  We consider the gradient flow structure of the porous medium equations with
nonnegative constant boundary conditions. We prove that weak solutions to the
equations can be obtained by the variational steepest descent scheme by
considering an entropy functional with respect to Wb2 distance, which is a
modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures
Appl. 94, (2010), pp. 107-130]. In addition, we establish an energy dissipation
inequality.
