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Global solution and smoothing eect for a non-local
regularization of an hyperbolic equation
Jerome Droniou, Thierry Gallouët, Julien Vovelle
To cite this version:
Jerome Droniou, Thierry Gallouët, Julien Vovelle. Global solution and smoothing eect for a non-
local regularization of an hyperbolic equation. Journal of Evolution Equations, Springer Verlag, 2003,
3 (3), pp.499-521. �10.1007/s00028-003-0503-1�. �hal-00003438�
Global solution and smoothing effect for a
non-local regularization of an hyperbolic equation
J. Droniou
1
, T. Gallou¨et
2
, J. Vovelle
2
03/10/2002
1 Introduction
We study the problem
(
∂
t
u(t, x) + ∂
x
(f(u))(t, x) + g[u(t, ·)](x) = 0 t ∈]0, ∞[ , x ∈ R
u(0, x) = u
0
(x) x ∈ R,
(1.1)
where f ∈ C
∞
(R) is such that f(0) = 0 (there is not loss of generality in assuming
this), u
0
∈ L
∞
(R) and g is the non-local (in general) operator defined through the
Fourier transform by
F(g[u(t, ·)])(ξ) = |ξ|
λ
F(u(t, ·))(ξ) , with λ ∈]1, 2].
Remark 1.1 We could also very well study a multi-dimensional scalar equation,
that is to say on R
N
instead of R. All the methods and results presented below
would apply; but this would lead to more technical manipulations so, for the sake
of clarity, we have chosen to fully describe only the mono-dimensional case.
The interest of such an equation (namely Equation (1.1)) was pointed out to us
by Paul Clavin in the context of pattern formation in detonation waves. The study
of detonations leads, in a first approximation, to nonlinear hyperbolic equations.
As it is well known, the solutions of such equations may develop discontinuities
in finite time. A theory of existence and uniqueness of (entropy weak) solutions
to Equation (1.1) with g = 0, in the L
∞
framework, is known since the work of
Krushkov ([Kru70], see also [Vol67]). The case of a parabolic regularization (of a
nonlinear hyperbolic equation) is often considered and used to prove the Krushkov
result; it corresponds to (1.1) with λ = 2. In this case, existence and uniqueness of
a solution is also well known along with a regularizing effect. However, it appears
that the choice of λ = 2 is not quite natural, at least for the problem of detonation
(see [CD02], [CH01], [CD01]) where it seems more natural to consider a nonlocal
term as g[u] with λ close to 1 but greater than 1 (although the case λ = 1 is also of
1
D´epartement de Math´em atiques , CC 051, Universit´e Montpellier II, Place Eug`ene Bataillon,
34095 Montpellier cedex 5, France. email: droniou@math.univ-montp2.fr
2
CMI, Universit´e de Provence, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France. email:
gallouet@cmi.univ-mrs.fr, vo vell e@cm i.un iv- mrs. fr
1
interest but more complicated). This term corresponds to some spatial fractional
derivative of u of order λ. The main motivation of this paper is therefore to prove
existence and uniqueness of the solution to (1.1) in the L
∞
framework as well as
a regularizing effect (a regularizing effect which is well-known in the case λ = 2,
as it is said above). In particular, the solution will be C
∞
in space and time for
t > 0. We also prove the so called “maximum principle”, namely the fact that the
solution takes values between the maximum and the minimum values of the initial
data, and a property of “L
1
contraction” on the solutions, which is the fact that,
for any time, the L
1
norm of the difference of two solutions with different initial
data is bounded by the L
1
norm (if it exists) of the difference of the initial data.
A major difficulty is due to the nonlocal character of g[u] if λ ∈]1, 2[; this
prevents the classical way to prove the maximum principle, which leads to an L
∞
a priori bound on the solution (a crucial estimate to obtain global solutions). It
is interesting to notice that the hypothesis λ ≤ 2 is necessary for the maximum
principle. Indeed, the maximum principle is no longer true in general for λ > 2.
