From delayed minimization to the harmonic map heat equation

TL;DR

This paper rigorously proves the convergence of a microscopic adhesion model with delays to a macroscopic harmonic map heat equation, extending gradient flow techniques to handle delay terms and non-linear constraints.

Contribution

It introduces a novel mathematical framework extending gradient flow methods to delayed, constrained systems in cell motility modeling.

Findings

Proved convergence from microscopic adhesion dynamics to a harmonic map heat equation.

Extended gradient flow techniques to include delay terms in the energy functional.

Demonstrated convergence under weaker hypotheses despite the lack of traditional stability estimates.

Abstract

In the context of cell motility modelling and more particularly related to the Filament Based Lamelipodium Model [Manhart et al 2015 & 2017], this work deals with a rigorous mathematical proof of convergence between solutions of two problems : we start from a microscopic description of adhesions using a delayed and constrained vector valued equation with spacial diffusion and show the convergence towards the corresponding friction limit. The convergence is performed with respect to the bond characteristic lifetime $\varepsilon$ whose inverse is also proportional to the stifness of the bonds. The originality of this work is the extension of gradient flow techniques to our setting. Namely, the discrete finite difference term in the gradient flow energy is here replaced by a delay term which complicates greatly the mathematical analysis. Contrarily to the standard approach [Oelz SeMa 2011], compactness in time is not provided by the energy minimization process~: a series of past times are taken into account in our discrete energy. A supplementary equation on the time derivative is obtained requiring uniform estimate with respect to $\varepsilon$ of the Lagrange multiplier and provides compactness. Due to the non-linearity induced by the constraint, a specific stability estimate useful in our previous works, is not at hand here. Numerical simulations even showed that this estimate does not hold. Nevertheless, transposing our delay operator, we succeed in proving convergence under slightly weaker hypotheses. The result relies on a careful initial layer analysis, extending [Milisic Esaim Proc 2018] to the space dependent setting.

Equations

$${\mathbf{z}}_{\epsilon }(x,t)=\underset{w\in \mathcal{A}}{\mathrm{argmin}}{\mathcal{E}}_t(w),$$

$$\mathcal{A}:=\left\{\mathbf{w}\in {\mathbf{H}}^1(\Omega )\text{ s.t. }|\mathbf{w}(x){|}^2=1,\text{a.e.}x\in \Omega \right\}.$$

$${\mathcal{E}}_t(\mathbf{w}(\cdot )):=\frac{1}{2\epsilon }{\int }_{\Omega }{\int }_{{\mathbb{R}}_{+}}\frac{|\mathbf{w}(x)-{\mathbf{z}}_{\epsilon }(x,t-\epsilon a){|}^2}{\epsilon }{\rho }_{\epsilon }(x,t,a)dadx+\frac{1}{2}{\int }_{\Omega }|{\partial }_x\mathbf{w}{|}^{2}dx.$$

$$\{\begin{array}{rlr}& \epsilon {\partial }_t{\rho }_{\epsilon }+{\partial }_a{\rho }_{\epsilon }+{\zeta }_{\epsilon }{\rho }_{\epsilon }=0,& x\in \Omega ,a>0,t>0,\\ & {\rho }_{\epsilon }(x,a=0,t)={\beta }_{\epsilon }(x,t)\left(1-{\mu }_{0,\epsilon }(t,x)\right),& x\in \Omega ,a=0,t>0,\\ & {\rho }_{\epsilon }(x,a,t=0)={\rho }_I(x,a),& x\in \Omega ,a>0,t=0,\end{array}$$

$$\{\begin{array}{rlr}& {\mathcal{L}}_{\epsilon }-{\partial }_{xx}{\mathbf{z}}_{\epsilon }+{\lambda }_{\epsilon }{\mathbf{z}}_{\epsilon }=0,& \text{ a.e. }(x,t)\in \Omega \times (0,T),\\ & |{\mathbf{z}}_{\epsilon }(x,t)|=1& \text{ a.e.}(x,t)\in \Omega \times (0,T),\\ & {\partial }_x{\mathbf{z}}_{\epsilon }=0& \forall (x,t)\in \{0,1\}\times (0,T),\\ & {\mathbf{z}}_{\epsilon }(x,t)={\mathbf{z}}_p(x,t)& \text{ a.e. }(x,t)\in \Omega \times {\mathbb{R}}_{-},\end{array}$$

$$\{\begin{array}{rlr}& {\mu }_{1,0}{\partial }_t{\mathbf{z}}_0-{\partial }_{xx}{\mathbf{z}}_0-|{\partial }_x{\mathbf{z}}_0{|}^2{\mathbf{z}}_0=0,& \text{ a.e. }(x,t)\in \Omega \times (0,T),\\ & |{\mathbf{z}}_0(x,t)|=1& \text{ a.e.}(x,t)\in \Omega \times (0,T),\\ & {\partial }_x{\mathbf{z}}_0=0& \forall (x,t)\in \{0,1\}\times (0,T),\\ & {\mathbf{z}}_0(x,0)={\mathbf{z}}_p(x,0)& \text{ a.e. }(x,t)\in \Omega \times \{0\}.\end{array}$$

$$\{\begin{array}{rlr}& {\partial }_a{\rho }_0+{\zeta }_0{\rho }_0=0,& x\in \Omega ,a>0,t>0,\\ & {\rho }_0(x,a=0,t)={\beta }_0(x,t)\left(1-{\mu }_{0,0}(x,t)\right),& x\in \Omega ,a=0,t>0.\end{array}$$

$${\Vert f\Vert }_{X_T}:=\underset{(a,t)\in {\overline{Q}}_T}{\sup }{\Vert f(\cdot ,a,t)\Vert }_{L^1(\Omega )}.$$

$${\Vert f\Vert }_{Y_T}:={\int }_{Q_T}{\Vert f(\cdot ,a,t)\Vert }_{L^{\infty }(\Omega )}dadt.$$

$${\mathrm{D}}_t^{\tau }f:=\frac{f(x,a,t+\tau )-f(x,a,t)}{\tau },{\mathrm{D}}_a^{\alpha }f:=\frac{f(x,a+\alpha ,t)-f(x,a,t)}{\alpha },$$

$$U_T:=\left\{f\in Y_T\text{ s.t. }\underset{\sigma \to 0}{\mathrm{limsup}}\left({\Vert {\mathrm{D}}_a^{\sigma }f\Vert }_{Y_{(T-\sigma )}}+{\Vert {\mathrm{D}}_t^{\sigma }f\Vert }_{Y_{(T-\sigma )}}\right)<\infty \right\}$$

$${\Vert f\Vert }_{U_T}:={\Vert f\Vert }_{Y_T}+\underset{\sigma \to 0}{\mathrm{limsup}}\left({\Vert {\mathrm{D}}_a^{\sigma }f\Vert }_{Y_{(T-\sigma )}}+{\Vert {\mathrm{D}}_t^{\sigma }f\Vert }_{Y_{(T-\sigma )}}\right).$$

$$|{\partial }_{t}f|:=\underset{\begin{array}{c}\varphi \in \mathcal{D}(Q_T;L^1(\Omega ))\\ {\Vert \varphi \Vert }_{L_{a,t}^{\infty }{L}_x^1}\le 1\end{array}}{\sup }{\int }_{{\mathbb{R}}_{+}\times (0,T)}⟨f,{\partial }_t\varphi ⟩dadt,|{\partial }_{a}f|:=\underset{\begin{array}{c}\varphi \in \mathcal{D}(Q_T;L^1(\Omega ))\\ {\Vert \varphi \Vert }_{L_{a,t}^{\infty }{L}_x^1}\le 1\end{array}}{\sup }{\int }_{{\mathbb{R}}_{+}\times (0,T)}⟨f,{\partial }_a\varphi ⟩dadt$$

$${\Vert {\zeta }_{\epsilon }-{\zeta }_0\Vert }_{L_{x,a,t}^{\infty }}\to 0\text{and}{\Vert {\beta }_{\epsilon }-{\beta }_0\Vert }_{L_{x,t}^{\infty }}\to 0$$

$$0<{\zeta }_{\min }\le {\zeta }_{\epsilon }(x,a,t)\le {\zeta }_{\max }\text{and}0<{\beta }_{\min }\le {\beta }_{\epsilon }(x,t)\le {\beta }_{\max }$$

$$M\ge {\rho }_I(x,a)\ge 0,\text{ a.e. }(x,a)\in \Omega \times {\mathbb{R}}_{+},$$

$$0<{\int }_{{\mathbb{R}}_{+}}{\rho }_I(x,a)da<1$$

$$0<{\mu }_I:={\int }_{{\mathbb{R}}_{+}}{\rho }_I(x,a)da,\text{for a.e.}x\in \Omega ,$$

$${\int }_{{\mathbb{R}}_{+}}\underset{x\in \Omega }{\sup }{\rho }_I(x,a)a^{p}da<\infty ,\text{for }p\in \{0,1,2\},$$

$$\underset{\sigma \to 0}{\mathrm{limsup}}{\int }_{{\mathbb{R}}_{+}}\underset{x\in \Omega }{\sup }\left|{\mathrm{D}}_a^{\sigma }{\rho }_I(x,a)\right|da<\infty .$$

$$|{\mathbf{z}}_p(x,t_2)-{\mathbf{z}}_p(x,t_1)|\le C_{{\mathbf{z}}_p}(x)|t_2-t_1|,\forall (t_2,t_1)\in ({\mathbb{R}}_{-}{)}^2$$

$${\varrho }_{\epsilon ,i}^{n+1}(x):={\varrho }_{\epsilon ,i-1}^n(x)/\left(1+\frac{\Delta t}{\epsilon }{\zeta }_{\epsilon ,i}^{n+1}(x)\right),i\in {\mathbb{N}}^{\ast },n\in \mathbb{N}\cup \{-1\},$$

$${\varrho }_{\epsilon ,0}^{n+1}:={\varrho }_{\epsilon ,b}^{n+1}/\left(1+\frac{\Delta t}{\epsilon }{\zeta }_{\epsilon ,0}^{n+1}\right),n\in \mathbb{N}\cup \{-1\},$$

$${\varrho }_{\epsilon ,b}^{n+1}:={\beta }_{\epsilon }^{n+1}(1-{\mu }_{\epsilon }^{n+1}),{\mu }_{\epsilon }^{n+1}:=\sum _{i=0}^{\infty }{\varrho }_{\epsilon ,i}^{n+1}\Delta a.$$

$${\varrho }_{\epsilon ,0}^{n+1}:=\frac{{\beta }_{\epsilon }^{n+1}}{1+\Delta a({\beta }_{\epsilon }^{n+1}+{\zeta }_{\epsilon ,0}^{n+1})}\left(1-\sum _{i=1}^{\infty }{\varrho }_{\epsilon ,i}^{n+1}\Delta a\right).$$

$${\varrho }_{\epsilon ,i}^{-1}:=\frac{1}{\Delta a}{\int }_{i\Delta a}^{(i+1)\Delta a}{\rho }_I(a)da,\forall i\in \mathbb{N}.$$

$${\mu }_{\epsilon }^{n+1}+\Delta a\sum _{i=0}^{\infty }\frac{\Delta t}{\epsilon }{\zeta }_{\epsilon ,i}^{n+1}{\varrho }_{\epsilon ,i}^{n+1}={\mu }_{\epsilon }^n+{\varrho }_{\epsilon ,b}^{n+1}={\mu }_{\epsilon }^n+\Delta a{\beta }_{\epsilon }^{n+1}(1-{\mu }_{\epsilon }^{n+1}).$$

$${\rho }_{\epsilon ,\Delta }(x,a,t):=\sum _{i,j\in {\mathbb{N}}^2}{\varrho }_{\epsilon ,j}^n(x){\chi }_{(i\Delta a,(i+1)\Delta a)\times (j\Delta t,(j+1)\Delta t)}(a,t).$$

$${\mathbf{Z}}_{\epsilon }^n:=\underset{\mathbf{w}\in \mathcal{A}}{\mathrm{argmin}}{\mathcal{E}}_n(\mathbf{w}),$$

$${\mathcal{E}}_n(\mathbf{w}):=$$

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35Q92

1 Introduction

Cell motility is at heart of important biological/medical concerns (cancer metastasis, wound healing, etc.) [3]. Among models describing spontaneous motion of cells, two types appear : those who heuristically mimic macroscopic features and models based on a microscopic description that are in some sense homogenized. The Filament Based Lamelipodium Model (FBLM) [18] belongs to the second category and has reached a certain level of maturity [10, 11].

