Some convergences results on the stochastic Cahn-Hilliard-Navier-Stokes equations with multiplicative noise

Abstract

In this paper, we prove that the sequence (u_n, ϕ_n) of the Galerkin approximation of the solution (u, ϕ) to a stochastic 2D Cahn-Hilliard-Navier-Stokes model verifies the following convergence result

$ \underset {n\to \infty }{\lim } \mathbb {E}\left [\underset {t\in [0,T]}{\sup } \tilde {\psi }\left (\|(u_{n}(t)),\phi _{n}(t)-(u(t),\phi (t))\|_{\mathbb {V}}^{2}\right )\right ]=0 $

for any deterministic time T > 0 and for a specified moment function $\tilde {\psi }(x)$. Also, we provide a result on uniform boundedness of the moment

$\mathbb {E}\underset {t\in [0,T]}{\sup } \psi (\|(u(t),\phi (t))\|_{\mathbb {V}}^{2})$

where ψ grows as a single logarithm at infinity and furthermore, we establih the results on convergence of the Galerkin approximation up to a deterministic time T when the $\mathbb {V}$-norm is replaced by the $\mathbb {H}$-norm.