Model of the Maxwell Compressible Fluid
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MODEL OF THE MAXWELL COMPRESSIBLE FLUID D. A. Zakora
UDC 517.984.48:532.135
Abstract. A model of viscoelastic barotropic Maxwell fluid is investigated. The unique solvability theorem is proved for the corresponding initial-boundary value problem. The associated spectral problem is studied. We prove statements on localization of the spectrum, on the essential and discrete spectra, and on the asymptotic behavior of the spectrum.
CONTENTS 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Problem Formulation . . . . . . . . . . . . . . . . . . . . Unique Solvability Theorem . . . . . . . . . . . . . . . . Spectrum Problem for Viscoelastic Compressible Fluids References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
We study the Maxwell model of viscoelastic barotropic fluids. For the first time, models of incompressible fluids taking into account the prehistory of the flow (they are called linear viscoelastic fluids nowadays) were proposed in [8, 13, 14, 25, 26]. In [17, 18], these models are substantially developed. Further, they and more general models were studied by many authors (see, e.g., [20, 28] and references therein). In [3, 16, 22, 24] (see references therein as well), the spectral analysis of models of viscoelastic incompressible fluids is provided. In this work, we investigate the problem of small motions of the Maxwell viscoelastic compressible fluid filling a bounded uniformly rotating region. In the third section, the solvability of the corresponding system of integrodifferential equations, boundary conditions, and initial conditions is investigated. The corresponding Cauchy problem for the system of integrodifferential equations is reduced to the Cauchy problem dξ = −Aξ + F(t), ξ(0) = ξ 0 , dt in a Hilbert space H. The operator A is an operator block-matrix and is a maximal accretive operator. This yields the solvability result for the original initial-boundary value problem. In the fourth section, we investigate the problem on the spectrum of the operator A associated with the spectral problem for the original system of integrodifferential equations. We prove that the spectrum of the operator A is located in the right-hand open half-plane; additionally, if there is no rotation, then it is separated from the imaginary axis. In the general case, the essential spectrum of the operator A consists of a finite set of points and segments of the real positive axis. The discrete spectrum is located in a vertical band, is condensed at infinity, and has a power asymptotic distribution. If the system does not rotate, then there are conditions for the physical parameters of the system such that the discrete spectrum of the operator A located in a neighborhood of the real line is real. Translated from Sovremennaya Matematika. Fundamental’nye Napravle
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