However, the regularizing effect is still true for λ > 2, a property which is probably
not verified if λ < 1. The case λ = 1 is not so clear and needs an additional
work. Indeed, for the study of detonation waves, our result has to be viewed as a
preliminary result or, at least, as a study of a very simplified case. Realistic models
are much more complicated. In particular, it seems that λ is actually depending
on the unknown and, even if λ > 1, λ is probably not bounded from below by some
λ
0
> 1. The possibility to generalize our result to such a case is not manifest.
We first prove (Section 4) the uniqueness of a “weak” solution (solution in the
sense of Definition 3.1 below). Then, assuming the existence of a “weak” solution,
we prove (Section 5) the regularizing effect (the equation is then satisfied in a
classical sense). The results of these two sections are in fact true for any λ > 1.
In Section 6, the existence result is given, using a splitting method. The use of
splitting methods is classical, in particular to define numerical schemes, but is not
usual to prove an existence result as it is done here. In this section, the central
argument is the proof of the maximum principle (which is limited to λ ≤ 2).
Here is our main result.
Theorem 1.1 If u
0
∈ L
∞
(R), then there exists a unique solution u to (1.1) on
]0, ∞[ (in the sense of Definition 3.1, see below). Moreover, this solution satisfies:
i) u ∈ C
∞
(]0, ∞[×R) and all its derivatives are bounded on ]t
0
, ∞[×R for all
t
0
> 0,
ii) for all t > 0, ||u(t)||
L
∞
(R)
≤ ||u
0
||
L
∞
(R)
and, in fact, u takes its values between
the essential lower and upper bounds of u
0
,
iii) u satisfies ∂
t
u + ∂
x
(f(u)) + g[u] = 0 in the classical sense (g[u] being properly
defined by Proposition 5.2).
2
iv) u(t) → u
0
, as t → 0, in L
∞
(R) weak-∗ and in L
p
loc
(R) for all p ∈ [1, ∞[.
Remark 1.2 In the course of our study of (1.1), we will also see that, if u
0
∈
L
∞
(R) ∩ L
1
(R), then the solution u to (1.1) satisfies, for all t > 0: ||u(t)||
L
1
(R)
≤
||u
0
||
L
1
(R)
.
We will also see that (1.1) has a L
1
contraction property: if (u
0
, v
0
) ∈ L
∞
(R)
are such that u
0
− v
0
∈ L
1
(R), then, denoting by u and v the solutions to (1.1)
corresponding to initial conditions u
0
and v
0
, we have, for all t > 0: ||u(t) −
v(t)||
L
1
(R)
≤ ||u
0
− v
0
||
L
1
(R)
.
2 Properties of the kernel of g
Using the Fourier transform, we see that the semi-group generated by g is formally
given by the convolution with the kernel (defined for t > 0 and x ∈ R)
K(t, x) = F
−1
e
−t|·|
λ
(x) =
Z
R
e
2iπ xξ
e
−t|ξ|
λ
dξ = F
e
−t|·|
λ
(x).
The function ξ ∈ R → e
−t|ξ|
λ
being real-valued and even, K is real-valued (in the
sequel, we consider only real-valued solutions to (1.1)).
The most important property of K is its nonnegativity. For the sake of com-
pleteness, we give here a sketch of the proof of this result, but notice that it is a
well-known result since a rather long time now. We refer to the work of L´evy for
example [L´ev25]. Also notice that we study the question of the non-negativity of
the kernel K because it is the issue at stake in the analysis of a maximum principle
for the equation u
t
+ g[u] = 0. From this point of view, we shall make reference
to the work of Courr`ege and coworkers (see [BCP68] and references therein) who
give a characterization of a large class of pseudo-differential operators satisfying
the positive maximum principle and also, more recently, to the work of Farkas,
Jacob, Schilling [FJS01] (see also Hoh [Hoh95]).
Lemma 2.1 If λ ∈]0, 2] then, for all (t, x) ∈]0, ∞[×R, we have K(t, x) ≥ 0.
Proof of Lemma 2.1
If λ = 2, it is well-known that K(t, x) = (π/t)
1/2
e
−
π
2
t
ξ
2
, which implies the
result. Assume now that λ ∈]0, 2[ and let f(x) = A|x|
−1−λ
1
R\]−1,1[
(x), with A > 0
such that
R
R
f(x) dx = 1. Since f is even with integral equal to 1, we have
F(f )(ξ) = 1 +
Z
R
(cos(2πxξ) − 1)f(x) dx = 1 + A|ξ|
λ
Z
|y |≥| ξ|
cos(2πy) − 1
|y|
1+λ
dy.
