Quasistatic evolution and congested crowd transport
Abstract
We consider the relationship between HeleShaw evolution with drift, the porous medium equation with drift, and a congested crowd motion model originally proposed by Maury et al (2010 Math. Models Methods Appl. Sci. 20 1787821). We first use viscosity solutions to show that the porous medium equation solutions converge to the HeleShaw solution as m → ∞ provided the drift potential is strictly subharmonic. Next, using the gradientflow structure of both the porous medium equation and the crowd motion model, we prove that the porous medium equation solutions also converge to the congested crowd motion as m → ∞. Combining these results lets us deduce that in the case where the initial data to the crowd motion model is given by a patch, or characteristic function, the solution evolves as a patch that is the unique solution to the HeleShaw problem. While proving our main results we also obtain a comparison principle for solutions with the minimizing movement scheme based on the Wasserstein metric, of independent interest.
 Publication:

Nonlinearity
 Pub Date:
 April 2014
 DOI:
 10.1088/09517715/27/4/823
 arXiv:
 arXiv:1304.3072
 Bibcode:
 2014Nonli..27..823A
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 36 pages, 5 Figures