Euler-Lagrange equation for a minimization problem over monotone transport maps

Abstract

A variational time discretization for the compressible Euler equations has been introduced recently. It involves a minimization problem over the cone of monotone transport maps in each timestep. A matrix-valued measure field appears in the minimization as a Lagrange multiplier for the monotonicity constraint. We show that the absolutely continuous part of this measure field vanishes in the support of the density.