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

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.