpha_r(1 - \alpha_m)}, \quad m_3(z) = -\frac{\alpha_{\ell}(1 + \alpha_{\ell})}{\alpha_r^2(1 - \alpha_r)(1 - \alpha_m)}(2(z - 1) + \alpha_r(\alpha_r + \alpha_m)), \end{aligned}$$

we notice that

$$\begin{aligned} \frac{1}{\kappa_{\ell}^{u}(z)} \frac{m_4(z) - \kappa_r(z)m_2(z)}{\alpha_{\ell}(1 - \alpha_{\ell})\Delta(z)} &= -\frac{1}{\alpha_{\ell}(1 + \alpha_{\ell})\Delta(z)} (-\Delta(z) + \kappa_r(z)m_1(z) - m_3(z)) \\ &= \frac{1}{\alpha_{\ell}(1 + \alpha_{\ell})} + \frac{1}{\alpha_r(1 - \alpha_m)\Delta(z)} \left( -\kappa_r(z) + \frac{2(1 - z) - \alpha_r(\alpha_r + \alpha_m)}{\alpha_r(1 - \alpha_r)} \right) \\ &= -\frac{1}{\alpha_{\ell}(1 - \alpha_{\ell})\kappa_{\ell}(z)\kappa_{\ell}^{u}(z)} + \frac{\alpha_r - \alpha_m - (1 + \alpha_r)\kappa_r(z)^{-1}}{\alpha_r(1 - \alpha_r)(1 - \alpha_m)\Delta(z)}. \end{aligned}$$

We end up with the following expression

$$\mathcal{G}_j^{j_0}(z) = \begin{cases} \frac{2 \kappa_{\ell}(z)^{j+1} \kappa_{\ell}^{u}(z)^{-j_0}}{\alpha_r(1 - \alpha_r)(1 - \alpha_m)\Delta(z)} (\alpha_r - \alpha_m - (1 + \alpha_r)\kappa_r(z)^{-1}) & j \le j_0, \\ + \frac{2(\kappa_{\ell}(z)^{j-j_0} - \kappa_{\ell}(z)^j \kappa_{\ell}^{u}(z)^{-j_0})}{\alpha_{\ell}(1 - \alpha_{\ell})(\kappa_{\ell}^{u}(z) - \kappa_{\ell}(z))}, & j \le j_0, \\ \frac{2 \kappa_{\ell}(z)^{j+1} \kappa_{\ell}^{u}(z)^{-j_0}}{\alpha_r(1 - \alpha_r)(1 - \alpha_m)\Delta(z)} (\alpha_r - \alpha_m - (1 + \alpha_r)\kappa_r(z)^{-1}) & j_0 \le j \le 0, \\ + \frac{2(\kappa_{\ell}^{u}(z)^{j-j_0} - \kappa_{\ell}(z)^j \kappa_{\ell}^{u}(z)^{-j_0})}{\alpha_{\ell}(1 - \alpha_{\ell})(\kappa_{\ell}^{u}(z) - \kappa_{\ell}(z))}, & j_0 \le j \le 0. \end{cases}$$

We then use the expression of $\Delta(z)$ given in Lemma 3.3 and derive the expression (3.13) of $\mathcal{G}_j^0(z)$ given in Proposition 3.1 (for $j_0 \le -1$ and $j \le 0$). The proof of Proposition 3.1 is therefore complete. $\square$

The expression of the spatial Green's function given in Proposition 3.1 gives us in a straightforward way the following estimates away from the point 1. From inspection of the expressions (3.12) and (3.13),
it is useful to introduce a tiny modification of the spatial Green's function that we define as follows. For any couple of integers $(j_0, j) \in \mathbb{Z}^2$, we define the following function:

$$ \tilde{G}_j^{j_0}(z) := \begin{cases} G_j^{j_0}(z) + \frac{2 \kappa_r^u(z)^{j-j_0}}{\alpha_r(1-\alpha_r)(\kappa_r^u(z)-\kappa_r(z))}, & \text{if } 1 \le j \le j_0, \\ G_j^{j_0}(z) + \frac{2 \kappa_r(z)^{j-j_0}}{\alpha_r(1-\alpha_r)(\kappa_r^u(z)-\kappa_r(z))}, & \text{if } 1 \le j_0 < j, \\ G_j^{j_0}(z) + \frac{2 \kappa_\ell^u(z)^{j-j_0}}{\alpha_\ell(1-\alpha_\ell)(\kappa_\ell^u(z)-\kappa_\ell^u(z))}, & \text{if } j_0 \le j \le 0, \\ G_j^{j_0}(z) + \frac{2 \kappa_\ell(z)^{j-j_0}}{\alpha_\ell(1-\alpha_\ell)(\kappa_\ell^u(z)-\kappa_\ell^u(z))}, & \text{if } j < j_0 \le 0, \\ G_j^{j_0}(z), & \text{otherwise.} \end{cases} \quad (3.20) $$

Under the assumptions made in Proposition 3.1, Proposition 3.1 and Lemma 3.1 show that $\tilde{G}_j^{j_0}$ is well-defined on $\mathcal{W} \setminus \{1\}$. Furthermore, this function is holomorphic on $\mathcal{W}$ and can be holomorphically extended in the neighborhood of any point of $\mathbb{S}^1 \setminus \{1\}$. The interest for defining this reduced function $\tilde{G}_j^{j_0}$ will be made clear in Chapter 4. Our result is the following.

**Corollary 3.3.** Let the weak solution (2.2) satisfy the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. Then for any $\varepsilon_\star > 0$, there exist constants $\eta_\star > 0$, $C > 0$ and $c > 0$ such that, if we define the set:

$$ \mathcal{L}_{\varepsilon_\star, \eta_\star} := \{ \zeta \in \mathbb{C} \mid e^{-\eta_\star} \le |\zeta| \le 2 \} \setminus \{ \zeta = e^\tau \in \mathbb{C} \mid \tau \in \mathbf{B}_{\varepsilon_\star}(0) \}, \quad (3.21) $$

then, for any couple $(j, j_0) \in \mathbb{Z}^2$, the function $\tilde{G}_j^{j_0}$ defined in (3.20) depends holomorphically on $z$ on $\mathcal{L}_{\varepsilon_\star, \eta_\star}$ and it satisfies the uniform bound:

$$ \forall z \in \mathcal{L}_{\varepsilon_\star, \eta_\star}, \quad \forall (j_0, j) \in \mathbb{Z}^2, \quad \left| \tilde{G}_j^{j_0}(z) \right| \le C \exp(-c(|j| + |j_0|)). $$

The region $\mathcal{L}_{\varepsilon_\star, \eta_\star}$ is schematically depicted in Figure 3.2.

**Proof of Corollary 3.3.** The proof of Corollary 3.3 directly follows from the expressions (3.12) and (3.13) and the definition (3.20). Indeed, let $\varepsilon_\star > 0$ be given. Then the set:

$$ \{ \zeta \in \mathbb{C} \mid 1 \le |\zeta| \le 2 \} \setminus \{ \zeta = e^\tau \in \mathbb{C} \mid \tau \in \mathbf{B}_{\varepsilon_\star}(0) \}, $$

is a compact subset of $\mathcal{O}$ and also of $\mathcal{W} \setminus \{1\}$. Moreover, thanks to Assumption 1, we know that the Lopatinskii determinant $\Delta$ does not vanish on that set. Lemma 3.1 also shows that the dispersion relation (3.1a), resp. (3.1b), has two distinct roots $\kappa_\ell$ and $\kappa_\ell^u$, resp. $\kappa_r$ and $\kappa_r^u$, for any $z$ in that set. By using Lemma 3.1 (for the holomorphy properties of the roots of (3.1a) and (3.1b)), Lemma 3.2 (for the holomorphy properties of $\Delta$) and Assumption 1, we can thus choose some $\eta_\star > 0$ such that:

* the set $\mathcal{L}_{\varepsilon_\star, \eta_\star}$ defined in (3.21) is a compact subset of $\mathcal{O}$ so that $\Delta$ is holomorphic on $\mathcal{L}_{\varepsilon_\star, \eta_\star}$,
<figure><img src="image_12.png" /><figcaption>Figure 3.2: The region $\mathcal{L}_{\varepsilon_*, \eta_*}$ (in red) of Corollary 3.3. In grey: the set $\{\zeta = e^\tau \in \mathbb{C} \mid \tau \in \mathbf{B}_{\varepsilon_*}(0)\}$. In blue: the unit circle $\mathbb{S}^1$.</figcaption></figure>

* $\kappa_{\ell}(z), \kappa_r^u(z)$ belong to $\mathcal{U}$, $\kappa_r(z), \kappa_{\ell}^u(z)$ belong to $\mathbb{D}$ for any $z \in \mathcal{L}_{\varepsilon_*, \eta_*}$ and those four functions depend holomorphically on $z$ on $\mathcal{L}_{\varepsilon_*, \eta_*}$,

* $\kappa_{\ell}(z)$ is different from $\kappa_{\ell}^u(z)$, resp. $\kappa_r(z)$ is different from $\kappa_r^u(z)$, for any $z \in \mathcal{L}_{\varepsilon_*, \eta_*}$,

* $\Delta$ is holomorphic and does not vanish on $\mathcal{L}_{\varepsilon_*, \eta_*}$.

All the above properties imply that for any couple $(j, j_0) \in \mathbb{Z}^2$, the function $\tilde{G}_j^{j_0}$ defined in (3.20) extends to a holomorphic function on $\mathcal{L}_{\varepsilon_*, \eta_*}$.

We now consider $j_0 \ge 1$ and look at the expression (3.12) for the spatial Green's function and the definition (3.20) for $\tilde{G}_j^{j_0}(z)$. Since $\mathcal{L}_{\varepsilon_*, \eta_*}$ is compact, we can find some constants $C$ such that for any $z \in \mathcal{L}_{\varepsilon_*, \eta_*}$, there holds:

$$|\tilde{G}_j^{j_0}(z)| \le \begin{cases} C |\kappa_r^u(z)|^{-j_0} |\kappa_\ell(z)|^j, & \text{if } j \le 0, \\ C |\kappa_r^u(z)|^{-j_0} |\kappa_r(z)|^j, & \text{if } j \ge 1. \end{cases}$$

It remains to use uniform lower or upper bounds:

$$|\kappa_r^u(z)| \ge e^{-c}, \quad |\kappa_\ell(z)| \ge e^{-c}, \quad |\kappa_r(z)| \le e^{-c},$$

with a uniform constant $c > 0$, and the conclusion of Corollary 3.3 follows in the case $j_0 \ge 1$. The case $j_0 \le 0$ follows from similar arguments by using the expression (3.13) for the spatial Green's function and the definition (3.20) for $\tilde{G}_j^{j_0}(z)$. $\square$
## 3.3 Spectral stability. Proof of Theorem 2.3

This short paragraph is devoted to the proof of Theorem 2.3 (which is a more general version of Theorem 2.2) using all above ingredients, that is our analysis of the Lopatinskii determinant and the construction of the spatial Green's function for $z \in \overline{\mathcal{W}} \setminus \{1\}$.

We shall only give the proof of Theorem 2.3 and leave the analogous analysis for either convex or concave fluxes to the interested reader. Let us first quickly show that 1 is an eigenvalue of $\mathcal{L}$, which is reminiscent of the fact that the Lopatinskii determinant vanishes at 1 (Lemma 3.2), see [15, 5, 14, 28, 27]. The fact that 1 is an eigenvalue for $\mathcal{L}$ was already proven in [28, Theorem 2.3]. We just reproduce a proof here, with our notation, for the sake of completeness.

We consider the expressions (3.12) and (3.13) of the spatial Green's function. Since $\Delta$ vanishes at 1, these expressions incorporate a (simple) pole at $z = 1$. We thus introduce the sequence $(\mathcal{H}_j)_{j \in \mathbb{Z}}$ that is defined by:

$$ \mathcal{H}_j := \lim_{z \to 1} (z-1) \mathcal{G}_j^{j_0}(z), $$

and whose precise expression is given by:

$$ \mathcal{H}_j = \begin{cases} -\frac{2(1-\alpha_m)}{\alpha_\ell \Delta'(1)} \kappa_\ell(1)^j, & \text{if } j \le 0, \\ \frac{2(1+\alpha_m)}{\alpha_r \Delta'(1)} \kappa_r(1)^{j-1}, & \text{if } j \ge 1, \end{cases} \quad (3.22) $$

the expression being independent of $j_0 \in \mathbb{Z}$. The sequence given in (3.22) is nonzero and it belongs to any $\ell^q(\mathbb{Z}; \mathbb{C})$ since it has exponential decay at infinity (recall that $\kappa_r(1)$ belongs to $\mathbb{D}$ and $\kappa_\ell(1)$ belongs to $\mathcal{U}$). It is also a mere algebra exercise to verify that the sequence given in (3.22) belongs to the kernel of the operator $\mathrm{Id} - \mathcal{L}$, as expected from the above formal analysis. This means that 1 is an eigenvalue for $\mathcal{L}$ in any $\ell^q(\mathbb{Z}; \mathbb{C})$.

