Mild Solutions of Time Fractional Navier-Stokes Equations Driven by Finite Delayed External Forces

In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external forces involving finite delay. Equations are considered on a bounded domain Ω ⊂ R3 having sufficiently smooth boundary. We transform the system of equations (NSE) to an abstract Cauchy problem and then investigate local existence and uniqueness of the mild solutions for the initial datum (Formula Presented), where r > 0 and A is the Stokes operator. With some suitable condition on initial datum we establish the global continuation and regularity of the mild solutions.We use semigroup theory, tools of fractional calculus and Banach contraction mapping principle to establish our results