| Abstract: |
| Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent m, we prove that solutions are Lipschitz continuous with respect to m, providing a quantitative version of the result of Benilan and Crandall. This result can be extended to a general aggregation-diffusion equation. |
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