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Global entropy weak solutions for general non-local
trac ow models with anisotropic kernel
Felisia Angela Chiarello, Paola Goatin
To cite this version:
Felisia Angela Chiarello, Paola Goatin. Global entropy weak solutions for general non-local trac
ow models with anisotropic kernel. ESAIM: Mathematical Modelling and Numerical Analysis, EDP
Sciences, 2018, 52, pp.163-180. �hal-01567575�
Global entropy weak solutions for general non-local traffic flow
models with anisotropic kernel
Felisia Angela Chiarello
∗
Paola Goatin
∗
July 24, 2017
Abstract
We prove the well-posedness of entropy weak solutions for a class of scalar conservation laws
with non-local flux arising in traffic modeling. We approximate the problem by a Lax-Friedrichs
scheme and we provide L
∞
and BV estimates for the sequence of approximate solutions. Stabil-
ity with respect to the initial data is obtained from the entropy condition through the doubling
of variable technique. The limit model as the kernel support tends to infinity is also studied.
Key words: Scalar conservation laws, Anisotropic non-local flux, Lax-Friedrichs scheme, Traffic
flow models.
1 Introduction
We consider the following scalar conservation law with non-local flux
∂
t
ρ + ∂
x
f(ρ)v(J
γ
∗ ρ)
= 0, x ∈ R, t > 0, (1.1)
where
J
γ
∗ ρ(t, x) :=
Z
x+γ
x
J
γ
(y − x)ρ(t, y)dy, γ > 0. (1.2)
In (1.1), (1.2), we assume the following hypotheses:
(H)
f ∈ C
1
(I; R
+
), I = [a, b] ⊆ R
+
,
v ∈ C
2
(I; R
+
) s.t. v
0
≤ 0,
J
γ
∈ C
1
([0, γ]; R
+
) s.t. J
0
γ
≤ 0 and
R
γ
0
J
γ
(x)dx := J
0
, ∀γ > 0, lim
γ→∞
J
γ
(0) = 0.
This class of equations includes in particular some vehicular traffic flow models [4, 10, 15, 18], where
γ > 0 is proportional to the look-ahead distance and the integral J
0
is the interaction strength (here
assumed to be independent of γ). In this setting, the non-local dependence of the speed function v
can be interpreted as the reaction of drivers to a weighted mean of the downstream traffic density.
Unlike similar non-local equations [2, 3, 5, 6, 7, 11, 19], these models are characterized by the
presence of an anisotropic discontinuous kernel, which makes general theoretical results [1, 2, 3]
inapplicable as such. On the other side, the specific monotonicity assumptions on the speed function
v and the kernel J
γ
ensure nice properties of the corresponding solutions, such as a strong maximum
principle (both from below and above) and the absence of unphysical oscillations due to a sort of
monotonicity preservation, which make the choice (1.2) interesting and justified from the modeling
perspective.
Adding an initial condition
ρ(0, x) = ρ
0
(x), x ∈ R, (1.3)
with ρ
0
∈ BV(R; I), entropy weak solutions of problem (1.1), (1.3), are intended the following sense
[2, 3, 13].
∗
Inria Sophia Antipolis - M´editerran´ee, Universit´e Cˆote d’Azur, Inria, CNRS, LJAD, 2004 route des Lucioles - BP
93, 06902 Sophia Antipolis Cedex,France. E-mail: {felisia.chiarello, paola.goatin}@inria.fr
1
Definition 1. A function ρ ∈ (L
1
∩ L
∞
∩ BV)(R
+
× R; I) is an entropy weak solution of (1.1),
(1.3), if
Z
+∞
0
Z
R
n
|ρ − κ|ϕ
t
+ sgn(ρ − κ)(f(ρ) − f(κ))v(J
γ
∗ ρ)ϕ
x
− sgn(ρ − κ)f(κ)v
0
(J
γ
∗ ρ)∂
x
(J
γ
∗ ρ)ϕ
o
dxdt +
Z
R
ρ
0
(x) − κ
ϕ(0, x)dx ≥ 0 (1.4)
for all ϕ ∈ C
1
c
(R
2
; R
+
) and κ ∈ R.
The main results of this paper are the following.