Let us now show that the set $\overline{\mathcal{W}} \setminus \{1\}$ lies in the resolvent set for $\mathcal{L}$. We consider $\mathbf{h} \in \ell^q(\mathbb{Z}; \mathbb{C})$ and we wish to construct a solution in $\ell^q(\mathbb{Z}; \mathbb{C})$ to the resolvent equation:

$$ (z \mathrm{Id} - \mathcal{L}) \mathbf{v}(z) = \mathbf{h}. \quad (3.23) $$

The solution will necessarily be unique because of Corollary 3.2. For $z \in \overline{\mathcal{W}} \setminus \{1\}$ and $j \in \mathbb{Z}$, we define:

$$ v_j(z) := \sum_{j_0 \in \mathbb{Z}} \mathcal{G}_j^{j_0}(z) h_{j_0}, $$

with $\mathcal{G}_j^{j_0}(z)$ given in Proposition 3.1. Let us first show that the sequence $\mathbf{v}(z) = (v_j(z))_{j \in \mathbb{Z}}$ thus defined belongs indeed to $\ell^q(\mathbb{Z}; \mathbb{C})$. The arguments below are given for a fixed $z \in \overline{\mathcal{W}} \setminus \{1\}$ and the constants may depend on $z$. From the expressions (3.12), (3.13) and using Lemma 3.1 and Assumption 1, we obtain bounds of the form:

$$ \forall (j_0, j) \in \mathbb{Z}^2, \quad |\mathcal{G}_j^{j_0}(z)| \le C_z \exp(-c_z (|j| + |j_0|)) + C_z \exp(-c_z |j - j_0|), $$

<footnote n="9">⁹In the case of a convex or concave flux, one can show that the whole set $\mathcal{O}$ lies in the resolvent set of $\mathcal{L}$ since the Lopatinskii determinant $\Delta$ does not vanish on $\mathcal{O}$ and Proposition 3.1 extends to any $z \in \mathcal{O}$ in that case.</footnote>
<figure><img src="image_13.png" /><figcaption>Figure 3.3: An example of flux f that yields spectrally unstable shock profiles. The graph of the flux is depicted in blue and the chord between $u_r = -1$ and $u_\ell = +1$ is depicted in red. The Rankine-Hugoniot condition (2.3) and Oleinik’s entropy condition are satisfied.</figcaption></figure>

with positive constants $C_z, c_z$ that may depend on $z$ but that do not depend on $(j_0, j)$. Applying either the Hölder or the Young inequality, we obtain that the above defined sequence $v(z) := (v_j(z))_{j \in \mathbb{Z}}$ belongs to $\ell^q(\mathbb{Z}; \mathbb{C})$. It is then a mere exercise to verify that $v(z)$ is a solution to the resolvent equation (3.23) (this is rather easy in this framework since $\mathcal{L}$ involves a finite stencil). We have thus shown that any point $z \in \overline{\mathcal{W}} \setminus \{1\}$ belongs to the resolvent set of $\mathcal{L}$, which completes the proof of Theorem 2.3.

## 3.4 Instability cases

This whole article will be devoted to stable discrete shock profiles, but let us just take a little time to discuss two unstable cases, just to show that spectral instabilities may occur. We go back to the expression (3.3) of the Lopatinskii determinant. As explained in Remark 3.2, in the symmetric case $f'(u_\ell) = -f'(u_r)$, the Lopatinskii determinant $\Delta$ is independent of the mid-point derivative $f'((u_\ell + u_r)/2)$ and spectral stability<fnref n="10" /> always holds.

Let us therefore assume from now on $f'(u_\ell) \neq -f'(u_r)$ so that one has $\alpha_\ell + \alpha_r \neq 0$. We first go back to the expression (3.4) of the derivative $\Delta'(1)$. We observe from this expression that $\Delta'(1)$ can be zero if $\alpha_m$ is given by:

$$ \alpha_m = \frac{\alpha_r - \alpha_\ell + 2\alpha_\ell\alpha_r}{\alpha_\ell + \alpha_r}. \qquad (3.24) $$

This gives for instance the value $\alpha_m = 13/3$ in the case $(\alpha_\ell, \alpha_r) = (1/3, -2/3)$. The case $\Delta'(1) = 0$ is a weak form of instability that we do not study here but that can be achieved with a non-convex flux

<footnote n="10">¹⁰Spectral stability means here that Assumption 1 is satisfied.</footnote>
function $f$. An example of such a function $f$ is depicted in Figure 3.3 with the choice $u_r = -1$, $u_\ell = +1$ and $f(u_r) = f(u_\ell) = 1$ so that the Rankine-Hugoniot condition (2.3) is satisfied. The entropy inequalities (2.4) are also satisfied and we can choose the CFL parameter $\lambda$ sufficiently small in such a way that the condition (2.12) is met. We can then tune the “small amplitude” oscillations near the origin in such a way that the derivative $f'(0)$ satisfies (3.24) (recall the relation $\alpha_m = \lambda f'(0)$, see (2.9)) and the graph of $f$ between $-1$ and $1$ lies below the horizontal chord of height $1$. This means that not only the Lax shock entropy inequalities (2.4) are satisfied but also Oleinik’s entropy condition which is stronger, see [12, 26].

A more severe instability scenario corresponds to finding a root of $\Delta$ in the instability region $\mathcal{U}$. For concreteness, we still assume that the end points of the shock are $u_r = -1$, $u_\ell = +1$, that the flux $f$ satisfies $f(u_r) = f(u_\ell) = 1$, $f'(u_r) = -2 f'(u_\ell)$ and that the parameter $\lambda$ has been chosen in such a way that $(\alpha_\ell, \alpha_r) = (1/3, -2/3)$. We then compute:

$$ \kappa_\ell(2) = -5 - 3\sqrt{3}, \quad \kappa_r(2) = \frac{13 - 3\sqrt{21}}{10}. $$

We then see on the expression (3.3) that $\Delta$ vanishes at $z = 2 \in \mathcal{U}$ provided that we have$^{11}$:

$$ \alpha_m \left( \frac{3}{2} + 2\sqrt{3} - \frac{\sqrt{21}}{2} \right) = \frac{7}{2} + 3 \left( \sqrt{3} + \sqrt{7} + \frac{\sqrt{21}}{2} \right). $$

Once again, this can be achieved by tuning small amplitude oscillations near the origin to have the desired value for $f'(0)$, as shown in Figure 3.3. Spectral instabilities may therefore occur for stationary shock profiles of the Lax-Wendroff scheme even though Oleinik’s entropy condition is satisfied.

## 3.5 Decomposing the spatial Green’s function

The detailed expression of the spatial Green’s function $(G_j^{j_0}(z))_{j \in \mathbb{Z}}$ is given in Proposition 3.1. For later use, we need to decompose the expression of $G_j^{j_0}(z)$ by isolating several parts in it and specifically its singular behavior near $z = 1$. A convenient way to do so is to introduce yet two other Green’s functions which correspond to that of the Lax-Wendroff scheme for a constant coefficient transport operator on the whole real line. The chosen velocity will be either $f'(u_\ell)$ or $f'(u_r)$ depending on the sign of the initial position $j_0$. The choice of the velocity $f'(u_\ell)$ corresponds to the expression of the operator $\mathcal{L}$ in the region $\{j \le -1\}$, see (2.11), and the choice of the velocity $f'(u_r)$ corresponds to the expression of the operator $\mathcal{L}$ in the region $\{j \ge 2\}$, see again (2.11). We thus devote the following paragraph to recalling several facts on the Green’s function for the Lax-Wendroff on the whole real line. Additional material in this case can be found in [10] and [9].

### 3.5.1 The free Green’s function on the whole real line

We pause for a while and go back to the definition (2.11) of the linearized operator $\mathcal{L}$. In the regions $\{j \ge 2\}$ and $\{j \le -1\}$, the coefficients in the operator $\mathcal{L}$ are independent of the spatial index, meaning that $\mathcal{L}$ reduces to a convolution operator that corresponds to the linearization of the Lax-Wendroff scheme

<footnote>$^{11}$This gives the value $\alpha_m \approx 8, 79$.</footnote>
at the constant state $u_\ell$ or $u_r$. We thus introduce the convolution operators $\mathcal{L}_\ell, \mathcal{L}_r$ that are defined on complex valued sequences $\mathbf{v} = (v_j)_{j \in \mathbb{Z}}$ defined on $\mathbb{Z}$ as follows:

$$ \forall j \in \mathbb{Z}, \quad (\mathcal{L}_\ell \mathbf{v})_j := v_j - \frac{\alpha_\ell}{2} (v_{j+1} - v_{j-1}) + \frac{\alpha_\ell^2}{2} (v_{j+1} - 2v_j + v_{j-1}), \quad (3.25a) $$

$$ (\mathcal{L}_r \mathbf{v})_j := v_j - \frac{\alpha_r}{2} (v_{j+1} - v_{j-1}) + \frac{\alpha_r^2}{2} (v_{j+1} - 2v_j + v_{j-1}). \quad (3.25b) $$

We recall that $\alpha_\ell$ and $\alpha_r$ are defined in (2.9). The operators in (3.25) are nothing but the operators arising from the Lax-Wendroff scheme applied to the transport equation with velocity equal to either $f'(u_\ell)$ or $f'(u_r)$. Thanks to the Lévy-Wiener Theorem [22], the spectrum of the operators $\mathcal{L}_\ell$ and $\mathcal{L}_r$ on any space $\ell^q(\mathbb{Z}; \mathbb{C})$ is completely known (see [30] for more on the spectral analysis of convolution operators). We have:

$$ \sigma(\mathcal{L}_\ell) = \left\{ 1 - 2\alpha_\ell^2 \sin^2 \frac{\xi}{2} + \mathbf{i}\alpha_\ell \sin\xi \mid \xi \in \mathbb{R} \right\}, $$

$$ \sigma(\mathcal{L}_r) = \left\{ 1 - 2\alpha_r^2 \sin^2 \frac{\xi}{2} + \mathbf{i}\alpha_r \sin\xi \mid \xi \in \mathbb{R} \right\}, $$

where the result is independent of $q \in [1, +\infty]$. For instance, the spectrum of $\mathcal{L}_\ell$ is represented as the black curve (an ellipse, actually) in Figure 3.1. Due to the restriction (2.12), the spectrum of both $\mathcal{L}_\ell$ and $\mathcal{L}_r$ is included in the closed unit disk $\bar{\mathbb{D}}$ with a single tangency point at 1 with the unit circle.

We can then proceed as in Section 3.2 above and introduce the spatial Green's function for either $\mathcal{L}_\ell$ or $\mathcal{L}_r$. Because of the spatial invariance in (3.25a) and (3.25b), it is sufficient to look at the case where the Dirac mass $\delta_{j_0}$ is located at $j_0 = 0$. Hence, for $z$ in the exterior $\mathcal{O}$ of the curve (2.13), we have that $z$ lies in the resolvent set of both $\mathcal{L}_\ell$ and $\mathcal{L}_r$, and we can therefore introduce the solutions $\bar{\mathcal{G}}_\ell(z) = (\bar{\mathcal{G}}_{\ell,j}(z))_{j \in \mathbb{Z}}$ and $\bar{\mathcal{G}}_r(z) = (\bar{\mathcal{G}}_{r,j}(z))_{j \in \mathbb{Z}}$ to the resolvent problems:

$$ (z \operatorname{Id} - \mathcal{L}_\ell) \bar{\mathcal{G}}_\ell(z) = \delta_0, \quad (z \operatorname{Id} - \mathcal{L}_r) \bar{\mathcal{G}}_r(z) = \delta_0. \quad (3.26) $$

The computations that lead to the precise expressions for $(\bar{\mathcal{G}}_{\ell,j}(z))_{j \in \mathbb{Z}}$ and $(\bar{\mathcal{G}}_{r,j}(z))_{j \in \mathbb{Z}}$ follow from the same methodology as in Sections 3.1 and 3.2 (we refer to [10] for an even more general analysis in the case of finite difference schemes with arbitrary, possibly infinite, stencils). The analysis is actually much simpler for pure convolution operators than what we did in Sections 3.1 and 3.2 since there is no Lopatinskii determinant involved. We thus feel free to state without proof the following result that is in the same vein as Proposition 3.1.

**Proposition 3.2.** Under the condition (2.12) on $\lambda$, for any $z$ in the exterior $\mathcal{O}$ of the spectral curve (2.13), there exist unique solutions $\bar{\mathcal{G}}_\ell(z) = (\bar{\mathcal{G}}_{\ell,j}(z))_{j \in \mathbb{Z}} \in \ell^q(\mathbb{Z}; \mathbb{C})$ and $\bar{\mathcal{G}}_r(z) = (\bar{\mathcal{G}}_{r,j}(z))_{j \in \mathbb{Z}} \in \ell^q(\mathbb{Z}; \mathbb{C})$ to the equations (3.26). These sequences are given by:

$$ \bar{\mathcal{G}}_{r,j}(z) = \begin{cases} \frac{-2 \kappa_r^u(z)^j}{\alpha_r(1 - \alpha_r)(\kappa_r^u(z) - \kappa_r(z))}, & \text{if } j \le 0, \\ \frac{-2 \kappa_r(z)^j}{\alpha_r(1 - \alpha_r)(\kappa_r^u(z) - \kappa_r(z))}, & \text{if } j \ge 0, \end{cases} \quad (3.27) $$
and

$$ \bar{\mathcal{G}}_{\ell,j}(z) = \begin{cases} \frac{-2 \kappa_{\ell}(z)^{j}}{\alpha_{\ell}(1 - \alpha_{\ell})(\kappa_{\ell}(z) - \kappa_{\ell}^{u}(z))}, & \text{if } j \le 0, \\ \frac{-2 \kappa_{\ell}^{u}(z)^{j}}{\alpha_{\ell}(1 - \alpha_{\ell})(\kappa_{\ell}(z) - \kappa_{\ell}^{u}(z))}, & \text{if } j \ge 0. \end{cases} \qquad (3.28) $$

The expressions in (3.28) and (3.27) can be recognized in (3.12) and (3.13). This is made more explicit in the following paragraph.

### 3.5.2 Decomposing the spatial Green's function

The decomposition proceeds as follows. From inspection of the expressions (3.12) and (3.13) and the result of Proposition 3.2, the definition (3.20) can be recast as follows:

$$ \tilde{\mathcal{G}}_{j}^{j_0}(z) = \begin{cases} \mathcal{G}_{j}^{j_0}(z) - \mathbb{1}_{j \ge 1} \bar{\mathcal{G}}_{r,j-j_0}(z), & \text{if } j_0 \ge 1, \\ \mathcal{G}_{j}^{j_0}(z) - \mathbb{1}_{j \le 0} \bar{\mathcal{G}}_{\ell,j-j_0}(z), & \text{if } j_0 \le 0, \end{cases} \qquad (3.29) $$

where the notation $\mathbb{1}_{j \ge 1}$ is used to denote 1 if $j \ge 1$ and 0 otherwise (and similarly for $\mathbb{1}_{j \le 0}$). It is also useful to introduce the following two fourth degree polynomial functions $\varphi_r$ and $\varphi_\ell$:

$$ \forall \tau \in \mathbb{C}, \quad \varphi_{r,\ell}(\tau) := -\frac{1}{\alpha_{r,\ell}} \tau + \frac{1 - \alpha_{r,\ell}^2}{6 \alpha_{r,\ell}^3} \tau^3 - \frac{1 - \alpha_{r,\ell}^2}{8 \alpha_{r,\ell}^3} \tau^4. \qquad (3.30) $$

With such notation, our result is the following.

