Well posedness for multi-dimensional junction problems with Kirchoff-type conditions

TL;DR

This paper establishes the uniqueness of viscosity solutions for multi-dimensional junction problems with Kirchoff-type conditions, extending previous one-dimensional results and ensuring convergence of approximations without convexity assumptions.

Contribution

It introduces the first comprehensive analysis of multi-dimensional junctions with Kirchoff conditions, proving uniqueness and convergence of viscosity solutions without convexity constraints.

Findings

Uniqueness of viscosity solutions for multi-dimensional junctions.

Convergence of viscosity-type approximations to a unique limit.

Extension of one-dimensional junction results to higher dimensions.

Abstract

We consider multi-dimensional junction problems for first- and second-order pde with Kirchoff-type Neumann boundary conditions and we show that their generalized viscosity solutions are unique. It follows that any viscosity-type approximation of the junction problem converges to a unique limit. The results here are the first of this kind and extend previous work by the authors for one-dimensional junctions. The proofs are based on a careful analysis of the behavior of the viscosity solutions near the junction, including a blow-up argument that reduces the general problem to a one-dimensional one. As in our previous note, no convexity assumptions and control theoretic interpretation of the solutions are needed.

Equations

$$\{\begin{array}{l}{u}_{i,t}+H_i(D{u}_i,u_i,x,t)=0\text{in}{\Pi }_i\times (0,T],\\ \min \left({\Sigma }_i{u}_{i,x_i}-B,{\min }_i(u_{i,t}+H_i(D{u}_i,u_i,x,t))\right)\le 0\text{on}L\times (0,T],\\ \max \left({\Sigma }_i{u}_{i,x_i}-B,{\max }_i(u_{i,t}+H_i(D{u}_i,u_i,x,t))\right)\ge 0\text{on}L\times (0,T],\end{array}$$

$$B\in \mathbb{R}\text{and}u(\cdot ,0)=u_0\text{on}\overline{\Pi },$$

$$u_0\in BUC(\overline{\Pi }),$$

$$\{\begin{array}{l}{H}_i\text{is coercive in p uniformly on x,t and bounded u, Lipshitz continuous in u and t,}\\ \text{and uniformly continuous in p,u,x,t for bounded p and u.}\end{array}$$

$$H_i\text{is strictly increasing in u.}$$

$$\{\begin{array}{l}{u}_{i,t}+H_i(x_i{D}^2{u}_i,D{u}_i,u_i,x,t)=0\text{in}{\Pi }_i\times (0,T],\\ \min \left({\Sigma }_i{u}_{i,x_i}-B,{\min }_i(u_{i,t}+H_i(0,D{u}_i,u_i,x,t))\right)\le 0\text{on}L\times (0,T],\\ \max \left({\Sigma }_i{u}_{i,x_i}-B,{\max }_i(u_{i,t}+H_i(0,D{u}_i,u_i,x,t))\right)\ge 0\text{on}L\times (0,T],\\ u(\cdot ,0)=u_0\text{on}\overline{\Pi },\end{array}$$

$$\{\begin{array}{l}{u}_{i,ϵ,t}-ϵ\Delta u_{i,ϵ}+H_i(D{u}_{i,ϵ},u_{i,ϵ},x,t)=0\text{in}{\Pi }_i\times (0,T],\\ {\Sigma }_i{u}_{i,ϵ,x_i}=B,\text{on}L\times (0,T],\\ u_{i,ϵ}(\cdot ,0)=u_{0,i}\text{on}{\overline{\Pi }}_i,\end{array}$$

$$\{\begin{array}{l}{u}_{i,t}+H_i(u_{i,x_i},x,t)=0\text{in}I\times (0,T],\\ \min \left({\Sigma }_i{u}_{i,x_i}-B,{\min }_i(u_{i,t}+H_i(u_{i,x_i},0,t))\right)\le 0\text{on}\{0\}\times (0,T],\\ \max \left({\Sigma }_i{u}_{i,x_i}-B,{\max }_i(u_{i,t}+H_i(u_{i,x_i},0,t))\right)\ge 0\text{on}\{0\}\times (0,T],\\ u(\cdot ,0)=u_0\text{on}\overline{I}.\end{array}$$

