Inertial Manifolds for a Smoluchowski Equation on the Unit Sphere

Abstract

The existence of inertial manifolds for a Smoluchowski equationa nonlinear Fokker-Planck equation on the unit sphere which arises in modeling of colloidal suspensions -is investigated. A nonlinear and nonlocal transformation is used to eliminate the gradient from the nonlinear term.

Inertial manifolds for the hyperviscous Navier–Stokes equations

“…Indeed, spectral gap conditions have been widely used in the literature to establish the existence of inertial manifolds for many dissipative evolution equations (cf. [1,13,17,18,19,32,34]). However, for a system that lacks the spectral gap condition, in their pioneering work [26], Mallet-Paret and Sell have introduced the so-called spatial averaging method to prove the existence of inertial manifolds for a three-dimensional reaction-diffusion equation.…”

The Onsager Equation for Corpora

“…Some of the results concerning equilibria and kinetics for the Maier-Saupe potential can be found in [1] - [4], [6], [7], [10] - [14] and [19]. Recently it was shown that the system has inertial manifolds in both S 1 and S 2 ([17]- [18]).…”

Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability

“…Using the Cole-Hopf transform, a large class of diffusive Burgers equations including BSE and QSE will be shown to possess inertial manifolds in both one and two space dimensions. The idea to transform the equation was also developed and used by the author to prove the existence of inertial manifolds for a class of nonlinear Fokker-Planck equations, which appear in the modeling of nematic polymers (see [28,29,30]). Rather than the Cole-Hopf transform, the author developed a nonlinear nonlocal transform which also eliminates the first-order derivatives from the equation, thus allowing the equation to satisfy the spectral-gap condition.…”