Entropy formulation for fractal conservation laws
Nathaël Alibaud
3/03/2006
Université Montpellier II, Département de mathématiques, CC 051, Place E. Bataillon, 34 095
Montpellier cedex 5, France
E-mail:alibaud@math.univ-montp2.fr
Abstract. Using an integral formula of Droniou and Imbert (2005) for the fractional Laplacian,
we define an entropy formulation for fractal conservation laws with pure fractional diffusion of order
λ ∈]0, 1]. This allows to show the existence and the uniqueness of a solution in the L
∞
framework.
We also establish a result of controled speed of propagation that generalizes the finite propagation
sp eed result of scalar conservation laws. We finally let the non-local term vanish to approximate
solutions of scalar conservation laws, with optimal error estimates for BV initial conditions as
Kuznecov (1976) for λ = 2 and Droniou (2003) for λ ∈]1, 2].
Keywords: Fractional Laplacian, fractal conservation laws, entropy formulation, vanishing vis-
cosity method, error estimates.
Mathematics Subject Classification: 35B30, 35L65, 35L82, 35S10, 35S30.
1 Introduction
We study the fractal conservation law
(
∂
t
u(t, x) + div(f(u))(t, x) + g[u(t, .)](x) = 0 t > 0, x ∈ R
N
,
u(0, x) = u
0
(x) x ∈ R
N
,
(1.1)
where f = (f
1
, ..., f
N
) is locally Lipschitz-continuous from R to R
N
, u
0
∈ L
∞
(R
N
) and g is the
fractional power of order λ/2 of the Laplacian with λ ∈]0, 1]. That is to say, g is the non-local
operator defined through the Fourier transform by
F(g[u(t, .)])(ξ) = |ξ|
λ
F(u(t, .))(ξ).
Remark 1.1. We could also very well study equations with source term h and such that f and
h depend o n (t, x, u). All the methods used in this paper would apply, but this would lead to more
technical difficulties and for the sake of clarity, we have chosen to present only the framework
above.
1
The well-posedness of the pure scalar conservation law, namely the Cauchy proble m
(
∂
t
u(t, x) + div(f(u))(t, x) = 0 t > 0, x ∈ R
N
,
u(0, x) = u
0
(x) x ∈ R
N
,
(1.2)
is well-known since the work of Kruzhkov [11], thanks to the notion of entropy solution. In
this paper, we define a notion of entropy solution for (1.1) that allows to solve (1.1) in the L
∞
framework. We then consider the problem
(
∂
t
u
ε
(t, x) + div(f(u
ε
))(t, x) + εg[u
ε
(t, .)](x) = 0 t > 0, x ∈ R
N
,
u
ε
(0, x) = u
0
(x) x ∈ R
N
(1.3)
and we show that u
ε
converges, as ε → 0, to the (entropy) solution u of (1.2).
The interest of Equation (1.1) was pointed out to us by two papers of Droniou et al. [7, 6]
which deal with the case λ ∈]1, 2] (the results of these papers are recalled below). One of their
motivation s was a preliminary study of equations involved in the theory of detonation in gases
[3, 4]. In fact, λ depends on the unknown in the realistic models and is probably not bounded
from below by any λ
0
> 1. Thus, the case λ = 1 is also of interest. Moreover, the general case
λ ∈]0, 1] ha s many other applications to hydrodynamics, molecular biology, etc [1, 2].