$$\{\begin{array}{l}(i)p_i\ge q_i\text{and}a+H_i(p_i)\le 0\le b+H_i(q_i),\\ (ii)\min \left({\Sigma }_i{p}_i^{\prime },{\min }_i(a+H_i(p_i^{\prime }))\right)\le 0\text{for all}{p}_i^{\prime }\le p_i,\\ (iii)\max \left({\Sigma }_i{q}_i^{\prime },{\max }_i(b+H_i(q_i^{\prime }))\right)\ge 0\text{for all}{q}_i^{\prime }\ge q_i.\end{array}$$

$$p_i>q_i\text{and}a+\underset{i}{\max }{H}_i(p_i)<0<b+\underset{i}{\min }{H}_i(q_i).$$

$$B=0.$$

$$\{\begin{array}{l}a+H_i(U_{x_i},x_i,t_0)\le 0\text{in}{\overline{I}}_i\text{and }\min \left({\Sigma }_i{U}_{x_i}(0),a+{\min }_i{H}_i(U_{x_i}(0),0,t_0)\right)\le 0,\\ b+H_i(V_{x_i},x_i,t_0)\ge 0\text{in}{\overline{I}}_i\text{and }\max \left({\Sigma }_i{V}_{x_i}(0),b+{\max }_i{H}_i(V_{x_i}(0),0,t_0)\right)\ge 0,\end{array}$$

$$u(0,t_0)-v(0,t_0)-\delta t_0=\underset{(x,t)\in \overline{I}\times [0,T]}{\max }\left[(u-v)(x,t)-\delta t\right].$$

$$a\ge b+\delta >b.$$

$$\underset{(x,t)\to (0,t_0)}{\lim }\underset{p\in \partial z(x,t)}{\sup }|p-\bar{a}|=0.$$

$$a+H_i(u_{x_i},0,t_0)\le {\eta }_i(x_i,t)\text{and}b+H_i(v_{x_i},0,t_0)\ge {\eta }_i(x_i,t).$$

$$u_i^{ϵ}(x_i,t)=\frac{u(ϵx_i,t_0+ϵ(t-t_0))-u(0,t_0)}{ϵ}\text{and}{v}_i^{ϵ}(x_i,t)=\frac{v(ϵx_i,t_0+ϵ(t-t_0))-v(0,t_0)}{ϵ}.$$

$$u_i^{ϵ}\le v_i^{ϵ}+\delta (t-t_0).$$

$$u_{i,t}^{ϵ}(x_i,t)=u_t(ϵx_i,t_0+ϵ(t-t_0))\to a\text{and}{v}_{i,t}^{ϵ}(x_i,t)=v_t(ϵx_i,t_0+ϵ(t-t_0))\to b.$$

$$\tilde{w}(x_i,t)-\tilde{w}{x}_i,t_0)=\frac{w(ϵx_i,t_0+ϵ(t-t_0))-w(ϵx_i,t_0)}{ϵ}={\int }_0^1{w}_{\tau }(ϵx_i+\lambda ϵ(t-t_0))d\lambda (t-t_0),$$

$$u_i^{ϵ}(x_i,t)-u_i^{ϵ}(x_i,t_0)\to a(t-t_0)\text{and}{v}_i^{ϵ}(x_i,t)-v_i^{ϵ}(x_i,t_0)\to b(t-t_0).$$

$$u_i^{{ϵ}_n}(x_i,t)\to U_i(x_i)+a(t-t_0)$$

$$a+\overline{H}(U_{i,x_i})\le 0,\min \left({\Sigma }_i{U}_{i,x_i},\underset{i}{\min }(a+{\overline{H}}_i(U_{i,x_i}))\right)\le 0\text{and}{U}_i(0)=0,$$

$$b+H_i(M)\ge 0\text{and}{u}_i^{ϵ}(x_i,t_0)\le M{x}_i.$$

$${\tilde{v}}_i^{ϵ}(x_{,}t)=v_i^{ϵ}(x,t)\wedge (M{x}_i+b(t-t_0))$$

$$V_{i{,}_{\star }}(x_i,t)=\underset{ϵ\to 0,(y_i,s)\to (x_i,t)}{\mathrm{liminf}}{\tilde{v}}^{ϵ}(y_s,s),$$

