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

Benjamin Jourdain

Université Paris-Est, CERMICS (ENPC), Inria, 77455 Marne-la-Vallée, France

E-mail Address: jourdain@cermics.enpc.fr

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and

Julien Reygner

Université Paris-Est, CERMICS (ENPC), Inria, 77455 Marne-la-Vallée, France

E-mail Address: julien.reygner@enpc.fr

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https://doi.org/10.1142/S0219891616500144Cited by:1 (Source: Crossref)

Abstract

This paper is devoted 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 multi-type version of the sticky particle dynamics and we obtain the existence of global weak solutions via a compactness argument. We then derive a $L^{p}$ $L^{p}$ stability estimate on the particle system which is uniform in the number of particles. This allows us to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161(1) (2005) 223–342]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all order, which extends the classical $L^{1}$ $L^{1}$ estimate and generalizes to diagonal systems a result by Bolley, Brenier and Loeper [Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ.2(1) (2005) 91–107] in the scalar case. Our results are established without any smallness assumption on the variation of the data, and we only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic.

Communicated by P. G. LeFloch

Keywords:

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