Galerkin Approximations of the Problem for One-Dimensional Barotropic Equations of Compressible Viscous Multicomponent Media Dynamic

УДК 517.95

  • Vladimir A. Makarov Novosibirsk State University, Novosibirsk, Russia Email: v.makarov2@g.nsu.ru
  • Dmitriy A. Prokudin Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia; Altai State University, Barnaul, Russia Email: prokudin@hydro.nsc.ru
DOI_disc_logo https://doi.org/10.14258/izvasu(2026)1-16
Keywords: compressible viscous medium, multicomponent flows, Galerkin approximations

Abstract

This paper considers an initial-boundary value problem for a one-dimensional model of barotropic dynamics of compressible viscous multicomponent media described by a system of equations that are a generalization of the Navier — Stokes equations. Unlike the Navier — Stokes equations, where viscosity is a scalar quantity, the viscosities of a multicomponent case form a viscosity matrix that reflects the composite structure of viscous stress tensors. This leads to the appearance of senior derivatives of the velocities of all components in the equations under study and significantly complicates the mathematical analysis. The diagonal elements of the viscosity matrix are responsible for viscous friction within each component, while the off-diagonal elements are responsible for the intercomponent viscous interaction. Such a structure excludes the possibility of directly extrapolating the existing theoretical results obtained for the Navier — Stokes equations to multicomponent models. The case of a diagonal viscosity matrix significantly simplifies the situation because the components are related only through junior terms. This study is devoted to substantiating the existence of Galerkin approximations of the initial-boundary value problem in a more general case with the non-diagonal structure of the viscosity matrix.

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Author Biographies

Vladimir A. Makarov, Novosibirsk State University, Novosibirsk, Russia

Undergraduate Student

Dmitriy A. Prokudin, Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia; Altai State University, Barnaul, Russia

Doctor of Sciences in Physics and Mathematics, Leading Researcher; Chief Researcher

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Published
2026-04-08
How to Cite
Makarov V. A., Prokudin D. A. Galerkin Approximations of the Problem for One-Dimensional Barotropic Equations of Compressible Viscous Multicomponent Media Dynamic // Izvestiya of Altai State University, 2026, № 1(147). P. 114-119 DOI: 10.14258/izvasu(2026)1-16. URL: https://izvestiya.asu.ru/article/view/%282026%291-16.
Issue
No 1(147) (2026): Известия Алтайского государственного университета
Section
Математика и механика

Copyright (c) 2026 Владимир Александрович Макаров, Дмитрий Алексеевич Прокудин

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