A gradient flow for the Porous Medium Equations with Dirichlet boundary condition

In this talk, 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. This is a joint work with Dongkwang Kim (Yonsei) and Dowan Koo (Yonsei).