ON THE STOCHASTIC CAHN-HILLIARD-NAVIER-STOKES SYSTEMS WITH SURFACTANT

We consider the stochastic Cahn-Hilliard-Navier-Stokes with surfactant on a bounded domain, driven by a multiplicative noise. The resulting system of partial differential equations consists of a Navier-Stokes systems for the (average) velocity, a sixth-order Cahn-Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn-Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier-Stokes system with a multiplicative noise of Gaussian type for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no -slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients in both two and three dimensional bounded domains. 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 classical result to derive the existence of a strong solution (in probabilistic sense).