**Proposition 3.3.** Let the weak solution (2.2) satisfy the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. Then there exists $\varepsilon_0 > 0$, there exist constants $C > 0$ and $c > 0$, there exist two complex valued sequences $(\gamma_j^r)_{j \in \mathbb{Z}}$ and $(\gamma_j^\ell)_{j \in \mathbb{Z}}$, there exist two bounded holomorphic functions $\Psi_r$ and $\Psi_\ell$ on the square $\mathbf{B}_{\varepsilon_0}(0)$, and there exist sequences $(\Phi_{r,j})_{j \in \mathbb{Z}}$, $(\Phi_{\ell,j})_{j \in \mathbb{Z}}$, $(\Theta_{r,j})_{j \in \mathbb{Z}}$, $(\Theta_{\ell,j})_{j \in \mathbb{Z}}$ and $(\Theta_{1,j})_{j \in \mathbb{Z}}$ of bounded holomorphic functions on the square $\mathbf{B}_{\varepsilon_0}(0)$ such that the following hold:

* the sequences $(\gamma_j^r)_{j \in \mathbb{Z}}$ and $(\gamma_j^\ell)_{j \in \mathbb{Z}}$ satisfy the estimates:

$$ \forall j \in \mathbb{Z}, \quad |\gamma_j^r| + |\gamma_j^\ell| \le C \exp(-c|j|); $$

* the sequences $(\Phi_{r,j})_{j \in \mathbb{Z}}$, $(\Phi_{\ell,j})_{j \in \mathbb{Z}}$, $(\Theta_{r,j})_{j \in \mathbb{Z}}$ and $(\Theta_{\ell,j})_{j \in \mathbb{Z}}$ satisfy the estimates:

$$ \forall j \in \mathbb{Z}, \quad \forall \tau \in \mathbf{B}_{\varepsilon_0}(0), \quad |\Phi_{r,j}(\tau)| + |\Phi_{\ell,j}(\tau)| + |\Theta_{r,j}(\tau)| + |\Theta_{\ell,j}(\tau)| + |\Theta_{1,j}(\tau)| \le C \exp(-c|j|); $$

* for any couple of integers $(j_0, j) \in \mathbb{Z}^2$, the function:

$$ \tau \in \mathbf{B}_{\varepsilon_0}(0) \cap \{\zeta \in \mathbb{C} \mid \mathrm{Re} \, \zeta > 0\} \mapsto \tilde{\mathcal{G}}_{j}^{j_0}(e^{\tau}), $$

whose expression is given in (3.29), has a meromorphic extension to the square $\mathbf{B}_{\varepsilon_0}(0)$ with a first order pole at 0 only, and there holds:

$$ e^{\tau} \tilde{\mathcal{G}}_{j}^{j_0}(e^{\tau}) = \begin{cases} \left( \frac{\mathcal{H}_j}{\tau} + \gamma_j^r + \tau \Phi_{r,j}(\tau) \right) \exp(-j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)), & \text{if } j_0 \ge 1, \\ \left( \frac{\mathcal{H}_j}{\tau} + \gamma_j^\ell + \tau \Phi_{\ell,j}(\tau) \right) \exp(-j_0 \varphi_\ell(\tau) + j_0 \tau^5 \Psi_\ell(\tau)), & \text{if } j_0 \le 0, \end{cases} \qquad (3.31) $$
for any $\tau \in \mathbf{B}_{\varepsilon_0}(0) \setminus \{0\}$, where we recall that $\mathcal{H}_j$ is defined in (3.22);

• for any couple of integers $(j_0, j) \in \mathbb{Z}^2$, the function:

$$ \tau \in \mathbf{B}_{\varepsilon_0}(0) \cap \{\zeta \in \mathbb{C} | \mathrm{Re} \, \zeta > 0\} \mapsto \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) - \tilde{\mathcal{G}}_j^{j_0-1}(e^{\tau}) $$

has a holomorphic extension to the square $\mathbf{B}_{\varepsilon_0}(0)$, and there holds:

$$ e^{\tau} \left( \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) - \tilde{\mathcal{G}}_j^{j_0-1}(e^{\tau}) \right) = \begin{cases} \left( \frac{\mathcal{H}_j}{\alpha_r} + \tau \Theta_{r,j}(\tau) \right) \exp \left( -j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right), & \text{if } j_0 \ge 2, \\ \frac{\mathcal{H}_j}{\alpha_r} + \gamma_j^r - \gamma_j^\ell + \tau \Theta_{1,j}(\tau), & \text{if } j_0 = 1, \\ \left( \frac{\mathcal{H}_j}{\alpha_\ell} + \tau \Theta_{\ell,j}(\tau) \right) \exp \left( -j_0 \varphi_\ell(\tau) + j_0 \tau^5 \Psi_\ell(\tau) \right), & \text{if } j_0 \le 0, \end{cases} \quad (3.32) $$

for any $\tau \in \mathbf{B}_{\varepsilon_0}(0)$.

*Proof.* We give the proof in the case $j_0 \ge 1$ and construct all quantities associated with the “right” state $u_r$. The proof in the case $j_0 \le 0$ is entirely similar and is left to the interested reader. We thus always consider from on some $j_0 \ge 1$ and some arbitrary integer $j \in \mathbb{Z}$. We also define, for later use, the function:

$$ z \mapsto \Theta(z) := \frac{\Delta(z)}{z-1}, $$

where the Lopatinskii determinant $\Delta$ is defined in (3.3). Thanks to Lemma 3.2 and Assumption 1, we know that $\Theta$ can be holomorphically extended to some set of the form $\{\zeta \in \mathbb{C} | e^{-\delta_0} < |\zeta|\}$ for an appropriate $\delta_0 > 0$. This is because $\Delta$ has a simple zero at 1. Furthermore, we know that, up to restricting $\delta_0$, $\Theta$ does not vanish on the set $\{\zeta \in \mathbb{C} | e^{-\delta_0} < |\zeta|\}$.

From the definition (3.20) and the expressions (3.12), (3.27), we get the factorization:

$$ \forall z \in \mathcal{U}, \quad \tilde{\mathcal{G}}_j^{j_0}(z) = \chi_j^r(z) \bar{\mathcal{G}}_{r,1-j_0}(z), $$

where we have set

$$ \chi_j^r(z) := \begin{cases} \frac{\chi_r^-(z)}{z-1} \kappa_\ell(z)^j, & \text{if } j \le 0, \\ \left( \frac{\chi_r^+(z)}{z-1} - 1 \right) \kappa_r(z)^{j-1}, & \text{if } j \ge 1, \end{cases} \quad (3.33) $$

with

$$ \chi_r^+(z) := \frac{(\alpha_\ell - \alpha_m + (1-\alpha_\ell)\kappa_\ell(z))(1-\alpha_r)(\kappa_r^u(z) - \kappa_r(z))}{\Theta(z)}, $$

$$ \chi_r^-(z) := \frac{\alpha_r (1 - \alpha_r) (1 - \alpha_m) (\kappa_r^u(z) - \kappa_r(z))}{\alpha_\ell \Theta(z)}. $$

It appears from those expressions and from Lemma 3.1 that, up to restricting $\delta_0$ again, both functions $\chi_r^\pm$ have a holomorphic extension to the set $\{\zeta \in \mathbb{C} | e^{-\delta_0} < |\zeta|\}$ and there is no loss of generality in
assuming that both $\kappa_\ell$ and $\kappa_r$ are also holomorphic functions and do not vanish on that set (see Lemma 3.1). Moreover, we compute:

$$ \chi_r^+(1) = - \frac{2(1 + \alpha_m)}{\underline{\Delta}'(1)}, \quad \chi_r^-(1) = \frac{2\alpha_r(1 - \alpha_m)}{\alpha_\ell \underline{\Delta}'(1)}. $$

For $z$ in a neighborhood of 1, we may then write (see (3.33)):

$$ \chi_j^r(z) = \frac{\xi_j^r}{z-1} + \Gamma_j^r + \Xi_j^r(z), $$

where $\xi_j^r$ and $\Gamma_j^r$ are complex numbers defined by:

$$ \xi_j^r := \lim_{z \to 1} (z-1) \chi_j^r(z) = \begin{cases} \chi_r^-(1) \kappa_\ell(1)^j, & \text{if } j \le 0, \\ \chi_r^+(1) \kappa_r(1)^{j-1}, & \text{if } j \ge 1, \end{cases} $$

$$ \Gamma_j^r := \lim_{z \to 1} \left( \chi_j^r(z) - \frac{\xi_j^r}{z-1} \right), $$

and the function $(z \mapsto \Xi_j^r(z))$ is defined by:

$$ \Xi_j^r(z) := \chi_j^r(z) - \frac{\xi_j^r}{z-1} - \Gamma_j^r, $$

so that $\Xi_j^r$ has a holomorphic extension to the set $\{\zeta \in \mathbb{C} | e^{-\delta_0} < |\zeta|\}$ and vanishes at 1. In other words, we have isolated the first order pole at 1 in $\chi_j^r$. Moreover, it is not difficult to compare the expression for $\xi_j^r$ and the defining equation (3.22) for $\mathcal{H}_j$ and to find the relation $\xi_j^r = -\alpha_r \mathcal{H}_j$.

We know from Lemma 3.1 and from Section 3.3 that $\kappa_\ell(1)$ belongs to $\mathcal{W}$ and $\kappa_r(1)$ belongs to $\mathbb{D}$. Hence we can infer from the above definitions the exponentially decaying bounds:

$$ |\xi_j^r| + |\Gamma_j^r| \le C e^{-c|j|}, $$

as well as the local bound in $z$ close to 1:

$$ |\Xi_j^r(z)| \le C |z-1| e^{-c|j|}. $$

In the same way, we find from the expression (3.27) that for $j_0 \ge 1$, the function $\bar{\mathcal{G}}_{r,1-j_0}$ has a holomorphic extension to the set $\{\zeta \in \mathbb{C} | e^{-\delta_0} < |\zeta|\}$ (up to restricting $\delta_0$ one more time). Since $\kappa_r^u(1)$ equals 1, for $\tau$ in a sufficiently small square $\mathbf{B}_{\varepsilon_0}(0)$ centered at the origin, we can write:

$$ \kappa_r^u(e^\tau) = \exp(\omega_r(\tau)), $$

where $\omega_r$ is holomorphic and bounded on $\mathbf{B}_{\varepsilon_0}(0)$. It is then a mere algebra exercise to infer from (3.1b) the Taylor expansion of $\omega_r$ at 0 and we get:

$$ \omega_r(\tau) = \varphi_r(\tau) + \mathcal{O}(\tau^5), $$

with the fourth degree polynomial $\varphi_r$ defined in (3.30). We can thus write, for any $\tau \in \mathbf{B}_{\varepsilon_0}(0)$:

$$ e^\tau \bar{\mathcal{G}}_{r,1-j_0}(e^\tau) = \left( -\frac{1}{\alpha_r} + \mu_r \tau + \tau^2 \tilde{\Phi}_r(\tau) \right) \exp \left( -j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right), $$

where $\tilde{\Phi}_r$ and $\Psi_r$ are two holomorphic bounded functions on $\mathbf{B}_{\varepsilon_0}(0)$ and $\mu_r$ is a complex number. It then remains to perform the Taylor expansion of $\chi_j^r(e^\tau)$ at $\tau = 0$ and to multiply with the above expansion of $e^\tau \bar{\mathcal{G}}_{r,1-j_0}(e^\tau)$ to get the result of Proposition 3.3. $\square$

We now turn to the proof of our main estimates for the Green's function of the operator $\mathcal{L}$ (see Theorem 4.1 below). This will lead to time decay estimates as stated in Theorem 2.4.
# Chapter 4

## Linear stability

The goal of this chapter is to derive sharp bounds for the temporal Green's function, that is, for any given $j_0 \in \mathbb{Z}$, the solution $(\mathcal{G}^n(j, j_0))_{(n,j) \in \mathbb{N} \times \mathbb{Z}}$ to the recurrence relation:

$$ \left\{ \begin{array}{l} \mathcal{G}^{n+1}(j, j_0) = (\mathcal{L} \mathcal{G}^n(\cdot, j_0))_j, \quad (n, j) \in \mathbb{N} \times \mathbb{Z}, \\ \mathcal{G}^0(\cdot, j_0) = \delta_{j_0}. \end{array} \right. \qquad (4.1) $$

The relevance of this sequence is motivated by the fact that for any initial condition $\mathbf{h} \in \ell^q(\mathbb{Z}; \mathbb{R})$, the solution to the recurrence relation:

$$ \begin{array}{l} \forall n \in \mathbb{N}, \quad \mathbf{v}^{n+1} = \mathcal{L} \mathbf{v}^n, \\ \mathbf{v}^0 = \mathbf{h}, \end{array} $$

can be decomposed into:

$$ \forall (n, j) \in \mathbb{N} \times \mathbb{Z}, \quad v_j^n = \sum_{j_0 \in \mathbb{Z}} \mathcal{G}^n(j, j_0) h_{j_0}, $$

and we expect that sharp bounds on $\mathcal{G}^n(j, j_0)$ will quantify the decay properties of the semigroup of operators $(\mathcal{L}^n)_{n \in \mathbb{N}}$. We first decompose the temporal Green's function by following the decomposition given in Proposition 3.3 for the (reduced) spatial Green's function. We then analyze each contribution in the decomposition and derive bounds that are meant to be as sharp as possible in order to obtain large time decaying bounds for the semigroup $(\mathcal{L}^n)_{n \in \mathbb{N}}$.