$$V_{i{,}_{\star }}(x_i,t)=V_i(x_i)+b(t-t_0),U_i\le V_i\text{in}(-\infty ,0]\text{and}{V}_i(0)=0.$$

$$b+\overline{H}(V_{i,x_i})\ge 0\text{and}\max \left({\Sigma }_i{V}_{i,x_i},\underset{i}{\max }(b+{\overline{H}}_i(V_{i,x_i}))\right)\ge 0.$$

$$a>b.$$

$${\underline{p}}_i:=\underset{x_i\to 0}{\mathrm{liminf}}\frac{U_i(x_i)}{x_i},{\overline{p}}_i:=\underset{x_i\to 0}{\mathrm{limsup}}\frac{U_i(x_i)}{x_i},{\underline{q}}_i:=\underset{x_i\to 0}{\mathrm{liminf}}\frac{V_i(x_i)}{x_i}\text{ and}{\overline{q}}_i:=\underset{x_i\to 0}{\mathrm{limsup}}\frac{V_i(x_i)}{x_i},$$

Full text

Key words and phrases Hamilton-Jacobi equations, networks, discontinuous Hamiltonians, junction problesm, stratification problems, comparison principle, viscosity solutions.

AMS Class. Numbers. 35F21, 49L25, 35B51, 49L20.

1 Notation and Terminology

Given $x\in\mathbb{R}^{d}$ we write $x=(x^{\prime},x_{d})$ with $x^{\prime}\in\mathbb{R}^{d-1}$. For $i=1,\ldots,K$, $\Pi_{i}:=\{(x^{\prime},x_{d,i})\in\mathbb{R}^{d}:x^{\prime}\in\mathbb{R}^{d-1},x_{d,i}\leq 0\}$ are half-planes intersecting along the line $L:=\{(x^{\prime},0):x^{\prime}\in\mathbb{R}^{d-1}\}$ and set $\Pi:=\bigcup_{i=1}^{K}\Pi_{i}$. For simplicity we write $x_{i}$ instead of $x_{d,i}.$ Given $u\in C(\overline{\Pi};\mathbb{R})$, if $(x^{\prime},x_{i})\in\overline{\Pi}_{i}$, we write $u_{i}(x^{\prime},x_{i}):=u(0,\ldots,x^{\prime},x_{i},0.\ldots);$ when possible, to simplify the notation, we drop the subscript on $u_{i}$ and simply write $u(x^{\prime},x_{i}).$ In this setting $u_{x_{i}}(0,x_{i})$ is the exterior normal derivative of $u_{i}:\Pi\to\mathbb{R}$ on $L$. We consider $K$-junction one dimensional problems in the domain $I:=\bigcup_{i=1}^{K}I_{i}$ with junction $\{0\}$, where, for $i=1,\ldots,K$, $I_{i}:=(-a_{i},0)$ and $a_{i}\in[-\infty,0)$. We work with functions $u\in C({\overline{I}};\mathbb{R})$ and, for $x=(x_{1},\ldots,x_{K})\in\bar{I}$, we write $u_{i}(x_{i})=u(0,\ldots,x_{i},\dots,0)$; when possible, to simplify the writing, we drop the subscript on $u_{i}$ and write $u(x_{i})$. We also use the notation $u_{x_{i}}$ and $u_{x_{i}x_{i}}$ for the first and second derivatives of $u_{i}$ in $x_{i}$. For $w\in C(\overline{I}\times[0,T])$ and $t_{0}\in(0,T]$, $J^{+}w(0,t_{0})$ and $J^{-}w(0,t_{0})$ denote respectively the super- and sup-jets of $w$ at $(0,t_{0})$, which may be, of course, empty. If $(p_{1},\dots,p_{K},a)\in J^{+}w(0,t_{0})$, then, for all $(x,t)\in\overline{I}\times[0,T]$, $w(x_{i},t)\leq w(0,t_{0})+p_{i}x_{i}+a(t-t_{0})+\text{o}(|x|+|t-t_{0}|).$ If $(p_{1},\dots,p_{K},a)\in J^{-}w(0,t_{0}),$ then $w(x_{i},t)\geq w(0,t_{0})+p_{i}x_{i}+a(t-t_{0})+\text{o}(|x|+|t-t_{0}|).$

Throughout the paper we work with viscosity sub- and super-solutions. In most cases, however, we will not be using the term viscosity. Also we will not keep repeating that $i\in\{1,\ldots,K\}$ but rather we will say for all $i$.

