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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2026. Vol 57

Existence and Uniqueness Theorem for a Weak Solution to the Initial-Boundary Value Problem for the Kelvin–Voigt Mathematical Model of the Motion of a Mixture of Fluids

Author(s)

Dmitry A. Zakora1, Dmitry A. Prokudin2

1 Vernadsky Crimean Federal University, Simferopol, Russian Federation

2 Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russian Federation

Abstract
Equations describing three-dimensional unsteady motions of Kelvin–Voigt mixture of viscous incompressible liquids are considered. The theorem of existence and uniqueness of a weak solution of an initial boundary value problem corresponding to flows in a limited domain is proved. To prove the existence of a solution, an approximation problem is considered, its solvability and a priori estimates independent of the approximation parameter are established. After that, a transition is made to the limit on the approximation parameter and it is shown that the solutions of the approximation problem weakly converge to the solution of the original problem. The uniqueness of the solution is established using the Gronwall inequality.
About the Authors

Dmitry A. Zakora, Dr. Sci. (Phys.–Math.), Assoc. Prof., Vernadsky Crimean Federal University, Simferopol, 295007, Russian Federation, dmitry.zkr@gmail.com

 Dmitry A. Prokudin, Dr. Sci. (Phys.-Math.), Assoc. Prof., Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, 630090, Russian Federation, prokudin@hydro.nsc.ru,

For citation
Zakora D. A., Prokudin D. A. Existence and Uniqueness Theorem for a Weak Solution to the Initial-Boundary Value Problem for the Kelvin–Voigt Mathematical Model of the Motion of a Mixture of Fluids. The Bulletin of Irkutsk State University. Series Mathematics, 2026, vol. 57, pp. 66–80. (in Russian) https://doi.org/10.26516/1997-7670.2026.57.66
Keywords
existence and uniqueness theorem, initial boundary value problem, incompressible fluid, multicomponent medium, Kelvin–Voigt model
UDC
517.9
MSC
35A01, 76D03
DOI
https://doi.org/10.26516/1997-7670.2026.57.66
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