A gradient flow for the porous medium equations with Dirichlet boundary conditions

We consider the gradient flow structure of the porous medium equations with non-negative constant Dirichlet boundary conditions. We construct weak solutions to the equations via the minimizing movement scheme by considering an entropy functional with respect to W b 2 distance, which is a modified Wasserstein distance introduced by Figalli and Gigli (2010) [9] . Furthermore, the constructed solutions are characterized as curves of maximal slope in a suitable sense.