2 Introduction

We study the well-posedness of the generalized viscosity solutions to time dependent multi-dimensional junction problems satisfying a Kirchoff-type Neumann condition at the junction. We prove that the solutions satisfy a comparison principle and, hence, are unique. It is then immediate that viscosity approximations satisfying the same boundary condition converge to the unique solution. Our results, which are the first of this kind, are simple, self-contained and depend on elementary considerations about viscosity solutions and, we emphasize, do not require any convexity assumptions and the control theoretical interpretation of the solutions.

This work is a continuation of our previous paper (Lions and Souganidis [10]) where we introduced the notion of state constraint solution to one-dimensional junction problems, proved its well-posedness, and considered, for the first time, the limit of Kirchoff-type viscosity approximations.

We also show that the so-called flux limiter solutions introduced and studied in the references below for convex problems reduce to Kirchoff-type generalized viscosity solutions. Hence, uniqueness follows immediately by the simple arguments in this note.

Among the long list of references on this topic with convex Hamiltonians we refer to Achdou and Tchou [1], Barles, Briani and Chasseigne[2, 3], Barles, Briani, Chasseigne and Imbert [4], Barles and Chasseigne [5], Bressan and Hong [6], Imbert and Monneu [7] and Imbert and Nguen [8].

We are interested in the well-posedeness of continuous solutions $u:\overline{\Pi}\to\mathbb{R}$ to the Kirchoff -type initial boundary value problem

[TABLE]

with

[TABLE]

where

[TABLE]

and, for each $i$,

[TABLE]

As always for time-independent problems the Lipshitz continuity of $H_{i}$ in $u$ is replaced by

[TABLE]

We remark that, as it will be clear from the proofs, the particular choice of the Neumann condition in (1) is by no means essential. The arguments actually apply to more general boundary conditions of the form $G(u_{x_{1}},\ldots,u_{x_{K}},u),$ with the map $(p_{1},\ldots,p_{k},u)\to G(u_{x_{1}},\ldots,u_{x_{K}},u)$ strictly increasing with respect to all its arguments.

The main result is:

Theorem 2.1

Assume (4). If $u,v\in BUC(\overline{\Pi}\times[0,T])$ are respectively a sub-and super-solution to (1) with $u_{i}(\cdot,0)\leq v_{i}(\cdot,0)$ on $\overline{\Pi}_{i}$, then $u\leq v$ on $\overline{\Pi}\times[0,T].$ Moroever, for each $u_{0}\in BUC(\overline{\Pi})$, the initial boundary value problem (2), (3) has a unique solution $u\in BUC(\overline{\Pi}\times[0,T]).$

As it will become apparent from the proof, it is possible to generalize the result to the problem

[TABLE]

when, in addition to (4), each $H_{i}$ is degenerate elliptic with respect to the Hessian. Since the arguments are almost identical to the ones for the proof of Theorem 2.1, we do not present any details.

Next we state the result about the convergence of viscosity approximations to (1). The claim is immediate from the fact that any limit is solution to (1) and, hence, we do not write the proof. We remark that we can easily use “more complicated” second-order approximations than the one below.

For $\epsilon>0$ consider the initial boundary value problem

[TABLE]

which, in view of the (4), if $u_{0}$ satisfies (3) has a unique solution $u\in BUC(\overline{\Pi}\times[0,T]).$

Theorem 2.2

Assume (3) and (4). Then $u=\lim_{\epsilon\to 0}u_{\epsilon}$ exists and $u$ is the unique solution to (1).

Since the proof is an immediate consequence of well known estimates and the uniqueness result, we will not discuss it any further.

We also show that, in the context of the one-dimensional time dependent junction problems, the flux-limiter solutions put forward in [7] are actually generalized viscosity solution to (8) with appropriate choice of B in the Kirchoff condition, and, hence, are unique. This provides a simple and straightforward proof of the uniqueness without the need to consider cumbersome test functions and invoke any convexity.

