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Uniqueness issues for evolution equations with density constraints

Simone Di Marino

Laboratoire des Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France

E-mail Address: simone.dimarino@math.u-psud.fr

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and

Alpár Richárd Mészáros

Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA

E-mail Address: alpar@math.ucla.edu

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https://doi.org/10.1142/S0218202516500445Cited by:12 (Source: Crossref)

Abstract

In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first-order systems modeling crowd motion with hard congestion effects, introduced recently by Maury et al. The monotonicity of the velocity field implies that the $2$ $2$ -Wasserstein distance along two solutions is $λ$ $λ$ -contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an $L^{1}$ $L^{1}$ -contraction property. In this case, by the regularization effect of the nondegenerate diffusion, the result follows even if the given velocity field is only $L^{\infty}$ $L^{\infty}$ as in the standard Fokker–Planck equation.

Communicated by L. Ambrosio

Keywords:

AMSC: 35A02, 35K61, 35F25, 49J45

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