Browsing by Author "Sell, George R."
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Item Approximation dynamics and the stability of invariant sets(1996-03) Pliss, Victor A.; Sell, George R.Item Construction of Inertial Manifolds by Elliptic Regularization(1988) Fabes, Eugene; Luskin, Mitchell; Sell, George R.Item The Construction of Inertial Manifolds for Reaction-Diffusion Equations by Elliptic Regularization(1989) Luskin, Mitchell; Sell, George R.Item Dichotomies for linear evolutionary equations in Banach spaces(1991-08) Sacker, Robert J.; Sell, George R.Item Ensemble dynamics and bred vectors(University of Minnesota. Institute for Mathematics and Its Applications, 2010-11) Balci, Nusret; Mazzucato, Anna; Restrepo, Juan M.; Sell, George R.Item Ergodic Properties of Linear Dynamical Systems(1984) Johnson, Russell; Palmer, Kenneth; Sell, George R.The Multiplicative Ergodic Theorem give information about the dynamical structure of a cocycle , or a linear skew product flow , over a suitable base space M. In typical applications the base space M is either an attractor; a compact invariant set; or the space of coefficients for a diffeomorphism, a differential equation, or a vector field. This theorem asserts that for every invariant probability measure on M there is a measurable decomposition of the vector bundle over M into invariant measurable subbundles, and that every solution with initial conditions in any of these subbundles has strong Lyapunov exponets. These exponents, or growth rates, depend on the measure , and when is ergodic, they are constant (almost everywhere) on M and form a finite set meas(), the measurable (Millionscikov-Oseledec) spectrum. The main objective in this paper is to study the connection between the measurable spectrum meas() and the dynamical spectrum dyn introduced by Sacker and Sell (1975, 1978, 1980). (Also see Daletskii and Krein (1974), as well as Selgrade (1975). The dynamical spectrum dyn consists of those values R for which the shifted flow fails to have an exponential dichotomy over M. It follows from the Spectral Theorem, Sacker and Sell (1978), that the dynamical spectrum is the finite union of disjoint compact intervals when M is compact and dynamically connected.Item Global attractors for the 3D Navier-Stokes equations(1994-12) Sell, George R.Item Hausdorff and Lyapunov Dimensions for Gradient Systems(1988) Sell, George R.Item Homoclinic Orbits and Bernoulli Bundles in Almost Periodic Systems(1986) Meyer, Kenneth R.; Sell, George R.Item Inertial Manifolds for Nonlinear Evolutionary Equations(1986) Foias, C.; Sell, George R.; Temam, R.Item Inertial Manifolds for Reaction Diffusion Equations in Higher Space Dimensions(1987) Mallet-Paret, J.; Sell, George R.Item Inertial Manifolds for the Kuramoto-Sivashinsky Equation and an Estimate of their Lowest Dimension(1986) Foias, C.; Nicolaenko, B.; Sell, George R.; Temam, R.Item Linearization and Global Dynamics(1983) Sell, George R.In this paper we show how the spectral theory of linear skew-product flows may be used to study the following three questions in the qualitative theory of dynamical systems: (1) when is an -limit set or an attractor a manifold? (2) Under which conditions will a dynamical system undergo a Hopf-Landau bifurcation from a k-dimensional torus to a (k + 1)-dimensional torus? (3) When is a vector field i the vicinity of a compact invariant manifold smoothly conjugate to the linearized vector field and how smooth is the conjugacy?Item Local dissipativity and attractors for the Kuramoto--Sivashinsky equation in thin 2D domains(1991-02) Sell, George R.; Taboada, MarioItem Melnikov Transforms, Bernoulli Bundles, and Almost Periodic Perturbations(1987) Meyer, Kenneth R.; Sell, George R.Item Navier-Stokes equations in thin 3d domains: Global regularity of solutions(1990-06) Raugel, Geneviéve; Sell, George R.Item Navier-Stokes equations with Navier boundary conditions in nearly flat domains(University of Minnesota. Institute for Mathematics and Its Applications, 2008-11) Hoang, Luan Thach; Sell, George R.Item On the existence of global attractors(1998-04) Sell, George R.; You, YunchengItem Perturbations of attractors of differential equations(1990-07) Pliss, Victor A.; Sell, George R.Item Perturbations of normally hyperbolic manifolds with applications to the Navier-Stokes equations(1999-10) Pliss, Victor A.; Sell, George R.