Uniqueness issues for evolution equations with density constraints

Di Marino, Simone; Mészáros, Alpár Richárd

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
• Public URL: https://durham-repository.worktribe.com/output/1284877
• Related Public URLs: https://arxiv.org/abs/1507.02900