*Preprint*

**Inserted:** 22 aug 2024

**Year:** 2024

**Abstract:**

We investigate an infinite dimensional partial differential equation of Isaacsâ€™ type, which arises
from a zero-sum differential game between two masses. The evolution of the two masses is described by a controlled transport*continuity equation, where the control is given by the vector velocity 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.*

**Keywords:**
Viscosity solutions, differential games, Mass transportation, infinite-dimensional Isaacs equation, zero-sum games