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Uniqueness issues for evolution equations with density constraints.

Di Marino, Simone and Mészáros, Alpár Richárd (2016) 'Uniqueness issues for evolution equations with density constraints.', Mathematical models and methods in applied sciences., 26 (09). pp. 1761-1783.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1142/S0218202516500445
Publisher 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
Date accepted:13 May 2016
Date deposited:28 February 2020
Date of first online publication:13 July 2016
Date first made open access:28 February 2020

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