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Analysis of Two Linearized Problems Modeling Viscous Two‐Layer Flows
Author(s) -
Pileckas Konstantinas,
Socolowsky Jürgen
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200211)245:1<129::aid-mana129>3.0.co;2-n
Subject(s) - mathematics , linearization , sobolev space , mathematical analysis , boundary value problem , domain (mathematical analysis) , eigenvalues and eigenvectors , flow (mathematics) , boundary layer , nonlinear system , boundary (topology) , geometry , mechanics , physics , quantum mechanics
Two problems that appear in the linearization of certain free boundary value problems of the hydrodynamics of two viscous fluids are studied in the strip‐like domain Π = { x = ( x 1 , x 2 ) ∈ ℝ 2 : x 1 ∈ ℝ 1 , (0 < x 2 < h * ) ∨ ( h * < x 2 < 1)}. The first problem arises in the linearization of a two‐layer flow down a geometrically perturbed inclined plane. The second one appears after the linearization of a two‐layer flow in a geometrically perturbed inclined channel with one moving (smooth) wall. For this purpose the unknown flow domain was mapped onto the double strip Π. The arising linear elliptic problems contain additional unknown functions in the boundary conditions. The paper is devoted to the investigation of these boundary problems by studying the asymptotics of the eigenvalues of corresponding operator pencils. It can be proved that the boundary value problems are uniquely solvable in weighted Sobolev spaces with exponential weight. The study of the full (nonlinear) free boundary value problems will be the topic of a forthcoming paper.