Theorem 1. Let hypotheses (H) hold and ρ
0
∈ BV(R; I). Then the Cauchy problem (1.1), (1.3),
admits a unique weak entropy solution ρ
γ
in the sense of Definition 1, such that
min
R
{ρ
0
} ≤ ρ
γ
(t, x) ≤ max
R
{ρ
0
}, for a.e. x ∈ R, t > 0. (1.5)
Moreover, for any T > 0 and τ > 0, the following estimates hold:
TV(ρ
γ
(T, ·)) ≤ e
C(J
γ
)T
TV(ρ
0
), (1.6a)
ρ
γ
(T, ·) − ρ
γ
(T − τ, ·)
L
1
≤ τe
C(J
γ
)T
f
0
kvk + J
0
kfk
v
0
TV(ρ
0
), (1.6b)
with C(J
γ
) := J
γ
(0)
v
0
f
0
kρ
0
k + 2kfk
+
7
2
J
0
kfk
v
00
.
Above, and in the sequel, we use the compact notation k·k for k·k
L
∞
.
Corollary 2. Let hypotheses (H) hold and ρ
0
∈ BV(R; I). As γ → ∞, the solution ρ
γ
of (1.1),
(1.3) converges in the L
1
loc
-norm to the unique entropy weak solution of the classical Cauchy problem
(
∂
t
ρ + ∂
x
f(ρ)v(0)
= 0, x ∈ R, t > 0
ρ(0, x) = ρ
0
(x), x ∈ R.
(1.7)
In particular, we observe that C(J
γ
) → 0 in (1.6a) and (1.6b), allowing to recover the classical
estimates.
The paper is organized as follows. Section 2 is devoted to the proof of the stability of solutions
with respect to the initial data, based on a doubling of variable argument [13]. We observe that, for
a close class of non-local equations, uniqueness of solutions has been recently derived in [12] relying
on characteristics method and a fixed-point argument, thus avoiding the use of entropy conditions.
In our setting, we prefer to keep the classical approach to pass to the limit γ → ∞.
In Section 3 we derive existence of solutions through an approximation argument based on
a Lax-Friedrichs type scheme. In particular, we prove accurate L
∞
and BV estimates on the
approximate solutions, which allow to derive (1.5) and (1.6). We remark once again that these
estimates heavily rely on the monotonicity properties of J
γ
, and do not hold for general kernels,
see [2, 4]. Note that, regarding the Arrhenius look-ahead model [18], our result allows to establish
a global well-posedness result and more accurate L
∞
estimates with respect to previous studies
[15]. Moreover, to our knowledge, Corollary 2 provides the first convergence proof of a limiting
procedure on the kernel support. We present some numerical tests illustrating this convergence in
Section 4. Unfortunately, the limit γ → 0, which was investigated numerically in [2, 4, 10], remains
unsolved, since in this case the constants in (1.6) blow up.
2
2 Uniqueness and stability of entropy solutions
The Lipschitz continuous dependence of entropy solutions with respect to initial data can be derived
using Kruˇzkov’s doubling of variable technique [13] as in [3, 4, 10].
Theorem 3. Under hypotheses (H), let ρ, σ be two entropy solutions to (1.1) with initial data
ρ
0
, σ
0
respectively. Then, for any T > 0 there holds
ρ(t, ·) − σ(t, ·)
L
1
≤ e
KT
kρ
0
− σ
0
k
L
1
∀t ∈ [0, T ], (2.1)
with K given by (2.5).
Proof. The functions ρ and σ are respectively entropy solutions of
∂
t
ρ(t, x) + ∂
x
(f(ρ(t, x))V (t, x)) = 0, V := v(ρ ∗ J
γ
), ρ(0, x) = ρ
0
(x),
∂
t
σ(t, x) + ∂
x
(f(σ(t, x))U(t, x)) = 0, U := v(σ ∗ J
γ
), σ(0, x) = σ
0
(x).
V and U are bounded measurable functions and are Lipschitz continuous w.r. to x, since ρ, σ ∈
(L
1
∩ L
∞
∩ BV)(R
+
× R; R). In particular, we have
kV
x
k ≤ 2J
γ
(0)
v
0
kρk, kU
x
k ≤ 2J
γ
(0)
v
0
kσk.
Using the classical doubling of variables technique introduced by Kruzkov, we obtain the following
inequality:
ρ(T, ·) − σ(T, ·)
L
1
≤ kρ
0
− σ
0
k
L
1
(2.2)
+
f
0
Z
T
0
Z
R
ρ
x
(t, x)
U(t, x) − V (t, x)
dxdt
+
Z
T
0
Z
R
f(ρ(t, x))
U
x
(t, x) − V
x
(t, x)
dxdt.
