Abstract
In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero.
Highlights
Non-Newtonian fluids are difficult to model and to analyze because they display essentially nonlinear and even discontinuous flow properties
In the present work we consider an evolution problem generated by vanishing viscosity approximations and prove the existence of a global attractor for the corresponding semiflow with respect to the strong topology of the phase space
We show the upper semicontinuity of these attractors with respect to the set of all complete bounded trajectories of the original problem (1) when ε → 0
Summary
Non-Newtonian (or complex) fluids are difficult to model and to analyze because they display essentially nonlinear and even discontinuous flow properties. In this paper we consider an evolution problem which appears in the investigation of the model of concentrated suspensions proposed by Hebraud and Lequex [11]. In this model the system is divided in mesoscopic blocks. In the present work we consider an evolution problem generated by vanishing viscosity approximations and prove the existence of a global attractor for the corresponding semiflow with respect to the strong topology of the phase space. A solution of equation (2) on a finite time interval [τ, T ] is defined as follows. (weak) solution of equation on [τ, T ], if the following equality holds:. Let 0 < ε 1, 0 ≤ τ < T < ∞, pτ ∈ L1 ∩ L2, and p be a solution of equation (2) on [τ, T ]
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