Adhesion mechanisms are some of the pillars of the FBLM and appear as friction terms. In the pioneering paper [18], they are obtained as formal limits of memory terms inside the Euler-Lagrange equations associated to a minimization process. This limit is interpreted as quasi-instantaneous with respect to a dimensionless parameter $\varepsilon$. Our work deals with the rigorous mathematical justification of this asymptotic.

Previously, we introduced simplifications that allowed to fully understand, from the mathematical point of view, either the delay model, for fixed $\varepsilon$, or its convergence when $\varepsilon$ tends to zero [12, 13, 14, 16]. More specifically in [12] and [13], we studied adhesions of a single point submitted to an external force and proved convergence. In [14] we proved that a non-linear fully coupled model could either have global solutions or, if the external load exceeds the microscopic adhesion capacity, blow-up could occur. More recently [16], we extended these results adding space dependent adhesion and diffusion.

In the previous works, Euler-Lagrange equations were considered and in some cases [16] this is equivalent to the minimization of a convex energy. Here we consider the minimization process for which the energy functional contains adhesion terms and the Dirichlet energy. But it is set pointwisely on the sphere almost everywhere, leading to a non-linear saddle point problem at the Euler-Lagrange level. The mathematical tools previously introduced extend only very partially to this new problem.

Gradient flow techniques provide existence of solutions for complicated possibly non-linear energies complemented with a finite difference term in time [2, 8]. Here the delay term in the energy could be considered as a generalization of such a finite difference. Except that it provides neither existence of solutions nor compactness in time. Thus we are forced to discretize the energy with respect to time and age. Here the age accounts for the delay. First, we obtain new energy estimates similar to the minimization principle in gradient flow theory (cf. Lemma 3.10), but there is then an extra amount of work in order to prove compactness i.e. boundedness of the time derivative in an appropriate space (cf. Proposition 3.20). This estimate is made possible thanks to a closed equation obtained for the discrete time derivative of the position. This equation appears when taking finite differences with respect to time of the Euler-Lagrange equations of the minimization process. In [16], another estimate of the elongation provided extra compactness useful in the asymptotic of the variational formulation that is not at hand here. The reason will be made more precise below. Thus, we were forced to transpose the delay term in the Euler-Lagrange equations on the test function and the density of population of bonds $\rho_{\varepsilon}$. This latter unknown is singular : $\partial_{t}\rho_{\varepsilon}$ is a measure that converges, when $\varepsilon$ goes to zero, to $\partial_{t}\rho_{0}$, the time derivative of the population of linkages and to a Dirac mass located near the origin in time for all ages. Since the problem is here space dependent, these results extend and complete the initial layer analysis performed in [15].

To be more specific, we denote by $\Omega:=(0,1)$. The vector position in $\mathbb{R}^{d}$ of the moving binding site, ${\mathbf{z}}_{\varepsilon}({x},t)$, minimizes at each time $t\geq 0$ an energy functional :

[TABLE]

where the minimization is performed on the set

[TABLE]

The energy is defined for every ${\boldsymbol{w}}\in{\mathcal{A}}$ as

[TABLE]

Past positions are given by the function ${\mathbf{z}}_{\varepsilon}({x},t)={\mathbf{z}}_{p}({x},t)$ for $t<0$. The age distribution $\rho_{\varepsilon}=\rho_{\varepsilon}({x},a,t)$ is the solution of the structured model :

[TABLE]

where $\mu_{0,\varepsilon}({x},t):=\int_{0}^{\infty}\rho_{\varepsilon}({x},{\tilde{a}},t)\,d{\tilde{a}}$ and the on-rate of bonds is a given function $\beta_{\varepsilon}$ times a factor, that takes into account saturation of the moving binding site with linkages. When the off-rate $\zeta_{\varepsilon}$ is a prescribed function, we say that the problem is weakly coupled : first one exhibits $\rho_{\varepsilon}$ solving (3) which then becomes the weight in (2).

First, we discretize in time and age the minimization process (1) and the age structured system (3). For the transport problem (3) we use $i)$ the upwind scheme inside the domain, $ii)$ an implicit discretization of the off-rates, $iii)$ the non-local term is discretized using a piecewise constant approximation. This step provides, as in the gradient flow case (see for instance minimizing movements chap. 2 [2]), existence of a discrete pair of solutions $((\varrho^{n}_{\varepsilon,j})_{j\in\mathbb{N}},{\mathbf{Z}}^{n}_{\varepsilon})_{n\in\mathbb{N}}$. Then thanks to compactness arguments, we pass to the limit with respect to the discretization parameter ${\Delta a}$, and prove that there exists a unique couple $(\rho_{\varepsilon},{\mathbf{z}}_{\varepsilon})$. The bond population density $\rho_{\varepsilon}$ solves (3), whereas ${\mathbf{z}}_{\varepsilon}$ satisfies, almost everywhere in $(0,T)$, the weak formulation associated to the Euler-Lagrange equation :

[TABLE]

where ${\boldsymbol{\mathcal{L}}}_{\varepsilon}({x},t):=\int_{\mathbb{R}_{+}}\rho_{\varepsilon}({x},a,t)({\mathbf{z}}_{\varepsilon}({x},t)-{\mathbf{z}}_{\varepsilon}({x},t-\varepsilon a))/\varepsilon da$ and $\lambda_{\varepsilon}$ is the Lagrange multiplier associated to the constraint $|{\mathbf{z}}_{\varepsilon}({x},t)|=1$. We prove that when $\varepsilon$ goes to zero, $(\rho_{\varepsilon},{\mathbf{z}}_{\varepsilon})$ the solutions of the previous minimization problem converge to $(\rho_{0},{\mathbf{z}}_{0})$. These solve the limit problems reading :

• •

Find ${\mathbf{z}}_{0}\in C^{0}_{t}([0,T];{\mathbf{H}}^{1}_{x}({\Omega}))\cap H^{1}_{t}((0,T);{\mathbf{L}}^{2}_{x}({\Omega}))$ being the weak solution of the problem

[TABLE]

The convergence of ${\mathbf{z}}_{\varepsilon}$ towards ${\mathbf{z}}_{0}$ holds strongly in $C({\Omega}\times[0,T])$.

• •

The function $\mu_{k,0}:=\int_{{\mathbb{R}_{+}}}a^{k}\rho_{0}({x},a,t)\,da$ represents the moment of order $k$ of $\lim_{\varepsilon\to 0}\rho_{\varepsilon}=:\rho_{0}$ which solves in $C([0,T];L^{1}_{a}({\mathbb{R}_{+}};L^{\infty}_{x}({\Omega})))$

[TABLE]

The solution $(1+a)\rho_{\varepsilon}$ converges to $(1+a)\rho_{0}$ in $L^{1}({\mathbb{R}_{+}}\times(0,T);L^{\infty}_{x}({\Omega}))$ strong.

The article is structured as follows. In Section 2, we list the hypotheses used throughout the paper and set notations. In Section 3, we detail the discrete minimization process in age and time providing the piecewise solutions $((\varrho^{n}_{\varepsilon,j})_{j\in\mathbb{N}},{\mathbf{Z}}^{n}_{\varepsilon})_{n\in\mathbb{N}}$. In the same section, we provide stability estimates in the appropriate functional spaces. We underline that most of the results obtained therein are uniform with respect to $\varepsilon$ and the discretisation step $\Delta a$ : the same properties can be extended to the continuous model for fixed $\varepsilon$. This leads to study first this latter limit for $\varepsilon$ fixed and $\Delta a$ going to 0. This is done in Section 4. Then when $\varepsilon$ tends to zero, we prove, in Section 5, that indeed convergence occurs towards the limit heat harmonic map equation (5) . As $\partial_{t}{\mathbf{z}}_{\varepsilon}$ converges weakly in $L^{2}({\Omega}\times(0,T))$ and $\rho_{\varepsilon}$ converges strongly in $L^{1}({\mathbb{R}_{+}}\times(0,T);L^{\infty}({\Omega}))$, it is not possible to obtain directly the convergence of the delay term ${\boldsymbol{\mathcal{L}}}_{\varepsilon}({x},t)$ towards $\mu_{1,0}\partial_{t}{\mathbf{z}}_{0}$. Instead, as mentioned above, we transpose the delay operator on a test function and on $\rho_{\varepsilon}$, and then we pass to the limit with respect to $\varepsilon$.

2 Notations and hypotheses

We set ${Q_{T}}:=\mathbb{R}^{*}_{+}\times(0,T)$ and ${\overline{Q}_{T}}:={\mathbb{R}_{+}}\times[0,T]$. As stated in the introduction we consider a one dimensional space domain $\Omega:=(0,1)$. We set as well ${{\mathfrak{Q}}_{T}}:={\Omega}\times{\mathbb{R}_{+}}\times(0,T)$.

We define $X_{T}:=C^{0}({\overline{Q}_{T}};L^{1}(\Omega))$ to be the Banach space of continuous functions in age and time whose $L^{1}$ norm in space goes to zero when $a$ goes to infinity. We endow $X_{T}$ with the norm :

[TABLE]

$X_{T}$ is a Banach space [20]. It is also a closed subspace of $Z_{T}:=C_{b}({\overline{Q}_{T}};L^{1}({\Omega}))$ which is a non-separable Banach space. We define $Y_{T}:=L^{1}({Q_{T}};L^{\infty}(\Omega))$ which is also a Banach space endowed with the corresponding norm :

[TABLE]

We denote the discrete differences as

[TABLE]

and we define the space of Banach valued functions

[TABLE]

and one endows $U_{T}$ with the norm :

[TABLE]

If the same space is set on a time interval $(t_{1},t_{2})$ then the notation $U_{(t_{1},t_{2})}$ is well understood. In the rest of the paper we abbreviate the notation of function spaces writing the subscripts $t$ for function spaces on $t\in[0,T]$ and the subscript ${x}$ for function spaces on ${x}\in{\Omega}$, for instance $C_{t}L^{1}_{a}L^{\infty}_{x}$ denotes $C([0,T];L^{1}({\mathbb{R}_{+}};L^{\infty}({\Omega})))$.

One should notice that $U_{T}$ is in fact the space of functions of bounded variation with values in a Banach space $BV({\mathbb{R}_{+}}\times(0,T);L^{\infty}({\Omega}))$. The generic space $BV((0,T);Z^{*})$ is presented and studied in a very detailed way in [7], where $Z^{*}$ is the dual space of a Banach space $Z$. Since $L^{\infty}({\Omega})$ is the dual space of $L^{1}({\Omega})$ we are exactly in this framework. The semi norms with the discrete derivatives coincide with the total variation of the $L^{\infty}$-valued Radon measures corresponding to the time/age derivative *i.e. *

[TABLE]

where the brackets $\langle\cdot,\cdot\rangle$ denote the $L^{\infty}({\Omega}),L^{1}({\Omega})$ duality in the space variable. The proof of this equivalence can be found in [6] p. 12 in the proof of Theorem 1.7.1 and is easily extendable to the Banach valued case presented in [7].

Assumptions 2.1

The dimensionless parameter $\varepsilon>0$ is assumed to induce two families of chemical rate functions that satisfy:

1. (i)

For every $\varepsilon\geq 0$, the function $\beta_{\varepsilon}$ belongs to $W^{1,\infty}(Q_{T})$ and the off-rate $\zeta_{\varepsilon}$ is s.t. $\zeta_{\varepsilon}\in W^{1,\infty}(\Omega\times{\mathbb{R}_{+}}\times[0,T])$ moreover itr holds that

[TABLE]

as ${\varepsilon}\to 0$. 2. (ii)

We also assume that there are upper and lower bounds such that

[TABLE]

for all $\varepsilon>0$, ${x}\in\Omega$, $a\geq 0$ and $t>0$.