We observe that
U(t, x) − V (t, x)
≤ J
γ
(0)
v
0
ρ(t, ·) − σ(t, ·)
L
1
, (2.3)
and that for a.e. x ∈ R
U
x
(t, x) − V
x
(t, x)
≤
2(J
γ
(0))
2
v
00
ρ(t, ·)
+
v
0
J
0
γ
ρ(t, ·) − σ(t, ·)
L
1
+ J
γ
(0)
v
0
(|ρ − σ|(t, x + γ) + |ρ − σ|(t, x). (2.4)
Plugging (2.3) and (2.4) into (2.2), we get
ρ(T, ·) − σ(T, ·)
L
1
≤ kρ
0
− σ
0
k
L
1
+ K
Z
T
0
ρ(t, ·) − σ(t, ·)
L
1
dt
with
K = J
γ
(0)
v
0
f
0
sup
t∈[0,T ]
ρ(t, ·)
BV(R)
+ 2 sup
t∈[0,T ]
f(ρ(t, ·))
!
+ sup
t∈[0,T ]
f(ρ(t, ·))
1
2(J
γ
(0))
2
v
00
sup
t∈[0,T ]
ρ(t, ·)
+
v
0
J
0
γ
!
. (2.5)
By Gronwall’s lemma, we get the thesis.
3
3 Existence
3.1 Lax-Friedrichs numerical scheme
We discretize (1.1) on a fixed grid given by the cells interfaces x
j+
1
2
= j∆x and the cells centers
x
j
= (j − 1/2)∆x for j ∈ Z, taking a space step ∆x such that γ = N∆x for some N ∈ N,
and t
n
= n∆t the time mesh. Our aim is to construct a finite volume approximate solution
ρ
∆x
(t, x) = ρ
n
j
for (t, x) ∈ C
n
j
= [t
n
, t
n+1
[×]x
j−1/2
, x
j+1/2
]. We approximate the initial datum ρ
0
with the piecewise constant function
ρ
0
j
=
1
∆x
Z
x
j+1/2
x
j−1/2
ρ
0
(x)dx.
We denote J
k
γ
:= J
γ
(k∆x) for k = 0, ..., N − 1 and set
V
n
j
:= v(c
n
j
),
where
c
n
j
:= ∆x
N−1
X
k=0
J
k
γ
ρ
n
j+k
.
The Lax-Friedrichs flux adapted to (1.1) is given by
F
n
j+1/2
:=
1
2
f(ρ
n
j
)V
n
j
+
1
2
f(ρ
n
j+1
)V
n
j+1
+
α
2
(ρ
n
j
− ρ
n
j+1
), (3.1)
α ≥ 0 being the viscosity coefficient. In this way, we obtain the N + 2 points finite volume scheme
ρ
n+1
j
= H(ρ
n
j−1
, ..., ρ
n
j+N
), (3.2)
where
H(ρ
j−1
, ..., ρ
j+N
) := ρ
j
+
λ
2
α(ρ
j−1
− 2ρ
j
+ ρ
j+1
) +
λ
2
f(ρ
j−1
)V
n
j−1
− f (ρ
j+1
)V
n
j+1
, (3.3)
with λ = ∆t/∆x.
Assume ρ
i
∈ I for i = j − 1, ..., j + N, we can compute:
∂H
∂ρ
j−1
=
λ
2
α + V
j−1
f
0
(ρ
j−1
) + ∆x v
0
(c
j−1
)J
0
γ
f(ρ
j−1
)
, (3.4a)
∂H
∂ρ
j
= 1 − λ
α −
1
2
∆xf(ρ
j−1
)v
0
(c
j−1
)J
1
γ
≥ 1 − λ
α +
1
2
∆xJ
γ
(0) kf k
v
0
, (3.4b)
∂H
∂ρ
j+1
=
λ
2
α + ∆xf (ρ
j−1
)v
0
(c
j−1
)J
2
γ
− f
0
(ρ
j+1
)V
j+1
− ∆xf (ρ
j+1
)v
0
(c
j+1
)J
0
γ
, (3.4c)
∂H
∂ρ
j+k
= −
λ
2
∆x
f(ρ
j+1
)v
0
(c
j+1
)J
k−1
γ
− f (ρ
j−1
)v
0
(c
j−1
)J
k+1
γ
, k = 2, . . . , N − 2, (3.4d)
∂H
∂ρ
j+N−1
= −
λ
2
∆xf(ρ
j+1
)v
0
(c
j+1
)J
N−2
γ
, (3.4e)
∂H
∂ρ
j+N
= −
λ
2
∆xf(ρ
j+1
)v
0
(c
j+1
)J
N−1
γ
. (3.4f)
4
