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

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 Hölder regularity of mild solutions to the given problem under certain sufficient conditions, which depend on the fractional order α and Hurst parameter H. The results obtained in this study improve some results in existing literature.

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

“…Of importance for physics are also fractional kinetic equations with application to statistical mechanics and fractional stochastic PDEs. For these developments we refer to [37], [45] and [51], [55], [79], [80] and references therein.…”

Error estimates of finite element methods for fractional stochastic Navier–Stokes equations

“…Also we assume that the operator A is self-adjoint and there exist eigenvectors corresponding to eigenvalues such that (cf., [ 28 , 29 ]) In a standard way, the fractional powers , , of A are introduced by Let with its norm denoted by …”

“…At the same time, the stochastic perturbations cannot be avoided in a physical system, sometimes they even cannot be ignored. Hence fractional stochastic Navier–Stokes equations have been proposed, which display the behavior of a viscous velocity field of an incompressible liquid and have wide application value in the fields of physics, chemistry, population dynamics, and so on [ 26 – 28 ].…”

Caratheodory’s approximation for a type of Caputo fractional stochastic differential equations

“…Its worth mentioning that the Euler-Maruyama type approximate results for Caputo fractional stochastic differential equations have been established by [17]. For more related work, see [12,[18][19][20][21][22].…”