On the stochastic Cahn-Hilliard-Navier-Stokes systems with surfactant and jump noise

We consider the stochastic Cahn-Hilliard-Navier-Stokes system with surfactant on a bounded domain Od=2,3, driven by a multiplicative noise of jump type. The system consists of a Navier-Stokes system, coupled with a sixth-order and a fourth-order Cahn-Hilliard equation with singular potential. The existence of a global weak martingale solution is proved. In the two-dimensional case and when the viscosity of the mixture is assumed to be constant, we prove the pathwise uniqueness of the weak solution, and using the Yamada-Watanabe result to derive the existence of a strong probabilistic solution.