### 4.1 Preliminary facts

We always consider from now on that the weak solution (2.2) satisfies the entropy inequalities (2.4) and that the CFL parameter $\lambda$ satisfies the stability condition (2.12). We also assume that Assumption 1 is satisfied so that the analysis of Chapter 3 can be used. The value of the temporal Green's function $\mathcal{G}^n(j, j_0)$ is given by the so-called functional calculus (see [8]):

$$ \mathcal{G}^n(j, j_0) = \frac{1}{2\pi i} \int_{\tilde{\Gamma}} z^n \mathcal{G}_j^{j_0}(z) \, dz, \qquad (4.2) $$

where $\tilde{\Gamma}$ is any contour that encompasses the spectrum of the operator $\mathcal{L}$. In view of Theorem 2.3, one can choose for instance $\tilde{\Gamma} = (1 + \delta) \mathbb{S}^1 = \{\zeta \in \mathbb{C} \mid |\zeta| = 1 + \delta\}$ for any $\delta > 0$, since the spectrum of $\mathcal{L}$ is included in $\mathbb{D} \cup \{1\}$.
Two other key quantities of interest to us are the temporal Green's functions associated with the operators $\mathcal{L}_\ell$ and $\mathcal{L}_r$ that are defined in (3.25). The associated temporal Green's functions for these operators are defined as the solutions to the recurrences:

$$ \left\{ \begin{aligned} \bar{\mathcal{G}}_{\ell}^{n+1} &= \mathcal{L}_{\ell} \bar{\mathcal{G}}_{\ell}^{n}, & n \in \mathbb{N}, \\ \bar{\mathcal{G}}_{\ell}^{0} &= \delta_{0}, \end{aligned} \right. \qquad (4.3a) $$

$$ \left\{ \begin{aligned} \bar{\mathcal{G}}_{r}^{n+1} &= \mathcal{L}_{r} \bar{\mathcal{G}}_{r}^{n}, & n \in \mathbb{N}, \\ \bar{\mathcal{G}}_{r}^{0} &= \delta_{0}, \end{aligned} \right. \qquad (4.3b) $$

and the same functional calculus rules as before yield the expressions:

$$ \bar{\mathcal{G}}_{\ell}^{n}(j) = \frac{1}{2\pi i} \int_{\tilde{\Gamma}} z^{n} \bar{\mathcal{G}}_{\ell,j}(z) dz, \quad \bar{\mathcal{G}}_{r}^{n}(j) = \frac{1}{2\pi i} \int_{\tilde{\Gamma}} z^{n} \bar{\mathcal{G}}_{r,j}(z) dz, $$

where $\tilde{\Gamma}$ is once again any closed contour that encompasses the spectrum of both $\mathcal{L}_\ell$ and $\mathcal{L}_r$ in its interior. We can choose, for instance, $\tilde{\Gamma} = (1 + \delta)\mathbb{S}^1$ for any $\delta > 0$. A crucial observation for what follows is that the sequences $(\bar{\mathcal{G}}_\ell^n(j))_{(n,j) \in \mathbb{N} \times \mathbb{Z}}$ and $(\bar{\mathcal{G}}_r^n(j))_{(n,j) \in \mathbb{N} \times \mathbb{Z}}$ have been thoroughly studied in [9], though with a different point of view since the framework allows for the use of Fourier analysis. We shall feel free to use repeatedly several results from [9], which are themselves more accurate estimates for the free Green's functions (in the whole space) than previous bounds obtained in [16, 17]. The main results of [9] are gathered in Appendix A together with some supplementary material that is needed to carry out the analysis below.

We start our analysis with a first elementary observation. Since the Lax-Wendroff scheme as a finite stencil, we readily see from the recurrence equation (4.1) defining the Green's function that for all $n \in \mathbb{N}$ and $(j, j_0) \in \mathbb{Z}^2$:

$$ |j - j_0| > n \Rightarrow \mathcal{G}^n(j, j_0) = 0. $$

this fact is repeatedly used below in order to restrict the possible regimes for $j, j_0, n$. As a second crucial observation, we have the following Lemma.

**Lemma 4.1.** Let $j_0 \in \mathbb{Z}$ and $n \in \mathbb{N}$. If $j_0 \ge 1$ and $n \le j_0 - 1$, then there holds:

$$ \forall j \in \mathbb{Z}, \quad \mathcal{G}^n(j, j_0) = \bar{\mathcal{G}}_r^n(j - j_0). $$

If $j_0 \le 0$ and $n \le |j_0|$, then there holds:

$$ \forall j \in \mathbb{Z}, \quad \mathcal{G}^n(j, j_0) = \bar{\mathcal{G}}_\ell^n(j - j_0). $$

*Proof.* We give the proof in the case $j_0 \ge 1$ and leave the other situation to the interested reader.

The proof directly follows from the expression (2.11) of the operator $\mathcal{L}$ and from the definition (3.25b) of the convolution operator $\mathcal{L}_r$. Indeed, we easily see that if $w = (w_j)_{j \in \mathbb{Z}}$ is a sequence that is supported in $\{j \in \mathbb{Z} | j \ge j_{\min}\}$ with $j_{\min} \ge 2$, then $\mathcal{L}w$ equals $\mathcal{L}_r w$ and both sequences are supported in $\{j \in \mathbb{Z} | j \ge j_{\min} - 1\}$.

For $j_0 = 1$, the proof of Lemma 4.1 is obvious. We thus assume $j_0 \ge 2$. Then we can prove by induction that for any $n = 0, \dots, j_0 - 2$, the two sequences $\mathcal{G}^n(\cdot, j_0)$ and $\bar{\mathcal{G}}_r^n(\cdot - j_0)$ are equal and they are supported in $\{j \in \mathbb{Z} | j \ge j_0 - n\}$. Applying one last time the above fact when $n$ equals $j_0 - 2$, we see that $\mathcal{G}^{j_0-1}(\cdot, j_0)$ and $\bar{\mathcal{G}}_r^{j_0-1}(\cdot - j_0)$ are equal and they are supported in $\{j \in \mathbb{Z} | j \ge 1\}$. This is when we cannot use the above fact any longer. $\square$
Finally, as a last preparatory step, we shall change variable in the integral representation (4.2) to write instead

$$ \mathcal{G}^n(j, j_0) = \frac{1}{2\pi i} \int_{\Gamma} e^{n\tau} \mathcal{G}_j^{j_0}(e^{\tau}) e^{\tau} d\tau, $$

with $\Gamma = \{\tau = \rho + i\theta \mid \theta \in [-\pi, \pi]\}$, for any $\rho > 0$. This change of variable justifies the decomposition made in Proposition 3.3 (see (3.31)). We now introduce some further notations and clarify the bounds that we intend to prove on the Green's function of the operator $\mathcal{L}$.

## 4.2 Notation and bounds for the Green's function

We first introduce some constants:

$$ c_{3,\ell} := \frac{\alpha_{\ell}(1 - \alpha_{\ell}^2)}{6} > 0, \qquad c_{4,\ell} := \frac{\alpha_{\ell}^2(1 - \alpha_{\ell}^2)}{8} > 0, \qquad (4.4a) $$

$$ c_{3,r} := \frac{\alpha_r (1 - \alpha_r^2)}{6} < 0, \qquad c_{4,r} := \frac{\alpha_r^2 (1 - \alpha_r^2)}{8} > 0. \qquad (4.4b) $$

We then define the following two functions<fnref n="1" /> $\mathbf{A}_\ell$ and $\mathbf{A}_r$ on $\mathbb{R} \times \mathbb{R}^{+*}$ as follows. For $\eta > 0$, we set:

$$ \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad \mathbf{A}_{\ell}(x, y) := \frac{1}{2\pi} \int_{\mathbb{R}} e^{x(\eta+i\theta)} e^{-c_{3,\ell}y(\eta+i\theta)^3} e^{-c_{4,\ell}y(\eta+i\theta)^4} \frac{d\theta}{\eta+i\theta}, \qquad (4.5a) $$

$$ \mathbf{A}_r(x, y) := \frac{1}{2\pi} \int_{\mathbb{R}} e^{x(\eta+i\theta)} e^{-c_{3,r}y(\eta+i\theta)^3} e^{-c_{4,r}y(\eta+i\theta)^4} \frac{d\theta}{\eta+i\theta}, \qquad (4.5b) $$

where both definitions (4.5a) and (4.5b) make sense since $c_{4,\ell}$ and $c_{4,r}$ are positive. The definitions (4.5) are shown to be independent of $\eta > 0$ thanks to the Cauchy formula for holomorphic functions. Eventually, we introduce two other functions $\mathbf{M}_\ell$ and $\mathbf{M}_r$ on $\mathbb{R}^{+*} \times \mathbb{R} \times \mathbb{R}^{+*}$ as follows:

$$ \forall (c, x, y) \in \mathbb{R}^{+*} \times \mathbb{R} \times \mathbb{R}^{+*}, \quad \mathbf{M}_{\ell}(c, x, y) := \begin{cases} \frac{1}{y^{1/3}} \exp(-c|x|^{3/2}/y^{1/2}), & \text{if } x \ge 0, \\ \frac{1}{y^{1/3}}, & \text{if } -y^{1/3} \le x \le 0, \\ \frac{1}{|x|^{1/4} y^{1/4}} \exp(-c x^2/y), & \text{if } x \le -y^{1/3}, \end{cases} \qquad (4.6a) \\ \mathbf{M}_{r}(c, x, y) := \begin{cases} \frac{1}{y^{1/3}} \exp(-c|x|^{3/2}/y^{1/2}), & \text{if } x \le 0, \\ \frac{1}{y^{1/3}}, & \text{if } 0 \le x \le y^{1/3}, \\ \frac{1}{|x|^{1/4} y^{1/4}} \exp(-c x^2/y), & \text{if } y^{1/3} \le x. \end{cases} \qquad (4.6b) $$

These functions encode the bounds for the free Green's functions associated with the convolution operators $\mathcal{L}_\ell$ and $\mathcal{L}_r$, as recalled in Appendix A (see for instance Corollary A.1).

<footnote n="1">¹The letter A refers to “activation” in analogy with [6].</footnote>
A crucial property for what follows is that both $M_\ell$ and $M_r$ are non-increasing with respect to their first argument. When various positive constants $c_1, c_2, \dots$ appear, we can always use the largest function $M_r(\min_i c_i, \cdot, \cdot)$ as an upper bound for all functions $M_r(c_i, \cdot, \cdot)$. This will be used repeatedly in order to avoid using specific notations for the various small positive constants $c$ that appear.

Our main result for the Green's function of the operator $\mathcal{L}$ reads as follows.

**Theorem 4.1** (The Green's function of the linearized numerical scheme). Let the weak solution (2.2) satisfy the Rankine-Hugoniot condition (2.3) and the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. Then there exist some positive constants $C$ and $c$ such that, for any $j_0 \le 0$, there holds:

$$ \forall (n, j) \in \mathbb{N}^* \times \mathbb{Z}, \quad \left| \mathcal{G}^n(j, j_0) - \mathcal{H}_j \mathbf{A}_\ell(j_0 + n \alpha_\ell, n) \right| \le C \mathbf{M}_\ell(c, j_0 - j + n \alpha_\ell, n) \mathbf{1}_{j \le 0} \\ + C e^{-c|j|} \mathbf{M}_\ell(c, j_0 + n \alpha_\ell, n) \\ + C e^{-cn} e^{-c|j|} e^{-c|j_0|}, \quad (4.7) $$

and for any $j_0 \ge 1$, there holds:

$$ \forall (n, j) \in \mathbb{N}^* \times \mathbb{Z}, \quad \left| \mathcal{G}^n(j, j_0) - \mathcal{H}_j \mathbf{A}_r(-j_0 + n |\alpha_r|, n) \right| \le C \mathbf{M}_r(c, j - j_0 + n |\alpha_r|, n) \mathbf{1}_{j \ge 1} \\ + C e^{-c|j|} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) \\ + C e^{-cn} e^{-c|j|} e^{-c|j_0|}, \quad (4.8) $$

where in both (4.7) and (4.8), $\mathcal{H}_j$ is defined in (3.22).

## 4.3 Decomposing the temporal Green's function

Throughout this section, we assume that $n \ge 1$ is an integer and $(j, j_0) \in \mathbb{Z}^2$ satisfy $|j - j_0| \le n$. We will mainly focus on the case $j_0 \ge 1$, and present at the end of the section the corresponding results for the case $j_0 \le 0$. From Lemma 4.1, we shall further assume that $1 \le j_0 \le n$ since for $n \le j_0 - 1$ we have already obtained an explicit expression for the temporal Green's function in terms of the free Green's function associated to $\mathcal{L}_r$ (so the final derivation of a bound for the Green's function will be simpler).

The starting point of our analysis is to exploit our expression for the spatial Green's function to get

$$ \mathcal{G}^n(j, j_0) = \overline{\mathcal{G}}_r^n(j - j_0) \mathbf{1}_{j \ge 1} + \frac{1}{2\pi i} \int_{\Gamma} e^{n\tau} \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) e^{\tau} d\tau, \quad (4.9) $$

and define

$$ \tilde{\mathcal{G}}^n(j, j_0) := \frac{1}{2\pi i} \int_{\Gamma} e^{n\tau} \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) e^{\tau} d\tau. $$

At this stage, $\Gamma$ denotes the segment $\{\tau = \rho + i\theta \mid \theta \in [-\pi, \pi]\}$ for any $\rho > 0$, but we shall be allowed to deform $\Gamma$ thanks to Cauchy's formula.

