On a pursuit-evasion like game between two populations

We investigate an infinite dimensional partial differential equation of Isaacs’ type, which arises from a pursuit-evasion game between two populations of agents. The evolution of the two populations is described by a controlled transport/continuity equation, where the control is given by the velocity vector field. Our study is set in the framework of the viscosity solutions theory in Hilbert spaces, and we prove the uniqueness of the value functions as solutions of the Isaacs equation.