Mild Solutions to Time Fractional Stochastic 2D-Stokes Equations with Bounded and Unbounded Delay

Abstract

In this paper, the well-posedness of stochastic time fractional 2D-Stokes equations of order α ∈ (0, 1) containig finite or infinite delay with multiplicative noise is established, respectively, in the spaces C([−h, 0]; L2(Ω; L2 σ )) and C((−∞, 0]; L2(Ω; L2 σ )). 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.