Let $\varepsilon_0 > 0$ be given by Proposition 3.3 and let $C_0 > 0$ be such that for all $\tau \in \mathbf{B}_{\varepsilon_0}(0)$ one has the uniform bound:

$$ |\tau^5 \Psi_r(\tau)| \le C_0 (|\mathrm{Re}(\tau)|^5 + |\mathrm{Im}(\tau)|^5). \quad (4.10) $$

Then, let $\varepsilon_* \in (0, \varepsilon_0)$ be such that for all $\varepsilon \in (0, \varepsilon_*)$ the following conditions are satisfied:

$$ 0 < \varepsilon < \min \left[ \frac{1 - \alpha_r^2}{16 \alpha_r^2 C_0}, \left(\frac{1}{3}\right)^{1/5} \right], \quad (4.11) $$
<figure><img src="image_14.png" /><figcaption>Figure 4.1: Schematic illustration of the contour Γ and its decomposition into Γ<sub>out</sub> (in red) and Γ<sub>in</sub> (in blue). The contour Γ<sub>in</sub> can be any path joining −η − **i**ε to −η + **i**ε, which remains within B<sub>ε</sub>(0) and passes to the right of the origin. The black bullets represent the end points of the contours.</figcaption></figure>

$$
\frac{4 |\alpha_r|}{1 - \alpha_r^2} \left[ \max \left( \frac{|\alpha_r|}{2}, 1 - \alpha_r^2 \right) + |\alpha_r| C_0 \right] \varepsilon + \varepsilon^3 + \frac{3}{2} \varepsilon^8 + \frac{1}{3} \varepsilon^{11} + \frac{4 \alpha_r^2}{1 - \alpha_r^2} C_0 \varepsilon^{21} < \frac{1}{8}, \quad (4.12)
$$

$$
\frac{1 - \alpha_r^2}{4|\alpha_r|} \left( 1 + \frac{3}{2}\varepsilon^5 \right) \varepsilon^2 + \frac{\alpha_r^2}{2} C_0 \varepsilon^{20} < \frac{|\alpha_r|}{4}. \quad (4.13)
$$

Finally, once for all, we fix ε ∈ (0, ε*), and we let ηε > 0 be provided by Corollary 3.3 associated to this
ε > 0 and we also set 0 < η < min(ηε, ε⁵).

We can now proceed by choosing an appropriate contour Γ in our integral defining $\tilde{\mathcal{G}}^n(j, j_0)$. We would like to choose the segment {τ = −η + **i**θ | θ ∈ [−π, π]} but this is not possible right away because of the pole at the origin, so we shall make a detour on the right of the origin. We thus decompose Γ into two pieces Γ<sub>out</sub> and Γ<sub>in</sub> where

$$
\Gamma_{\text{out}} := \{-\eta + \mathbf{i}\theta \mid \varepsilon \le |\theta| \le \pi\},
$$

and Γ<sub>in</sub> is any path joining −η − **i**ε to −η + **i**ε, which remains within B<sub>ε</sub>(0) and passes to the right of
the origin. These contours are depicted in Figure 4.1. From Corollary 3.3 in Chapter 3, we infer that

$$
\left| \frac{1}{2\pi i} \int_{\Gamma_{\text{out}}} e^{n\tau} \tilde{\mathcal{G}}_{j}^{j_0}(e^{\tau}) e^{\tau} d\tau \right| \le C e^{-\eta n} e^{-c(|j|+|j_0|)}. \quad (4.14)
$$

Since Γ<sub>in</sub> ⊂ B<sub>ε</sub>(0) ⊂ B<sub>ε<sub>0</sub></sub>(0), we can use Proposition 3.3 and the expression (3.31) to decompose the
remaining integral as

$$
\frac{1}{2\pi i} \int_{\Gamma_{\text{in}}} e^{n\tau} \tilde{\mathcal{G}}_{j}^{j_0}(e^{\tau}) e^{\tau} d\tau = \tilde{\mathcal{G}}_{1,r}^{n}(j, j_0) + \tilde{\mathcal{G}}_{2,r}^{n}(j, j_0) + \mathcal{R}_{1,r}^{n}(j, j_0),
$$
where we have set

$$
\begin{align}
\tilde{\mathcal{G}}_{1,r}^n(j, j_0) &:= \mathcal{H}_j \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} \exp \left( n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right) \frac{d\tau}{\tau}, \\
\tilde{\mathcal{G}}_{2,r}^n(j, j_0) &:= \gamma_j^r \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} \exp \left( n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right) d\tau, \tag{4.15}
\end{align}
$$

and

$$
\mathcal{R}_{1,r}^{n}(j, j_0) := \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} \exp (n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)) \tau \Phi_{r,j}(\tau) \, d\tau. \quad (4.16)
$$

Thank's to Cauchy formula for holomorphic functions, it is important to remark that the above integrals
do not depend on $\Gamma_{\text{in}}$, and we shall later on in this section choose specific contours $\Gamma_{\text{in}}$ depending on
various regimes between $n$ and $j_0$. In fact, for $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$ and $\mathcal{R}_{1,r}^n(j, j_0)$, we can select any contour $\Gamma_{\text{in}}$
that joins $-\eta - \mathbf{i}\varepsilon$ to $-\eta + \mathbf{i}\varepsilon$ and which remains within $\overline{\mathbf{B}}_{\varepsilon}(0)$ since both integrand are holomorphic
functions in $\overline{\mathbf{B}}_{\varepsilon}(0)$, while for $\tilde{\mathcal{G}}_{1,r}^n(j, j_0)$ we need to keep the constraint that the contour $\Gamma_{\text{in}}$ passes to the
right of the origin because of the pole at the origin of the integrand.

Our next task is to extract the leading order term of $\tilde{\mathcal{G}}_{1,r}^n(j, j_0)$. For that, we may simply further
-decompose

$$
\tilde{\mathcal{G}}_{1,r}^{n}(j, j_0) = \mathcal{H}_j \mathcal{A}_r^{n}(j_0) + \tilde{\mathcal{G}}_{3,r}^{n}(j, j_0) + \mathcal{R}_{2,r}^{n}(j, j_0),
$$

where we have set

$$
\mathcal{A}_r^n(j_0) := \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau}, \quad (4.17)
$$

$$
\tilde{\mathcal{G}}_{3,r}^n(j, j_0) := \mathcal{H}_j \times j_0 \times \Psi_r(0) \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau - j_0 \varphi_r(\tau)} \tau^4 d\tau, \quad (4.18)
$$

and

$$
\mathcal{R}_{2,r}^{n}(j, j_0) := \mathcal{H}_j \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau - j_0 \varphi_r(\tau)} \left( \frac{\exp(j_0 \tau^5 \Psi_r(\tau)) - 1 - j_0 \tau^5 \Psi_r(0)}{\tau} \right) d\tau. \quad (4.19)
$$

For convenience, we further define $\mathcal{B}_r^n(j, j_0) := \tilde{\mathcal{G}}_{2,r}^n(j, j_0) + \tilde{\mathcal{G}}_{3,r}^n(j, j_0)$ (with the definitions (4.15) and (4.18)) and $\mathcal{R}_r^n(j, j_0) = \mathcal{R}_{1,r}^n(j, j_0) + \mathcal{R}_{2,r}^n(j, j_0)$, such that we have obtained the intermediate decomposition

$$
\frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau} \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) e^{\tau} d\tau = \mathcal{H}_j \mathcal{A}_r^n(j_0) + \mathcal{B}_r^n(j, j_0) + \mathcal{R}_r^n(j, j_0). \quad (4.20)
$$

For future reference, we gather the previous definitions of $\mathcal{B}_r^n(j, j_0)$ and $\mathcal{R}_r^n(j, j_0)$:

$$
\begin{equation}
\begin{split}
\mathcal{B}_r^n(j, j_0) ={}& \gamma_j^r \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} \exp \left( n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right) d\tau \\
& + \Psi_r(0) \times \mathcal{H}_j \times j_0 \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau - j_0 \varphi_r(\tau)} \tau^4 d\tau,
\end{split}
\tag{4.21}
\end{equation}
$$

$$
\begin{equation}
\begin{split}
\mathcal{R}_r^n(j, j_0) ={}& \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} \exp(n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)) \tau \Phi_{r,j}(\tau) \, d\tau \\
& + \mathcal{H}_j \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau - j_0 \varphi_r(\tau)} \left( \frac{\exp(j_0 \tau^5 \Psi_r(\tau)) - 1 - j_0 \tau^5 \Psi_r(0)}{\tau} \right) \, d\tau.
\end{split}
\tag{4.22}
\end{equation}
$$

We shall now study each term appearing in (4.20) separately.
<figure><img src="image_15.png" /><figcaption>Figure 4.2: In blue: the contour $\Gamma_{in}$ used in the regime $\frac{n|\alpha_r|}{2} \le j_0 \le n$ in Lemma 4.2 and its decomposition into three parts $(\Gamma_-, \Gamma_\eta$ and $\Gamma_+)$. In red dash: the contour $\Gamma_{-\eta}$. The black bullets represent the end points of the contours.</figcaption></figure>

### 4.3.1 Estimates of the leading order term $\mathcal{A}_r^n(j_0)$

In this subsection, we obtain a sharp estimate on the term $\mathcal{A}_r^n(j_0)$ in (4.17) and prove that it behaves like an activation function. This clarifies the behavior of the first term in the right-hand side of the decomposition (4.20). We shall consider here and in all the remainder of this Section, two different regimes for $n$ and $j_0$:

(i) the main regime : $\frac{n|\alpha_r|}{2} \le j_0 \le n$;

(ii) the tail : $1 \le j_0 < \frac{n|\alpha_r|}{2}$.

The main result of this subsection is the following Lemma.

**Lemma 4.2.** Let $n \ge 1$ and let $1 \le j_0 \le n$. Let $\mathcal{A}_r^n(j_0)$ be defined in (4.17). Then there exist some positive constants $C$ and $c$ that are uniform with respect to $n$ and $j_0$ such that with the functions $\mathbf{A}_r$ defined in (4.5b) and $\mathbf{M}_r$ defined in (4.6b), there holds:

$$|\mathcal{A}_r^n(j_0) - \mathbf{A}_r(-j_0 + n|\alpha_r|, n)| \le C \mathbf{M}_r(c, -j_0 + n|\alpha_r|, n) + C e^{-cn-cj_0}. \quad (4.23)$$

*Proof. Case (i).* We assume that $j_0$ and $n$ satisfy $\frac{n|\alpha_r|}{2} \le j_0 \le n$. The contour $\Gamma_{in}$ is decomposed into three parts as depicted in Figure 4.2. From the definition (4.17), we thus have:

$$\mathcal{A}_r^n(j_0) = \frac{1}{2\pi i} \int_{\Gamma_- \cup \Gamma_+} e^{n \tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau} + \frac{1}{2\pi i} \int_{\Gamma_\eta} e^{n \tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau},$$

where $\Gamma_\eta = \{\eta + i\theta \mid |\theta| \le \varepsilon\}$ and $\Gamma_\pm$ are the two horizontal paths joining $-\eta \pm i\varepsilon$ to $\eta \pm i\varepsilon$. Upon denoting

$$\omega := \frac{j_0 - n|\alpha_r|}{n} \in [-\frac{|\alpha_r|}{2}, 1 - |\alpha_r|] \quad \text{and} \quad \zeta := \frac{j_0}{n} \frac{1 - \alpha_r^2}{2\alpha_r^2} \in \left[\frac{1 - \alpha_r^2}{4|\alpha_r|}, \frac{1 - \alpha_r^2}{2\alpha_r^2}\right], \quad (4.24)$$
we get

$$n\tau - j_0 \varphi_r(\tau) = \frac{n}{|\alpha_r|} \left[ -\omega\tau + \frac{\zeta}{3}\tau^3 - \frac{\zeta}{4}\tau^4 \right]. \quad (4.25)$$

Let now $\tau \in \Gamma_-$ so that $\tau$ reads $\tau = t - \mathbf{i}\epsilon$ with $t \in [-\eta, \eta]$. We compute

$$\mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 \right) = -\omega t - \zeta \epsilon^2 t + \frac{\zeta}{3} t^3 + \frac{3\zeta}{2} \epsilon^2 t^2 - \frac{\zeta}{4} t^4 - \frac{\zeta}{4} \epsilon^4.$$

Recalling that $\eta \le \epsilon^5$ by assumption, we obtain that

$$\mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 \right) \le -\frac{\zeta}{4} \epsilon^4 + |\omega| \epsilon^5 + \zeta \epsilon^7 + \frac{3\zeta}{2} \epsilon^{12} + \frac{\zeta}{3} \epsilon^{15} \le -\frac{\zeta}{8} \epsilon^4, \quad (4.26)$$

thanks to our smallness assumption (4.12) on $\epsilon$. Thus, we get the estimate

$$\left| \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{-}} e^{n\tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau} \right| \le C e^{-n \frac{1-\alpha_r^2}{32\alpha_r^2} \epsilon^4} = C e^{-cn}.$$

There is of course a similar estimate on $\Gamma_+$. Since $n$ dominates $j_0$, one deduces that

$$\left| \mathcal{A}_r^n(j_0) - \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_\eta} e^{n\tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau} \right| \le C e^{-cn-cj_0},$$

for constants $C, c > 0$ that are independent of both $n$ and $j_0$.

Next, we complete the remaining integral along $\Gamma_\eta$ as follows:

$$\frac{1}{2\pi\mathbf{i}} \int_{\Gamma_\eta} e^{n\tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau} = \frac{1}{2\pi\mathbf{i}} \int_{\eta+i\mathbb{R}} e^{n\tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau} - \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_\eta^c} e^{n\tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau},$$

where $\Gamma_\eta^c := \{\eta + \mathbf{i}\theta \mid |\theta| > \epsilon\}$. The reader can verify indeed that the integral along the line $\eta + i\mathbb{R}$ converges. This is due to the form of the function $\varphi_r$ and the fact that $j_0$ is positive. We now estimate the integral along $\Gamma_\eta^c$ and show that it is a remainder term just like several other contributions that we have already estimated.

