Invariant measures for a class of stochastic third grade fluid equations in $2D$ and $3D$ bounded domains

This work aims to investigate the well-posedness and the existence of ergodicinvariant measures for a class of third grade fluid equations in bounded domain$D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First,we show the existence of a martingale solution by coupling a stochasticcompactness and monotonicity arguments. Then, we prove a stabilty result, whichgives the pathwise uniqueness of the solution and therefore the existence ofstrong probabilistic solution. Secondly, we use the stability result to showthat the associated semigroup is Feller and by using "Krylov-BogoliubovTheorem" we get the existence of an invariant probability measure. Finally, weshow that all the invariant measures are concentrated on a compact subset of$L^2$, which leads to the existence of an ergodic invariant measure.