Inertial manifolds of incompressible, nonlinear bipolar viscous fluids

Abstract

Abstract.The existence of an inertial manifold is established for the nonlinear system of equations describing the motion of a bipolar incompressible viscous fluid. In this paper we consider only the case of a spatially periodic velocity field. Existence of an inertial manifold for the model complements earlier work on the existence of compact global attractors for bipolar viscous fluids and serves to further highlight the differences between the bipolar model and the usual model based on the linear Stokes constitutive relation.

Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow

“…The existence of an inertial manifold for the nonlinear system of equations describing the motion of a bipolar incompressible viscous fluid is taken up in Sect. 6.2; we show, following the analysis in [BH3] that, unlike the current situation with regard to the Navier-Stokes equations, an inertial manifold does exist for the case of the space-periodic problem, in both dimensions n D 2 and n D 3, provided 0 IJ< 1 (1 < p Ä 2). In the H. Bellout and F. Bloom, Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-00891-2__6, © Springer International Publishing Switzerland 2014 435 course of establishing the existence of an inertial manifold a squeezing property is proven for the orbits of the semigroup S 1 .t/; while this particular squeezing property is naturally adopted to establishing the existence of the inertial manifold for incompressible bipolar fluid flow, a more basic L 2 form of the squeezing property is shown to hold in Sect.…”

“…The essential content of Sect. 5.3, including the estimates for the Hausdorff and fractal dimensions of the global attractor A 1 , viscous, bipolar fluid flow is based on the Ph.D. thesis [Hao] and may also be found in [BH3]. The work in Sect.…”

On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow

“…It will be seen in §3 that the form of the higher-order boundary condition delineated in (1.9) depends, in a crucial manner, on a deep result of Heron in [16], which was written under the direction of Roger Temam at Orsay. Various studies of the incompressible bipolar fluid flow model and its specialization to the non-Newtonian case, in which μ 1 = 0, have appeared in the recent literature: special types of flows have been analyzed in [17]- [21], exterior flow and flow over nonsmooth boundaries have been studied, respectively, in [22] and [23], general existence and uniqueness results have appeared in [24]- [27], and theorems related to asymptotic stability and the existence of global attractors and inertial manifolds may be found in [28]- [34]. A complete overview of the entire body of work cited above will appear in the forthcoming monograph [35].…”

Cone invariance and squeezing properties for inertial manifolds for nonautonomous evolution equations

“…Hence the inertial form systems describes the dynamics of the slow modes. For applications of the concept of inertial manifolds see for example [HR92,RB95,SK95,BH96,MSZ00].…”