数理解析研究所講究録
第 1651 巻 2009 年 161-178

161

NONLOCAL HAMILTON-JACOBI EQUATIONS RELATEDTO DISLOCATION DYNAMICS AND A
FITZHUGH-NAGUMO SYSTEMOLIVIER LEY

ABSTRACT. We describe recent existence and uniqueness results obtained for nonlocal nonmonotone Eikonal equations modelling the evolution of interfaces. We focus on two model cases. The first one arises
in dislocation dynamics and the second one comes from a FitzHughNagumo system. The equation is nonlocal since, in both case, the velocity at a point of the boundary of the interface depends on the wholeenclosed set via a convolution. In these models, the evolution is nonmonotone since we do not expect to have an inclusion principle.

1. INTRODUCTION
This text is the proceeding of conference given a the RIMS Meeting Viscosity solutions of differential equations and related topics in Kyoto in 2008.
I would like to thank Professors H. Ishii and S. Koike for the invitation.
The aim is to describe recent results for nonlocal and nonmonotone Eikonal
equations obtained in [11, 8, 10, 9] in collaboration with Guy Barles, Pierre
Cardaliaguet, R\’egis Monneau and Aur\’elien Monteillet. I also refer the reader
to [33] and [35]. I have chosen to sacrify some generality and to present themost significant results in two model cases: the dislocation dynamics and a
FitzHugh-Nagumo system.
We are interested in the following equation

(1.1)

where

$\{\begin{array}{ll}\frac{\partial u}{\partial t}(x, t)=c[I_{\{u\geq 0\}}](x, t)|Du(x, t)| in \mathbb{R}^{N}\cross[0, T],u(\cdot, 0)=u_{0} in \mathbb{R}^{N},\end{array}$$u_{0}$:$\mathbb{R}^{N}arrow \mathbb{R}$(1.2)

is Lipschitz continuous and, for all open subset$c[I_{\overline{\Omega}}](x, t)$$:=\alpha(k\star I_{\overline{\Omega}}(x, t))+c_{1}(x, t)$$\Omega\subset \mathbb{R}^{N}$,.

The functions ::andare Lipschitz continuous,”denotes some convolution between a kernel and the indicator functionof St. More precise assumptions will be given later. This expression ofencompasses the two model cases.
$\alpha$$\star$$\mathbb{R}arrow \mathbb{R}$$c_{1}$$\mathbb{R}^{N}\cross[0, T]arrow \mathbb{R}$$k$$c[\cdot]$

1991 Mathematics Subject Classification. $49L25;35F25;35A05;45G10;35K65;35D05$ .
Key words and phrases. Nonlocal Hamilton-Jacobi Equations, dislocation dynamics,
Fitzhugh-Nagumo system, nonlocal front propagation, level-set approach, lower-boundgradient estimate, viscosity solutions, eikonal equation,-dependence in time.
$L^{1}$

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OLIVIER LEY

The paper is organized as follows. In Section 2, we briefly recall somefacts about the level set approach to study front propagation problems. It
is the motivation to study (1.1). Then, in Sections 3 and 4, we introduce
the dynamics of dislocations and a FitzHugh-Nagumo system. Ront propagation problems corresponding to these problems lead to nonlocal nonmonotone speed like (1.2). In Section 5, we introduce a notion of weaksolutions for (1.1). Before stating some existence results of weak solutions,
we recall some properties of the solutions of the classical Eikonal equation(Section 6). The last three sections are devoted to uniqueness results. As
shown by a counter-example (Section 8), weak solutions are not unique ingeneral. Uniqueness holds when the velocity is positive. Our results in thisdirection are stated in Section 9 and a sketch of proof is given in Section 10.

2. PRELIMINARIES ON THE LEVEL SET APPROACH AND NONLOCALNONMONOTONE FRONT PROPAGATION PROBLEMS

Consider the following front propagation problem: we want to find asuch that every point of the boundaryof open subsets of(called the ”front”) evolves with a prescribed normal velocity

family$\mathbb{R}^{N}$$(\Omega_{t})_{t>0}$$x$$\Gamma_{t};=\partial\Omega_{t}$given by
(2.1)$\vec{\mathcal{V}}_{\Omega_{t}}(x)=h(x, t,\overline{\Omega}_{t})\vec{n}\Omega_{t}(x)$,(it means thatisis the outer unit normal ofatwhere“oriented” by its interior ) and is a given evolution law.
The idea of Osher&Sethian [38] for the level set approach is to introduce
whose 0-level set represents thean auxiliary function :front . We therefore define such that, for all $t\geq 0$ ,$u(\cdot, t)>0$ in$u(\cdot, t)=0$ on(2.2)
and $u(\cdot, t)<0$ otherwise.
$\vec{n}\Omega_{t}(x)$$\Gamma_{t}$$x$$\Gamma_{t}$$h$$\Omega_{t}$$\mathbb{R}^{N}\cross[0, T]arrow \mathbb{R}$

$u$

$u$

$\Gamma_{t}$$\Omega_{t}$$\Gamma_{t},$Straightforward computations give:$\vec{n}_{\Omega_{t}}(x)=-\frac{Du(x,t)}{|Du(x_{l}t)|}$$\mathbb{R}om(2.1)$

(2.3)

and

$\vec{\mathcal{V}}_{\Omega_{t}}(x)=\frac{\tau_{t}\partial u(x,t)}{|Du(x,t)|}\vec{n}_{\Omega_{t}}(x)$for all$x\in\Gamma_{t}$.

, we obtain the level set PDE$\frac{\partial u}{\partial t}(x, t)=h(x, t, \{u(\cdot, t)\geq 0\})|Du(x, t)|$for all$x\in\Gamma_{t}$.

