Operator Pencil in a Domain with Concentrated Masses. A Scalar Analog of Linear Hydrodynamics

Chechkin, Gregory A.

doi:10.1007/978-1-4419-1343-2_3

Cited by 1 publication

(3 citation statements)

References 34 publications

(42 reference statements)

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“…As was already noted in [4], it was an idea of Professor S. A. Stepin to use the scheme due to the Kopachevskii-Krein consistent with methods of homogenization theory to a scalar counterpart of the linear hydrodynamics in the case of microinhomogeneities on the boundary.…”

Section: Acknowledgmentmentioning

confidence: 97%

“…Therefore, we are restrict ourselves to homogenization results for such problems. Note that the problem (1.2) leads to homogenization of operator pencils (cf., for example, [2]- [4]). …”

Section: Statement Of the Problemmentioning

confidence: 99%

“…Such problems with surfaces of microinhomogeneous periodic structure were considered in [2]- [4], where homogenization of an operator pencil was studied. This study was based on homogenization methods (cf., for example, [5]- [13] and the references therein), asymptotic analysis, and matching asymptotic expansions (cf., for example, [14]).…”

Section: Dedicated To Nina Nikolaevna Uraltsevamentioning

confidence: 99%

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Plane Scalar Analog of Linear Degenerate Hydrodynamic Problem with Nonperiodic Microstructure of Free Surface

Чечкин

Chechkina

2015

J Math Sci

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We consider the degenerate problem on low-frequency oscillations of a heavy viscous incompressible fluid in a vessel with free surface of inhomogeneous nonperiodic microstructure. For the obtained quadratic operator pencil we construct the limit (homogenized) pencil which turns out to be degenerate. We prove a homogenization theorem. Bibliography: 40 titles. Illustration: 2 figures. Dedicated to Nina Nikolaevna UraltsevaA scalar analog of linear hydrodynamics was first described in [1] in detail. Such problems with surfaces of microinhomogeneous periodic structure were considered in [2]-[4], where homogenization of an operator pencil was studied. This study was based on homogenization methods (cf., for example, [5]-[13] and the references therein), asymptotic analysis, and matching asymptotic expansions (cf., for example, [14]). were used. Owing to this approach, it becomes possible to construct effective models of inhomogeneous media or study singular perturbations of problems without microinhomogeneities; moreover, these methods provide rigorous justifications of constructions and accurate proofs. In this direction, various problems with singular perturbation were considered, in particular, perturbation of geometry (cf., for example, [7,12], [15]-[17]), coefficients (cf. [18, 19]), and the type of boundary condition (cf., for example, [20]-[27] and [13]). Similar methods were used in hydrodynamics for studying strongly inhomogeneous fluids with different rheology (cf., for example, [28]-[39]).In this paper, we study a plane analog of the problem on low-frequency oscillations of a heavy viscous incompressible fluid in a vessel with free surface covered by a perforated cap with nonperiodic structure (cf. Figure 1).

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Section: Acknowledgmentmentioning

confidence: 97%

“…Therefore, we are restrict ourselves to homogenization results for such problems. Note that the problem (1.2) leads to homogenization of operator pencils (cf., for example, [2]- [4]). …”

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Dedicated To Nina Nikolaevna Uraltsevamentioning

confidence: 99%

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