Nonlinear Diﬀer. Equ. Appl. (2017) 24:53
2017 Springer International Publishing AG
published online August 14, 2017
Nonlinear Diﬀerential Equations
and Applications NoDEA
A “forward-in-time” quadratic potential
for systems of conservation laws
Abstract. A quadratic interaction potential t → Υ(t) for hyperbolic sys-
tems of conservation laws is constructed, whose value Υ(
depends only on the present and the future proﬁles of the solution and
not on the past ones. Such potential is used to bound the change of the
speed of the waves at each interaction.
Mathematics Subject Classiﬁcation. 35L65.
Keywords. Conservation laws, Interaction estimates, Quadratic potential.
Let us consider the Cauchy problem for the system of conservation laws
+ F (u)
where F : R
is a smooth (C
) function, which is assumed to be strictly
hyperbolic, i.e. its diﬀerential DF(u)hasn distinct real eigenvalues in each
point of its domain, and ¯u is a BV function with “small” total variation.
As usual, we denote by λ
(u) < ··· <λ
(u) the eigenvalues of DF(u)
and by r
(u) its eigenvectors, DF(u)r
(u). The d-
dimensional Lebesgue measure on R
is denoted by L
The k-th characteristic ﬁeld is said to be genuinely non linear (GNL) if
is strictly increasing in the direction of r
, while it is said to be linearly
This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Appli-
cations: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona,
Stefano Bianchini, Pierangelo Marcati, Andrea Marson.
The author would like to thank Prof. Stefano Bianchini for many helpful discussions
about the topic of this paper.
The paper was submitted while the author was a Post-Doc at the MPI for Mathematics
in the Sciences in Leipzig.