This PDE holds a priore on . The main work of Chen, Giga&Goto [18]and Evans&Spruck [22], who were the first to develop rigorously the level.
set approach, was to prove that (2.3) can be set and solved on
which represents theThis PDE is complemented with an initial data$(2.2)$$i.e.,$$t=0$
given). One recovers
a
and
with
holds atinitial front (by setting$\Gamma_{t}$$\mathbb{R}^{N}\cross(0, T]$$u_{0}$$u_{0}$$\Omega_{0}$$\Gamma_{t}$for all $t\geq 0$ .
It is worth mentioning that, even for very simple velocities, the front
may develop singularities in finite time and some changes of topology may$\Gamma_{t}:=\{u(\cdot, t)=0\}$163
NONLOCAL HAMILTON-JACOBI EQUATIONS

happen. Similarly, one cannot hope to find smooth solutions of (2.3). We
will use the notion of viscosity solutions which are well adapted to these
nonlinear problems. We refer the reader to Crandall, Ishii&Lions [20] forviscosity solutions and the book of Giga [25] for an overview of the level set
approach.
Let us introduce some evolution laws we will be interested in. The first
and the simplest one is $h=c(x, t)$ (no dependence with respect to ). In
this case, (2.3) becomes the classical Eikonal equation$h$$\Omega_{t}$(2.4)$\frac{\partial u}{\partial t}(x, t)=c(x, t)|Du(x, t)|$in$\mathbb{R}^{N}\cross[0, T]$(see Barles [6] and Bardi &Capuzzo Dolcetta [5] for instance). We recall
some properties about this equation in Section 6 and we need to develop
fine estimates for its solutions to prove uniqueness results for the more complicated velocities which follows.
We are mainly interested in nonlocal velocities which can be written(2.5)$h(x, t, Stt)=c[I_{D_{t}}](x, t)=\alpha(k\star I_{\overline{\Omega}_{t}}(x, t))+c_{1}(x, t)$.

They lead to the level set PDE (1.1). Notice that this PDE is nonlocal
since the velocity does not depend only on local properties of at but on
the whole set . This brings some difficulties to study (1.1). This nonlocal
dependence is enlighted by the use of the notation.
The first typical case that we consider is the dislocation dynamics where$\Gamma_{t}$$x$$\Omega_{t}$$c[\cdot]$(2.6)$c[I_{\overline{\Omega}_{t}}](x, t)=c_{0}\star I_{\overline{\Omega}_{t}}(x)+c_{1}(x, t)$with a space convolution:(2.7)$c_{0}\in L^{1}(\mathbb{R}^{N})$,and$c_{0} \star I_{\Pi_{t}}(x)=\int_{\mathbb{R}^{N}}c_{0}(x-y)I_{\overline{\Omega}_{t}}(y)dy$.

The second case is a velocity which govems the asymptotics of a FitzHughNagumo system, namely$c[I_{\overline{\Omega}_{t}}](x, t)=\alpha(v(x, t))$

where
of

$\alpha$,is a real valued Lipschitz continuous function and(2.8)$\frac{\partial v}{\partial t}-\Delta v=I_{\Pi_{t}}$in$\mathbb{R}^{N}\cross(0, T)$$v$

is the solution.

Using the representation formula for the heat equation (with a zero initial
data), we have
(2.9)$v(x, t)=G*I_{\overline{\Omega}_{t}}(x, t)$,where ” is the usual space-time convolution and $G$ is the classical Green
kernel. Therefore,$*$(2.10)$c[I_{\{u\geq 0\}}](x, t)=\alpha(G*I_{\{u\geq 0\}}(x, t))$is also in the form (2.5).

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OLIVIER LEY

Before giving some details about dislocations and FitzHugh-Nagumo systems, let us discuss the monotonicity properties of the evolutions underconsideration.
Ront propagation problems (local and nonlocal ones) can be classify into
two categories: the monotone evolutions and the nonmonotone ones.
We say that a front propagation problem is monotone if the inclusion
principle is satisfied. Otherwise it is called a nonmonotone problem. Inclusion principle can be described as follows. Start with initial sets
andand let them evolves with the same velocity. They satisfy the inclusion$\Omega_{0}^{1}$$\Omega_{0}^{2}$principle if(2.11)$\Omega_{0}^{1}\subset\Omega_{0}^{2}$$\Rightarrow$$\Omega_{t}^{1}\subset\Omega_{t}^{2}$for all $t\geq 0$ .

At least formally, the inclusion principle holds when
(2.12)$\Omega\subset\Omega’\subset \mathbb{R}^{N}$and$x\in\partial\Omega\cap\partial\Omega’$$\Rightarrow$$\mathcal{V}_{\Omega}(x)\leq \mathcal{V}_{\Omega’}(x)$.

For instance, this latter property is true for the mean curvature motion andfor $h=c(x, t)$ . Using the level set approach, where is the solution of (2.3)
corresponding tofor $i=1,2$ , the inclusion principle implies$u^{i}$$\Omega^{i}$$\{u_{0}^{1}\geq 0\}\subset\{u_{0}^{2}\geq 0\}$$\Rightarrow$$\{u^{1}(\cdot, t)\geq 0\}\subset\{u^{2}(\cdot, t)\geq 0\}$

for $aUt\geq 0$ .

Since the level set PDE (2.3) holds for all level sets (and not only the 0-levelset), we get. It means that one expects a comparison
principle for (2.3) in the monotone case. It allows to apply Perron’s method(see Ishii [31]) to build solutions for all times to (2.1).
On the contrary, for nonmonotone evolutions, (2.11)-(2.12) are violated
and one cannot expect to have a comparison principle for (2.3). It is a serious
obstacle to build solutions and prove uniqueness results. It happens that our
typical cases (2.6) and (2.10) are nonmonotone front propagation problems.
Indeed, in the case of dislocation dynamics, a physical assumption is$u^{1}\leq u^{2}$(2.13)$(if u_{0}^{1}\leq u_{0}^{2})$$\int_{\mathbb{R}^{N}}c_{0}=0$.

In consequence, (2.12) cannot be satisfied. In the FitzHugh-Nagumo model,
is merely Lispchitz continuous and this is not sufficient to ensure (2.12).
$\alpha$3. DISLOCATION DYNAMICS
Dislocations are lines of defects which propagate in crystals. It is themain microscopic explanation of their macroscopic properties (see the booksof Nabarro [36] and Hirth&Lothe [28] for the physics of dislocations andLardner [32] for a mathematical exposition of the model). In our work, we
consider a special mathematical model due to Rodney, Le Bouar&Finel [39].

