Dynamics of 2D Stochastic non-Newtonian fluids driven by fractional Brownian motion

TL;DR

This paper investigates the behavior of 2D stochastic non-Newtonian fluids driven by fractional Brownian motion, establishing existence, regularity, and attractor properties under various assumptions.

Contribution

It introduces Wiener-type stochastic integrals for infinite-dimensional fractional Brownian motion and proves existence, regularity, and attractor results for the non-Newtonian fluid system.

Findings

Existence and regularity of stochastic convolution under multiple assumptions

Mild solutions obtained via a modified fixed point theorem

Existence of a random attractor for the system on square domains

Abstract

A 2D Stochastic incompressible non-Newtonian fluids driven by fractional Bronwnian motion with Hurst parameter $H \in (1/2,1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Four groups of assumptions, including the requirement of Nuclear operator or Hilbert-Schmidt operator, are discussed. The existence and regularity of stochastic convolution for the corresponding additive linear stochastic equation are obtained under each group of assumptions. Mild solution are then obtained for the non-Newtonian systems by the modified fix point theorem in the selected intersection space. When the domain is square, the random dynamical system generated by non-Newtonian systems has a random attractor under some condition on the spectrum distribution of the corresponding differential operator.