We keep the notation (4.24) and use the relation (4.25). For $\tau \in \Gamma_\eta^c$, we write $\tau = \eta + \mathbf{i}\theta$ with $|\theta| > \epsilon$, so that we have

$$\mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 \right) = -\omega \eta + \frac{\zeta}{3} \eta^3 - \frac{\zeta}{4} \eta^4 - \frac{\zeta}{4} \theta^4 - \zeta \eta \left( 1 - \frac{3}{2} \eta \right) \theta^2.$$

Since $0 < \eta \le \epsilon^5$ and $\epsilon$ is assumed to be small enough to satisfy (4.11) and (4.12), we have that

$$\begin{aligned} \mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 \right) &\le -\omega \eta + \frac{\zeta}{3} \eta^3 - \frac{\zeta}{4} \theta^4 - \frac{\zeta \eta}{2} \theta^2 \\ &\le -\frac{\zeta}{4} \epsilon^4 + |\omega| \epsilon^5 + \frac{\zeta}{3} \epsilon^{15} - \frac{\zeta \eta}{2} \theta^2 \le -\frac{\zeta}{8} \epsilon^4 - \frac{\zeta \eta}{2} \theta^2, \end{aligned}$$

and as a consequence, we obtain

$$\left| \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_\eta^c} e^{n\tau - j_0 \varphi_r(\tau)} \frac{d\tau}{\tau} \right| \le C e^{-n \frac{1-\alpha_r^2}{32\alpha_r^2} \epsilon^4} \int_{\epsilon}^{+\infty} e^{-n \frac{\zeta\eta}{2} \theta^2} d\theta \le C e^{-cn} \le C e^{-cn-cj_0},$$
where we have used once again that $n$ dominates $j_0$. As a partial summary, we have obtained that

$$ \left| \mathcal{A}_r^n(j_0) - \frac{1}{2\pi i} \int_{\eta+i\mathbb{R}} e^{n\tau-j_0\varphi_r(\tau)} \frac{d\tau}{\tau} \right| \le C e^{-cn-cj_0}, $$

for positive constants $C$ and $c$ that are independent of $n$ and $j_0$ and for integers that satisfy $\frac{n|\alpha_r|}{2} \le j_0 \le n$. Cauchy's formula shows that the value of the integral:

$$ \frac{1}{2\pi i} \int_{\eta+i\mathbb{R}} e^{n\tau-j_0\varphi_r(\tau)} \frac{d\tau}{\tau} $$

does not depend on $\eta > 0$ so choosing the parametrization $\tau = |\alpha_r|(\eta + i\theta)$ in the integral, we get:

$$ \frac{1}{2\pi i} \int_{\eta+i\mathbb{R}} e^{n\tau-j_0\varphi_r(\tau)} \frac{d\tau}{\tau} = \frac{1}{2\pi} \int_{\mathbb{R}} e^{-(j_0+n\alpha_r)(\eta+i\theta)} e^{j_0 \frac{1-\alpha_r^2}{6}(\eta+i\theta)^3} e^{j_0 \alpha_r \frac{1-\alpha_r^2}{8}(\eta+i\theta)^4} \frac{d\theta}{\eta+i\theta}. $$

Recalling the definitions (4.4b) and (4.5b), we may write the above integral as

$$ \frac{1}{2\pi i} \int_{\eta+i\mathbb{R}} e^{n\tau-j_0\varphi_r(\tau)} \frac{d\tau}{\tau} = \mathbf{A}_r \left( -j_0 + n|\alpha_r|, \frac{j_0}{|\alpha_r|} \right), $$

so that we have obtained the estimate:

$$ \left| \mathcal{A}_r^n(j_0) - \mathbf{A}_r \left( -j_0 + n|\alpha_r|, \frac{j_0}{|\alpha_r|} \right) \right| \le C e^{-cn-cj_0}. $$

With the definition (4.6b), Corollary A.7 in Appendix A can be rewritten as:

$$ \left| \mathbf{A}_r \left( -j_0 + n |\alpha_r|, \frac{j_0}{|\alpha_r|} \right) - \mathbf{A}_r \left( -j_0 + n |\alpha_r|, n \right) \right| \le C \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n), $$

for suitable constants $C$ and $c$, since we have assumed $\frac{n|\alpha_r|}{2} \le j_0 \le n$. We can combine the previous two estimates and obtain the estimate (4.23) that we were aiming at. This completes the analysis of case (i).

**Case (ii).** For $1 \le j_0 < \frac{n|\alpha_r|}{2}$, we rather use the residue theorem. In other words, as suggested in Figure 4.2, we close the contour $\Gamma_{in}$ by using the segment $\Gamma_{-\eta}$ and we thus get:

$$ \mathcal{A}_r^n(j_0) = 1 + \frac{1}{2\pi i} \int_{\Gamma_{-\eta}} e^{n\tau-j_0\varphi_r(\tau)} \frac{d\tau}{\tau}, $$

with $\Gamma_{-\eta} := \{-\eta + i\theta \mid |\theta| \le \varepsilon\}$. Along $\Gamma_{-\eta}$, for each $\tau = -\eta + i\theta$ with $|\theta| \le \varepsilon$, we keep the notation (4.24) and compute (recall (4.25)):

$$ \mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 \right) = \omega \eta - \frac{\zeta}{3} \eta^3 - \frac{\zeta}{4} \eta^4 - \frac{\zeta}{4} \theta^4 + \zeta \eta \left( 1 + \frac{3}{2} \eta \right) \theta^2, $$

where $\omega$ and $\zeta$ are defined in (4.24) and now satisfy

$$ \omega \in \left( -|\alpha_r|, -\frac{|\alpha_r|}{2} \right), \quad \text{and} \quad \zeta \in \left( 0, \frac{1-\alpha_r^2}{4|\alpha_r|} \right). $$
As a consequence, we have

$$ \mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 \right) \leq \eta \left( -\frac{|\alpha_r|}{2} + \zeta \left( 1 + \frac{3}{2} \varepsilon^5 \right) \varepsilon^2 \right) \leq -\frac{|\alpha_r|}{4} \eta, $$

thanks to our smallness assumption (4.13) on $\varepsilon$. Thus, we get

$$ \left| \frac{1}{2\pi i} \int_{\Gamma-\eta} e^{n\tau-j_0} \varphi_r(\tau) \frac{d\tau}{\tau} \right| \leq C e^{-cn-cj_0}, $$

which implies that

$$ |\mathcal{A}_r^n(j_0) - 1| \leq C e^{-cn-cj_0}, $$

for positive constants $C$ and $c$ that are independent of $n$ and $j_0$, and $1 \leq j_0 < \frac{n|\alpha_r|}{2}$.

Using now Corollary A.6 in Appendix A, we have the estimate:

$$ |1 - \mathbf{A}_r(-j_0 + n|\alpha_r|, n)| \leq C e^{-cn} \leq C e^{-cn-cj_0}, $$

where the second inequality comes from the fact that $n$ dominates $j_0$. Adding the two previous estimates, we get

$$ |\mathcal{A}_r^n(j_0) - \mathbf{A}_r(-j_0 + n|\alpha_r|, n)| \leq C e^{-cn-cj_0}. $$

This completes the analysis of case (ii). □

### 4.3.2 Estimates of the next order term $\mathcal{B}_r^n(j, j_0)$

We now focus our attention to the next term $\mathcal{B}_r^n(j, j_0)$ appearing in the decomposition (4.20). Our main result is the following.

**Lemma 4.3.** Let $n \geq 1$ and let $1 \leq j_0 \leq n$. Let $\mathcal{B}_r^n(j, j_0)$ be defined in (4.21). Then there exist some positive constants $C$ and $c$ that are uniform with respect to $n$, $j \in \mathbb{Z}$ and $j_0$ such that with the function $\mathbf{M}_r$ defined in (4.6b), there holds:

$$ |\mathcal{B}_r^n(j, j_0)| \leq C e^{-c|j|} \mathbf{M}_r(c, -j_0 + n|\alpha_r|, n) + C e^{-cn-c|j|-cj_0}. \quad (4.27) $$

*Proof.* Once again, we shall consider two different regimes identified in the previous section, namely:

(i) $\frac{n|\alpha_r|}{2} \leq j_0 \leq n$;

(ii) $1 \leq j_0 < \frac{n|\alpha_r|}{2}$.

From its definition, $\mathcal{B}_r^n(j, j_0)$ splits into two parts:

$$ \mathcal{B}_r^n(j, j_0) = \tilde{\mathcal{G}}_{2,r}^n(j, j_0) + \tilde{\mathcal{G}}_{3,r}^n(j, j_0), $$

with quantities $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$ and $\tilde{\mathcal{G}}_{3,r}^n(j, j_0)$ defined in (4.15) and (4.18). We shall estimate separately each of these two terms below for either case (i) or case (ii).
**Case (i).** We shall first focus on the term $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$ whose expression is given (see (4.15)) by:

$$ \tilde{\mathcal{G}}_{2,r}^n(j, j_0) = \gamma_j^r \times \frac{1}{2\pi i} \int_{\Gamma_{in}} \exp(n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)) \, d\tau, $$

where, at first, $\Gamma_{in}$ is a contour that joins $-\eta - i\varepsilon$ to $-\eta + i\varepsilon$ and that passes to the right of the origin (as depicted in Figure 4.1). Since we now integrate a holomorphic function (there is no longer a pole at the origin !), we shall feel free to deform $\Gamma_{in}$ and choose any contour that joins $-\eta - i\varepsilon$ to $-\eta + i\varepsilon$ and remains within the closed square $\bar{\mathbf{B}}_\varepsilon(0)$ on which $\Psi_r$ is a holomorphic function. We shall only focus on the integral that depends on $j_0$ and $n$ since we already know that the factor $\gamma_j^r$ satisfies an exponential estimate (see Proposition 3.3).

For $\frac{n|\alpha_r|}{2} \le j_0 \le n$, we may take $\Gamma_{in}$ as the union of the following paths:

$$ \Gamma_{in} = \Gamma_{-} \cup \Gamma_{+} \cup \Gamma_{0}, $$

where $\Gamma_0 := \{i\theta \mid |\theta| \le \varepsilon\}$ and $\Gamma_\pm$ are horizontal paths joining $-\eta \pm i\varepsilon$ to $\pm i\varepsilon$ (this is rather similar to what is depicted in Figure 4.2 except that we have shifted the right segment to the abscissa 0 instead of $+\eta$). Upon writing as usual now (see (4.24) and (4.25)):

$$ n\tau - j_0\varphi_r(\tau) + j_0\tau^5\Psi_r(\tau) = \frac{n}{|\alpha_r|} \left( -\omega\tau + \frac{\zeta}{3}\tau^3 - \frac{\zeta}{4}\tau^4 + \frac{j_0}{n}|\alpha_r|\tau^5\Psi_r(\tau) \right), $$

and noticing that for any $\tau \in \Gamma_\pm \subset \mathbf{B}_{\varepsilon_0}(0)$ there holds (see (4.10)):

$$ |\tau^5 \Psi_r(\tau)| \le C_0 (\varepsilon^5 + \eta^5), $$

we have for each $\tau \in \Gamma_\pm$:

$$ \begin{aligned} \operatorname{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 + \frac{j_0}{n} |\alpha_r| \tau^5 \Psi_r(\tau) \right) &\le -\frac{\zeta}{4} \varepsilon^4 + |\omega| \eta + \zeta \eta \varepsilon^2 + \frac{3\zeta}{2} \eta^2 \varepsilon^2 + |\alpha_r| C_0 (\varepsilon^5 + \eta^5) \\ &\le -\frac{\zeta}{4} \varepsilon^4 + (|\omega| + |\alpha_r| C_0) \varepsilon^5 + \zeta \varepsilon^7 + \frac{3\zeta}{2} \varepsilon^{12} + |\alpha_r| C_0 \varepsilon^{25} \\ &\le -\frac{\zeta}{8} \varepsilon^4, \end{aligned} $$

thanks to condition (4.12) on $\varepsilon$. As a consequence, we have

$$ \left| \frac{1}{2\pi i} \int_{\Gamma_{\pm}} \exp \left( n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right) d\tau \right| \le C e^{-cn} \le C e^{-cn-cj_0}, $$

since $n$ dominates $j_0$ in this regime.

For the remaining integral along $\Gamma_0$, we use the parametrization $\tau = i|\alpha_r|\theta$ and we thus get the expression:

$$ \begin{aligned} & \frac{1}{2\pi i} \int_{\Gamma_0} \exp \left( n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right) d\tau \\ = & \frac{|\alpha_r|}{2\pi} \int_{-\frac{\varepsilon}{|\alpha_r|}}^{\frac{\varepsilon}{|\alpha_r|}} \exp \left( i(-j_0 + n|\alpha_r|)\theta + i \frac{j_0}{|\alpha_r|} c_{3,r} \theta^3 - \frac{j_0}{|\alpha_r|} c_{4,r} \theta^4 + j_0 i \theta^5 |\alpha_r|^5 \Psi_r(i|\alpha_r|\theta) \right) d\theta. \end{aligned} $$
We can now apply Theorem A.2 from Appendix A, and obtain that there exists some small enough $\varepsilon > 0$ with $\varepsilon \le \varepsilon/|\alpha_r|$ some constants C and c such that for any $j_0, n \in \mathbb{N}^*$ with $\frac{n|\alpha_r|}{2} \le j_0 \le n$, there holds<fnref n="2" />:

$$ \left| \int_{-\varepsilon}^{\varepsilon} \exp \left( \mathbf{i}(-j_0 + n|\alpha_r|)\theta + \mathbf{i} \frac{j_0}{|\alpha_r|} c_{3,r} \theta^3 - \frac{j_0}{|\alpha_r|} c_{4,r} \theta^4 + j_0 \mathbf{i} \theta^5 |\alpha_r|^5 \Psi_r(\mathbf{i}|\alpha_r|\theta) \right) d\theta \right| \le C \mathbf{M}_r \left( c, -j_0 + n|\alpha_r|, \frac{j_0}{|\alpha_r|} \right). $$

Since we are in the regime $\frac{n|\alpha_r|}{2} \le j_0 \le n$, we deduce that we always have an upper estimate

$$ \mathbf{M}_r \left( c, -j_0 + n|\alpha_r|, \frac{j_0}{|\alpha_r|} \right) \le C \mathbf{M}_r(c, -j_0 + n|\alpha_r|, n), $$

so that the latter estimate also reads:

$$ \left| \int_{-\varepsilon}^{\varepsilon} \exp \left( \mathbf{i}(-j_0 + n|\alpha_r|)\theta + \mathbf{i} \frac{j_0}{|\alpha_r|} c_{3,r} \theta^3 - \frac{j_0}{|\alpha_r|} c_{4,r} \theta^4 + j_0 \mathbf{i} \theta^5 |\alpha_r|^5 \Psi_r(\mathbf{i}|\alpha_r|\theta) \right) d\theta \right| \le C \mathbf{M}_r(c, -j_0 + n|\alpha_r|, n). $$

The only remaining task is to control the integral with respect to $\theta$ on the two intervals $[-\varepsilon/|\alpha_r|, -\varepsilon]$ and $[\varepsilon, \varepsilon/|\alpha_r|]$. This is entirely similar to what we have already done on the segments $\Gamma_{\pm}$ so we feel free to skip the details. The final estimate on these segments is of exponential type with respect to both n and $j_0$ (that are of comparable sizes in case (i)). Combining all above estimates on the segments $\Gamma_{\pm}$ and on $\Gamma_0$, we have obtained the following estimate for $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$:

$$ \left| \tilde{\mathcal{G}}_{2,r}^n(j, j_0) \right| \le C e^{-c|j|} \mathbf{M}_r(c, -j_0 + n|\alpha_r|, n) + C e^{-cn-c|j|-c|j_0|}, $$

for suitable constants C and c and $\frac{n|\alpha_r|}{2} \le j_0 \le n$ (the integer $j \in \mathbb{Z}$ is arbitrary).