Dislocation lines move preferentially in a crystallographic plane. The dynamics is given by a normal velocity proportional to the Peach-Koehler force

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NONLOCAL HAMILTON-JACOBI EQUATIONS

acting on this line. This Peach-Koehler force may have two possible contributions: the first one is the self-force created by the elastic field generated
by the dislocation line itself (i.e., this self-force is a nonlocal function of the
shape of the dislocation line); the second one is the force created by everything exterior to the dislocation line, like the exterior stress applied on thematerial, or the force created by other defects. It follows that the velocity
but we canis given by (2.6) and it leads to (1.1) (a priori in$N\geq2$).
consider any
A mathematical study of this model was started by Monneau and his
collaborators (see [3, 1, 2, 15] and the references therein). Here we focus on
long-time existence and uniqueness results for (1.1). Recall that the motion
is nonmonotone because of (2.13). The pioneer work in this direction is due
to Alvarez, Hoch, Le Bouar&Monneau [3] where existence and uniquenesswere proved for short time. The first uniqueness result was obtained by
Alvarez, Cardaliaguet et Monneau [1] under the assumption that the velocity
is regular enough $(C^{1,1})$ and nonnegative ( $i.e.$ , the front is expanding) whenstarting with initial setshaving an interior ball property. In [11], we
provide a new simpler proof of this fact. The techniques we introduced(lower bound gradient estimates, semiconvexity,
estimates for the levelsets of the solution, etc.) were re-used to obtain the results of [8, 10]. Let
us finally mention the work of Cardaliaguet &Marchi [16] for dislocations
with Neumann boundary conditions.
Several sets of assumptions on
were used in the different works underconsideration. We start with the basic ones.
$c_{1}\in C(\mathbb{R}^{N}\cross[0, T])$(dislo-l)
and there exist
such that, for all$t\in[0, T]$ ,$x,$$\mathbb{R}^{2}\cross[0, T]$$\Omega_{0}$$L^{1}$$c_{0},$ $c_{1}$$\overline{c},\overline{C}>0$$c_{0},$$y\in \mathbb{R}^{N},$$|c_{0}(x, t)|+|c_{1}(x,t)|\leq\overline{c}$,$|c_{0}(x, t)-c_{0}(y,t)|+|c_{1}(x, t)-c_{1}(y, t)|\leq\overline{C}|x-y|$.

Moreover, $c_{0}\in C([0, T], L^{1}(\mathbb{R}^{N}))$ .
Notice that this assumption ensures that the velocity is bounded:$c[I_{\{u(\cdot,t)\geq 0\}}](x,t)$$=$$\int_{\mathbb{R}^{N}}c_{0}(x-y)$

I $\{u(\cdot,t)\geq 0\}(y)dy+c_{1}(x, t)$$\leq$$\sup_{0\leq t\leq T}|c_{0}(\cdot, t)|_{L^{1}(\mathbb{R}^{N})}+\overline{c}$.

4. A FITZHUGH-NAGUMO TYPE SYSTEM
Consider(4.1)$\{\begin{array}{ll}u_{t}=\alpha(v)|Du| in \mathbb{R}^{N}\cross(0, T),v_{t}-\Delta v=g^{+}(v)I_{\{u\geq 0\}}+g^{-}(v)(1-I_{\{u\geq 0\}}) in \mathbb{R}^{N}\cross(0, T),u(\cdot, 0)=u_{0}, v(\cdot, 0)=v_{0} in \mathbb{R}^{N}.\end{array}$This system yields a front $\Gamma_{t}=\{u(\cdot, t)=0\}$ which evolves with normal, the function being itself the solution of a reaction-diffusion
velocity$\alpha(v)$

$v$

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OLIVIER LEY

equation whose coefficients change according to the regions determined by$\Gamma_{t}$.
This system appears when taking the asymptotics, as$\epsilonarrow 0$, to theFitzHugh-Nagumo system(4.2)$\{\begin{array}{ll}u_{t}^{\epsilon}-\epsilon\triangle u^{\epsilon}=\frac{1}{\epsilon}f(u^{\epsilon}, v^{\epsilon}) in \mathbb{R}^{N}\cross(0, T),v_{t}^{\epsilon}-\Delta v^{\epsilon}=g(u^{\epsilon}, v^{\epsilon}) in \mathbb{R}^{N}\cross(0, T),\end{array}$where$\{\begin{array}{l}f(u, v)=u(1-u)(u-a)-v (0<a<1),g(u, v)=u-\gamma v (\gamma>0).\end{array}$andin (4.1) are Lipschitz continuous and depend
The functions$f,$. Moreoverin . Initialon
are bounded and satisfy
and.
are Lipschitz continuous and is bounded anddata
andThese equations are related to wave propagation phenomena in excitable
media. There exist a lot of works on this subject in biology, chemistry,
physics and mathematics, see for instance [24, 37, 23, 41, 27, 17].
The issues we are interested in are the same as for dislocations. We want
to define long-time solutions and prove some uniqueness properties. Giga,
Goto&Ishii [26] obtained some weak solutions of (4.1). Wheras Soravia&
Souganidis [40] established rigorously the convergence of (4.2) towards the. In particular,
limit problem (4.1) and proved the properties ofandthey found some conditions under which $\alpha>0$ . Until [10, Theorem 4.1],
uniqueness was an open problem. We proved uniqueness for (4.1) when$\alpha>0$ .
To simplify, here we will choose $g^{+}\equiv 1,$ $g^{-}\equiv 0$ and $v_{0}=0$ (see [10]for the general case). To sum up, we consider (1.1) with a velocity givenby (2.10), where is the solution of (2.8) and thus may be written as (2.9).
The following properties of are straightforward.
$\alpha,$$g$$u_{0}$$g^{+}$$g^{-}$$g^{-}$$g^{+}$$g^{-}\leq g^{+}$$v_{0}$$v_{0}$$\mathbb{R}$$C^{1}$$\alpha,$$g^{+}$$g^{-}$$v$$v$

Lemma 4.1. [10, Lemma 4.2] For all$v$$\chi\in L^{\infty}(\mathbb{R}^{N}\cross[0, T];[0,1])$, the solution

of

(4.3)

$\frac{\partial v}{\partial t}-\Delta v=\chi$is continuous,$|v(x, t)|\leq t,$

where$\gamma_{N}$$v(\cdot, t)$in$\mathbb{R}^{N}\cross(0, T)$,$v(x, 0)=0$ ,is $C^{1,\beta}(\beta<1)$ and, for all$|Dv(x, t)|\leq\gamma_{N}\sqrt{t}$$x\in \mathbb{R}^{N},$$0\leq s\leq t\leq T$,

and $|v(x, t)-v(x, s)|\leq\gamma_{N}\sqrt{s}\sqrt{t-s}+t-s$ ,is a constant which depends only on the dimension.

