Well-posedness of stochastic time fractional 2D-Stokes models with finite and infinite delay

Abstract

We analyze the well-posedness of two versions of a stochastic time delay fractional 2D-Stokes model with nonlinear multiplicative noise. The main tool to prove the existence and uniqueness of mild solutions is a fixed point argument. The results for the first model can only be proved for \(\alpha\in (1/2,1)\), and the global existence in time is shown only when the noise is additive. As for the second model, all results are true for \(\alpha \in (0,1)\), and the global solutions in time is shown for general nonlinear multiplicative noise. The analyzes for the finite and infinite delay cases, follow the same lines, but they require different phase spaces and estimates. This article can be considered as a first approximation to the challenging model of stochastic time fractional Navier-Stokes (with or without delay) which so far remains as an open problem.