Equation (1.1) constitutes an extension of th e classical parabolic equation
∂
t
u + div(f(u)) − △u = 0, (1.4)
which corresponds (up to a multiplicative constant) to the case λ = 2. In this case, it is well-
known that the Cauchy problem is well-posed and that the operator ∂
t
− △ has a reg ulariz ing
effect; (1.3) then is called the parabolic regularization of (1.2) and the use of such a regularization
allows to prove th e Kruzhkov result. Dep end ing on the value of λ, (1.1) should share properties
of (1.4) and/or the non-linear hyperbolic equation (1.2). Most of the studies (well-posed ness,
asymptotic behaviour, etc) are concerned with the range of exponent λ ∈]1, 2] (see [1, 2, 7, 6, 8,
10]). In this case, the operator ∂
t
+ g[.] still has a regularizin g effect. The first results on this
subject are probably due to Biler et al. [1] and these results have recently been strengthened in
[7], where the existence and the uniqueness of a smooth solution is proved. Let us also refer the
reader to [9, 8] for the case of Hamilton-Jacobi equations. For λ ∈]0, 1], the order of the diffusive
part is lower than the order of the hyperbolic part; hence, we do not expect any regularizing effect,
since it is natural to think that (1.1) could behaves as (1.2). Let us recall that the possibilities of
loss of regularity in finite time and of non-uniqueness of weak solutions o f the Cauchy problem
(1.2) led to the notion of entropy solution of Kruzhkov. The numerical computations of Clavin
et al. [3, 4] lead to think that the solutions of (1.1) may also loose some regularity; but, this
point is still an open question whose answer does not seem obvious. Neit her there is answer to
the question of non-uniqueness of weak solutions in a general framework. To our best knowledge,
there is only one existence and uniqueness result for (1.1) with λ ≤ 1. It appears in a paper of
Biler et al. [1] which deals also with asymptotic behaviour of solutions. They have established
the local-in-time (or global with small initial data) existenc e and unique ness of a weak solution
of the monodimensional fractal Burgers equation (N = 1 and f = |.|
2
) with λ ∈]1/2, 1] and
u
0
∈ H
1
(R). The proof does not seem adaptable to other dimensions, parameters λ or other
initial conditions less regular, since the S obolev imbedd ings and the interpolations used to derive
the needed energy estimates would be no longer true.
2
Following these comments, a good formulation for (1.1), with λ ∈]0, 1], is probably an entropy
formulation, which we h ave to define. Let us mention that Carrillo [5] has also used an entropy
formulation t o study a scalar conservation law perturbed by a local degenerate diffusion operator
(of the form −△(b(u))). In our ca se, the operator g is non-local. Becau se of this, the inequality
(4.16), already mentioned in [8, Lemma 4.1], seems lead to a too weak formulation, namely
(4.17). We discuss this issue in Remark 4.2. To find a good formulation (see Definition 2.1), we
have used an integral formula for g (see papers of Imbert [9] and of Droniou and Imbert [8] or
Theorem 2.1 below).
This no tion of entropy solution has allowed to prove the following results for (1.1): well-
posedness in the L
∞
framework, maximum principle, controled speed of propagation (see The-
orem 3.2 which generalizes the finite propagation speed result of scalar conservation laws), L
1
contraction, non-increase of the L
1
norm and the BV semi-norm, etc. The existence is proved
by a splitting method, as in [7, 6] for λ ∈]1, 2], and the convergence of this method is proved
for general u
0
∈ L
∞
(R
N
) (in [7], the convergence of the splitting method has been established
only for u
0
∈ L
∞
(R
N
) ∩ L
1
(R
N
) ∩ BV (R
N
)). Note that the cla ssical parabolic r egula riza tion
could also work. As far as t he non-local vanishing viscosity method to (1.2) is concerned, the
convergence of u
ε
is obtained in the general case and optimal error estimates are stated for BV
initial conditions, as in [12] for the parabolic regularization of (1.2) and as in [6] for λ ∈]1, 2]. Let
us also refer the reader to [9, 8], which derive same error estimates (in an appropriate topology)
for Hamilton-Jacobi equations.
The rest of the paper is organized as follow. The entropy formulation is given in Section 2
and the main results are stated in Section 3. These results are finally proved in Sections 4-6
(uniqueness and existence for (1.1) and co nvergence for (1.3), respectively).
2 Entropy formulation
To present our formulation for (1.1), we have to recall th e following result on g.