The initial data for the density model (3) satisfies some hypotheses that we sum up here.

Assumptions 2.2

The initial condition $\rho_{I}\in L^{\infty}_{{x},a}(\Omega\times{\mathbb{R}_{+}})$ satisfies

• •

positivity and boundedness : there exists $M>\beta_{\max}$, s.t.

[TABLE]

moreover, one has also that the total initial population satisfies

[TABLE]

for almost every ${x}\in\Omega$.

• •

boundedness from below of the zero order moment,

[TABLE]

• •

initial integrability with respect to the limit problem :

[TABLE]

• •

the derivative with respect to age satisfies as well :

[TABLE]

Concerning the minimization problem (1), we assume

Assumptions 2.3

The past data satisfies :

1. i)

for every time $t\leq 0$, we assume that ${\mathbf{z}}_{p}(\cdot,t)$ is in ${\mathcal{A}}$, 2. ii)

there exists a Lipschitz constant which is $L^{2}$ in space s.t. :

[TABLE]

for a.e. ${x}\in{\Omega}$ where $C_{{\mathbf{z}}_{p}}({x})\in L^{2}(\Omega)$.

3 Existence of minimizers and a priori estimates : the discrete scheme

We discretize both (3) and the minimization process (1) in time and age, but not in space. We set $\Delta a$ a small parameter denoting the age discretization step, while the time step satisfies the CFL condition $\Delta t=\varepsilon\Delta a$. This provides $N:=\lfloor T/\Delta t\rfloor$, the number of times steps. We solve :

• •

for the $\rho_{\varepsilon}$ model, we use a first order upwind scheme and treat the source term implicitly, so we define inside the mesh

[TABLE]

while on the boundary we set

[TABLE]

where

[TABLE]

This definition provides explicitly $\varrho^{n+1}_{\varepsilon,0}$,

[TABLE]

The initial condition is defined as

[TABLE]

The zero order moment ${\mu}_{\varepsilon}^{n+1}:=\Delta a\sum_{i\in\mathbb{N}}\varrho^{n+1}_{\varepsilon,i}$ can be expressed in an inductive way :

[TABLE]

We define a piecewise constant function

[TABLE]

• •

whereas the minimization process is performed for each $n\in\mathbb{N}$

[TABLE]

where the discrete energy functional reads :

[TABLE]

for all $n\in\mathbb{N}$ and ${{\mathbf{Z}}}_{p}^{i}:=\int_{i\Delta t}^{(i+1)\Delta t}{\mathbf{z}}_{p}({x},t)dt/\Delta t$ for all $i\in\mathbb{Z}$, $i<0$, and we set ${\mathbf{Z}}^{n}_{\varepsilon}={{\mathbf{Z}}}_{p}^{n}$ for every $n<0$. We define the piecewise constant function

[TABLE]

The piecewise linear extension reads :

[TABLE]

where $\delta{\mathbf{Z}}^{n+{\frac{1}{2}}}_{\varepsilon}:={\mathbf{Z}}^{n+1}_{\varepsilon}-{\mathbf{Z}}^{n}_{\varepsilon}$, while in what follows we denote as well $\delta\varrho^{n}_{\varepsilon,i+{\frac{1}{2}}}:=\varrho^{n}_{\varepsilon,i+1}-\varrho^{n}_{\varepsilon,i}$ an so on.

3.1 Positivity and convergence of the discrete solution $\rho_{\varepsilon,\Delta}$

From Lemma 3.1 to Theorem 3.8, we extend results from previous works [12, 16, 15] to the discrete case. When needed, we characterize also some properties of $\rho_{\varepsilon}$, the continuous solution of (3).

Lemma 3.1**.**

For almost every $x\in{\Omega}$, under the CFL condition $\Delta t=\varepsilon{\Delta a}$, under hypotheses 2.1, if $\varrho^{-1}_{\varepsilon,i}\geq 0$ and ${\mu}_{\varepsilon}^{-1}\leq 1$ then

[TABLE]

Moreover if there exists a constant $0<\mu_{0,\min}<\min({\mu}_{\varepsilon}^{-1},\beta_{\min}/(\beta_{\min}+\zeta_{\max}))$, then ${\mu}_{\varepsilon}^{n}>\mu_{0,\min}$ for every $n\in\mathbb{N}$,

Proof 3.2**.**

The first result is proved by induction : by hypothesis, the claim is true for $k=-1$. We assume that for $k=n$, $\varrho^{n}_{\varepsilon,i}\geq 0$ and ${\mu}_{\varepsilon}^{n}\in[0,1]$. Since $\varrho^{n}_{\varepsilon,i}\geq 0$ for $i\in\mathbb{N}$, it is straightforward that $\varrho^{n+1}_{\varepsilon,i}\geq 0$ for all $i\in\mathbb{N}^{*}$. Thanks to (10), one writes :

[TABLE]

which gives :

[TABLE]

which rearranging terms on both sides provides

[TABLE]

This shows that $1-{\mu}_{\varepsilon}^{n+1}\geq 0$ since the coefficient in front of $(1-{\mu}_{\varepsilon}^{n+1})$ is always positive definite. In turn one concludes that $\varrho^{n+1}_{\varepsilon,0}\geq 0$.

We prove the last claim by induction, under the hypothesis on $\mu_{0,\min}$, the claim is true for $k=-1$. We suppose that the claim is true for $k=n$. Using (10) gives

[TABLE]

where the latter inequality holds since $\mu_{0,\min}\leq\beta_{\min}/(\beta_{\min}+\zeta_{\max})<1$. Because the right hand side is strictly positive, so is the left hand side. This shows the statement for $k=n+1$, and the recursion is complete.

Using the same Lyapunov functional ${\mathcal{H}}[u]:=\int_{\mathbb{R}_{+}}|u(a)|da+\left|\int_{\mathbb{R}_{+}}u(a)da\right|$, as in [12, 16], one proves that

Proposition 3.3**.**

Under hypotheses 2.1 and 2.2, there exists a unique solution $\rho_{\varepsilon}\in Y_{T}$, solving (3). Moreover :

[TABLE]

where the constant is independent of $\varepsilon$ and on $\tau$.

Proof 3.4**.**

For the existence and uniqueness part, one proceeds as in Theorem 3.1 in [16] : as ${x}$ is a mute parameter, for a.e. ${x}\in{\Omega}$, there exists a solution $\rho_{\varepsilon}({x},\cdot,\cdot)\in C_{t}([0,T];L^{1}_{a}({\mathbb{R}_{+}}))$. Then using Duhamel’s formula in order to commute the supremum with respect to ${x}$ with the integrals, one obtains the result in $Y_{T}$. Combining results from the proof of Lemma 5.1. p. 16 [16] and from Theorem 3.2 [15], one gets :

[TABLE]

Indeed, again, since ${x}$ is only a mute parameter, one obtains easily that

[TABLE]

which then integrated in time and taking the ess-sup on $\Omega$ proves this first step. Then we use the method of characteristics and write :

[TABLE]

where ${\mathcal{R}}_{\tau}({x},a,t):={D_{t}^{\tau}}\zeta_{\varepsilon}({x},a,t)\rho_{\varepsilon}({x},a,t)$. Now we define $q(a,t):={\operatorname*{ess\,sup}\limits_{{x}\in\Omega}\,}\left|{D_{t}^{\tau}}\rho_{\varepsilon}({x},a,t)\right|$. One has

[TABLE]

then

[TABLE]

For the term $I_{1}$, one has ${D_{t}^{\tau}}\rho_{\varepsilon}(x,0,t)=({D_{t}^{\tau}}\beta_{\varepsilon})(1-\mu_{0,\varepsilon})-\beta_{\varepsilon}{D_{t}^{\tau}}\mu_{0,\varepsilon}$ which gives

[TABLE]

so that

[TABLE]

These two estimates guarantee that $\int_{0}^{T}\int_{0}^{\frac{t}{\varepsilon}}q(a,t)dadt<C.$ In a similar way one writes that

[TABLE]

the latter term being under control, we focus on the first one, that we denote $I_{3}$.

[TABLE]

Since $\rho_{\varepsilon}$ is bounded uniformly in space and with respect to $\varepsilon$, the first term $I_{3,1}$ is smaller than $\exp(-\zeta_{\min}t/\varepsilon)M/\varepsilon$. Then using the method of characteristics, one splits $I_{3,2}$ in two parts :

[TABLE]

This shows that $\int_{0}^{T}\int_{\frac{t}{\varepsilon}}^{\infty}q(a,t)dadt<C$ which ends the proof.

Then using standard a priori estimates provides in a similar manner as in the previous proof :

Proposition 3.5**.**

Under the previous hypotheses, one has as well that

[TABLE]

where the constant is uniform with respect to $\varepsilon$. This result together with the previous proposition shows that $\rho_{\varepsilon}\in U_{T}$ uniformly with respect to $\varepsilon$.

One defines $C^{n}_{j}:=(j\Delta a,(j+1)\Delta a)\times(n\Delta t,(n+1)\Delta t)$, and one sets

[TABLE]

where $\rho_{\varepsilon}$ is the exact solution of (3) and

[TABLE]

in the same way one defines $\overline{\mu}_{\varepsilon}^{n}=\int_{n\Delta t}^{(n+1)\Delta t}\mu_{0,\varepsilon}({x},t)dt/\Delta t$, and so on. With these notations, we compute error estimates for the upwind scheme :

Lemma 3.6**.**

Under the same hypotheses as above, if $\rho_{\varepsilon}\in U_{T}$ solves (3), and $\rho_{\varepsilon,\Delta}$ is its piecewise constant approximation computed using the upwind scheme (8) with the non-local boundary term (9), then one has

[TABLE]

Proof 3.7**.**

Using the method of characteristics one gets :

[TABLE]

for all $j\geq 0$. In a similar fashion one derives for $n\geq 1$

[TABLE]

while if $n=0$,

[TABLE]

Setting $E^{n}=\Delta a\sum_{j\in\mathbb{N}}|\varrho^{n}_{\varepsilon,j}-\overline{\varrho}^{n}_{\varepsilon,j}|$, the previous estimates give for $n\geq 1$

[TABLE]

and

[TABLE]

where $\alpha:=1/(1+\Delta a\zeta_{\min})$ and by definition $E^{-1}=0$. Combining these estimates leads to

[TABLE]

which gives the first result. Using similar arguments as in Lemma B.5, one can show that

[TABLE]

which gives the second result.

Theorem 3.8**.**

Under hypotheses 2.1 and 2.2, one has

[TABLE]

strongly in $Y_{T}$ when $\Delta a$ goes to zero for $\varepsilon$ fixed.

3.2 Existence, uniqueness and stability of the discrete solution ${\mathbf{z}}_{\varepsilon,\Delta}$

Existence of minimizers relies on the convexity of the Dirichlet norm and is standard as the few properties listed below (see for instance Lemma 1 and 2, p. 973 [17]).

Theorem 3.9**.**

Under hypotheses 2.1, 2.2, 2.3, for every $n\geq 0$ there exists a minimizer ${\mathbf{Z}}^{n}_{\varepsilon}\in{\mathcal{A}}$ of (11), i.e. there exists a minimizing subsequence $({\mathbf{Z}}^{n,k}_{\varepsilon})_{k\in\mathbb{N}}$ s.t. as $k\to\infty$,

${\mathbf{Z}}^{n,k}_{\varepsilon}\rightharpoonup{\mathbf{Z}}^{n}_{\varepsilon}$* weak in ${\mathbf{H}}^{1}({\Omega})$,* 2. 2)

${\mathbf{Z}}^{n,k}_{\varepsilon}\to{\mathbf{Z}}^{n}_{\varepsilon}$* strong in ${\mathbf{L}}^{2}({\Omega})$,* 3. 3)

${\mathbf{Z}}^{n,k}_{\varepsilon}\to{\mathbf{Z}}^{n}_{\varepsilon}$* a.e. ${x}\in{\Omega}$,* 4. 4)

${\mathbf{Z}}^{n}_{\varepsilon}\in{\mathcal{A}}$* and thus ${\mathbf{Z}}^{n}_{\varepsilon}\neq 0$.*

A way to insure convergence, when $\varepsilon$ or ${\Delta a}$ go to zero, is to obtain some control on a discrete time derivative of ${\mathbf{z}}_{\varepsilon,\Delta}$, typically an ${\mathbf{L}}^{2}_{{x},t}$-bound is obtained in the case of a classical gradient flow directly from the minimization principle (cf Appendix in [19] and references therein). Here the result is less immediate : first, in the next lemma, we obtain a dissipation term in the energy estimates. These estimates provide a uniform bound on the dissipation term. It then appears as a source term in a closed equation (15), on $\delta{\mathbf{Z}}^{n+{\frac{1}{2}}}_{\varepsilon}$ that finally provides these key estimates (cf. Proposition 3.20).

