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

Jin Li; Jianhua Huang

doi:10.3934/dcdsb.2012.17.2483

Article Contents

2012, Volume 17, Issue 7: 2483-2508. Doi: 10.3934/dcdsb.2012.17.2483

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

Jin Li^1, and
Jianhua Huang^2,

1.
Department of Mathematics, National University of Defense Technology, Changsha 410073
2.
Department of Mathematics, National University of Defense Technology, Changsha, 410073

Received: June 30, 2011

Revised: January 31, 2012

Published: September 2012

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Abstract

A 2D Stochastic incompressible non-Newtonian fluid driven by fractional Brownian motion with Hurst index $H \in (\frac{1}{2},1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Including the requirements of Nuclear and Hilbert-Schmidt operators, three kinds of condition, which ensure the existence and regularity of infinite-dimensional stochastic convolution for the corresponding additive linear stochastic equation, are summarized. Without the requirements of compact parameters, another condition is proposed for the existence and regularity of stochastic convolution. By any of the four kinds of condition, the existence and uniqueness of mild solution are obtained for the stochastic non-Newtonian fluid through a modified fixed point theorem in the selected intersection space. Existence of a random attractor is then obtained for the random dynamical system generated by non-Newtonian fluid.

Keywords:

Infinite-dimensional fractional Brownian motion,
stochastic convolu- tion,
stochastic non-Newtonian fluid,
random attractor.

Mathematics Subject Classification: Primary: 35Q35, 35R60; Secondary: 60G22, 37L55.

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