On the Existence and long time behaviour of $H^1$-Weak Solutions for $2,3d$-Stochastic $3^{rd}$-Grade Fluids Equations

In the present work, we investigate stochastic third grade fluids equations in a $d$-dimensional setting, for $d=2,3$. More precisely, on a bounded and simply connected domain $\mathcal{D}$ of ${\mathbb{R}}^d$, $d=2,3$, with a sufficiently regular boundary $\mathrm{\partial }\mathcal{D},$ we consider incompressible third grade fluid equations perturbed by a multiplicative Wiener noise. Supplementing our equations by Dirichlet boundary conditions and taking initial data in the Sobolev space $H^1(\mathcal{D})$, we establish the existence of global stochastic weak solutions by performing a strategy based on the conjugation of stochastic compactness criteria and monotonicity techniques. Furthermore, we study the asymptotic behaviour of these solutions, as $t\to \mathrm{\infty }$.