Lemma 3.10**.**

If $(\varrho^{n}_{\varepsilon,i})_{(i,n)\in\mathbb{N}^{2}}$ and $({\mathbf{Z}}^{n}_{\varepsilon})_{n\in\mathbb{N}}$, are defined as above, one has :

[TABLE]

where the dissipation term reads :

[TABLE]

and we denote by ${\mathbf{U}}^{n}_{\varepsilon,j}$ the discrete elongation variable for $(j,n)\in\mathbb{N}^{2}$. The generic constant $C$ in (12) is independent either of $\varepsilon$ or ${\Delta a}$.

Proof 3.11**.**

By definition of the minimization process, one has

[TABLE]

since ${\mathbf{Z}}^{n+1}_{\varepsilon}$ minimises the energy at time step $t=(n+1)\Delta t$. This reads

[TABLE]

Changing the indices in the first summation of the latter right hand side provides

[TABLE]

for all $n\in\mathbb{N}$. In the last estimates we used the convexity of the square function, writing

[TABLE]

where $j\geq 1$, while for $j=0$, one has simply $\left({\mathbf{U}}^{n}_{\varepsilon,0}\right)^{2}\leq\left(\delta{\mathbf{Z}}^{n-{\frac{1}{2}}}_{\varepsilon}\right)^{2}/(2\varepsilon^{2})$. For ${\mathbf{Z}}^{0}_{\varepsilon}$, one has simply that

[TABLE]

Using (7), one has that for almost every ${x}\in{\Omega}$ and $j>1$

[TABLE]

Moreover one notices that $\varrho^{0}_{\varepsilon,j}\leq\varrho^{-1}_{\varepsilon,j-1}$ for $j\geq 1$. Together these facts allow to give a bound on ${\mathcal{E}}_{0}({\mathbf{Z}}^{0}_{\varepsilon})$ uniform with respect to $\varepsilon$ and ${\Delta a}$ :

[TABLE]

For a.e. ${x}\in{\Omega}$, we denote by ${\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}({x}):=\Delta a\sum_{j\in\mathbb{N}}\varrho^{n}_{\varepsilon,j}{\mathbf{U}}^{n}_{\varepsilon,j}.$

Lemma 3.12**.**

For every time $t^{n}=n\Delta t$, ${\mathbf{Z}}^{n}_{\varepsilon}$ solves :

[TABLE]

for all ${\boldsymbol{v}}\in{\mathbf{H}}^{1}({\Omega})$, and ${\lambda^{n}_{\varepsilon}}({x}):=-{\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}\cdot{\mathbf{Z}}^{n}_{\varepsilon}-|\partial_{x}{\mathbf{Z}}^{n}_{\varepsilon}|^{2}$, is a $L^{1}({\Omega})$ function.

Proof 3.13**.**

We take ${\boldsymbol{v}}\in{\mathbf{H}}^{1}({\Omega})$, and set

[TABLE]

because ${\mathbf{Z}}^{n}_{\varepsilon}\in{\mathcal{A}}$ for a $\tau$ small enough $\left|{\mathbf{Z}}^{n}_{\varepsilon}+\tau{\boldsymbol{v}}\right|$ is strictly positive and bounded, thus on this interval ${\boldsymbol{v}}(\tau)\in{\mathcal{A}}$. As ${\mathbf{Z}}^{n}_{\varepsilon}$ minimizes ${\mathcal{E}}_{n}$, $i(\tau):={\mathcal{E}}_{n}({\boldsymbol{v}}(\tau))$ admits a minimum in $\tau=0$. This leads to $i^{\prime}(0)=0$, as $\partial_{\tau}{\boldsymbol{v}}(0)=({\rm I}_{d}-{\mathbf{Z}}^{n}_{\varepsilon}\otimes{\mathbf{Z}}^{n}_{\varepsilon}){\boldsymbol{v}}$, this gives

[TABLE]

where the parentheses denote the ${\mathbf{L}}^{2}({\Omega})$ scalar product and ${\rm I}_{d}$ the identity matrix in $\mathbb{R}^{d}$. As $\partial_{x}{\mathbf{Z}}^{n}_{\varepsilon}\cdot{\mathbf{Z}}^{n}_{\varepsilon}=0$ for almost every ${x}\in{\Omega}$, the previous expression transforms into

[TABLE]

for all ${\boldsymbol{v}}\in{\mathbf{H}}^{1}({\Omega})$. Denoting ${\lambda^{n}_{\varepsilon}}:=-{\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}\cdot{\mathbf{Z}}^{n}_{\varepsilon}-|\partial_{x}{\mathbf{Z}}^{n}_{\varepsilon}|^{2}$, it is a Lagrange multiplier associated to the constraint. Thanks to Theorem 3.9 and Lemma 3.10, $\lambda^{n}\in L^{1}({\Omega})$. Thus, (14) together with the constraint $|{\mathbf{Z}}^{n}_{\varepsilon}|=1$ is the Euler-Lagrange system associated to the discrete minimization problem (11).

Proposition 3.14**.**

Under the previous hypotheses, one has the estimate

[TABLE]

where the constant is uniformly bounded with respect to $\varepsilon$,${\Delta a}$ and $\Delta t$.

Proof 3.15**.**

As ${\mathbf{Z}}^{n}_{\varepsilon}\in L^{\infty}_{t}{\mathbf{H}}_{x}^{1}$ uniformly in $\varepsilon$, it is already clear that $|\partial_{x}{\mathbf{Z}}^{n}_{\varepsilon}|^{2}$ belongs to $L^{\infty}_{t}L^{1}_{x}$. It remains to estimate ${\left\|{{\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}\cdot{\mathbf{Z}}^{n}_{\varepsilon}}\right\|}_{L^{1}({\Omega})}$. Since ${\mathbf{Z}}^{n-j}_{\varepsilon}\in{\mathcal{A}}$ for all $j\in\mathbb{N}$ (this statement uses the first assumption in hypotheses 2.3, in the case when $n-j<0$), a simple computation gives that, for every ${x}\in{\Omega}$,

[TABLE]

This in turn suggests that

[TABLE]

By Lemma 3.10, the first term is bounded for any $n\geq 0$. Thanks to the definition of $\lambda^{n}$, the claim follows.

Remark 3.16**.**

Proposition 3.14 shows as well that the energy minimization procedure provides a $L^{\infty}_{t}L^{1}_{x}$ bound, uniform in $\varepsilon$, on the Lagrange multiplier $\lambda_{\varepsilon,\Delta}$. Direct use of the energy estimates from Lemma 3.10 and Jensen’s inequality give ${\left\|{{\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}}\right\|}_{L^{2}({\Omega})}\lesssim\sqrt{{\mathcal{E}}_{n}({\mathbf{Z}}^{n}_{\varepsilon})/\varepsilon}\lesssim\varepsilon^{-1/2}$ which provides only

[TABLE]

Remark 3.17**.**

The previous result shows that the delay operator ${\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}$ points out of the unit sphere since by convexity of the square function, ${\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}\cdot{\mathbf{Z}}^{n}_{\varepsilon}>0$ for a.e. ${x}\in{\Omega}$. In the next proposition, we show that the scalar product is of order $\varepsilon$ with respect to the $L^{1}_{{x},t}$ norm, which makes sense. Indeed, when $\varepsilon$ is small, ${\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}$ approximates $\mu_{1,0}\partial_{t}{\mathbf{z}}_{0}$ which is tangent to the sphere, and thus orthogonal to ${\mathbf{z}}_{0}$.

Proposition 3.18**.**

Under hypotheses 2.1, 2.2 and 2.3, one can also show that

[TABLE]

where the constant does not depend on $\varepsilon$.

Proof 3.19**.**

Using again the same idea as in the previous proof, one writes :

[TABLE]

the latter estimate coming from the dissipation term in the proof of Lemma 3.10.

Here we show one of the key estimates of the paper.

Proposition 3.20**.**

Under hypotheses above, and for $\Delta t$ small enough, one has :

[TABLE]

where the constant does not depend neither on $\varepsilon$ nor on $\Delta t$.

Proof 3.21**.**

Recalling the definition of ${\mathbf{U}}^{n}_{\varepsilon,j}$ one checks easily that

[TABLE]

while ${\mathbf{U}}^{n}_{\varepsilon,0}=\delta{\mathbf{Z}}^{n-{\frac{1}{2}}}_{\varepsilon}/(2\varepsilon)$. Equivalently, because of the specific CFL condition, ${\mathbf{U}}^{n+1}_{\varepsilon,j+1}={\mathbf{U}}^{n}_{\varepsilon,j}+\delta{\mathbf{Z}}^{n+{\frac{1}{2}}}_{\varepsilon}/\varepsilon$ for all $j\geq 1$. Setting ${\mathbf{T}}^{n}_{\varepsilon,j}=\varrho^{n}_{\varepsilon,j}{\mathbf{U}}^{n}_{\varepsilon,j}$ for $j\in\mathbb{N}$, one obtains using (8) :

[TABLE]

which, summing over $j\in\mathbb{N}^{*}$, gives

[TABLE]

By definition,

[TABLE]

Adding both equations gives :

[TABLE]

since $\sum_{j\in\mathbb{N}}{\mathbf{T}}^{n}_{\varepsilon,j}\Delta a=\sum_{j\in\mathbb{N}}\varrho^{n}_{\varepsilon,j}{\mathbf{U}}^{n}_{\varepsilon,j}\Delta a={\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}$. Now we make the discrete difference of (14) between steps $n+1$ and $n$, in order to express $\delta{\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n+{\frac{1}{2}}}$ as a function of $\delta{\mathbf{Z}}^{n+{\frac{1}{2}}}_{\varepsilon}$. This reads :

[TABLE]

We now close the problem solved by $\delta{\mathbf{Z}}^{n+{\frac{1}{2}}}_{\varepsilon}$ :

[TABLE]

We rewrite the difference

[TABLE]

Applying $J^{n+{\frac{1}{2}}}$ to ${\boldsymbol{v}}=\delta{\mathbf{Z}}^{n+{\frac{1}{2}}}_{\varepsilon}$ and using that both ${\mathbf{Z}}^{n+1}_{\varepsilon}$ and ${\mathbf{Z}}^{n}_{\varepsilon}$ satisfy the constraint, reduces to :

[TABLE]

cancelling the term containing the finite differences $\delta\lambda^{n+{\frac{1}{2}}}$. Next we use the crucial estimates from Proposition 3.14, indeed :

[TABLE]

In one space dimension, the Gagliardo-Nirenberg estimates (cf. [1], p. 140, Theorem 5.9) provide

[TABLE]

Thus setting ${\boldsymbol{v}}=\delta{\mathbf{Z}}^{n+{\frac{1}{2}}}_{\varepsilon}$ in the weak formulation above and because there is an $\varepsilon$ in front of $J^{n+{\frac{1}{2}}}$ in (15) one writes finally :

[TABLE]

Using Young’s inequality on the right hand side above, for $\varepsilon$ small enough, one has :

[TABLE]

The previous argument provides uniqueness as well :

Proposition 3.22**.**

Under hypotheses 2.1, 2.2 and 2.3, there exists a unique solution ${\mathbf{z}}_{\varepsilon,\Delta}\in L^{\infty}((0,T);{\mathbf{H}}^{1}({\Omega}))\cap H^{1}((0,T);{\mathbf{L}}^{2}({\Omega}))$ solving (14).

