# Mild solutions of time fractional Navier-Stokes equations driven by finite delayed external forces

## TL;DR

This paper studies the existence, uniqueness, and regularity of mild solutions for time-fractional Navier-Stokes equations with delayed external forces on bounded 3D domains, using advanced mathematical tools.

## Contribution

It introduces a framework for analyzing time-fractional Navier-Stokes equations with finite delays, extending classical results to fractional and delayed settings.

## Key findings

- Established local existence and uniqueness of mild solutions.
- Proved conditions for global continuation and regularity.
- Applied semigroup theory and fractional calculus techniques.

## Abstract

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 in 3-D space 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. In particular, 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.