Following the last remark, we emphasize that Kirchoff-type conditions appear to be the “correct” ones, that is the only conditions that are compatible with the maximum principle. This can be easily seen, for example, at the level of second-order equations by considering affine solutions in each branch.

In a forthcoming paper [11], we discuss problems with more general dependence on the Hessian both in the equations and along the junctions. We also consider “stratification”-type problems, that is junctions with branches of different dimension, and, we present results about the convergence of semi-discrete in time approximations with error estimates. Finally, we consider solutions which are not necessarily Lipschitz.

In this note, to simplify the notation and explain the ideas better, we present all the arguments in the special case $d=1$, in which case (1) reduces to

[TABLE]

Organization of the paper.

In the next section we state and prove an elementary lemma which is the basic tool for the proof of the comparison principle which is presented in Section 4. Section 5 is about the relation with the the flux limiters.

3 A general lemma

In this section we state and prove a general lemma which is the basic tool for the proof of Theorem 2.1. It applies to problems of one-dimensional junctions with Kirchoff condition and expands the class of “gradients” that can be used in the inequalities at the junction.

Lemma 3.1

Assume that $H_{1},\ldots H_{K}\in C(\mathbb{R})$, $p_{1},\ldots,p_{K},q_{1},\dots,q_{K}\in\mathbb{R}$ and $a,b\in R$ are such that, for all $i\in\{1,\ldots,K\}$,

[TABLE]

Then $a\leq b$.

Proof. We argue by contradiction and assume that $a>b$.

Modifying $p_{1},\ldots,p_{K},q_{1},\dots,q_{K},a$ and $b$ by small amounts and using the continuity of $H_{1},\ldots,H_{K}$, we may assume that

[TABLE]

If $\Sigma_{i}q_{i}\geq 0$, choose $\epsilon>0$ small enough so that $q_{i}+\epsilon/K<p_{i}$. Then $p_{i}^{\prime}=q_{i}+\epsilon/K$ in (9)(ii) yields $\min_{i}(a+H_{i}(q_{i}+\epsilon/K))\leq 0,$ and, after letting $\epsilon\to 0$, $\min_{i}(a+H_{i}(q_{i}))\leq 0,$ which, in view of (10), is not possible given that it assumed that $a>b$. A similar argument yields a contradiction if $\Sigma_{i}p_{i}\leq 0$.

Next we assume that $\Sigma_{i}p_{i}>0>\Sigma_{i}q_{i},$ and let $c\in(b,a)$ and $r_{i}\in(q_{i},p_{i})$ be such that $H_{i}(r_{i})+c=0.$ If $\Sigma_{i}r_{i}\geq 0$ (resp. $\Sigma_{i}r_{i}\leq 0$), we choose $p_{i}^{\prime}=r_{i}$ (resp. $q_{i}^{\prime}=r_{i}$) and argue as before.

4 One-dimensional time dependent junctions

Here we prove Theorem 2.1 for the initial value problem (8). The proof in the multi-dimensional setting is almost identical and we leave it up to the reader to fill in the details. The existence of solutions is immediate from Perron’s method or Theorem 2.2.

To simplify the presentation here we take

[TABLE]

Although the proof is not long, to clarify the strategy and highlight the new ideas, we present first a heuristic description of the argument assuming that $u_{i},v_{i}\in C^{1}(\overline{I}_{i}\times[0,T])$ with possible discontinuities in the spatial derivative as $i$ changes; note that, since it also assumed that $u,v\in C(\overline{I}\times[0,T]),$ the previous assumption also gives that $u_{t}(0,t),v_{t}(0,t)$ exist for all $t\in(0,T].$

Following the proof of the classical maximum principle, we assume that, for $\delta>0$, the

$\max_{x\in\overline{I}\times[0,T]}\left[(u-v)(x,t)-\delta t\right]$ is achieved at $(x_{0},t_{0})\in\overline{I}\times[0,T]$ with $t_{0}>0.$ If $x_{0}\neq 0$, we argue as in the classical uniqueness proof. Hence, we continue assuming that $x_{0}=0.$ Let $a=u_{t}(0,t_{0})$ and $b=v_{t}(0,t_{0})$. It follows that $a\geq b+\delta>b.$

The functions $U(x_{i})=u(x_{i},t_{0})$ and $V(x_{i})=v(x_{i},t_{0})$ are smooth sub-and super-(viscosity) solutions to

[TABLE]

while $U(x_{i})-V(x_{i})\leq U(0)-V(0)$, which in turn implies that $U_{x_{i}}(0)\geq V_{x_{i}}(0)$.

