Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises

TL;DR

This paper investigates the stochastic Navier-Stokes equations with Caputo derivatives driven by fractional Brownian motion, analyzing regularity, existence, and uniqueness of solutions, and improving upon existing results in the literature.

Contribution

It introduces a novel analysis of stochastic Navier-Stokes equations with fractional derivatives and fractional noise, providing new regularity and solution existence results.

Findings

Derived pathwise spatial and temporal regularity of the generalized Ornstein-Uhlenbeck process.

Proved existence and uniqueness of mild solutions under certain conditions.

Established Hölder regularity depending on fractional order and Hurst parameter.

Abstract

In this paper, we consider the extended stochastic Navier-Stokes equations with Caputo derivative driven by fractional Brownian motion. We firstly derive the pathwise spatial and temporal regularity of the generalized Ornstein-Uhlenbeck process. Then we discuss the existence, uniqueness, and H\"{o}lder regularity of mild solutions to the given problem under certain sufficient conditions, which depend on the fractional order $\alpha$ and Hurst parameter $H$. The results obtained in this study improve some results in existing literature.