Uniqueness issues for evolution equations with density constraints

作者：Di Marino Simone; Meszaros Alpar Richard^*

来源：Mathematical Models and Methods in Applied Sciences, 2016, 26(9): 1761-1783.

DOI：10.1142/S0218202516500445

• Crowd motion models
• uniqueness results
• density constraints
• optimal transport
• quasilinear elliptic-parabolic system

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 lambda-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-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-infinity as in the standard Fokker-Planck equation.

• 出版日期2016-8