In the sequel, we will assume(FN-1) :is Lipschitz continuous.
From Lemma 4.1 and (FN-1), we obtain some properties of the velocity$c[\chi]=\alpha(v)$ . In particular, it is bounded (because
is bounded in $[0, T]$
independently of ).
$\alpha$$\mathbb{R}arrow \mathbb{R}$$v$$\chi$

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NONLOCAL HAMILTON-JACOBI EQUATIONS

The main features of the FitzHugh-Nagumo problem are the following.
On the one hand, the motion is nonmonotone since there is no monotonicity
assumption on . On the other hand, even if is smooth, the regularity ofthe velocity is limited by the regularity of which is, at the best,for
all $\beta<1$ (it comes from the regularity properties for the heat equation withcoefficients). This lack of regularity is a major difficulty and it prevents
us to use the techniques of [8] which require avelocity (see Section 9).
$\alpha$$\alpha$$C^{1,\beta}$$v$$L^{\infty}$$C^{1,1}$

5. DEFINITION OF WEAK SOLUTIONSIn [8] and [9], we introduce a new notion of weak solution.

Definition 5.1. [8, 9] A continuous function
solution of (1.1) if there exists(1)
is a
viscosity solution of$\chi\in L^{\infty}(\mathbb{R}^{N}\cross[0, T];[0,1])$

$u$

(5.1)

is a weaksuch that$u:\mathbb{R}^{N}\cross[0, T]arrow \mathbb{R}$$L^{1}$$\{\begin{array}{ll}\frac{\partial u}{\partial t}(x, t)=c[\chi](x, t)|Du(x, t)| in \mathbb{R}^{N}\cross[0, T],u(\cdot, 0)=u_{0} in \mathbb{R}^{N}.\end{array}$(2) For almost all $t\in[0, T]$ ,$I_{\{u(\cdot,t)>0\}}\leq\chi(\cdot, t)\leq I_{\{u(\cdot,t)\geq 0\}}$almost everywhere in

Moreover, we say that the weak solution
almost all $t\in[0, T]$ ,(5.2)$I_{\{u(\cdot,t)>0\}}=I_{\{u(\cdot,t)\geq 0\}}$$u$$\mathbb{R}^{N}$.

of (1.1) is classical if, foralmost everywhere in$\mathbb{R}^{N}$.

The main difficulty to define solutions of geometrical equations like (1.1)
is the fattening phenomenon which may appear (See Giga [25] and the references therein). In this case, the set $\{u(\cdot, t)=0\}$ has positive Lebesgue
measure andis discontinuous from $[0,$ $T]$ into.
When there is no fattening, is uniquely determined by$t\mapsto c[I_{\{u(\cdot,t)\geq 0\}}]$$L^{1}(\mathbb{R}^{N})$$\chi$$\chi(\cdot, t)=I_{\{u(\cdot,t)>0\}}=I_{\{u(\cdot,t)\geq 0\}}$.

This definItion makes interest for equations which are well-posed when the
non-local term is frozen. More precisely, the point is to be able to solve (5.1)
in the sense ofviscosity solutions for a fixed . Notice thatviscosity
solutions appear naturally since, in the dislocation case for instance, theconvolution regularizes the velocity in space but not in time, namely $(x, t)\mapsto$$c[\chi](x, t)$ is merely
measurable. The generalization of the notion of viscosity
solutions for equations with measurable in time coefficients is due to Ishii[30]. For further references see [8, Appendix ] where the results we needare collected.
This notion of solutions is very weak. In general, there is no uniqueness(see Section 8) but it provides general existence results. When the velocity
is positive, we prove that the solutions are in fact classical and we obtainsome uniqueness results (see Section 9).
$L^{1}$$L^{1}$$\chi$$A$

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OLIVIER LEY

6. PRELIMINARIES ON THE CLASSICAL EIKONAL EQUATION AND LOWER
BOUND GRADIENT ESTIMATE

Consider (2.4) with an initial data$c$$u_{0}$are:. Classical assumptions on the speed(eikonal) $c\in C(\mathbb{R}^{N}\cross[0, T])$ and there exist$t\in[0, T]$ ,$x,$$\overline{c},\overline{C}>0$

such that, for all$y\in \mathbb{R}^{N},$$0\leq c(x, t)\leq\overline{c}$,$|c(x, t)-c(y, t)|\leq\overline{C}|x-y|$.

Assume moreover that(lower-bound) (Lower bound gradient estimate on the initial front)
is Lipschitz continuous and there exists $\eta_{0}>0$ such that$u_{0}$:$\mathbb{R}arrow \mathbb{R}$(6.1)$-|u_{0}|-|Du_{0}|+\eta_{0}\leq 0$in$\mathbb{R}^{N}$in the viscosity sense.

Some comments about this latter hypothesis are given below. The firstpart of the following theorem is classical, see Crandall &Lions [21] andIshii [29]. The second part comes from Ley [34] and still holds in the contextof$L^{1}$

viscosity solutions.

Theorem 6.1. [34](i) (Lipschitz regularity) Under the assumption (eikonal), (2.4) has a
is Lipschitz continuous, then is
unique viscosity solution . If
$t\in[0, T]$ ,Lipschitz continuous and, for all
$u$

$u$

$u_{0}$$x\in \mathbb{R}^{N},$$|Du(x, t)|\leq e^{\overline{C}T}|Du_{0}|_{\infty}$,$|u_{t}(x, t)|\leq\overline{c}e^{\overline{C}T}|Du0|_{\infty}$.

(ii) (Preservation of the lower bound gradient estimate) Assume that(eikonal) and (lower-bound) hold true. Then there exists $\eta=$$\eta(T,\overline{C},\overline{c}, \eta_{0})>0$

such that$-|u(x, t)|-|Du(x, t)|+\eta\leq 0$ in $\mathbb{R}^{N}x[0, T]$(6.2)

in the viscosity sense.

