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.

Uniqueness issues for evolution equations with density constraints

Di Marino S.;
2016

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.
2016
Settore MAT/05 - Analisi Matematica
Crowd motion models; density constraints; optimal transport; quasilinear elliptic-parabolic system; uniqueness results
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/79426
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