Fractional Navier-Stokes Equations with Caputo Derivative Driven by Hermite Noise

TL;DR

This paper investigates time-fractional stochastic Navier-Stokes equations driven by Hermite noise, establishing existence, uniqueness, regularity, and limit theorems for solutions in a two-dimensional domain.

Contribution

It introduces a novel analysis of fractional Navier-Stokes equations driven by Hermite processes, extending stochastic calculus and regularity results to this class of noises.

Findings

Constructed Wiener integral with respect to Hermite noise.

Proved solution regularity under specific fractional and noise parameters.

Established a non-central limit theorem linking solutions to discrete approximations.

Abstract

We study time-fractional stochastic Navier-Stokes equations on a bounded domain of $\R^2$ (the restriction to dimension two is essential for the bilinear estimates via Sobolev embeddings) driven by a Hermite process $Z_H^k$ of order $k\ge1$ and Hurst parameter $H\in(1/2,1)$. This class of noises generalizes fractional Brownian motion ($k=1$) and the Rosenblatt process ($k=2$). We construct the Wiener integral with respect to $Z_H^k$ and establish sharp $L^p$ estimates via hypercontractivity, explicitly capturing the dependence on $k$. Using a refined Hilbert-Schmidt estimate for the Mittag-Leffler operator, we prove that the stochastic convolution belongs to $\dot{H}^\nu$ under the condition $\al(1-\nu)+2H>2$. A fixed-point argument in a weighted space yields the existence, uniqueness, and H\"older regularity of mild solutions. We also prove a non-central limit theorem linking the solution to discrete approximations.