In the context of the level set approach, (6.1) and (6.2) imply a lower. Indeed, suppose thatarebound gradient estimate on the front$|Du(x,t)|\geq$smooth. If is on the front, then $u(x, t)=0$ and (6.2) implies$\eta>0$ . It follows from the implicit function theorem that the front is a smooth
are not smooth in general and (6.1), (6.2) has to
hypersurface. But
sense (see [34] for details). Nevertheless theinageneralized
be understoodlower bound gradient estimate holds almost everywhere in a neighborhood
type estimates for level setsof the front. This is enough to prove some) which are crucial.
(with
like
At this step, let us make a very important remark. Since the velocity isbounded (cf. Sections 3 et 4), let us say by a constant , we have a finite
speed if propagation. With the notations of (2.2), if$u_{0},$ $u$$\Gamma_{t}$$x$$u_{0},$ $u$$L^{1}$$\{-\delta\leq u(\cdot, t)\leq\delta\}$$\delta\approx 0$$\overline{V}$(6.3)$\Gamma_{0}\cup\Omega_{0}=\{u_{0}\geq 0\}\subset\overline{B}(0, R_{0})$,169
NONLOCAL HAMILTON-JACOBI EQUATIONS

then

(6.4)

$\Gamma_{t}\cup\Omega_{t}=\{u(\cdot, t)\geq 0\}\subset\overline{B}(0, R_{0}+\overline{V}T)$

for all $t\geq 0$ .

It means that, starting with compact fronts, everything takes place in a big
fixed ball
. Thanks to the expression (1.2) for the velocity
together with the assumptions (dislo-l) and (FN-1), we deduce that thevelocitysatisfies (eikonal) with constants which are independent of.
as soon as is compactly supported in
It follows that the greater part of the results for the the classical eikonal
equation applies to our model problems.
$\overline{B}(0, R_{0}+\overline{V}T)$$c[\chi]$$\chi\in$$L^{\infty}(\mathbb{R}^{N}\cross[0, T];[0,1])$$\overline{B}(0, R_{0}+\overline{V}T)$$\chi$

7. EXISTENCE OF WEAK SOLUTIONS AND CLASSICAL SOLUTIONS
We have

Theorem 7.1. [8, 9] Under the assumptions (dislo-l) (dislocations case) or(FN-1) (FitzHugh-Nagumo case), for all Lipschitz continuous
such that(6.3) holds, Equation (1.1) admits at least a weak solution in.
$u_{0}$$\mathbb{R}^{N}\cross[0, T]$As said above, one does not have any comparison principle which allows to build visocsity solutions by Perron’s method. We need to use otherstrategies. In the case of dislocations, existence is proved in [8, Theorem
1.2 by approximation: the velocity
is regularized by replacing theindicator function by a continuous function. We can apply Schauder’s theorem to the perturbated equation and extract a convergent subsequence byAscoli’s theorem. To conclude, it remains to prove that the limit is a solution. This is not obvious because we are not in the classical frameworkof viscosity solutions. At this point, we need to use a new stability result
for measurable in time equations which was proved by Barles [7]. For the
FitzHugh-Nagumo system, existence was proved in [26] for a different notionof weak solutions. In [9], we introduce a general framework yielding weaksolutions (in the sense of Definition 5.1) for both model problems (and evenmore general cases). Our proof is based on Kakutani’s fixed point theorem(see [4]) which was already the main ingredient of the proof in [26]. We end
by recalling that, since the velocity
satisfies (eikonal) with constants
which are independent of , we can apply Theorem 6.1 (i) and (6.4) in our$c[1_{\{u\geq 0\}}]$$]$$c[\chi]$$\chi$

proof.
Let us state some additional assumptions in order to obtain classical solutions.
$t\in[0, T],$ $0\leq-|c_{0}(\cdot, t)|_{L^{1}(\mathbb{R}^{N})}+c_{1}(x,t)$ .
(dislo-2) For all(FN-2).
A consequence of these new assumptions is that $c[\chi](x,t)$ is nonnegative
for all $\chi\in L^{\infty}(\mathbb{R}^{N}x[0, T];[0,1]),$and $t\in[0, T]$ .
$x\in \mathbb{R}^{N},$$0\leq\alpha$$x\in \mathbb{R}^{N}$

Theorem 7.2. [8, 9] Under the assumptions $(dislo-1-2)$ (dislocations case)or (FN-1-2) (FitzHugh-Nagumo case), for all Lipschitz continuoussuch$u_{0}$

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OLIVIER LEY

that (6.3) and (lower-bound) hold, the weak solutions of (1.1) are classicalones.

The proof is straightforward using the preservation of the lower boundgradient estimate of Theorem 6.1 since this latter property implies that thefront has zero Lebesgue measure and therefore (5.2) holds.

8. A COUNTER-EXAMPLE TO UNIQUENESS
The following example comes from [8, Example 3.1] and is inspired by [12].
It takes profit of the fact that the velocity vanishes.
We set $N=1$ and consider the following PDE of type (1.1),(8.1)$\{\begin{array}{l}\frac{\partial U}{\partial t}= (1 \star I\{U(\cdot,t)\geq 0\}(x)+c_{1}(t))|DU| in \mathbb{R}\cross(0,2],U(\cdot, 0)=u_{0} in \mathbb{R},\end{array}$where we choose $c_{1}(x, t)$ $:=c_{1}(t)=2(t-1)(2-t)$ and $u_{0}(x)=1-|x|$ . Notice.
for all measurable subsetthat 1We start by studying auxiliary problems in the time intervals $[0,1]$ and[1, 2] which will be useful to build a family of weak solutions for (8.1) in$[0,2]$ .
1. Construction of a solution for $0\leq t\leq 1$ . The function $x_{1}(t)=(t-1)^{2}$ issolution of $\dot{x}_{1}(t)=c_{1}(t)+2x_{1}(t)$ on $(0,1)$ with $x(O)=1$ (note thatin $[0,1])$ . Consider$\star I_{A}=\mathcal{L}^{1}(A)$$A\subset \mathbb{R}$$\dot{x}_{1}\leq 0$(8.2)$\{\begin{array}{l}\frac{\partial u}{\partial t}=\dot{x}_{1}(t)|\frac{\partial u}{\partial x}| in \mathbb{R}\cross(0,1],u(\cdot, 0)=u_{0} in R.\end{array}$

From Theorem 6.1, there exists a unique continuous viscosity solutionof (8.2). By Lax-Oleinik formula, $u(x, t)=u_{0}(|x|-x_{1}(t)+1)$ . Hence, for$0\leq t\leq 1$ , we have
$u$

