Evolution Monge–Kantorovich equation

The paper deals with Monge–Kantorovich equation (MK for short) in an open bounded domain Ω with Dirichlet boundary condition. We study existence and uniqueness of a solution to the associated evolution problem (EMK for short) and we prove the convergence to a solution of MK, when time goes to ∞. A solution is a couple (u,Φ), where u is the potential and Φ is the transportation flux. We study the problem for a given Radon measure source term and we show how to use the numerical method of Dumont and Igbida (2009) [22] to provide numerical approximation of the solution of MK.