# On the uniqueness of mild solutions to the time-fractional Navier-Stokes equations in $L^{N} \left( \mathbb{R} ^{N}\right) ^{N} $

## TL;DR

This paper proves the uniqueness of mild solutions to the time-fractional Navier-Stokes equations in a specific function space, using maximum regularity and integral inequalities, advancing understanding of fractional fluid dynamics models.

## Contribution

It establishes the uniqueness of mild solutions for time-fractional Navier-Stokes equations in $L^{N}$ spaces, employing $L^{p}-L^{q}$ estimates and Gronwall inequality.

## Key findings

- Uniqueness of mild solutions in $C([0,
abla
ightarrow ext{infinity})$ space.
- Application of $L^{p}-L^{q}$ estimates to fractional Navier-Stokes.
- Use of Gronwall inequality to prove solution uniqueness.

## Abstract

In this paper, we present the result of maximum regularity of the mild solution of the fractional Cauchy problem. As our main result, we investigate the uniqueness of mild solutions for time-fractional Navier-Stokes equations in class $C\left([0,\infty);L^{N}\left( \mathbb{R}^{N}\right)^{N}\right)$ by means of the estimates $L^{p}-L^{q}$ of Giga-Shor inequality and the Gronwall inequality.