We finally turn our attention to the next term $\tilde{\mathcal{G}}_{3,r}^n(j, j_0)$ that enters the definition of $\mathcal{B}_r^n(j, j_0)$. We recall that $\tilde{\mathcal{G}}_{3,r}^n(j, j_0)$ is defined as follows:

$$ \tilde{\mathcal{G}}_{3,r}^n(j, j_0) = \mathcal{H}_j \times \Psi_r(0) \times j_0 \times \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau, $$

and we shall mainly focus our efforts on the above integral on the right-hand side since we already know that $\mathcal{H}_j$ decays exponentially with respect to $j$, and $\Psi_r(0)$ is just a constant factor. We still focus here on case (i), that is on the regime $\frac{n|\alpha_r|}{2} \le j_0 \le n$. It is important to observe that we need to absorb the factor $j_0$ that is unbounded.

Since we integrate a holomorphic function, we may use Cauchy's formula and deform the contour $\Gamma_{\text{in}}$. We therefore choose $\Gamma_{\text{in}}$ to be the union of the paths $\Gamma_0 := \{\mathbf{i}\theta \mid |\theta| \le \varepsilon\}$ and $\Gamma_{\pm}$ which are horizontal paths joining $-\eta \pm \mathbf{i}\varepsilon$ to $\pm \mathbf{i}\varepsilon$, just as we did above for the analysis of the term $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$. For the integrals

<footnote n="2">²With the notation of Theorem A.2, we use $x := -j_0 + n|\alpha_r|$ and $y := j_0/|\alpha_r|$. In case (i), we are in a regime where $|x|/y$ is bounded and $y$ is bounded from below so that we can tune the constant $C$.</footnote>
along $\Gamma_{\pm}$, we may use again the estimate (4.26) and combine with a uniform bound for the factor $\tau^4$. Since $n$ and $j_0$ are comparable in this first regime, we readily obtain the estimate:

$$ \left| j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_{\pm}} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau \right| \le C j_0 e^{-cn-cj_0} \le C e^{-cn-cj_0}. $$

For the remaining integral along $\Gamma_0$, we add and subtract to get:

$$ \begin{aligned} & j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_0} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau \\ &= j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\mathbf{i}\mathbb{R}} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau - j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_0^c} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau, \end{aligned} $$

with rather obvious notation. For $\tau = \mathbf{i}\theta \in \Gamma_0^c$, we have (see the definition (3.30)):

$$ \mathrm{Re} \left( n \tau - j_0 \varphi_r(\tau) \right) = -c j_0 \theta^4, $$

for some constant $c > 0$, so we have the straightforward estimate:

$$ \left| j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_0^c} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau \right| \le C j_0 \int_{\varepsilon}^{+\infty} \theta^4 e^{-cj_0 \theta^4} d\theta \le C j_0 e^{-cj_0} \le C e^{-cj_0}. $$

Since $j_0$ and $n$ are comparable in case (i), we can collect all the above estimates and already obtain the estimate:

$$ \left| \tilde{\mathcal{G}}_{3,r}^n(j, j_0) - \mathcal{H}_j \times \Psi_r(0) \times j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\mathbf{i}\mathbb{R}} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau \right| \le C e^{-cn-c|j|-cj_0}, $$

for suitable uniform constants $C$ and $c$, and integers $n$, $j_0$, $j$ that satisfy $\frac{n|\alpha_r|}{2} \le j_0 \le n$ and $j \in \mathbb{Z}$.

It thus only remains to estimate the integral over the imaginary axis for which we refer to the definition (A.49) in Appendix A of the function $\mathbf{G}_4$ (the constants $c_3$ and $c_4$ should be taken to be $c_{3,r}$ and $c_{4,r}$ since we deal here with the right state of the shock). Using the parametrization $\tau = \mathbf{i}|\alpha_r|\theta$ in the integral, we get the relation:

$$ j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\mathbf{i}\mathbb{R}} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau = |\alpha_r|^5 j_0 \mathbf{G}_4 \left( -j_0 + n|\alpha_r|, \frac{j_0}{|\alpha_r|} \right), $$

and we can then use the estimates provided by Theorem A.4 to get<fnref n="3" />:

$$ \left| j_0 \times \frac{1}{2\pi \mathbf{i}} \int_{\mathbf{i}\mathbb{R}} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau \right| \le C \mathbf{M}_r \left( c, -j_0 + n|\alpha_r|, \frac{j_0}{|\alpha_r|} \right) \le C \mathbf{M}_r (c, -j_0 + n|\alpha_r|, n), $$

since we have $\frac{n|\alpha_r|}{2} \le j_0 \le n$. Collecting all the above estimates for the various contributions in the decomposition of $\tilde{\mathcal{G}}_{3,r}^n(j, j_0)$, we end up with the estimate:

$$ \left| \tilde{\mathcal{G}}_{3,r}^n(j, j_0) \right| \le C e^{-c|j|} \mathbf{M}_r(c, -j_0 + n|\alpha_r|, n) + C e^{-cn-c|j|-cj_0}, $$

for $\frac{n|\alpha_r|}{2} \le j_0 \le n$, $j \in \mathbb{Z}$, and for suitable constants $C$ and $c$ that are uniform with respect to $n$, $j$ and $j_0$. This completes the analysis of case (i) by combining with the estimate for $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$.

<footnote n="3">³The fact that we consider the function $\mathbf{G}_4$ is crucial here since 4 is the first index where the gain in the estimates of Theorem A.4 is sufficient to absorb the factor $j_0$.</footnote>
**Case (ii).** In the case $1 \le j_0 \le \frac{n|\alpha_r|}{2}$, we simply take $\Gamma_{in} = \Gamma_{-\eta}$ (see Figure 4.2) for evaluating the integrals arising in the definition of both $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$ and $\tilde{\mathcal{G}}_{3,r}^n(j, j_0)$. This is legitimate because we integrate holomorphic functions on the closed square $\overline{\mathbf{B}}_\varepsilon(0)$ so we can apply Cauchy's formula. Reproducing similar computations as in the previous subsection, we get that for each $\tau = -\eta + \mathbf{i}\theta \in \Gamma_{-\eta}$ (with therefore $|\theta| \le \varepsilon$), there holds:

$$ \begin{aligned} \mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 + \frac{j_0}{n} |\alpha_r| \tau^5 \Psi_r(\tau) \right) &\le \eta \left( -\frac{|\alpha_r|}{2} + \zeta \left( 1 + \frac{3}{2} \varepsilon^5 \right) \varepsilon^2 + \frac{\alpha_r^2}{2} C_0 \varepsilon^{20} \right) \\ &\quad - \theta^4 \left( \frac{\zeta}{4} - |\alpha_r| C_0 |\theta| \right) \\ &\le -\frac{|\alpha_r|}{4} \eta, \end{aligned} $$

and similarly:

$$ \mathrm{Re} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 \right) \le -\frac{|\alpha_r|}{4} \eta, $$

where once again we have used conditions (4.11) and (4.12) on $\varepsilon$ and our choice for $\eta$. By applying the triangle inequality (with a uniform bound for $\tau \in \Gamma_{-\eta}$), we deduce that

$$ \left| \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{-\eta}} \exp(n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)) \, d\tau \right| \le C e^{-cn} \le C e^{-cn-cj_0}, $$

and

$$ \left| j_0 \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{-\eta}} \exp(n\tau - j_0 \varphi_r(\tau)) \tau^4 d\tau \right| \le C j_0 e^{-cn-cj_0} \le C e^{-cn-cj_0}, $$

where we use once again the fact that $n$ dominates $j_0$. As a consequence, using the exponential estimate of $\gamma_j^r$ from Proposition 3.3 and the fact that $\mathcal{H}_j$ is also exponentially decaying, we obtain the uniform exponential estimate:

$$ |\mathcal{R}_r^n(j, j_0)| C e^{-cn-c|j|-cj_0}, $$

for $1 \le j_0 \le \frac{n|\alpha_r|}{2}$ and $j \in \mathbb{Z}$. This completes the analysis of case (ii). $\square$

### 4.3.3 Estimates of the remainder term $\mathcal{R}_r^n(j, j_0)$

In this section, we shall prove some estimates on the remainder term $\mathcal{R}_r^n(j, j_0)$ whose expression is gathered in (4.22). We recall that $\mathcal{R}_r^n(j, j_0)$ is decomposed into $\mathcal{R}_r^n(j, j_0) = \mathcal{R}_{1,r}^n(j, j_0) + \mathcal{R}_{2,r}^n(j, j_0)$ with $\mathcal{R}_{1,r}^n(j, j_0)$ and $\mathcal{R}_{2,r}^n(j, j_0)$ defined in (4.16) and (4.19). In both (4.16) and (4.19), the path $\Gamma_{in}$ is any path joining $-\eta - \mathbf{i}\varepsilon$ to $-\eta + \mathbf{i}\varepsilon$ which remains in $\overline{\mathbf{B}}_\varepsilon(0)$ since the integrand is holomorphic with respect to $\tau$ on that set. We further recall that the sequence $(\mathcal{H}_j)_{j \in \mathbb{Z}}$ in (3.22) is exponentially decreasing and the sequence of holomorphic functions $(\Phi_{r,j})_{j \in \mathbb{Z}}$ satisfies the exponential bound stated in Proposition 3.3, uniformly with respect to $\tau \in \mathbf{B}_{\varepsilon_0}(0)$. Restricting to the smaller set $\overline{\mathbf{B}}_\varepsilon(0) \subset \mathbf{B}_{\varepsilon_0}(0)$, we thus have

$$ |\mathcal{H}_j| + |\Phi_{r,j}(\tau)| \le C e^{-c|j|}, $$

for all $j \in \mathbb{Z}$ and $\tau \in \overline{\mathbf{B}}_\varepsilon(0)$ with uniform constants $C, c > 0$. Our main result in this section is the following.
**Proposition 4.1.** There exist $C, c > 0$ such that for all $n \in \mathbb{N}^*$ and $(j, j_0) \in \mathbb{Z}^2$ such that $1 \le j_0 \le n$, one has:

$$|\mathcal{R}_r^n(j, j_0)| \le C \begin{cases} \frac{1}{n^{1/4}} \frac{e^{-c|j|}}{n^{1/3}} \exp\left(-c\left(\frac{j_0 - n|\alpha_r|}{n^{1/3}}\right)^{3/2}\right), & \text{if } j_0 - n|\alpha_r| \ge 0, \\ \frac{e^{-c|j|}}{n^{7/12}}, & \text{if } -n^{1/3} |\alpha_r| \le j_0 - n|\alpha_r| \le 0, \\ \frac{1}{n^{1/8}} \frac{e^{-c|j|}}{n^{1/2}} \exp\left(-c\left(\frac{|j_0 - n|\alpha_r|}{n^{1/2}}\right)^2\right), & \text{if } j_0 - n|\alpha_r| \le -n^{1/3} |\alpha_r|. \end{cases}$$

In particular, we have:

$$|\mathcal{R}_r^n(j, j_0)| \le C e^{-c|j|} \mathbf{M}_r(c, -j_0 + |\alpha_r|n, n),$$

with the function $\mathbf{M}_r$ defined in (4.6b).

*Proof.* We will mainly focus on the remainder term $\mathcal{R}_{1,r}^n(j, j_0)$ and briefly explain how to recover similar estimates for the second term $\mathcal{R}_{2,r}^n(j, j_0)$ in the last part of this section. We shall decompose the proof into several steps, which corresponds to different regimes for $\omega$, which we recall is defined as

$$\omega := \frac{j_0 - n|\alpha_r|}{n} \in (-|\alpha_r|, 1 - |\alpha_r|), \quad \text{when} \quad 1 \le j_0 \le n.$$

More precisely, we define the following three regimes:

$$(I) \quad |\omega| \le n^{-2/3}|\alpha_r|, \quad (II) \quad n^{-2/3}|\alpha_r| \le \omega \le 1 - |\alpha_r|, \quad (III) \quad -|\alpha_r| \le \omega \le -n^{-2/3}|\alpha_r|.$$

**Case (I) – Uniform bound.** For $|\omega| \le n^{-2/3}|\alpha_r|$, we provide a uniform bound for $\mathcal{R}_r^n(j, j_0)$ using classical results from oscillatory integrals (see Proposition A.1 in Appendix A for similar arguments).