Proof 3.23**.**

We use induction arguments to show the claim. We suppose that there exists two solutions ${\mathbf{z}}_{\varepsilon,\Delta,i}$ for $i\in\{1,2\}$. We denote by $\delta{\mathbf{z}}^{k}:={\mathbf{Z}}^{k}_{\varepsilon,2}-{\mathbf{Z}}^{k}_{\varepsilon,1}$, and we write the equation it satisfies for $k=0$ :

[TABLE]

Thus choosing ${\boldsymbol{v}}=\delta{\mathbf{z}}^{0}$ and using the same arguments as above implies that $\delta{\mathbf{z}}^{0}=0$. We suppose at this point that $\delta{\mathbf{z}}^{k}=0$ for $k\leq n$. Then a careful decomposition of ${\mathbf{\mathcal{L}}}^{n}_{\varepsilon,\Delta,2}-{\mathbf{\mathcal{L}}}^{n}_{\varepsilon,\Delta,1}$ leads to

[TABLE]

which again, thanks to the lower bound $\mu_{0,\min}$ established in Lemma (3.1), shows that

[TABLE]

proving the claim for $\varepsilon$ small enough and $k=n+1$. This ends the proof since ${\mathbf{z}}_{\varepsilon,\Delta,2}={\mathbf{z}}_{\varepsilon,\Delta,1}$.

Proposition 3.24**.**

Under hypotheses 2.1, 2.2 and 2.3, $\tilde{{\mathbf{z}}}_{\varepsilon,\Delta}$, the piecewise linear interpolation of $({\mathbf{Z}}^{n}_{\varepsilon})_{n\in\mathbb{Z}}$ satisfies

[TABLE]

for every $\gamma\in(0,1)$, the bound is uniform with respect to $\Delta t$ and $\varepsilon$. Thus $\tilde{{\mathbf{z}}}_{\varepsilon,\Delta}$ converges strongly in $C^{0}(\overline{{\Omega}}\times[0,T])$ when $\Delta t$ goes to zero. Moreover, ${\mathbf{z}}_{\varepsilon,\Delta}$ converges strongly in $L^{\infty}((0,T);C(\overline{{\Omega}}))$.

Proof 3.25**.**

Thanks to Lemma 3.10, $\tilde{{\mathbf{z}}}_{\varepsilon,\Delta}$ belongs to $L^{\infty}_{t}{\mathbf{H}}^{1}_{x}$ uniformly with respect to $\varepsilon$, which shows weak-$\star$ convergence in this space. Weak convergence in $H^{1}_{t}{\mathbf{L}}^{2}_{x}$ follows from Proposition 3.20. The interpolation inequality

[TABLE]

holds for every $u\in H^{1}({\Omega})$ and for every $\gamma\in(0,1)$. Combined with the $L^{\infty}_{t}{\mathbf{H}}^{1}_{x}$ bound provided by Lemma 3.10, this leads to :

[TABLE]

We complete the convergence proof for $\tilde{{\mathbf{z}}}_{\varepsilon,\Delta}$ by an application of the Ascoli-Arzela theorem.

Corollary 3.26**.**

Under the previous hypotheses, the same result can be derived for ${\mathbf{z}}_{\varepsilon}:=\lim_{\Delta t\to 0}{\mathbf{z}}_{\varepsilon,\Delta}$, i.e.

[TABLE]

for every $\gamma\in(0,1)$, the bound is uniform with respect to $\varepsilon$. This implies that ${\mathbf{z}}_{\varepsilon}$ converges to ${\mathbf{z}}_{0}$ strongly in $C^{0}(\overline{{\Omega}}\times[0,T])$ when $\varepsilon$ goes to zero.

Proof 3.27**.**

Considering $\tilde{{\mathbf{z}}}_{\varepsilon,\Delta}$, the piecewise continuous function in time, $\partial_{t}\tilde{{\mathbf{z}}}_{\varepsilon,\Delta}$ is bounded in ${\mathbf{L}}^{2}_{{x},t}$ uniformly with respect to $\varepsilon$, thus $\partial_{t}\tilde{{\mathbf{z}}}_{\varepsilon,\Delta}\rightharpoonup\partial_{t}{\mathbf{z}}_{\varepsilon}$ weakly in ${\mathbf{L}}^{2}_{{x},t}$ and one has that

[TABLE]

A similar argument provides an $L^{\infty}_{t}{\mathbf{H}}^{1}_{x}$ bound for ${\mathbf{z}}_{\varepsilon}$. One can then follow again the same steps as in the proof of Proposition 3.24.

4 Convergence when $\varepsilon$ is fixed and $\Delta a$ goes to 0.

Next, we consider the convergence of ${\boldsymbol{\mathcal{L}}}_{\varepsilon,\Delta}({x},t):=\sum_{n=0}^{N}\chi_{(n,n+1)\Delta t}(t){\boldsymbol{\mathcal{L}}}_{\varepsilon}^{n}({x})$.

Proposition 4.1**.**

Under hypotheses 2.1, 2.2 and 2.3, for every fixed $\varepsilon>0$, the discrete delay term converges to the continuous limit when $\Delta a$ goes to zero, i.e.

[TABLE]

for all ${\boldsymbol{\varphi}}\in C^{0}([0,T];L^{2}({\Omega}))$ and ${\boldsymbol{\varphi}}_{\Delta}({x},t):=\sum_{n=0}^{N}\chi_{(n,n+1)\Delta t}(t){\boldsymbol{\varphi}}^{n}({x})$ where ${\boldsymbol{\varphi}}^{n}({x}):=$ $\int_{n\Delta t}^{(n+1)\Delta t}$ ${\boldsymbol{\varphi}}({x},t)dt/\Delta t$.

Proof 4.2**.**

In what follows the terms that we handle are integrable on the domain ${\Omega}\times{\mathbb{R}_{+}}\times(0,T)$ so the systematic use of Fubini’s Theorem is implicitly assumed and we freely commute integrals with respect to space, age and time. We set $I_{\Delta}:=\int_{\Omega}\int_{0}^{T}{\boldsymbol{\mathcal{L}}}_{\varepsilon,\Delta}({x},t)$ ${\boldsymbol{\varphi}}_{\Delta}({x},t)dtd{x}$ that we split in two parts :

[TABLE]

By Lemma 3.1, $\mu_{0,\varepsilon,\Delta}$ is uniformly bounded with respect to $\varepsilon$, $\Delta t$ and $\Delta a$, $\mu_{0,\varepsilon,\Delta}\stackbin{\star}{\rightharpoonup}\mu_{0,\varepsilon}$ in the weak-$\star$ topology in $L^{\infty}((0,T)\times{\Omega})$. Moreover since $L^{2}((0,T);L^{2}({\Omega}))$ is a separable space, the step functions in time with values in $L^{2}({\Omega})$ are dense. Thus ${\boldsymbol{\varphi}}_{\Delta}$ tends to ${\boldsymbol{\varphi}}$ strongly in ${\mathbf{L}}^{2}_{{x},t}$ and the product ${\mathbf{z}}_{\varepsilon,\Delta}{\boldsymbol{\varphi}}_{\Delta}$ converges strongly in $L^{1}({\Omega}\times(0,T))$. All this gives :

[TABLE]

For the second term, one first defines

[TABLE]

and then one has :

[TABLE]

We consider the convergence of the term ${\mathbf{z}}_{\varepsilon,\Delta}({x},t-\varepsilon a){\boldsymbol{\varphi}}_{\Delta}({x},t)$ on ${\Omega}\times\{(a,t)\in{\mathbb{R}_{+}}\times(0,T)\text{ s.t. }t>\varepsilon a\}$ :

[TABLE]

and thus in a similar manner, as for $I_{1,\Delta}$, one proves the convergence of $I_{2,1,\Delta}:=\int_{0}^{T}\int_{\Omega}\int_{0}^{\frac{t}{\varepsilon}}\rho_{\varepsilon,\Delta}({x},a,t)$ ${\mathbf{z}}_{\varepsilon,\Delta}({x},t-\varepsilon a)\cdot{\boldsymbol{\varphi}}_{\Delta}dad{x}dt$. On the other hand, on ${\Omega}\times\{(a,t)\in{\mathbb{R}_{+}}\times(0,T)\text{ s.t. }t<\varepsilon a\}$, one has that :

[TABLE]

A simple computation shows that if ${\tilde{a}}\in(n,n+1)\Delta t$ then

[TABLE]

which gives then that

[TABLE]

In a similar way, one proves, thanks to hypotheses 2.3, that

[TABLE]

For the last part, on ${\Omega}\times\{(a,t)\in{\mathbb{R}_{+}}\times(0,T)\text{ s.t. }t<\varepsilon a\}$, one has that :

[TABLE]

which proves that

[TABLE]

and ends the proof.

Theorem 4.3**.**

Under hypotheses above, there exists a unique ${\mathbf{z}}_{\varepsilon}\in H^{1}((0,T);{\mathbf{L}}^{2}({\Omega}))$ $\cap$ $L^{\infty}((0,T);$ ${\mathbf{H}}^{1}({\Omega}))$ solving, for almost every $t\in(0,T)$,

[TABLE]

*where the brackets denote the $L^{2}({\Omega})$ scalar product and the Lagrange multiplier $\lambda_{\varepsilon}=-{\boldsymbol{\mathcal{L}}}_{\varepsilon}\cdot{\mathbf{z}}_{\varepsilon}-|\partial_{x}{\mathbf{z}}_{\varepsilon}|^{2}$ is an $L^{\infty}((0,T);L^{1}({\Omega}))$ function uniformly with respect to $\varepsilon$. Moreover, for every $({x},t)\in{\Omega}\times[0,T]$, $|{\mathbf{z}}_{\varepsilon}({x},t)|=1$. *

Proof 4.4**.**

By Proposition 4.1, the convergence of ${\boldsymbol{\mathcal{L}}}_{\varepsilon,\Delta}$ is proved. Since $\partial_{x}{\mathbf{z}}_{\varepsilon,\Delta}\in L^{\infty}_{t}{\mathbf{L}}^{2}_{x}$ uniformly with respect to $\varepsilon,\Delta a$ and $\Delta t$, one has

[TABLE]

where again ${\boldsymbol{\varphi}}\in C^{0}([0,T];{\mathbf{H}}^{1}({\Omega}))$ and ${\boldsymbol{\varphi}}_{\Delta}({x},t):=\sum_{n=0}^{N}\chi_{(n,n+1)\Delta t}(t){\boldsymbol{\varphi}}^{n}({x})$ where ${\boldsymbol{\varphi}}^{n}({x}):=$ $\int_{n\Delta t}^{(n+1)\Delta t}$ ${\boldsymbol{\varphi}}({x},t)dt/\Delta t$. Since $L^{\infty}_{t}M_{x}$ is the dual space of $L^{1}_{t}C_{x}$ which is separable, the bounded sets in $L^{\infty}_{t}M_{x}$ are compact for the weak-$\star$ topology $\sigma(L^{\infty}_{t}M_{x},L^{1}_{t}C_{x})$, we denote by $\langle\cdot,\cdot\rangle$, the duality brackets associated to this dual paring. Defining $J_{\varepsilon}$ to be

[TABLE]

The first term tends to zero thanks to the density of valued step functions in $L^{1}_{t}L^{2}_{x}$, the second term is small due to the strong convergence of ${\mathbf{z}}_{\varepsilon,\Delta}$ established above, the last one tends to zero thanks to the weak-$\star$ convergence of $\lambda_{\varepsilon,\Delta}$ in $L^{\infty}_{t}\mathcal{M}_{x}$. At that point, the solution pair $({\mathbf{z}}_{\varepsilon},\lambda_{\varepsilon})$ solves :

[TABLE]

for every ${\boldsymbol{\varphi}}\in C^{0}([0,T];{\mathbf{H}}^{1}({\Omega}))$. Setting ${\boldsymbol{\varphi}}({x},t)={\mathbf{z}}_{\varepsilon}({x},t)\theta({x},t)$ with $\theta\in{\mathcal{D}}({\Omega}\times(0,T))$, in (17), proves that for almost every $({x},t)\in{\Omega}\times(0,T)$, $\lambda_{\varepsilon}=-{\boldsymbol{\mathcal{L}}}_{\varepsilon}\cdot{\mathbf{z}}_{\varepsilon}-|\partial_{x}{\mathbf{z}}_{\varepsilon}|^{2}<0$ and the right hand side is a $L^{\infty}_{t}L^{1}_{x}$ function.