(8.3)

$\{u(\cdot, t)>0\}=(-x_{1}(t), x_{1}(t))$

et$\{u(\cdot, t)\geq 0\}=[-x_{1}(t), x_{1}(t)]$ ,In Step 3, we will establish that is a weak solution of (8.1) on $[0,1]$ .
2. Construction of solutions for $1\leq t\leq 2$ . For all measurable functions$0\leq\gamma(t)\leq 1$ , let
be the unique solution of $\dot{y}_{\gamma}(t)=c_{1}(t)+2\gamma(t)y_{\gamma}(t)$ on$y_{\gamma}(1)=0$. By comparison, one has $0\leq y_{0}(t)\leq y_{\gamma}(t)\leq y_{1}(t)$(1,2) with
for $1\leq t\leq 2$ , where $y0,$
are the solutions of the previous equation within [1, 2]. Then, consider
and 1. Note that$u$$y_{\gamma}$$y_{1}$$\gamma(t)\equiv 0$$\dot{y}_{\gamma}\geq 0$$\{\begin{array}{l}\frac{\partial u_{\gamma}}{\partial t}=\dot{y}_{\gamma}(t)|\frac{\partial u_{\gamma}}{\partial x}| in \mathbb{R}\cross(1,2],u_{\gamma}(\cdot, 1)=u(\cdot, 1) in \mathbb{R},\end{array}$is the solution of (8.2). Again, this problem has a unique continuous
and $u_{\gamma}(x, t)=u(|x|-y_{\gamma}(t), 1)$
viscosity solution
which is zero if

where

$u$

$u_{\gamma}$$|x|\leq y_{\gamma}(t)$171
NONLOCAL HAMILTON-JACOBI EQUATIONS

otherwise (note that, since $u(\cdot, 1)\leq 0$ , by the maximum principle,. It follows$u_{\gamma}\leq 0$in$\mathbb{R}\cross[1,2])$

(8.4)

and

$\{u_{\gamma}(\cdot, t)>0\}=\emptyset$$\{u_{\gamma}(\cdot, t)’\geq 0\}=\{u_{\gamma}(\cdot, t)=0\}=[-y_{\gamma}(t), y_{\gamma}(t)]$.

3. There are several weak solutions to (8.1). For $0\leq\gamma(t)\leq 1$ , set$c_{\gamma}(t)=c_{1}(t)+2x_{1}(t)$,$c_{\gamma}(t)=c_{1}(t)+2\gamma(t)y_{\gamma}(t)$Then, from Steps 1 and 2,$U_{\gamma}(x, t)=u(x, t)$,$U_{\gamma}$$U_{\gamma}(x, t)=u_{\gamma}(x, t)$if $(x, t)\in \mathbb{R}\cross[0,1]$ ,if $(x, t)\in \mathbb{R}\cross[1,2]$ .

is the continuous viscosity solution of$\{\begin{array}{l}\frac{\partial U_{\gamma}}{\partial t}=c_{\gamma}(t)|\frac{\partial U_{\gamma}}{\partial x}| in \mathbb{R}\cross(0,2],U_{\gamma}(\cdot, 0)=u_{0} in \mathbb{R}.\end{array}$Takingobtain$\chi_{\gamma}(\cdot, t)=\gamma(t)I_{[-y_{\gamma}(t),y_{\gamma}(t)]}$for $1\leq t\leq 2$ , from (8.3) et (8.4), we$I_{\{U_{\gamma}(\cdot,t)>0\}}\leq\chi_{\gamma}(\cdot, t)\leq$

I$\{U_{\gamma}(\cdot,t)\geq 0\}$(see Figure 1). This implies that all the functions$\gamma(t)\leq 1$ , areweak solutions of (8.1).

$U_{\gamma}$, for measurable $0\leq$$\backslash -\underline{\lrcorner}\overline{\alpha,<0}\overline{\alpha,>0}$

FIGURE 1

172
OLIVIER LEY

9. UNIQUENESS RESULTS
We obtained several uniqueness results under different assumptions in
the case of dislocations. I focus on the most recent one here (establishedin [10], see Theorem 9.1). This result requires the weakest assumption on theregularity of the velocity which needs to be merely Lipschitz continuous. I
describe simultaneously the case of dislocations dynamics and the FitzHughNagumo system. A sketch of proof is given in Section 10.
When the velocity is only Lipschitz continuous, we assume that it is positive. In order to get this property, we need to reinforce (dislo-2) and(FN-2):(dislo-3) There exists$\underline{c}>0$$-|c_{0}(\cdot, t)|_{L^{1}(\mathbb{R}^{N})}+c_{1}(x, t)$(FN-3) There exists
We have.

$\underline{c}>0$

such that, for all

such that$x\in \mathbb{R}^{N},$$0<\underline{c}\leq\alpha$$t\in[0, T],$$0<\underline{c}\leq$.

Theorem 9.1. [10, Theorems 3.1 and 4.1] Assume (dislxl-3) (dislocations case) or (FN-1-3) (FitzHugh-Nagumo case) and suppose that
sat$\Gamma_{0}:=\{u_{0}=0\}$(lower-bound),
(6.3)
and
that
is
. Then there
isfiesexists a unique (classical) viscosity solution for (1.1).
Let us now explain what are the results we obtained previously for dislocations when the velocity is more regular, namely
or semiconvex. We,
recall that :is semiconvex if, for all $x,$$f(x+y)+\tilde{f}(x-y)-2f(x)\geq-L|y|^{2}$ .
(9.1)$u_{0}$$C^{2}$$C^{1,1}$$f$$\mathbb{R}^{N}arrow \mathbb{R}$$y\in \mathbb{R}^{N}$

We refer the reader to the book of Cannarsa&Sinestrari [14] for details
about semiconcavity (a function is semiconcave if $-f$ is semiconvex. If
is both semiconvex and semiconcave then it is).
Theorem 9.2. (Dislocations case) Suppose that (dislo-l) holds, thatsatisfies (6.3) and (lower-bound) and thatandare semiconvex uniformly with respect to $t\in[0, T]$ .
(1) [1, Theorem 4.3] and [11, Theorem 4.2] If (dislo-2) holds and
$f$

$f$

$C^{1,1}$$u_{0}$$c_{0}(\cdot, t)$$c_{1}(\cdot, t)$$u_{0}$is semiconvex, then there exists unique (classical) viscosity solutionfor (1.1).
(2) [8, Theorem 1.3] If (dislo-3) holds, then there enists unique (classical) viscosity solution for (1.1).