**Lemma 4.4.** There exist constants $C, c > 0$ such that for all $n \ge 1$, $1 \le j_0 \le n$ with $|j - j_0| \le n$ one has

$$|\mathcal{R}_{1,r}^n(j, j_0)| \le \frac{C}{n^{7/12}} e^{-c|j|},$$

for $|\omega| \le n^{-2/3}|\alpha_r|$.

*Proof.* Let us first observe that for $|\omega| \le n^{-2/3}|\alpha_r|$, there holds:

$$\left| \frac{j_0}{n|\alpha_r|} - 1 \right| \le n^{-2/3},$$

so for $n \ge 2$, we get a uniform bound from below $j_0/n \ge c > 0$. The case $n = 1$ should be dealt with separately but it is far easier since we have $j_0 = 1$ and $j \in \{0, 1, 2\}$ so we only deal with finitely many integrals then. We shall therefore assume $n \ge 2$ from now on and use that the quantity $\zeta$ defined in (4.24) is uniformly bounded from below by a positive constant.

We take $\Gamma_{in}$ as the union of the following paths:

$$\Gamma_{in} = \Gamma_{-} \cup \Gamma_{+} \cup \Gamma_{0},$$
where $\Gamma_0 = \{\mathbf{i}\theta \mid |\theta| \le \varepsilon\}$ and $\Gamma_{\pm}$ are horizontal paths joining $-\eta \pm \mathbf{i}\varepsilon$ to $\pm \mathbf{i}\varepsilon$. Upon writing as usual now

$$n \tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) = \frac{n}{|\alpha_r|} \left( -\omega \tau + \frac{\zeta}{3} \tau^3 - \frac{\zeta}{4} \tau^4 + \frac{j_0}{n} |\alpha_r| \tau^5 \Psi_r(\tau) \right),$$

we have already proved that for each $\tau \in \Gamma_{\pm}$, there holds:

$$\mathrm{Re} \left( -\omega\tau + \frac{\zeta}{3}\tau^3 - \frac{\zeta}{4}\tau^4 + \frac{j_0}{n}|\alpha_r|\tau^5\Psi_r(\tau) \right) \leq -\frac{\zeta}{8}\varepsilon^8,$$

thanks to the smallness condition (4.12) on $\varepsilon$. As a consequence, we get

$$\left| \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\pm}} \exp(n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)) \tau \Phi_{r,j}(\tau) d\tau \right| \le C e^{-c|j|-c j_0}.$$

But since $|\omega| \le n^{-2/3}|\alpha_r|$, we readily get $e^{-j_0} \le C e^{-cn}$, from which we deduce that

$$\left| \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\pm}} \exp(n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)) \tau \Phi_{r,j}(\tau) d\tau \right| \le C \frac{e^{-c|j|}}{n^{7/12}}.$$

Along the remaining integral on the vertical segment $\Gamma_0$, we have

$$\frac{1}{2\pi\mathbf{i}} \int_{\Gamma_0} \exp(n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)) \tau \Phi_{r,j}(\tau) d\tau \\ = \frac{1}{2\pi} \int_{-\varepsilon}^{\varepsilon} \exp \left( \frac{n}{|\alpha_r|} \left[ -\omega \mathbf{i} \theta - \frac{\zeta}{3} \mathbf{i} \theta^3 - \frac{\zeta}{4} \theta^4 + \frac{j_0}{n} |\alpha_r| \mathbf{i} \theta^5 \Psi_r(\mathbf{i} \theta) \right] \right) \theta \Phi_{r,j}(\mathbf{i} \theta) d\theta.$$

We introduce two functions (that depend on $(j, j_0, n)$):

$$h(\theta) := \exp \left( -n\mathbf{i} \left( \frac{\omega}{|\alpha_r|} \theta + \frac{\zeta}{3|\alpha_r|} \theta^3 \right) \right), \quad g(\theta) := \exp \left( -n \frac{\zeta}{4|\alpha_r|} \theta^4 + n\mathbf{i} \frac{2\zeta\alpha_r^2}{1-\alpha_r^2} \theta^5 \Psi_r(\mathbf{i}\theta) \right) \theta \Phi_{r,j}(\mathbf{i}\theta).$$

Using [24, Lemma 3.1], we have the estimate

$$\left| \int_{-\varepsilon}^{\varepsilon} h(\theta) g(\theta) d\theta \right| \le \left( \sup_{x \in [-\varepsilon, \varepsilon]} \left| \int_{-\varepsilon}^{x} h(\theta) d\theta \right| \right) \left( \|g\|_{L^{\infty}([-\varepsilon, \varepsilon])} + \|g'\|_{L^1([-\varepsilon, \varepsilon])} \right). \quad (4.28)$$

By an application of the van der Corput Lemma, there exists a constant $C > 0$, independent of $\omega$ and $n$, such that$^4$

$$\forall x \in [-\varepsilon, \varepsilon], \quad \left| \int_{-\varepsilon}^{x} h(\theta) d\theta \right| \le \frac{C}{n^{1/3}}.$$

Furthermore, with our choice (4.11) of $\varepsilon > 0$, we have

$$\forall \theta \in [-\varepsilon, \varepsilon], \quad |g(\theta)| \le C |\theta| e^{-n \frac{\zeta}{8|\alpha_r|} \theta^4} e^{-c|j|}.$$

Differentiating the expression for $g(\theta)$, we also get the bound

$$\forall \theta \in [-\varepsilon, \varepsilon], \quad |g'(\theta)| \le C (1 + n|\theta|^4) e^{-n \frac{\zeta}{8|\alpha_r|} \theta^4} e^{-c|j|},$$

<footnote>$^4$This holds because the parameter $\zeta$ is uniformly bounded from below in the considered regime.</footnote>
such that we get that

$$ \|g'\|_{L^1([-\varepsilon, \varepsilon])} \le \frac{C}{n^{1/4}} e^{-c|j|}, $$

since $\zeta$ is uniformly positive in the considered regime. Using estimate (4.28), we arrive at the final bound

$$ |\mathcal{R}_{1,r}^n(j, j_0)| \le \frac{C}{n^{1/3+1/4}} e^{-c|j|}, $$

for some constants $C, c > 0$ independent of $n, j_0$ and $j$. This concludes the proof of the lemma. □

**Case (II) – Fast decaying tail.** We now turn our attention to the second regime (II) where $n^{-2/3}|\alpha_r| < \omega \le 1 - |\alpha_r|$ implying that necessarily $j_0 - n|\alpha_r| > 0$ where we expect to observe a fast decaying bound for $\mathcal{R}_{1,r}^n(j, j_0)$.

**Lemma 4.5.** There exist constants $C, c > 0$ such that for all $n \ge 1$, $1 \le j_0 \le n$ with $|j - j_0| \le n$ one has

$$ |\mathcal{R}_{1,r}^n(j, j_0)| \le C \frac{e^{-c|j|}}{n^{2/3}} \left( \frac{j_0 - n|\alpha_r|}{n^{1/3}} \right)^{-1/2} \exp \left( -c \left( \frac{j_0 - n|\alpha_r|}{n^{1/3}} \right)^{3/2} \right), $$

as long as $n^{-2/3}|\alpha_r| \le \omega \le 1 - |\alpha_r|$.

*Proof.* We first note that when $\omega > 0$, one has $\frac{1-\alpha_r^2}{2|\alpha_r|} < \zeta \le \frac{1-\alpha_r^2}{2\alpha_r^2}$ so $\zeta$ is uniformly positive. With the constant $C_0 > 0$ in (4.10) (associated to $\varepsilon_0$ given by Proposition 3.3), we choose $\omega_\varepsilon \in (0, 1 - |\alpha_r|)$ small enough such that the following inequalities are satisfied

$$ \omega_\varepsilon \le \frac{1 - \alpha_r^2}{2|\alpha_r|} \varepsilon^2, \quad \sqrt{\omega_\varepsilon} \le \frac{1}{8|\alpha_r|C_0} \left( \frac{1 - \alpha_r^2}{2|\alpha_r|} \right)^{3/2}, \quad \sqrt{\omega_\varepsilon} \le \frac{1}{3} \left( \frac{1 - \alpha_r^2}{2|\alpha_r|} \right)^{1/2}. \quad (4.29) $$

Next, we fix $n^{-2/3}|\alpha_r| \le \omega \le \omega_\varepsilon$. We introduce a family of parametrized curves (see<fnref n="5" /> Figure 4.3 for an illustration) indexed by $\omega$ as follows

$$ \Gamma_{-} := \{t - \mathbf{i}\varepsilon \mid t \in [-\eta, 0]\}, \quad \Gamma_{+} := \{-t + \mathbf{i}\varepsilon \mid t \in [0, \eta]\}, \quad \Gamma_{\omega} := \left\{\sqrt{\frac{\omega}{\zeta}} + \mathbf{i}\theta \mid \theta \in [-\varepsilon, \varepsilon]\right\}, $$

together with

$$ \Gamma_{>}^{\omega} := \left\{ t - \mathbf{i}\varepsilon \mid t \in [0, \sqrt{\frac{\omega}{\zeta}}] \right\}, \quad \Gamma_{<}^{\omega} := \left\{ \sqrt{\frac{\omega}{\zeta}} - t + \mathbf{i}\varepsilon \mid t \in [0, \sqrt{\frac{\omega}{\zeta}}] \right\}. $$

Along $\Gamma_\pm$, we have already proved that

$$ \left| \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\pm}} \exp \left( n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right) \tau \Phi_{r,j}(\tau) \, d\tau \right| \le C e^{-c|j|-c j_0}. $$

Next, we remark that since $0 < \omega \le 1 - |\alpha_r|$ we deduce that

$$ -1 \le -\frac{1}{\sqrt{1 - |\alpha_r|}} \left( \frac{j_0 - n |\alpha_r|}{n} \right)^{1/2}, $$

<footnote n="5">⁵The reader may compare our choice with the one made in [9] that corresponds to the parametrization of the Green's function for the Cauchy problem. See also Appendix A. The difference here is that we rather parametrize all curves in terms of the time frequency rather than with respect to the space frequency.</footnote>
<figure><img src="image_16.png" /><figcaption>Figure 4.3: In blue: the contour $\Gamma_{\text{in}}$ and its decomposition into five parts ($\Gamma_{-}$, $\Gamma_{>}^{\omega}$, $\Gamma_{\omega}$, $\Gamma_{>}^{\omega}$ and $\Gamma_{+}$) in the regime where $n^{-2/3}|\alpha_r| \le \omega \le 1 - |\alpha_r|$. The black bullets represent the end points of the contours.</figcaption></figure>

and thus

$$-j_0 = -n |\alpha_r| - (j_0 - n |\alpha_r|) \le -n |\alpha_r| - \frac{1}{\sqrt{1 - |\alpha_r|}} \left( \frac{j_0 - n |\alpha_r|}{n^{1/3}} \right)^{3/2}.$$

As a consequence, we have obtained

$$\left| \frac{1}{2\pi i} \int_{\Gamma_{\pm}} \exp \left( n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau) \right) \tau \Phi_{r,j}(\tau) d\tau \right| \le C e^{-cn-c|j|} \exp \left( -c \left( \frac{j_0 - n |\alpha_r|}{n^{1/3}} \right)^{3/2} \right),$$

which can be subsumed into our desired bound.

Next, we handle the contributions along $\Gamma_{>}^{\omega}$ and $\Gamma_{>}^{\omega}$. For example, in the former case, upon denoting $\Lambda_r(\tau) := n\tau - j_0\varphi_r(\tau) + j_0\tau^5\Psi_r(\tau)$, we have that for each $\tau = t - i\epsilon \in \Gamma_{>}^{\omega}$ with $t \in [0, \sqrt{\omega/\zeta}]$,

$$\begin{aligned}
\operatorname{Re} (\Lambda_r (t - i\epsilon)) &\le \frac{n}{|\alpha_r|} \left[ -\omega t + \frac{\zeta}{3} (t^3 - 3\epsilon^2 t) - \frac{\zeta}{4} (t^4 + \epsilon^4 - 6\epsilon^2 t^2) + |\alpha_r| C_0 (t^5 + \epsilon^5) \right] \\
&\le \frac{n}{|\alpha_r|} \left[ -t \left( \frac{2}{3}\omega + \zeta\epsilon^2 \left(1 - \frac{3}{2}\sqrt{\frac{\omega}{\zeta}}\right) \right) - t^4 \left( \frac{\zeta}{4} - |\alpha_r|C_0t \right) - \epsilon^4 \left( \frac{\zeta}{4} - |\alpha_r|C_0\epsilon \right) \right] \\
&\le \frac{n}{|\alpha_r|} \left[ -t \left( \frac{2}{3}\omega + \frac{\zeta}{2}\epsilon^2 \right) - \frac{\zeta}{8}\epsilon^4 \right],
\end{aligned}$$

since from (4.11) one has $8|\alpha_r|C_0\epsilon < \zeta$ and from (4.29) one has $\sqrt{\omega} < \sqrt{\zeta}/3$ and $8|\alpha_r|C_0\sqrt{\omega} \le \zeta^{3/2}$. Hence,

$$\begin{aligned}
\left| \frac{1}{2\pi i} \int_{\Gamma_{>}^{\omega}} \exp \left( n\tau - j_0\varphi_r(\tau) + j_0\tau^5\Psi_r(\tau) \right) \tau\Phi_{r,j}(\tau) d\tau \right| &\le C e^{-n\frac{\zeta\epsilon^4}{8|\alpha_r|} - c|j|} \int_0^{\sqrt{\frac{\omega}{\zeta}}} e^{-nt\frac{\zeta}{2|\alpha_r|}\epsilon^2} dt \\
&\le C e^{-nc-c|j|} \exp \left( -c \left( \frac{j_0 - n |\alpha_r|}{n^{1/3}} \right)^{3/2} \right).
\end{aligned}$$