Taking now ${\boldsymbol{\varphi}}({x},t)={\boldsymbol{v}}({x})\psi(t)$ for any ${\boldsymbol{v}}\in{\mathbf{H}}^{1}({\Omega})$ and $\psi\in{\mathcal{D}}(0,T)$ shows that (16) holds a.e. $t\in(0,T)$ for any ${\boldsymbol{v}}\in{\mathbf{H}}^{1}({\Omega})$.

An easy computation shows that

[TABLE]

which gives that

[TABLE]

thanks to Proposition 3.20. Then a triangular inequality gives :

[TABLE]

As the right hand side is arbitrary small, the left hand side is zero. Thus the constraint is fulfilled a.e. in ${\Omega}\times(0,T)$. Thanks to Corollary 3.26, ${\mathbf{z}}_{\varepsilon}$ is a continuous function in time and in space, so the result holds true everywhere in ${\Omega}\times(0,T)$.

In order to prove uniqueness we assume that there exists two distinct solutions $({\mathbf{z}}_{\varepsilon}^{i})_{i\in\{1,2\}}$ sharing the same ${\mathbf{z}}_{p}$ condition for negative times, the same kernel $\rho_{\varepsilon}$ and both solving (16) for a.e. $t\in(0,T)$. We denote by ${\hat{\mathbf{z}}_{\varepsilon}}:={\mathbf{z}}_{\varepsilon}^{2}-{\mathbf{z}}_{\varepsilon}^{1}$ and it solves :

[TABLE]

Setting ${\boldsymbol{v}}={\hat{\mathbf{z}}_{\varepsilon}}(t)$ one obtains thanks to the same Gagliardo-Niremberg estimates as in Proposition 3.20 that

[TABLE]

Making the first term in the left hand side above explicit one writes :

[TABLE]

In the above equality this contributes to obtain :

[TABLE]

since the second term in the left hand side is positive we omit it, setting $q(t):={\frac{1}{2}}{\left\|{{\hat{\mathbf{z}}_{\varepsilon}}(\cdot,t)}\right\|}_{L^{2}({\Omega})}^{2}$ and using that $\rho_{\varepsilon}({x},a,t)\leq\beta_{\varepsilon}({x},t)\exp(-\zeta_{\min}a)$, for a.e. ${x}\in{\Omega}$ and $t\geq\varepsilon a$, one obtains that $q$ satisfies :

[TABLE]

which after some easy computations provide that $\int_{0}^{t}\exp(\zeta_{\min}\tau/\varepsilon)q(\tau)d\tau=0$ which in turn gives that $\exp(\zeta_{\min}t)q(t)\leq 0$ and since $q(t)$ is positive by definition this gives that ${\left\|{{\hat{\mathbf{z}}_{\varepsilon}}(\cdot,t)}\right\|}_{L^{2}({\Omega})}=0$ for a.e. $t$, which shows uniqueness.

Proposition 4.5**.**

Under the previous hypotheses, one has

[TABLE]

where the constant is independent of $\varepsilon$.

Proof 4.6**.**

Using the same arguments as in Proposition 4.1, one shows that ${\boldsymbol{\mathcal{L}}}_{\varepsilon,\Delta}\cdot{\mathbf{z}}_{\varepsilon,\Delta}$ tends to ${\boldsymbol{\mathcal{L}}}_{\varepsilon}\cdot{\mathbf{z}}_{\varepsilon}$ in $L^{1}_{{x},t}$ as $\Delta\to 0$. Then using the estimate established in Proposition 3.18, one concludes.

5 Convergence when $\varepsilon$ goes to zero in the continuous framework

5.1 Convergence of the population of bonds $\rho_{\varepsilon}$

For sake of conciseness we recall here the main result of Section 5.1 [16] in which the convergence of $\rho_{\varepsilon}$ towards $\rho_{0}$ solving (6) is fully established.

Theorem 5.1**.**

Under assumptions 2.1 and 2.2, one has

[TABLE]

with $\mathcal{R}_{\varepsilon}({x},a,t):=-\varepsilon\partial_{t}\rho_{0}({x},a,t)-\rho_{0}({x},a,t)(\zeta_{\varepsilon}({x},a,t)-\zeta_{0}({x},a,t))$ and $\mathcal{M}_{\varepsilon}({x},t):=(\beta_{\varepsilon}({x},t)-\beta_{0}({x},t))$ $\left(1-\int_{0}^{\infty}\right.$ $\left.\rho_{0}({x},a,t)\,da\right)$. These estimates imply $\sup_{{x}\in\Omega}|\rho_{\varepsilon}({x},a,t)-\rho_{0}({x},a,t)|$ converges strongly in $L^{1}((0,T)\times{\mathbb{R}_{+}},(1+a))$ which give strong convergence in $L^{1}({\mathbb{R}_{+}}\times(0,T);L^{\infty}({\Omega}),(1+a))$ when $\varepsilon$ goes to zero.

5.2 Study of the initial layer and convergence of continuous and descrete time derivative of $\rho_{\varepsilon}$

Here we perform a preliminary analysis in order to obtain limits when $\varepsilon$ goes to zero of the transposition of the delay operator. For this sake we introduce the initial layer and show to what limit $\partial_{t}\rho_{\varepsilon}$ converges in a second step (cf Theorem 5.15). Finally we exhibit the limit to which the delayed part transfered on $\rho_{\varepsilon}$ of ${\boldsymbol{\mathcal{L}}}_{\varepsilon}$ i.e. $(\rho_{\varepsilon}({x},a,t+\varepsilon a)-\rho_{\varepsilon}({x},a,t))/\varepsilon$ tends as a Banach valued Radon measure (cf Proposition 5.22).

Proposition 5.2**.**

If $f$ is in $U_{T}$ then its weak derivatives $\partial_{a}f$ and $\partial_{t}f$ are in $X_{T}^{\prime}$. One defines the corresponding duality brackets as

[TABLE]

for any $\varphi\in X_{T}$.

Proposition 5.3**.**

If $f$ is in $U_{T}$ then its weak derivatives $\partial_{a}f$ and $\partial_{t}f$ are in $Z_{T}^{\prime}$. One defines the corresponding duality brackets as

[TABLE]

for any $\varphi\in Z_{T}$.

For sake of conciseness, the proofs of these propositions are postponed in B. Using then these one shows :

Proposition 5.4**.**

Under hypotheses 2.1 and 2.2, the previous convergence result can be extended to $Z_{T}^{\prime}$ where $Z_{T}:=C_{b}({\overline{Q}_{T}};L^{1}({\Omega}))$. Namely for any $\varphi$ in $Z_{T}$, there exists a subsequence $\tau_{k}$ s.t.

[TABLE]

when $k\to\infty$.

In order to identify the limit to which $\partial_{t}\rho_{\varepsilon}$ tends when $\varepsilon$ goes to zero, (part of the main ingredients were presented in Proposition 3.2 p. 10, [15], but the space variable was not taken in account), we define an initial layer, as in [15]. Setting $\tilde{t}=t/\varepsilon$, we look for $\tilde{\rho}_{0}$ solution of

[TABLE]

and we define $\tilde{\rho}_{0,\varepsilon}({x},a,t):=\tilde{\rho}_{0}({x},a,t/\varepsilon)$. As in [15], we obtain at the microscopic level global existence and a priori bounds :

Theorem 5.5**.**

Under hypotheses 2.1 and 2.2, there exists a unique solution $\tilde{\rho}_{0}$ belonging to $C^{0}({\mathbb{R}_{+}};L^{1}_{a}({\mathbb{R}_{+}};L^{\infty}_{x}({\Omega}))$ $\cap$ $L^{\infty}({\Omega}\times{\mathbb{R}_{+}}\times{\mathbb{R}_{+}})$. Moreover, one has

[TABLE]

[TABLE]

and there exists a subsequence s.t. $D^{\tau_{k}}_{\tilde{t}}\tilde{\rho}_{0}\stackbin{\star}{\rightharpoonup}\partial_{\tilde{t}}\tilde{\rho}_{0}$ weak-$*$ in the $\sigma(Z_{\infty}^{\prime},Z_{\infty})$ topology.

Proof 5.6**.**

The proof of the existence and uniqueness part is easy and follows the same ideas as in [12, 15] where one shall only manage the ${x}$ dependence in addition. A priori estimates on ${D_{t}^{\tau}}\tilde{\rho}_{0}$ are obtained as in Theorem 2.2 p. 6 [15]. The last part follows the same ideas as in Propositions 5.2 and 5.3 below.

Corollary 5.7**.**

Under the same hypotheses, one has the scaling

[TABLE]

Proof 5.8**.**

We start from the change of variable $\tilde{t}=t/\varepsilon$, which gives

[TABLE]

where $\tilde{\tau}=\tau/\varepsilon$, then the right hand side (resp. left hand side) converges up to a subsequence to the right hand side (resp. left hand side) of the claim by the same arguments as in Propositions 5.2 and 5.3.

Theorem 5.9**.**

Under hypotheses 2.1 and 2.2, one has for any $\varphi\in C_{b}({\mathbb{R}_{+}};L^{1}_{x}({\Omega}))$,

[TABLE]

and we underline that here $\varphi$ does not depend on time.

Proof 5.10**.**

Using a priori estimates (20), one has

[TABLE]

where we recall that $U_{\infty}:={\rm BV}({\mathbb{R}_{+}}\times{\mathbb{R}_{+}};L^{\infty}({\Omega}))$. This shows that $q(t):=\int_{{\Omega}\times{\mathbb{R}_{+}}}\varphi({x},a)\tilde{\rho}_{0}({x},a,t)d{x}da$ is a function of bounded variation. Thus there exists a signed Radon measure $\nu_{\partial_{t}q}$ associated to the time derivative of $q$.

[TABLE]

Indeed, the integral $\int_{0}^{T/\varepsilon}d\nu_{\partial_{t}q}$ coincides with the Riemann-Stieltjes integral, thus integration by parts holds. Moreover one has that

[TABLE]

as $\tau$ (up to a subsequence) goes to zero. On the other hand

[TABLE]

in the weak-$\star$ topology $\sigma(Z_{T}^{\prime},Z_{T})$ (as in the proof of Proposition 5.3). Because

[TABLE]

and the arguments above, one has finally :

[TABLE]

One concludes since $|q(T/\varepsilon)|\lesssim\exp(-\zeta_{\min}T/\varepsilon)$ thanks to (19).

Proposition 5.11**.**

Under assumptions 2.1 and 2.2, one has

[TABLE]

which implies that :

[TABLE]

for all $\varphi\in Z_{T}$.

Proof 5.12**.**

As ${x}$ is a mute variable in the $\rho_{\varepsilon}$ model, we first establish that :

[TABLE]

using exactly the same arguments as in Proposition 3.2 p.10 [15]. The method of characteristics gives, under the hypotheses above :

[TABLE]

as the bounds do not depend on $\tau$, the first claim follows. One writes then in the $Z_{T}^{\prime},Z_{T}$ duality pairing, that

[TABLE]

Thanks to the first statement above, for any fixed $\delta>0$ and any fixed $\varphi\in Z_{T}$, there exists $\varepsilon_{0}$ s.t. $\varepsilon<\varepsilon_{0}$ implies

[TABLE]

By Proposition 5.3, there exists $\tau_{0}$ s.t. $\tau<\tau_{0}$ implies

[TABLE]

which ends the proof.