We get a contradiction using Lemma 3.1 provided we verify that (9) holds for the obvious choices of $H_{1},\ldots,H_{K},p_{1},\dots,p_{K},q_{1},\ldots,q_{K}.$ And this is immediate since (9)(i) is part of (11), while (9)(ii),(iii) follow from the observation that $J^{+}U_{i}(0)=(-\infty,U_{x_{i}}(0)]$ and $J^{-}V_{i}(0)=[V_{x_{i}}(0),\infty)$ and the fact that inequalities must hold in the viscosity sense.

We continue now with the actual proof of Theorem 2.1 for (8), which consists of making the above rigorous for $u,v\in\text{BUC}(\overline{I}\times[0,T]).$ For the beginning of the proof, we assume that the reader is familiar with the “standard” arguments of the proof of the comparison of viscosity solutions.

Proof. Arguing by contradiction we suppose that, for some $\delta>0$ and $t_{0}\in(0,T)$,

[TABLE]

We leave it up to the reader to convince him/herself that, if the above does not hold as $\delta\to 0$, then the conclusion of the theorem holds.

Replacing $u$ and $v$ respectively by the classical $\sup$- and $\inf$-convolutions in time, we may assume that (12) still holds, perhaps for a different $t_{0}$ and shorter time interval, and that $u$ and $v$ are respectively semiconvex and semiconcave and, hence, Lipschitz continuous with respect to $t$. Moreover, in view of the assumed uniform coercivity of the Hamiltonians, it follows that $u$ is also Lipschitz continuous in $x$; note that we do not make a similar claim for $v$.

Of course, this means that we need to consider (8) in a smaller time interval and evaluate $H$ at a different time. It is, however, standard that this does not alter the outcome, and, hence, we omit the details.

In view of the assumed semiconvexity and semiconcavity of the $u$ and $v$ respectively, both of them are differentiable with respect to $t$ at $(0,t_{0})$. Let $a=u_{t}(0,t_{0})$ and $b=v_{t}(0,t_{0})$. It follows that

[TABLE]

The next step is an observation, which, heuristically speaking, establishes a $C^{1}$-type property for the sub- and super-differentials of semiconvex and semiconcave functions near points of differentiability. Since the claim may be useful in other contexts, we state it as a separate lemma.

Lemma 4.1

Let $z:(c,0]\times(0,T)\to\mathbb{R}$ be uniformly continuous in space, and semiconvex in time and assume that, for some $t_{0}\in(0,T)$, $\bar{a}=z_{t}(0,t_{0})$ exists. If $\partial z(x,t)$ is the subdifferential of $z$ with respect to $t$ at $(x,t)$, then

[TABLE]

A similar statement is true for the superdifferential in $t$, if $z:(c,0]\times(0,T)\to\mathbb{R}$ is uniformly continuous in space, semiconcave in time and differentiable at some $(0,t_{0})$.

The claim follows from the classical fact that, for semiconvex functions, derivatives converge to derivatives, and the uniform continuity in $x$.

In what follows, all claims and statements hold for $i=1,\ldots,K$.

Continuing the ongoing proof we observe that Lemma 4.1 yields $\eta_{i}:\overline{I}_{i}\times[0,T]\to\mathbb{R}$ such that $\lim_{(x_{i},t)\to(0,t_{0})}\eta_{i}(x_{i},t)=0$ and, in the viscosity sense and in a neighborhood of $(0,t_{0})$,

[TABLE]

Indeed, if $(p_{i},\bar{p}_{i})\in J^{+}u(x,t)$, then $\bar{p}_{i}\in\partial u(x,t)$, and the claim follows from the previous observations and the continuity properties of $H_{i}$.

Next we use a blow up argument at $(0,t_{0})$ on all branches to reduce the problem to a time independent setting to which we can apply Lemma 3.1.

