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 ${\mathbb{R}}^2$, driven by a cylindrical Hermite noise of order $k\ge 1$ and Hurst parameter $H\in (1/2,1)$, built mode-by-mode on the eigenbasis of the Stokes operator. This class of noises generalizes fractional Brownian motion ($k=1$) and the Rosenblatt process ($k=2$) to arbitrary non-Gaussian order. Combining Wiener-chaos hypercontractivity with Hilbert-Schmidt estimates for the Mittag-Leffler operator, we establish the regularity of the associated stochastic convolution. A Da Prato-Debussche decomposition, carried out in a scale of Lebesgue spaces $L_{\sigma }^q(D)$ on which the bilinear term is controlled by a Calderón-Zygmund estimate, then yields existence and uniqueness of a maximal mild solution, and Hölder regularity of the time-frozen solution under the condition $\al+2H>2$ together with standard regularity assumptions on the initial datum. We further obtain moment bounds for the stochastic convolution and for the quadratic self-interaction of the noise, whose growth in the moment order is governed by $k$, and a functional non-central limit theorem giving a radially localised approximation of the solution by systems driven by functionals of Gaussian sequences with long-range dependence.