Weak solution of a stochastic 3D nonlocal Cahn–Hilliard–Navier–Stokes systems with shear-dependent viscosity

ABSTRACT We consider the stochastic nonlocal Cahn–Hilliard–Navier–Stokes system with shear-dependent viscosity on a bounded domain , d = 2, 3, driven by a multiplicative noise of Lévy and Gaussian types. The velocity u is governed by a Navier–Stokes system with a shear-dependent viscosity controlled by a power p>2. This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn–Hilliard equation for the order parameter φ. The existence of a global weak martingale solution is proved. In the 2D case, we prove the pathwise uniqueness of the weak solution, when .