A gradient flow for the Porous Medium Equations with Dirichlet boundary conditions
Abstract
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 Wb2 distance, which is a modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130]. Furthermore, the constructed solutions are characterized as curves of maximal slope in a suitable sense.
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