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

TL;DR

This paper develops a multitype sticky particle method to construct Wasserstein stable semigroups for one-dimensional diagonal hyperbolic systems with large monotonic initial data, ensuring existence, stability, and generalizing previous scalar results.

Contribution

It introduces a multitype sticky particle construction for hyperbolic systems, proving existence and stability of solutions without smallness constraints on initial data variation.

Findings

Constructed global weak solutions via multitype sticky particles.

Established Wasserstein distance stability estimates for the semigroup.

Extended scalar stability results to diagonal systems without small data assumptions.

Abstract

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.