A new family of transportation costs with applications to   reaction-diffusion and parabolic equations with boundary conditions
  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. 
Javier Morales
math.AP