  This article is dedicated to the study of diagonal hyperbolic systems in one
space dimension, with cumulative distribution functions, or more generally
nonconstant monotonic bounded functions, as initial data. Under a uniform
strict hyperbolicity assumption on the characteristic fields, we construct a
multitype version of the sticky particle dynamics and obtain existence of
global weak solutions by compactness. We then derive a $L^p$ stability estimate
on the particle system uniform in the number of particles. This allows to
construct nonlinear semigroups solving the system in the sense of Bianchini and
Bressan [Ann. of Math. (2), 2005]. We also obtain that these semigroup
solutions satisfy a stability estimate in Wasserstein distances of all orders,
which encompasses the classical $L^1$ estimate and generalises to diagonal
systems the results by Bolley, Brenier and Loeper [J. Hyperbolic Differ. Equ.,
2005] in the scalar case. Our results are obtained without any smallness
assumption on the variation of the data, and only require the characteristic
fields to be Lipschitz continuous and the system to be uniformly strictly
hyperbolic.
