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Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

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  • Gennadii Alekseev

    (Institute of Applied Mathematics, FEB RAS, 7, Radio St., 690041 Vladivostok, Russia
    Department of Mathematical and Computer Modelling, Far Eastern Federal University,690922 Vladivostok, Russia
    These authors contributed equally to this work.)

  • Olga Soboleva

    (Institute of Applied Mathematics, FEB RAS, 7, Radio St., 690041 Vladivostok, Russia
    Department of Mathematical and Computer Modelling, Far Eastern Federal University,690922 Vladivostok, Russia
    These authors contributed equally to this work.)

Abstract

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

Suggested Citation

  • Gennadii Alekseev & Olga Soboleva, 2024. "Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer," Mathematics, MDPI, vol. 12(3), pages 1-24, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:391-:d:1326495
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    References listed on IDEAS

    as
    1. Stepanova, Irina V., 2019. "Group analysis of variable coefficients heat and mass transfer equations with power nonlinearity of thermal diffusivity," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 57-66.
    2. Gennadii Alekseev, 2023. "Analysis of Control Problems for Stationary Magnetohydrodynamics Equations under the Mixed Boundary Conditions for a Magnetic Field," Mathematics, MDPI, vol. 11(12), pages 1-29, June.
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