A new family of transportation costs with applications to reaction-diffusion and parabolic equations with boundary conditions

TL;DR

This paper develops a new family of transportation costs that enable the derivation of reaction-diffusion and parabolic equations with drift and Dirichlet boundary conditions, allowing for more flexible modeling of boundary effects.

Contribution

It introduces a novel family of transportation costs that decouple drift and boundary conditions, extending previous work on gradient flows for parabolic equations.

Findings

New transportation costs enable boundary condition flexibility

Application to reaction-diffusion equations with drift

Extension of gradient flow framework for PDEs

Abstract

This paper introduces a family of transportation costs between non-negative measures. This family is used to obtain parabolic and reaction-diffusion equations with drift, subject to Dirichlet boundary condition, as the gradient flow of the entropy functional $\int_{\Omega}\rho\log\rho+V\rho+1\hspace{1mm}dx$. In 2010, Alessio Figalli and Nicola Gigli introduced a transportation cost that can be used to obtain parabolic equations with drift subject to Dirichlet boundary condition. However, the drift and the boundary condition are coupled in their work. The costs in this paper allow the drift and the boundary condition to be detached.