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

A multitype sticky particle construction of Wasserstein stable  semigroups solving one-dimensional diagonal hyperbolic systems with large  monotonic data