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Uniqueness issues for evolution equations with density constraints

Di Marino, Simone; Mészáros, Alpár Richárd

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Authors

Simone Di Marino



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-Wasserstein distance along two solutions is 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 L1-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∞ as in the standard Fokker–Planck equation.

Citation

Di Marino, S., & Mészáros, A. R. (2016). Uniqueness issues for evolution equations with density constraints. Mathematical Models and Methods in Applied Sciences, 26(09), 1761-1783. https://doi.org/10.1142/s0218202516500445

Journal Article Type Article
Acceptance Date May 13, 2016
Online Publication Date Jul 13, 2016
Publication Date Aug 31, 2016
Deposit Date Oct 1, 2019
Publicly Available Date Feb 28, 2020
Journal Mathematical Models and Methods in Applied Sciences
Print ISSN 0218-2025
Electronic ISSN 1793-6314
Publisher World Scientific Publishing
Peer Reviewed Peer Reviewed
Volume 26
Issue 09
Pages 1761-1783
DOI https://doi.org/10.1142/s0218202516500445
Related Public URLs https://arxiv.org/abs/1507.02900

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Copyright Statement
Electronic version of an article published as Di Marino, Simone & Mészáros, Alpár Richárd (2016). Uniqueness issues for evolution equations with density constraints. Mathematical Models and Methods in Applied Sciences 26(09): 1761-1783 - 10.1142/S0218202516500445] © copyright World Scientific Publishing Company - https://www.worldscientific.com/worldscinet/m3as





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