Since cos(2πy) − 1 = O(|y|
2
) on the neighborhood of 0 and λ < 2, the dominated
convergence theorem gives
Z
|y |≥| ξ|
cos(2πy) − 1
|y|
1+λ
dy → I :=
Z
R
cos(2πy) − 1
|y|
1+λ
dy < 0 as ξ → 0.
3
Hence, F(f )(ξ) = 1 − c|ξ|
λ
(1 + ω(ξ)) with c = −AI > 0 and lim
ξ→0
ω(ξ) = 0.
Define f
n
(x) = n
1/λ
f ∗ f ∗· · · ∗ f(n
1/λ
x), the convolution product being taken n
times. By the properties of the Fourier transform with res pect to the convolution
product, we have, for all ξ ∈ R,
F(f
n
)(ξ) =
F(f )(n
−1/λ
ξ)
n
=
1 − cn
−1
|ξ|
λ
(1 + ω(n
−1/λ
ξ))
n
→ e
−c|ξ|
λ
as n → ∞. Since (F(f
n
))
n≥1
is bounded by 1 (the L
1
-norm of f
n
for all n ≥ 1),
this convergence is also true in S
0
(R) and, taking the inverse Fourier transform, we
see that f
n
→ F
−1
(e
−c|·|
λ
) = K(c, ·) in S
0
(R) as n → ∞. f
n
being nonnegative
for all n, we deduce that K is nonnegative on {c} × R; the homogeneity property
(2.1) be low concludes then the proof of the lemma.
Here are some other important properties of K:
∀(t, x) ∈]0, ∞[×R , K(t, x) =
1
t
1/λ
K
1,
x
t
1/λ
.
(2.1)
K is C
∞
on ]0, ∞[×R and, for all m ≥ 0, there exists B
m
such that
∀(t, x) ∈]0, ∞[×R , |∂
m
x
K(t, x)| ≤
1
t
(1+m)/λ
B
m
(1 + t
−2/λ
|x|
2
)
.
(2.2)
(K(t, ·))
t>0
is, as t → 0, an approximate unit
(in particular, ||K(t, ·)||
L
1
(R)
= 1 for all t > 0).
(2.3)
∃K
1
such that, for all t > 0, ||∂
x
K(t, ·)||
L
1
(R)
= K
1
t
−1/λ
.
(2.4)
∀(a, b) ∈]0, ∞[ , K(a, ·) ∗ K(b, ·) = K(a + b, ·)
and K(a, ·) ∗ ∂
x
K(b, ·) = ∂
x
K(a + b, ·).
(2.5)
Proof of these properties
Equation (2.1) is obtained thanks to the change of variable ξ = t
−1/λ
η in the
integral defining K.
The regularity of K is an immediate application of the theorem of derivation
under the integral sign. To prove the second part of (2.2), we write ∂
m
x
K(1, x) =
R
R
(2iπξ)
m
e
−|ξ|
λ
e
2iπ xξ
dξ; since λ > 1, the first two derivatives of ξ → ξ
m
e
−|ξ|
λ
are
integrable on R and we can make two integrations by parts to obtain ∂
m
x
K(1, x) =
O(1/x
2
) on R; ∂
m
x
K(1, ·) being bounded on R , we deduce the estimate of (2.2) for
t = 1; the general case t > 0 comes from the case t = 1 and (2.1).
Since K(1, ·) ≥ 0, we have ||K(1, ·)||
L
1
(R)
=
R
R
K(1, x) dx = F(K(1, ·))(0) =
e
−|0|
λ
= 1 and (2.3) is thus a consequence of (2.1).
The estimate (2.4) come s from the derivation of (2.1) and from the change of
variable y = t
−1/λ
x in the computation of ||∂
x
K(1, ·/t
1/λ
)||
L
1
(R)
.
The identity (2.5), which translates the fact that the convolution with K(t) is
the semi-group generated by g, can be directly checked via Fourier transform.
4