1,721,002 research outputs found
On extended eigenvalues of operators
No full textA complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λXA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A* has trivial kernel.The first author was partially supported by NSF grant number DMS – 0074460 and by the Junior Summer Research Fellowship (2003) from University of North Carolina at Charlotte. The second author was supported in part by the FRACASF grant from the Western Michigan University.https://link.springer.com/article/10.1007/s00020-005-1381-
Higher order regularity, long term dynamics, and data assimilation for magnetohydrodynamic flows
No full textFirst we consider various inviscid equations of fluid dynamics and show that if the initial data is analytic in the space variables, then the resulting flows extend as analytic functions of both space and time variables, with explicit estimates of the analyticity radius. We then consider higher order regularity of the viscous magnetohydrodynamic equations for an incompressible conductive fluid. We establish the Gevrey regularity of solutions when the initial data is in a Sobolev class, of possibly negative order, in two and three spatial dimensions. In particular, we show that solutions evolving from singular initial data instantaneously become analytic, with the analyticity radius eventually expanding in time. This in turn allows us to establish decay in higher order Sobolev norms. Finally, using a recently developed data assimilation algorithm based on linear feedback control, we show that when the initial data is unknown, sparse measurement data is sufficient for accurate reconstruction of magnetohydrodynamic flows. This algorithm convergences exponentially in time to the reference solution and moreover, the reconstruction is exact on the attractor
Local existence and Gevrey regularity of 3-D Navier–Stokes equations with ℓp initial data
No full textAbstractWe obtain local existence and Gevrey regularity of 3-D periodic Navier–Stokes equations in case the sequence of Fourier coefficients of the initial data is in ℓp(p<3/2). The ℓp norm of the sequence of Fourier coefficients of the solution and its analogous Gevrey norm remains bounded on a time interval whose length depends only on the size of the body force and the ℓp norm of the Fourier coefficient sequence of the initial data. The control on the Gevrey norm produces explicit estimates on the analyticity radius of the solution as in Foias and Temam (J. Funct. Anal. 87 (1989) 359–369). The results provide an alternate approach in estimating the space-analyticity radius of solutions to Navier–Stokes equations than the one presented by Grujić and Kukavica (J. Funct. Anal. 152 (1998) 447–466)
Gevrey regularity for a class of dissipative equations with applications to decay
No full textAbstractIn this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via “caloric extension”. We apply our result to the Navier–Stokes equations, the surface quasi-geostrophic equations, the Kuramoto–Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier–Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation
Gevrey regularity for the supercritical quasi-geostrophic equation
No full textIn this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in “critical” Sobolev spaces. Moreover, the Gevrey class that we obtain is “near optimal” and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier–Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the dyadic Littlewood–Paley operators, is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty.This research was partly supported by NSF grant DMS-1425877.https://www.sciencedirect.com/science/article/pii/S002203961400197
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Determining functionals and data assimilation and a novel regularity criterion for the three-dimensional navier–stokes equations
No full textIn this paper we present two results: (1) a data assimilation algorithm for the 3D Navier–Stokes equation (3D NSE) using nodal data and, as a consequence, (2) a novel regularity criterion for the 3D NSE based on finitely many observations of the velocity. The data assimilation algorithm we employ utilizes nudging, a method based on a Newtonian relaxation scheme motivated by feedback control. The observations, which may be either modal, nodal or volume elements, are drawn from a weak solution of the 3D NSE and are collected almost everywhere in time over a finite grid, and our results, including the regularity criterion, hold for data of any of the aforementioned forms. The regularity criterion we propose follows from our data assimilation algorithm and is hence intimately connected to the notion of determining functionals (modes, nodes and volume elements). To the best of our knowledge, all existing regularity criteria require knowing the solution of the 3D NSE almost everywhere in space. Our regularity criterion is fundamentally different from any preexisting regularity criterion as it is based on finitely many observations (modes, nodes and volume elements). We further prove that the regularity criterion we propose is both a necessary and sufficient condition for regularity. Thus, our result can be viewed as a natural generalization of the notion of determining modes, nodes and volume elements as well as the asymptotic tracking property of the nudging algorithm for the 2D NSE to the 3D setting.https://link.springer.com/article/10.1007/s40687-025-00530-
Infinite Dimensional Dynamical Systems In Fluid Dynamics And Fluid-Structure Interaction
No full textIn Part 1 of this thesis, three results are presented : (1) A sufficient condition, \emph{based solely on the observed velocity data}, for the global well-posedness, regularity and the asymptotic tracking property of a data assimilation algorithm for the three-dimensional Boussinesq system employing nudging, (2) a data assimilation algorithm for the 3D Navier-Stokes equation (3D NSE) using \emph{nodal observations}, and, as a consequence (3) a novel regularity criterion for the 3D NSE \emph{based on finitely many observations} of the velocity. The observations are drawn from a Leray-Hopf weak solution of the of the underlying system. For the data assimilated 3D Boussinesq system the observations are comprised either of a finite-dimensional \emph{modal} projection or finitely many \emph{volume element observations}, whereas for the data assimilated 3D NSE, the observations could be a finite dimensional \emph{modal} projection, finitely many \emph{volume element observations} or finitely many \emph{nodal observations}. The proposed conditions on the data in each case are automatically satisfied for solutions that are globally regular and are uniformly bounded in the \h^1-norm. However, neither regularity nor any knowledge of a uniform \h^1-norm bound is {\it a priori} assumed on the solutions. To the best of our knowledge, this is the first such rigorous analysis of \emph{any} data assimilation algorithm for the \emph{three-dimensional} Boussinesq system for which global regularity and well-posedness is unknown. Our condition also guarantees the construction of the {\it determining map} for the 3D Boussinesq system, thus extending prior work on its existence for the two-dimensional NSE. Additionally, the regularity criterion for the 3D NSE is \emph{fundamentally different} from any preexisting regularity criterion as it is based on \emph{finitely many pointwise observations} and \emph{does not require knowing the solution almost everywhere in space}. Lastly, we show that the regularity criterion we propose is both a necessary and sufficient condition for regularity. In Part 2 of this thesis the {strong asymptotic stabilization} of 3D hyperbolic dynamics is achieved by a damped 2D elastic structure evolving on a bounded subset. The model is a Neumann wave-type equation with low regularity coupling conditions given in terms of a nonlinear von Karman plate. This problem is motivated by the elimination of aeroelastic instability (sustained oscillations of bridges, airfoils, etc.) in engineering applications. Empirical observations indicate that the subsonic wave-plate system to equilibria. Classical approaches which decouple the plate and wave dynamics have fallen short. Here, we operate on the model as it appears in the engineering literature with {no regularization} and achieve stabilization by microlocalizing the Neumann boundary data for the wave equation (given by the plate). We observe {a compensation} by the plate dynamics { precisely where the regularity of the 3D wave is compromised} (in the characteristic sector)
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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