The exponential behavior of a stochastic globally modified Cahn–Hilliard–Navier–Stokes model with multiplicative noise

The stochastic Cahn–Hilliard equation with degenerate mobility and logarithmic potential

“…Besides, let us point out the mathematical literature on stochastic two-phase flows has also been expanded in the last years, in the context of coupled stochastic systems of Cahn-Hilliard-Navier-Stokes and Allen-Cahn-Navier-Stokes type. In this direction, we refer the reader to the contributions [33,35,80,81] for existence of solutions, [3,36] about asymptotic long-time behaviour, [37] on large deviation limits, and [32,34] dealing with a nonlocal phase-field equation in the system instead.…”

The exponential behavior and stabilizability of quasilinear parabolic stochastic partial differential equation

“…Anh and Da [1] studied the exponential behaviour and stabilizability of a class of abstract nonlinear stochastic evolution equations, which include 2D Navier-Stokes equations. For more literatures, we refer the reader to [2,5,8,15,16] and the references.…”

Existence of a solution to the stochastic nonlocal Cahn–Hilliard Navier–Stokes model via a splitting-up method

“…The model (1.1) is obtained by the suitable coupling between the nonlocal Cahn-Hilliard equation through the transport term u • ∇φ and the stochastic Navier-Stokes model through the capillarity term (or the Korteweg force) μ∇φ with a multiplicative infinite-dimensional Gaussian type noise. Adding a stochastic force in the equation for the relative concentration in (1.1) is possible and the corresponding local version has been studied by some authors such as [13,15,40]. But, the presence of this noise in (1.1) will involve tedious calculations and will increase significantly the size of the paper.…”