10. SKETCH OF THE PROOF OF THE UNIQUENESS THEOREM 9.1
I give a sketch of the proof of Theorem 9.1 and I will point out how getting
the results of Theorem 9.2 for more regular velocities. For the whole proof,see [10, Proofs of Theorems 3.1 and 4.1].
Under the assumptions of Theorem 9.1, consider two classical solutionsof (1.1) with the same initial data
and(the existence is given by
7.2).
Theorems 7.1 and$u^{1}$$u^{2}$$u_{0}$

173
NONLOCAL HAMILTON-JACOBI EQUATIONS

1. Preliminary estimates. As explained at the end of Section 6, the subsets$i=1,2$ are contained in a ballandsatisfies (eikonal) with fixed constants. Therefore, the conclusions of Theorem 6.1 hold true for the. In particular, forsmall enough, wehave the lower bound gradient estimate$\{u^{i}(\cdot, t)\geq 0\},$$\overline{B}(0, R_{0}+\overline{V}T)$$u^{i}\prime s$(10.1)$|Du^{i}| \geq\frac{\eta}{2}$$c[I_{\{u^{i}(\cdot,t)\geq 0\}}]$$\overline{\delta}>0$

for almost all $(x, t)$ such that.

$x\in\{-\overline{\delta}\leq u^{i}(\cdot, t)\leq\overline{\delta}\}$For $0\leq\tau\leq T$, define$\delta_{\tau}=\sup_{\mathbb{R}^{N}\cross[0,\tau]}|u^{1}-u^{2}|$.

Since $\delta_{0}=0$ and using the continuity of , we can choose $\tau>0$ small.
enough in order to have
Sinoe the ’s satisfy (1.1), by a classical comparison result for eikonal
equations with different speeds (see [11, Lemma 2.2]), we have$u^{i}$$\delta_{\tau}<\overline{\delta}$$u^{i}$(10.2)$\delta_{\tau}\leq|Du_{0}|_{\infty}e^{\overline{C}\tau}\int_{0}^{\tau}|c[I_{\{u^{1}(\cdot,t)\geq 0\}}]-c[I_{\{u^{2}(\cdot,t)\geq 0\}}](\cdot, t)|_{\infty}dt$.

Now, the purpose is to bound the previous integral by a quantity like(10.3)$o_{\tau}(1)\delta_{\tau}$.

It follows
for small . By a step-by-step argument, we can conclude$\delta_{T}=0$ . At this step, we
have to distinguish the dislocations case and the
FitzHugh-Nagumo one. The difference lies in the convolution kernel whichappears in the velocity. For dislocations, this kernel is bounded whereas it is
not bounded in the FitzHugh-Nagumo case (the heat kernel is not boundedwith respect to time). In this latter case, we need fine perimeter estimatesin order to get (10.3).
2. Dislocations case. We continue the computation (10.2) by using (2.6).
$\delta_{\tau}=0$$\delta_{\tau}$since(10.4)$c_{0}$$\tau$$\leq$$|Du_{0}|_{\infty} e^{\overline{C}\tau}\int_{0}^{\tau}|c_{0}(\cdot,t)\star(I_{\{u^{1}(\cdot,t)\geq 0\}}-I_{\{u^{2}(\cdot,t)\geq 0\}})|_{\infty}dt$$\leq$$\overline{c}|Du_{0}|_{\infty}e^{\overline{C}\tau}\int_{0}^{\tau}\int_{\mathbb{R}^{N}}(1_{\{-\delta_{\tau}\leq u^{1}\leq 0\}}+I_{\{-\delta_{\tau}\leq u^{2}\leq 0\}})dxdt$,

is bounded (see (dislo-l)) and$|I_{\{u^{1}\geq 0\}}-I_{\{u^{2}\geq 0\}}|\leq$I$\{-\delta_{r}\leq u^{1}\leq 0\}+I_{\{-\delta_{\tau}\leq u^{2}\leq 0\}}$in$\mathbb{R}^{N}\cross[0, \tau]$.

It remains to deal with$\int_{0}^{\tau}\int_{\mathbb{R}^{N}}$I$\{-\delta_{\tau}\leq u^{t}\leq 0\}^{dxdt}$.

Depending on the type of assumptions, there are several ways to proceed toobtain one of the results of Theorems 9.1 or 9.2. Let us start by a heuristic
computation which enlights the interest of the lower bound gradient estimate

174
OLIVIER LEY

and the perimeter estimates in the proof of Theorem 9.2. By the coarea
formula, using (10.1), we have$\int_{\mathbb{R}^{N}}$I$=$$\{-\delta_{\tau}\leq u^{i}(\cdot,t)\leq 0\}^{dx}$$\int_{-\delta_{\tau}}^{0}\int_{\{u^{i}(\cdot,t)=s\}}|Du|^{-1}d\mathcal{H}^{N-1}ds$$\underline{2\delta_{\tau}}$$\leq$–$\sup$Per$(\{u(\cdot,$$t)=s\})$ .