Proposition 5.13**.**

Under the same hypotheses, there is a limit related to the initial layer : for any $\varphi\in Z_{T}$,

[TABLE]

Proof 5.14**.**

We set $\psi({x},a,t):=\varphi({x},a,t)-\varphi({x},a,0)$, and we use Corollary 5.7, giving that

[TABLE]

Next we write :

[TABLE]

We start with $I_{2}$ and write :

[TABLE]

Since ${\operatorname*{ess\,sup}_{{x}\in\Omega}\,}|{D_{t}^{\tau}}\tilde{\rho}_{0}|$ is a positive function in $L^{1}({Q_{T}})$, there exists $\nu$ a weak-$*$ limit in $\sigma(M^{1}({\overline{Q}_{T}}),C^{0}({\overline{Q}_{T}}))$ of the measure $\nu_{\tau}$ associated to it. Because $\nu_{\tau}$ is tight with respect to $\tau$, this convergence extends to the weak-$*$ topology in $\sigma(C_{b}({\overline{Q}_{T}})^{\prime},C_{b}({\overline{Q}_{T}}))$.

Since ${\left\|{\psi(\cdot,a,\varepsilon t)}\right\|}_{L^{1}({\Omega})}$ is a continuous bounded function on ${\overline{Q}_{T}}$, converging pointwisely to 0 a.e. $(a,t)\in{Q_{T}}$, there exists $\varepsilon_{0}$ s.t. for $\varepsilon<\varepsilon_{0}$,

[TABLE]

From here until the end of the proof, $\varepsilon$ is fixed. Thanks to the previous tight convergence result, there exists a $\tau_{0}$, s.t.

[TABLE]

and finally there exists $\tau_{1}$ s.t. $\tau<\tau_{1}$ implies

[TABLE]

thanks to the weak-$*$ convergence in topology $\sigma(Z_{T/\varepsilon}^{\prime},Z_{T/\varepsilon})$. Summing the three terms ends the proof.

Theorem 5.15**.**

Under hypotheses 2.1 and 2.2, one has

[TABLE]

for every $\varphi\in Z_{T}$.

Proof 5.16**.**

We set

[TABLE]

and split this difference adding and subtracting extra terms :

[TABLE]

Now for every fixed $\delta$ (small), there exists $\varepsilon_{0}$ s.t. $\varepsilon<\varepsilon_{0}$ implies $I_{1}<\delta/3$ thanks to Proposition 5.11, s.t. $I_{2}<\delta/3$ thanks to Proposition 5.13, and s.t. $I_{3}<\delta/3$ thanks to Theorem 5.9, which ends the proof.

We define

[TABLE]

where $P_{\varepsilon}:=\Omega\times\left\{(a,t)\in{Q_{T}}\text{ s.t. }a<\frac{T-t}{\varepsilon}\right\}$.

Theorem 5.17**.**

Under hypotheses 2.1 and 2.2, ${\cal K}_{\varepsilon}$ solves the weak problem : for all $\psi\in Z_{T}$

[TABLE]

where we set

[TABLE]

Proof 5.18**.**

In order to express the problem solved by ${\cal K}_{\varepsilon}$, we regularize the data. It gives a pointwise meaning to an approximation of $\partial_{t}\rho_{\varepsilon}$. For this sake, we regularize the boundary datum and the off-rate setting :

[TABLE]

where the cut-off function $\chi_{\delta}$ is monotone and $C^{\infty}({\mathbb{R}_{+}})$ s.t.

[TABLE]

and $\omega_{\delta,a}$ and $\omega_{\delta,t}$ are the standard mollifiers in the $a$ and $t$ variable. For the initial condition we use as in the Appendix, the specific regularisation of ${\rm BV}$ functions originally presented in [21, 5] for the real-valued case, and more recently adapted to the vector valued case in Theorem 2.21 [7]. This regularisation provides

[TABLE]

One solves (3) with initial, boundary and off-rate datum $(\rho_{I}^{\delta},\beta_{\varepsilon}^{\delta},\zeta_{\varepsilon}^{\delta})$, the solution is denoted $\rho_{\varepsilon}^{\delta}$. Together with assumptions 2.1 and 2.2, the time derivative $\partial_{t}\rho_{\varepsilon}^{\delta}$ solves :

[TABLE]

where $\mu_{0,\varepsilon}^{\delta}({x},t)=\int_{\mathbb{R}_{+}}\rho_{\varepsilon}^{\delta}({x},a,t)da$. Since the data of (23) is regular, for a fixed ${x}\in{\Omega}$, existence results follow from Theorem 2.1 p. 488 [12], and thanks to similar arguments as in Proposition 3.3, one proves as well that $\partial_{t}\rho_{\varepsilon}^{\delta}$ is a $C^{0}((0,T);L^{1}({\mathbb{R}_{+}};L^{\infty}({\Omega})))$ function. One obtains a priori estimates, uniform in $\varepsilon$, leading to $\partial_{t}\rho_{\varepsilon}^{\delta}\in Y_{T}$. In the same way as in Propositions 5.2 and 5.3, there is a limit $\partial_{t}\rho_{\varepsilon}$ in the weak-$*$ topology $\sigma(Z_{T}^{\prime},Z_{T})$, up to a subsequence. Moreover, using the Lyapunov functional ${\mathcal{H}}[\cdot]$, one has also that $\rho_{\varepsilon}^{\delta}-\rho_{\varepsilon}\sim o_{\delta}(1)$ in $Y_{T}$. Now as $\partial_{t}\rho_{\varepsilon}^{\delta}$ is regular enough, one derives the ODE solved by ${\cal K}_{\varepsilon}$,

[TABLE]

This can be integrated and gives :

[TABLE]

Tested against $\psi\in Z_{T}$ and integrated on ${{\mathfrak{Q}}_{T}}$, this becomes :

[TABLE]

Setting

[TABLE]

one recovers the regularized version of (21). Since $\zeta_{\varepsilon}^{\delta}\in W^{1,\infty}({{\mathfrak{Q}}_{T}})$, one has $\zeta_{\varepsilon}^{\delta}\to\zeta_{\varepsilon}$ strongly in $C(K)$ for any compact $K\subset{{\mathfrak{Q}}_{T}}$. Thus one has : $\varphi_{\varepsilon}^{\delta}\to\varphi_{\varepsilon}$ strongly in $L^{\infty}_{a,t}L^{1}_{x}$ when $\delta\to 0$. Since $\varphi_{\varepsilon}^{\delta}$ and $\varphi_{\varepsilon}$ are continuous and compactly supported, the strong convergence occurs as well in $Z_{T}$. There exists a subsequence ${\partial_{t}\rho_{\varepsilon}^{\delta}}$ converging in the $\sigma(Z_{T}^{\prime},Z_{T})$ topology to ${\partial_{t}\rho_{\varepsilon}}$ thus

[TABLE]

when $\delta$ goes to zero. Now other arguments using the strong convergence of $\rho_{\varepsilon}^{\delta}$ justify the claim. Moreover, $\varphi_{\varepsilon}$ is bounded uniformly with respect to $\varepsilon$. Indeed :

[TABLE]

which gives after taking the sup over ${\overline{Q}_{T}}$ that

[TABLE]

Corollary 5.19**.**

Under the previous hypotheses, one has that ${\left\|{\varphi_{\varepsilon}(\cdot,a,t)-\varphi_{0}(\cdot,a,t)}\right\|}_{L^{1}_{x}}$ tends to zero when $\varepsilon$ goes to zero, for every fixed $(a,t)\in{Q_{T}}$, where

[TABLE]

Proposition 5.20**.**

Under hypotheses 2.1 and 2.2, one has

[TABLE]

where $\varphi_{\varepsilon}$ is defined in (22).

Proof 5.21**.**

We set $\ell:=\int_{{\Omega}\times{\mathbb{R}_{+}}}\varphi_{0}({x},a,0)(\rho_{I}({x},a)-\rho_{0}({x},a,0))d{x}da$. As above one has

[TABLE]

the first term in the right hand side is $o_{\varepsilon}(1)$ thanks to Proposition 5.11. We focus on the second one : thanks to Corollary 5.19 and as ${\operatorname*{ess\,sup}_{{x}\in\Omega}\,}|\partial_{t}\rho_{0}({x},a,t)|$ is an integrable function on ${Q_{T}}$,

[TABLE]

By Lebesgue’s Theorem, the right hand side tends to zero. Using Corollary 5.7, one writes then

[TABLE]

Since $\chi_{(0,{\frac{T}{\varepsilon}})}(t){\left\|{(\varphi_{\varepsilon}-\varphi_{0})(\cdot,a,\varepsilon t)}\right\|}_{L^{1}({\Omega})}$ and $\chi_{(0,{\frac{T}{\varepsilon}})}(t){\left\|{\varphi_{0}(\cdot,a,\varepsilon t)-\varphi_{0}(\cdot,a,0)}\right\|}_{L^{1}({\Omega})}$ tend to zero for a.e. $(a,t)\in({\mathbb{R}_{+}})^{2}$, by the same arguments as in the proof of Proposition 5.13, one concludes that $J_{\varepsilon,1}$ and $J_{\varepsilon,2}$ vanish when $\varepsilon$ goes to 0. For the last term we use Theorem 5.9, and one concludes.

Now we are in the position to prove

Proposition 5.22**.**

Under hypotheses 2.1 and 2.2, when $\varepsilon$ goes to zero,

[TABLE]

for all $\psi\in C_{b}({\overline{{{\mathfrak{Q}}_{T}}}})$ and $\varphi_{0}$ is defined in (26) and depends on $\psi$.

Proof 5.23**.**

Considering the first term in (21), Proposition 5.20, shows that :

[TABLE]

On the other hand, hypotheses 2.1, standard arguments and the strong convergence of $\rho_{\varepsilon}$ imply that

[TABLE]

So that finally, one has

[TABLE]

As ${\cal K}_{0}({x},a,t):=a\partial_{t}\rho_{0}({x},a,t)$ is solving

[TABLE]

it is explicit and reads :

[TABLE]

It is then a matter of check to write

[TABLE]

which ends the proof.

5.3 Convergence of ${\mathbf{z}}_{\varepsilon}$

In [16], we derived, uniformly with respect to $\varepsilon$, $L^{\infty}_{t}L^{1}_{x}$ estimates for $\partial_{t}{\mathbf{z}}_{\varepsilon}$. Here we were not able to obtain this uniformity with respect to $\varepsilon$, and numerical simulations showed that these estimate do not hold true here. Thus the rest of the paper deals with the asymptotic when $\varepsilon$ goes to zero when only ${\mathbf{L}}^{2}_{{x},t}$ compactness for ${\mathbf{z}}_{\varepsilon}$ is available.

We consider

[TABLE]

and we want to express the limit of this operator when $\varepsilon$ goes to 0.

Theorem 5.24**.**

Under hypotheses 2.1, 2.2 and 2.3, when $\varepsilon$ goes to zero, one has that

[TABLE]

where $\mu_{1,0}({x},t):=\int_{{\mathbb{R}_{+}}}a\rho_{0}({x},a,t)da$ and for any test function ${\boldsymbol{\psi}}\in W^{1,\infty}({Q_{T}})$.

Proof 5.25**.**

Splitting the domain of integration,

[TABLE]

Thanks to A, $\ell_{1}$ and $\ell_{4}$ tend respectively to

[TABLE]

where ${\boldsymbol{\varphi}}_{0}({x},a,t):={\boldsymbol{\psi}}({x},t)\int_{a}^{\infty}\exp\left(-\int_{a}^{\tilde{a}}\zeta_{0}({x},s,t)ds\right)d{\tilde{a}}$. By strong convergence established for $\rho_{\varepsilon}$ and ${\mathbf{z}}_{\varepsilon}$, and the Lebesgue’s Theorem, one shows that

[TABLE]

Setting ${\hat{\mathbf{z}}_{\varepsilon}}={\mathbf{z}}_{\varepsilon}-{\mathbf{z}}_{0}$, and rewriting $\ell_{2}$ gives :

[TABLE]

Thanks to Corollary 3.26 and Proposition 5.22, one concludes that :

[TABLE]

then gathering the terms provides that

[TABLE]

where ${\boldsymbol{\varphi}}_{0}$ is defined in (26) as a function of ${\boldsymbol{\psi}}$. The last term of the previous right hand side can then be transformed into :

[TABLE]

since $\partial_{a}(a\rho_{0})+\zeta_{0}(a\rho_{0})=\rho_{0}$, one sees easily that the latter inner integral corresponds exactly to the integration of the latter ODE.