For $\epsilon>0$, let

[TABLE]

It is immediate that $u^{\epsilon}_{i}$ and $v^{\epsilon}_{i}$ are respectively sub- and super-solution of (8).

In view of the definition of $(0,t_{0})$, the properties of $u$ and $v$ and the observations above, it is immediate that

[TABLE]

Moreover, Lemma 4.1 implies that, as $\epsilon\to 0$ and locally uniformly in $(x,t)$,

[TABLE]

Since, for $\tilde{w}=u^{\epsilon}_{i}$ or $\tilde{w}=v^{\epsilon}_{i}$ and $w=u_{i}$ or $w=v_{i}$,

[TABLE]

it follows, again from Lemma 4.1 and (16), that

[TABLE]

The space-time Lipschitz continuity of the $u_{i}$’s, (17) and the stability properties of sub-solutions imply that, along subsequences $\epsilon_{n}\to 0$,

[TABLE]

and

[TABLE]

where, for notational simplicity, we write $\overline{H}_{i}(p)$ in place of $H_{i}(p,0,t_{0}).$

Since the $v_{i}$’s are not necessarily Lipschitz continuous in space, it is not possible to use a similar argument for the $v^{\epsilon}_{i}$’s. Instead, we need to consider their lower half-relaxed limit, which although finite from below, in view of (15), may be infinite from above.

We go around this potential difficulty as follows. The coercivity of the Hamiltonians and the Lipschitz continuity of $u_{i}$ yield $M>0$ such that

[TABLE]

Hence $Mx_{i}+b(t-t_{0})$ and, therefore,

[TABLE]

is a super-solution of (8).

It follows from the choice of $M$ and (15), (16) and (17) that the lower-half relaxed limit $V_{i,_{\star}}$ of the $\tilde{v}^{\epsilon}_{i}$ given by

[TABLE]

is finite from above and below, and that there exists a lower semicontinuous $V_{i}:\bar{I}_{i}\to\mathbb{R}$ such that

[TABLE]

The stability properties of the solutions also imply that

[TABLE]

Finally, we recall that

[TABLE]

Next we get a contradiction using Lemma 3.1. While the choice of the $H_{i}$’s is obvious, some work is necessary to identify $p_{1},\ldots,p_{K},q_{1},\ldots,p_{K}$ such that (9) holds.

Set

[TABLE]

and recall that $J^{+}U_{i}(0)=(-\infty,\underline{p}_{i}]$ and $J^{-}V_{i}(0)=[\overline{q}_{i},\infty)$.

Observe that $U_{i}\leq V_{i}$ does not necessarily yield $\underline{p}_{i}\geq\overline{q}_{i},$ the latter being enough to conclude using Lemma 3.1 with $p_{i}=\underline{p}_{i}$ and $q_{i}=\overline{q}_{i}$. Notice, however, that $U_{i}\leq V_{i}$ implies

[TABLE]

Although $\overline{p}_{i}\notin J^{+}U_{i}(0)$ and $\underline{q}_{i}\notin J^{-}V_{i}(0)$, unless $U_{i}$ and $V_{i}$ are respectively differentiable at [math], we claim that (9) holds for $p_{i}=\overline{p}_{i}$ and $q_{i}=\underline{q}_{i}.$

A classical blow-up argument (see, for example, Jensen and Souganidis [9]) shows that

[TABLE]

Moreover, if $p_{i}^{\prime}\leq\underline{p}_{i}$ for all $i$, then $p^{\prime}_{i}\in J^{+}U_{i}(0)$ and, hence, $\min\left(\Sigma_{i}p_{i}^{\prime},\min_{i}(a+\overline{H}_{i}(p_{i}^{\prime}))\right)\leq 0.$

If, for some fixed $i_{i_{0}}$, $p^{\prime}_{i_{0}}\in[\underline{p}_{i_{0}},\overline{p}_{i_{0}}],$ then, in view of (21), $a+H_{i_{0}}(p^{\prime}_{i_{0}},0,t_{0})\leq 0$, and again $\min\left(\Sigma_{i}p_{i}^{\prime},\max_{i}(a+\overline{H}_{i}(p_{i}^{\prime}))\right)\leq 0.$ It follows that (9)(ii) holds.