$\eta-\delta_{\tau}\leq s\leq 0$

It means that, if we may obtain a bound for the perimeter of $\{u(\cdot, t)=s\}$
for all $t\in[0, \tau]$ , then we are done. This bound was obtainedwithin [1] as a consequence of the propagation of the interior ball property for. In [11], we follow an equivalent strategy
the front when the velocity is(semiconvexity associated with a lower bound gradient estimate is equivalent
to the interior ball property, see [11, Lemma A.1] . This latter strategy has([11,
the advantage to require only volume estimates ofprove.
Section 3] and not perimeter estimates which are more delicate toMoreover, we could improve theseestimates for less regular velocities. In[8], we prove that, if (dislo-3) holds and the velocity is semiconvex, thenan interior ball property is created during the evolution and the desired
perimeter estimates follow.
:
be a
Let us come back to the proof of Theorem 9.1. Let(it suffices to take which iscontinuous function such that
. It
on
on
and linear with a slope
zero onfollows (see [10, Proposition 5.5]), using the lower bound gradient and theequation, that$s\approx 0$$C^{1,1}$$)$$\mathcal{L}^{N}(\{-\delta_{\tau}\leq u^{i}\leq 0\})$$)$$L^{1}$$\mathbb{R}arrow \mathbb{R}^{+}$$\varphi$$\delta_{\tau}\varphi’=n_{[-\delta_{\tau},0]}$$(-\infty, -\delta_{\tau}],$$1$$\int_{0}^{\tau}\int_{\mathbb{R}^{N}}$$\varphi$$\mathbb{R}^{+}$I$[-\delta_{\tau}, 0])$$1/\delta_{\tau}$$\{-\delta_{\tau}\leq u^{i}\leq 0\}^{dxdt}$$=$$\int_{0}^{\tau}\int_{\mathbb{R}^{N}}\delta_{\tau}\varphi’(u^{i}(x, t))dxdt$$\leq$$\int_{0}^{\tau}\int_{\mathbb{R}^{N}}\delta_{\tau}\varphi’(u^{i}(x, t))\frac{c[I_{\{u^{i}\geq 0\}}](x,t)}{\underline{c}}\frac{|Du^{i}|}{\eta}dxdt$$=$$\frac{\delta_{\tau}}{\underline{c}\eta}\int_{0}^{\tau}\int_{\mathbb{R}^{N}}\varphi’(u^{i}(x, t))\frac{\partial u^{i}}{\partial t}dxdt$$=$$\frac{\delta_{\tau}}{arrow c\eta}\int_{0}^{\tau}\int_{\mathbb{R}^{N}}\frac{\partial}{\partial t}\varphi(u^{i}(x, t))dxdt$$\leq$$\frac{\delta_{\tau}}{\underline{c}\eta}(\mathcal{L}^{N}(\{u^{i}(\cdot,$$\tau)\geq-\delta_{\tau}\})-\mathcal{L}^{N}(\{u0\geq 0\}))$.

is an upper-bound.
The dominated convergence theorem implies that
It completes the proof in the case of dislocations.
3. FitzHugh-Nagumo system. In this case, we estimate (10.2) as follows.
$o_{\tau}(1)\delta_{\tau}$$|c[I_{\{u^{1}(\cdot,t)\geq 0\}}]-c[I_{\{u^{2}(\cdot,t)\geq 0\}}](\cdot, t)|_{\infty}$$=$$|(\alpha(v_{1})-\alpha(v_{2}))(\cdot, t)|_{\infty}$$\leq$$\overline{C}|(v_{1}-v_{2})(\cdot, t)|_{\infty}$,

175
NONLOCAL HAMILTON-JACOBI EQUATIONS. We continue the computation (10.2) by taking profit of the formula for
given by Lemma 4.1
and (10.4):
where$v_{i}$

is the solution of (4.3) with$\chi=I_{\{u^{i}\geq 0\}}$$v_{i}$(10.5)$\delta_{\tau}\leq|Du_{0}|_{\infty}e^{\overline{C}\tau}$$\int_{0}^{r}\int_{0}^{t}\int_{\mathbb{R}^{N}}G(x-y, t-s)(I_{\{-\delta_{\tau}\leq u^{1}\leq 0\}}+I_{\{-\delta_{\tau}\leq u^{2}\leq 0\}})dydsdt$.

At this step, we cannot conclude as in the dislocation case sinoe
is notbounded. Moreover, the merely Lipschitz continu\’ity of the velocity does
not imply some interior ball properties. We overcome this difficulty by
establishing some interior cone properties which are more involved.
At first we have$G$$\{-\delta_{\tau}\leq u^{i}\leq 0\}\subset E_{i}(t):=(\{u^{i}(\cdot, t)\geq 0\}+\frac{2\delta_{\tau}B(0,1)}{\eta})\backslash \{u^{i}(\cdot, t)\geq 0\}$.

The above inclusion means that one can keep under control the size ofby broaden a bit the 0-level set. Notice this is hopelessin general; the lower bound gradient estimate is crucial.
Next step is devoted to show that the subsets $\{u^{i}(\cdot, t)\geq 0\}$ satisfy auniform interior cone property. Namely, for each, thereexists a conewith a degree of opening and a height whose vertex
is and such that(see Figure 2). The proof of this$\{-\delta_{\tau}\leq u^{i}\leq 0\}$$x\in\partial\{u^{i}(\cdot, t)\geq 0\}$$C_{x}^{\rho,\theta}$$\theta$$\rho$$C_{x}^{\rho,\theta}\subset\{u^{i}(\cdot, t)\geq 0\}$$x$

FIGURE 2
result is based on the positiveness of the velocity (FN-3) and the nonsmooth
Pontryagine maximum principle (see Clarke [19]). Such tools were alreadyused for proving the creation of the interior ball property in [8] (see also [13]and [1] .
Then, we prove that a bounded subset satisfying the uniform interior cone
property has a finite perimeter.
$)$

176
OLIVIER LEY

Theorem 10.1. [10, Theorem 5.8] Let $K$ be a compact subset of
sat$inte7\dot{v}or$
isfying the uniform
cone property with pammeters and . Then,
there exists
such that, for all $R>0$ ,$\mathbb{R}^{N}$$\theta$$\rho$$\Lambda=\Lambda(N, \rho, \theta)$(10.6)$\mathcal{H}^{N-1}(\partial K\cap\overline{B}(0, R))\leq\Lambda \mathcal{L}^{N}(K\cap\overline{B}(0, R+\rho/4))$.

The proof of this result is involved and used Besicovitch’s covering theorem.
From the two previous results, we get [10, Lemma 4.4]:$\int_{0}^{t}\int_{\mathbb{R}^{N}}G(x-y, t-s)I_{E_{i}(t)}dyds\leq\overline{\Lambda}\frac{2\delta_{\tau}}{\eta}$,

where A depends on the given data and (see (10.6)). Plugging this estimatein (10.5), we obtain an upper-bound like (10.3). The proof of the theorem
is complete.
$\Lambda$

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LABORATOIRE DE MATH\’EMATIQUES ET PHYSIQUE TH\’EORIQUE, F\’ED\’ERATION DENISPOISSON, UNIVERSIT\’E FRAN\caOIS RABELAIS TOURS, PARC DE GRANDMONT, 37200 TOURS,FRANCE, leyQlmpt. univ-tours. fr