Theorem 2.1 (Droniou, Imbert 2005). There exists a constant c
N
(λ) > 0 that only depends on
N and λ and such that for all ϕ ∈ S(R
N
), all r > 0 and all x ∈ R
N
,
g[ϕ](x) = −c
N
(λ)
Z
|z|≥r
ϕ(x + z) − ϕ(x)
|z|
N+λ
dz − c
N
(λ)
Z
|z|≤r
ϕ(x + z) − ϕ(x) − ∇ϕ(x).z
|z|
N+λ
. (2.1)
Moreover, when λ ∈]0, 1[ one can take r = 0.
Remark 2.1. In the s equel, g : C
∞
b
(R
N
) → C
∞
b
(R
N
) is d efined by this formula.
For a proof of this result, see [8, Theorem 2.1]. Here is our entropy formulation for (1.1).
Definition 2.1. Let u
0
∈ L
∞
(R
N
). We define an entrop y solution to (1.1) as a function u ∈
L
∞
(]0, ∞[×R
N
) and such that for all r > 0, all non-negative ϕ ∈ C
∞
c
([0, ∞[×R
N
), all smooth
3
convex function η : R → R and all φ = (φ
1
, ..., φ
N
) such th at φ
′
i
= η
′
f
′
i
(i = 1, ..., N )(
1
),
Z
∞
0
Z
R
N
(η(u)∂
t
ϕ + φ(u).∇ϕ)
+ c
N
(λ)
Z
∞
0
Z
R
N
Z
|z|≥r
η
′
(u(t, x))
u(t, x + z) − u(t, x)
|z|
N+λ
ϕ(t, x)dzdxdt
+c
N
(λ)
Z
∞
0
Z
R
N
Z
|z|≤r
η(u(t, x))
ϕ(t, x + z) − ϕ(t, x) − ∇ϕ(t, x).z
|z|
N+λ
dzdxdt+
Z
R
N
η(u
0
)ϕ(0, .) ≥ 0.
(2.2)
Remark 2.2. i) Notice that (2.2) for r > 0 implies (2.2) for all r
′
> r, but not necessarily for
0 < r
′
< r;
ii) when λ ∈]0, 1[, the gradient in the third integral term above can be taken out and this gives
an equivalent formulation.
Here are some properties of entropy solutions.
Proposition 2.1. i) Classical solutions to (1.1) are entropy solutions;
ii) ent ropy solutions to (1.1) are weak solutions in the sense that
Z
∞
0
Z
R
N
(u∂
t
ϕ + f(u).∇ϕ − ug[ϕ]) +
Z
R
N
u
0
ϕ(0, .) = 0,
for a ll ϕ ∈ C
∞
c
([0, ∞[×R
N
);
iii) entropy solutions are continuous with values in L
1
loc
(R
N
) (i.e. u is a.e. equal to a function
belonging to C([0, ∞[; L
1
loc
(R
N
)));
iv) if u is an entropy solution of (1.1) then u(0, .) = u
0
.
The proofs of ii) and iv) are similar to those used for the pure scalar conservation laws (see
[11]), thanks to Theorem 2.1 to treat the fractal part and thanks to iii) to deduce iv). Hence,
these proofs are left to the reader. The item iii) will be needed to prove uniqueness in Section 4.
For a first reading, this item could be assumed in Definition 2.1, since an approximating sequence
that converges in C([0, T ]; L
1
loc
(R
N
)) (for all T > 0) will be constructed in Section 5. Actually,
the formulation allows to find item iii) back without the use of this sequence. For the reader’s
interest, the proof is given in Appendix 7.2. Let us conclude this section with giving the proof
of i), which explains how we obtained our formulation (see also Remark 4 .2 about the treatment
of the fractal part).
