Time fractional stochastic Navier-Stokes equations driven by fractional Brownian motion

Abstract

A two-dimensional time fractional stochastic incompressible Navier-Stokes equations driven by fractional Brownian motion is studied with the Hurst parameter \(H\in(1/2,1)\) and time fractional differential operator of order \(\alpha\in(0,1)\) under the Dirichlet boundary condition. Without the requirement of compact parameters, the existence and regularity of the nonlocal stochastic convolution are obtained by combining the estimate on the spectrum of the Stokes operator under a square domain, an upper bound of a class of the generalized Mittag-Leffler functions, and the fractional calculus technique. Moreover, sufficient conditions are provided to ensure the local and global existence and uniqueness of mild solutions, as well as the square integrability and continuity of the solution's paths.