LAUSR

In this paper, the well-posedness of stochastic time fractional 2D-Stokes equations of order $$\alpha \in (0,1)$$ containig finite or infinite delay with multiplicative noise is established, respectively, in the spaces $$C([-h,0];L^2(\varOmega ;L^2_{\sigma }))$$ and $$C((-\infty ,0];L^2(\varOmega ;L^2_{\sigma }))$$. The existence and uniqueness of mild solution to such kind of equations are proved by using a fixed-point argument. Also the continuity with respect to initial data is shown. Finally, we conclude with several comments on future research concerning the challenging model: time fractional stochastic delay 2D-Navier–Stokes equations with multiplicative noise. Hence, this paper can be regarded as a first step to study this challenging topic.