Operator approach to linear problems of hydrodynamics. Vol. 2: Nonself-adjoint problems for viscous fluids.
type: Item
instance of: scholarly article
title: Operator approach to linear problems of hydrodynamics. Vol. 2: Nonself-adjoint problems for viscous fluids.
zbMATH Open document ID: 1048.76001
author: Selim G. Krein
author: Nikolay Dmitrievich Kopachevsky
published in: Operator Theory: Advances and Applications
publication date: 2003-11-05T00:00:00Z
review text: [For the review of Vol. 1 (2001) see Zbl 0979.76002.] Applications of the theory of linear operators are considered for linearized Navier-Stokes equations describing the flows of viscous incompressible fluids. The velocity vector field \(u(t,x)\) is treated as a time-dependent function \(u(t)\) which takes its values in the Hilbert space \(J_0 (\Omega)\), a set of functions of variable \(x\in\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^3\). The space \(J_0 (\Omega)\) includes all the functions \(v(x)\in L^2 (\Omega)^3\) such that \(\text{div}v=0\) in the distribution sense. After the linearization near a steady velocity field \(v_0 (x)\), the Navier-Stokes equations can be written as the ODE in Banach space \(J_0 (\Omega)\): \(du/dt= -\nu A_0 u - Ru+P_0 f,\) \(u(0)=u_0.\) Here \(P_0 :L^2 (\Omega)^3 \to J_0 (\Omega)\) is the orthoprojection operator, \(f(t,x)\) is the force vector field, \(\nu\) is the viscosity, \(A_0\) is the Stokes operator, and \(Ru=P_0 [(v_0 \cdot\nabla )u+(u\cdot\nabla)v_0 ]\). Restricted to smooth functions, the Stokes operator \(A_0\) coincides with the operator \(-P_0 \Delta\). A part of presentation is devoted to normal oscillations, which are the flows with the velocity law \(u(t,x)=e^{-\nu t\lambda}v(x)\). When \(v_0 (x)=0\) and \(f=0\), one arrives at the eigenvalue problem \(A_0 v=\lambda v\). Nonself-adjoint problems dominate in the volume. As a rule, these problems include time derivatives of unknown functions at the boundary \(\partial\Omega\). As far as the normal oscillations are concerned, the spectral problems include the spectral parameter both in the equation and in the boundary conditions. In their study the authors widely use the theory of nonself-adjoint operators acting in Hilbert space, and also the theory of operator pencils. In particular, the methods of operator pencil factorization and methods of operator theory in a space with indefinite metric find a wide application. The authors address both the cl…
Mathematics Subject Classification ID: 76-02
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 47N60
zbMATH DE Number: 2001178
zbMATH Keywords: linearlized Navier-Stokes equations
zbMATH Keywords: Hilbert space
zbMATH Keywords: Banach space
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: small oscillations
reviewed by: V. V. Shelukhin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2001178