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

2022/12/12 by Dongkwang ‍Kim, Kim, Dongkwang, Dowan Koo +3
Computer Science · Mathematics · #35A15 #49K20 #49Q22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2212.06092

openalex publication_date 2022/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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