A similar argument yields (9)(iii), while (9)(i) is obviously true, in view of (20) and (21).

5 Flux-limiter solutions are generalized Kirchoff solutions

We show here that the flux-limiter solutions to time-depending one dimension junction problems, which were introduced in [7], are actually generalized viscosity solutions to (8) for an appropriate choice of $B$ in the Kirchoff-condition.

We begin recalling the notion of flux-limiter solution. Following [7], we assume that, for all $i=1,\ldots,K$,

[TABLE]

and define $\widetilde{H}_{i}^{\pm}:\mathbb{R}\to\mathbb{R}$ by

[TABLE]

note that [7] considers quasiconvex $\widetilde{H}_{i}$’s but to simplify the presentation here we assume convexity. Finally, to simplify the presentation we assume that we deal with continuous solutions.

Fix $A\geq A_{0}=\max_{i}\min_{\mathbb{R}}\widetilde{H}$ and let $\widetilde{I}_{i}=(0,\infty)$ and $\widetilde{I}=\bigcup_{i=1}^{K}\widetilde{I}_{i}$. Then $\widetilde{u}\in BUC(\overline{\widetilde{I}}\times[0,T])$ is an $A$-limiter solution of the junction problem if

[TABLE]

For each $i$, let $p_{i}^{A}$ be the unique solution to $\widetilde{H}_{i}(p)=A$ such that $p_{i}^{A}\geq p_{i}^{0}$, which exists in view of (23).

Proposition 5.1

If $\widetilde{u}$ is an $A$-limiter solution, that it satisfies (24), then $u:\overline{I}\to R$ defined by $u(x)=\widetilde{u}(-x)$ is a generalized solution to (8) for $B=-\Sigma_{i=1}^{K}p_{i}^{A}$ and $H_{i}(p)=\widetilde{H}_{i}(-p).$

Proof. The conclusion follows once we show that $\widetilde{u}$ is a solution to

[TABLE]

Clearly we only need check the inequalities on $\{0\}\times(0,T].$ We begin with the sub-solution property and assume that, for some $t_{0}\in(0,T]$ and for each $i$, $(p_{i},a)\in J^{+}u_{i}(0,t_{0}).$

Since $\widetilde{u}$ is an $A$-limiter solution, for all $i$, we have

[TABLE]

Assume that

[TABLE]

It then follows that there exists $i_{0}$ such $p_{i_{0}}<p_{i_{0}}^{A}$. Then, if $p^{0}_{i_{0}}\leq p_{i_{0}}$, since we are in the increasing part of $\widetilde{H}_{i_{0}}$, we have $\widetilde{H}_{i_{0}}(p_{i_{0}})\leq\widetilde{H}_{i_{0}}(p_{i_{0}}^{A})=A$, and, hence

[TABLE]

If $p^{0}_{i_{0}}\geq p_{i_{0}},$ then $\widetilde{H}_{i_{0}}(p_{i_{0}})=\widetilde{H}^{-}_{i_{0}}(p_{i_{0}}),$ and again we have (28), and, hence, the sub-solution property.

For the super-solution property we assume that, for some $t_{0}\in(0,T]$ and for each $i$, $(q_{i},a)\in J^{-}u_{i}(0,t_{0}).$ It then follows from the definition of the $A$-limiter solution that

[TABLE]

If $\max_{i}\widetilde{H}^{-}_{i}(q_{i})\geq A$, then, since $\widetilde{H}_{i}(q_{i})\geq\widetilde{H}^{-}_{i}(q_{i})$,

[TABLE]

If $A>\max_{i}\widetilde{H}^{-}_{i}(q_{i})$, then, for all $i$,

[TABLE]

Assume that $-\Sigma_{i=1}^{K}q_{i}+\Sigma_{i=1}^{K}p_{i}^{A}\leq 0$, for otherwise (30) is satisfied.

Then there must exist some $i_{0}$ such that $q_{i_{0}}\geq p_{i_{0}}^{A}$, which implies that $\widetilde{H}_{i_{0}}(q_{i_{0}})\geq\widetilde{H}^{+}_{i_{0}}(p_{i_{0}}^{A})=A$, and (30) holds again.

The claim then follows using that $u(x)=\widetilde{u}(-x).$