Theorem 5.26**.**

Under hypotheses 2.1, 2.2 and 2.3, the unique solution ${\mathbf{z}}_{\varepsilon}$ of the problem 16 converges towards the unique solution pair ${\mathbf{z}}_{0}\in H^{1}((0,T);{\mathbf{L}}^{2}({\Omega}))\cap L^{\infty}((0,T);{\mathbf{H}}^{1}({\Omega}))$ satisfying :

[TABLE]

for every $\varphi\in H^{1}((0,T);{\mathbf{L}}^{2}({\Omega}))\cap L^{\infty}((0,T);{\mathbf{H}}^{1}({\Omega}))$.

Proof 5.27**.**

From Theorem 5.24, and stability results above one has that the solution ${\mathbf{z}}_{\varepsilon}$ satisfying the weak formulation (17) converges strongly in $C(\overline{{\Omega}}\times[0,T])$ to ${\mathbf{z}}_{0}\in H^{1}((0,T);{\mathbf{L}}^{2}({\Omega}))\cap L^{\infty}((0,T);{\mathbf{H}}^{1}({\Omega}))$ satisfying :

[TABLE]

together with the constraint $|{\mathbf{z}}_{0}|=1$ everywhere in $\overline{{\Omega}}\times[0,T]$ obtained in the same way as in Theorem 4.3. Then since in the first line of the latter equation all terms are well defined, one can perform an integration by parts in time and obtain that ${\mathbf{z}}_{0}$ solves :

[TABLE]

again as in Theorem 4.3, choosing the test function to be ${\boldsymbol{\varphi}}={\mathbf{z}}_{0}\theta$, with $\theta\in{\mathcal{D}}({\Omega}\times(0,T))$, one proves that for almost every $({x},t)$, $\lambda_{0}=-|\partial_{x}{\mathbf{z}}_{0}|^{2}$ which, because ${\mathbf{z}}_{0}$ belongs to $L^{\infty}_{t}{\mathbf{H}}^{1}_{x}$, is an $L^{1}_{{x},t}$ function. Uniqueness follows the same ideas as in the proof of Theorem 4.3, it is simpler since the first term is a derivative instead of being a delay term as in (16). Indeed, denoting ${\hat{\mathbf{z}}_{\varepsilon}}:={\mathbf{z}}_{0}^{2}-{\mathbf{z}}_{0}^{1}$, where $({\mathbf{z}}_{0}^{i})_{i\in\{1,2\}}$ are two distinct solutions, it solves for every ${\boldsymbol{v}}\in{\mathbf{H}}^{1}({\Omega})$ and for almost every $t\in(0,T)$,

[TABLE]

choosing ${\boldsymbol{v}}={\hat{\mathbf{z}}_{\varepsilon}}$ and using Gagliardo-Nirenberg as above, one recovers :

[TABLE]

For $\delta$ small enough, one neglects the second term in the left hand side. Thanks to Gronwall’s Lemma, one concludes, since ${\hat{\mathbf{z}}_{\varepsilon}}({x},0)=0$ for all $x\in{\Omega}$, that ${\left\|{{\hat{\mathbf{z}}_{\varepsilon}}(\cdot,t)}\right\|}_{L^{2}({\Omega})}^{2}=0$ a.e. $t\in(0,T)$ which shows the claim.

Appendix A Initial and final terms in the weak formulation

Proposition A.1**.**

Under hypotheses 2.1 and 2.2, one has

[TABLE]

where $\ell_{4}$ and ${\boldsymbol{\varphi}}_{0}$ are defined in the proof of Theorem 5.24

Proof A.2**.**

Using the characteristics, one writes :

[TABLE]

By the Lebesgue’s Theorem the latter term tends to

[TABLE]

where ${\boldsymbol{\varphi}}_{0}(a,t)$ is defined as

Proposition A.3**.**

Under hypotheses 2.1 and 2.2, one has

[TABLE]

where $\ell_{4}$ and ${\boldsymbol{\varphi}}_{0}$ are defined in the proof of Theorem 5.24.

Proof A.4**.**

Dividing $(0,T)$ in two equal parts $(0,T/2)$ and $(T/2,T)$, gives two terms $\ell_{1,1}$ and $\ell_{1,2}$. For the first one, we have

[TABLE]

Now, we estimate the latter term as

[TABLE]

applying Lebesgue’s Theorem shows that $\ell_{1,1,2}\to 0$ when $\varepsilon$ goes to zero.

[TABLE]

where in the third line, the function $(\exp(\zeta_{\min}a/2)-1)/(1+a)$ is monotone increasing provided that $\zeta_{\min}>0$, which then gives the result. The term $\ell_{1,2}$ can be split in two parts :

[TABLE]

Using Duhamel’s principle again for $t<\varepsilon a$ provides :

[TABLE]

Finally, we split $\ell_{1,2,1}$ in two terms

[TABLE]

In a straightforward manner, the latter term goes to zero since

[TABLE]

where we used that $\rho_{\varepsilon}({x},0,t-\varepsilon a)$ is bounded uniformly with respect to $\varepsilon$. The only term remaining and which should converge to a non-zero limit is

[TABLE]

We denote by $\omega_{\varepsilon}:=\Omega\times\{(a,t)\in{\mathbb{R}_{+}}\times(0,T)\text{ s.t. }t\in(T/2,T)\text{ and }a\in((T-t)/\varepsilon,T/\varepsilon)\}$.

[TABLE]

where $\hat{\rho}_{\varepsilon}=\rho_{\varepsilon}-\rho_{0}$. Thanks to the Duhamel’s principle applied to $\hat{\rho}_{\varepsilon}$, one has the estimate $|J_{1}|\sim o_{\varepsilon}(1)$. Indeed as in Lemma 5.2 [16], when $\varepsilon a<t$ one can show that $|\hat{\rho}_{\varepsilon}(a,t)|\lesssim\exp(-\zeta_{\min}t/\varepsilon)+(1+a)^{2}\exp(-\zeta_{\min}a)o_{\varepsilon}(1)$, which gives the result. By Lemma 3.4 [16], $|\rho_{0}({x},a,t)|\lesssim\exp(-\zeta_{\min}a)$ and ${\left\|{{\hat{\mathbf{z}}_{\varepsilon}}}\right\|}_{L^{\infty}((0,T);L^{2}(\Omega))}\sim o_{\varepsilon}(1)$, one has

[TABLE]

In order to prove the limit of $J_{3}$ we define $f(a,t)=a\int_{\Omega}\rho_{0}({x},a,t){\boldsymbol{\psi}}({x},t){\mathbf{z}}_{0}({x},t)d{x}$, and write :

[TABLE]

for all $a\in{\mathbb{R}_{+}}$ and all $\delta>0$ there exists $\eta_{i}$, $i\in\{1,2,3\}$ s.t.

[TABLE]

which means that $\forall a\in{\mathbb{R}_{+}}$, $\forall\delta>0$ there exists $\eta(a,\delta)=\min_{i\in\{1,2,3\}}\eta_{i}>0$ s.t.

[TABLE]

Now there exists $\varepsilon(a,\delta)$ small enough $\varepsilon<\eta(a,\delta)/(1+a)$ s.t. for all $t\in(T-\varepsilon a,T)$, $|t-T|<\eta$, thus

[TABLE]

Setting $g_{\varepsilon}(a)=\frac{1}{\varepsilon a}\int_{T-\varepsilon a}^{T}f(a,t)dt$ and $g_{0}(a)=f(a,T)$, this shows that there is pointwise convergence for every fixed $a$, $g_{\varepsilon}(a)\to g_{0}(a)$. Moreover,

[TABLE]

and by the Lebesgue’s Theorem the convergence holds :

[TABLE]

when $\varepsilon$ goes to zero.

Appendix B Functionnal analysis in Banach valued spaces

Proof B.1** (of Proposition 5.2).**

One has :

[TABLE]

for any $\varphi\in X_{T}$. This proves that ${\rm D}^{\sigma}_{a}f$ belongs to $X_{T}^{\prime}$. $X_{T}$ is a separable Banach space. Thus according to Corollary 3.30 p. 76 [4], for any bounded sequence in $X_{T}^{\prime}$ there exists a subsequence converging in the weak-$\star$ topology $\sigma(X_{T}^{\prime},X_{T})$. As the bound ${\left\|{{\rm D}^{\sigma}_{a}f}\right\|}_{Y_{T}}$ is uniform with respect to $\sigma$, there exists a weak limit in $X_{T}^{\prime}$ denoted $g$ and we define the $X_{T}^{\prime},X_{T}$ duality pairing as

[TABLE]

Moreover, $g=\partial_{a}f$ in the weak sense i.e. for any $\varphi\in{\mathcal{D}}({{\mathfrak{Q}}_{T}})$ one has

[TABLE]

The same results hold obviously for $\partial_{t}f$.

Proof B.2** (of Proposition 5.3).**

By Proposition 5.2, there exists a limit $\partial_{a}f\in X_{T}^{\prime}$ and a subsequence s.t.

[TABLE]

First, ${\rm D}^{\sigma}_{a}f$ defines a linear continuous functional on $Z_{T}$ :

[TABLE]

Moreover, one has tightness, for any continuous positive $\theta_{m}(a)$ s.t. $\theta_{m}\equiv 1$ on $[0,m]$ and $\hbox{\rm supp}\,\theta_{m}\in[0,m+1]$

[TABLE]

the latter term goes to zero as $m$ tends to infinity by the Lebesgue’s Theorem. Now one writes for any $\varphi$ in $Z_{T}$

[TABLE]

For every $\delta>0$ there exists $k_{0}$ s.t. for $k$ and $j$ greater than $k_{0}$ the first term on the right hand side is smaller than $\delta/3$ by weak-$\star$ convergence in $\sigma(X_{T}^{\prime},X_{T})*$, while the two latter terms can be made smaller than $2\delta/3$ due to the tightness proved above. This implies, because $\mathbb{R}$ is complete, that there exists a limit $L$ s.t.

[TABLE]

Since for every arbitrary fixed $\delta$ there exists $j_{0}$ s.t. $j>j_{0}$ implies

[TABLE]

$L$* is also a linear continuous form on $Z_{T}$ thanks to (27). By similar arguments as above we identify this limit with the weak derivative $\partial_{a}f$ and we denote $<\partial_{a}f,\varphi>_{Z_{T}^{\prime},Z_{T}}:=L(\varphi)$ for every $\varphi\in Z_{T}$. The same proof holds for the time derivative as well.*

Lemma B.3**.**

If $f\in U_{T}$ then

[TABLE]

Proof B.4**.**

Taking the test function $\varphi\in{\mathcal{D}}({Q_{T}};L^{1}({\Omega}))$, one has

[TABLE]

Lemma B.5**.**

If $f\in U_{T}$ then

[TABLE]

Proof B.6**.**

Thanks to Theorem 2.2.1 in [7], since $f\in U_{T}={\rm BV}({\mathbb{R}_{+}}\times(0,T);L^{\infty}({\Omega}))$, there exists a smooth function $f_{\delta}\in C^{\infty}({\mathbb{R}_{+}}\times(0,T);L^{\infty}({\Omega}))$, s.t. for every $\delta$ :

[TABLE]

*The first estimates imply directly that *

[TABLE]

as well as

[TABLE]

as $\delta\to 0$. Using that

[TABLE]

and

[TABLE]

one then writes that

[TABLE]

Setting $J^{\delta}({x},a,t):=f^{\delta}({x},a+\Delta a,t+\delta t)-\int_{0}^{1}f^{\delta}({x},a+s\Delta a,t+s\Delta t)ds$ and integrating in age and time gives that

[TABLE]

One concludes thanks to a triangular inequality.

Acknowledgements

I would like to thank warmly Ioan Ionescu for his encouragements and comments that helped me in order to finish and submit this work. I thank also the referee for the very attentive reading of this article and for her/his wise comments.