Proof of i). Let us assume that u
0
is smooth and that u ∈ C
∞
b
([0, ∞[×R
N
) satisfies (1.1). Since
η is convex, η(b) − η(a) ≥ η
′
(a)(b − a). Hence,
η(u(t, x + z)) − η(u(t, x)) − ∇(η(u))(t, x).z ≥ η
′
(u(t, x)) (u(t, x + z) − u(t, x) − ∇u(t, x).z)
1
Let us recall that such a couple (η, φ) is called an entropy-flux pair.
4
and
η
′
(u(t, x))g[u(t, .)](x) ≥ −c
N
(λ)η
′
(u(t, x))
Z
|z|≥r
u(t, x + z) − u(t, x)
|z|
N+λ
dz
− c
N
(λ)
Z
|z|≤r
η(u(t, x + z)) − η(u(t, x)) − ∇(η(u))(t, x)).z
|z|
N+λ
dz.
Let us multiply (1.1) by η
′
(u(t, x)) to get the following entropy inequality:
∂
t
(η(u))(t, x) + div(φ(u))(t, x) − c
N
(λ)η
′
(u(t, x))
Z
|z|≥r
u(t, x + z) − u(t, x)
|z|
N+λ
dz
− c
N
(λ)
Z
|z|≤r
η(u(t, x + z)) − η(u(t, x)) − ∇(η(u))(t, x)).z
|z|
N+λ
dz ≤ 0, t > 0, x ∈ R
N
.
Let us multiply by ϕ(t, x) and, thanks to an integration by parts, let us put the derivatives on
this function. Then,
Z
∞
0
Z
R
N
(η(u)∂
t
ϕ + φ(u).∇ϕ)
+ c
N
(λ)
Z
∞
0
Z
R
N
Z
|z|≥r
η
′
(u(t, x))
u(t, x + z) − u(t, x)
|z|
N+λ
ϕ(t, x)dzdxdt
+ c
N
(λ)
Z
∞
0
Z
R
N
Z
|z|≤r
η(u(t, x + z)) − η(u(t, x)) − ∇(η(u))(t, x).z
|z|
N+λ
ϕ(t, x)dzdxdt
+
Z
R
N
η(u
0
)ϕ(0, .) ≥ 0. (2.3)
Let us now put the fractional derivative on ϕ. We let I denote the third term of (2.3). By
Taylor’s Formula,
I = c
N
(λ)
Z
1
0
Z
∞
0
Z
R
N
Z
|z|≤r
(1 − τ)D
2
(η(u))(t, x + τz)z.z
|z|
N+λ
ϕ(t, x)dzdxdtdτ,
= c
N
(λ)
Z
1
0
Z
∞
0
Z
|z|≤r
(1 − τ)|z|
−N−λ
Z
R
N
D
2
(η(u))(t, x + τz)z.z ϕ(t, x)dx
dzdtdτ.
Note that D
2
(η(u))z.z = div
x
(F ) where F = (z
1
∇(η(u)).z, ..., z
n
∇(η(u)).z) (here, we let z
i
denote the coordinates of z w.r.t. the canonic basis of R
N
). So, an integration by parts gives
I = −c
N
(λ)
Z
1
0
Z
∞
0
Z
|z|≤r
(1 − τ)|z|
−N−λ
Z
R
N
∇(η(u))(t, x + τz).z ∇ϕ(t, x).zdx
dzdtdτ,
= −c
N
(λ)
Z
1
0
Z
∞
0
Z
R
N
Z
|z|≤r
(1 − τ)|z|
−N−λ
∇(η(u))(t, x + τz).z ∇ϕ(t, x).z dzdxdtdτ. (2.4)
Let us change the variables by (τ, t, x, z) → (τ, t, x + τz, −z) to get
I = −c
N
(λ)
Z
1
0
Z
∞
0
Z
R
N
Z
|z|≤r
(1 − τ)|z|
−N−λ
∇(η(u))(t, x).z ∇ϕ(t, x + τz).z dzdxdtdτ.
Computing I backward from (2.4) to (2.3) (exchanging the role of η(u) and ϕ) leads finally to
(2.2).
5