Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer
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 diffus...
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/3/391 |
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| author | Gennadii Alekseev Olga Soboleva |
| author_facet | Gennadii Alekseev Olga Soboleva |
| author_sort | Gennadii Alekseev |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T03:52:43Z |
| format | Article |
| id | doaj.art-63f45a6af052433eaf23043cfe707425 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T03:52:43Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-63f45a6af052433eaf23043cfe7074252024-02-09T15:18:11ZengMDPI AGMathematics2227-73902024-01-0112339110.3390/math12030391Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass TransferGennadii Alekseev0Olga Soboleva1Institute of Applied Mathematics, FEB RAS, 7, Radio St., 690041 Vladivostok, RussiaInstitute of Applied Mathematics, FEB RAS, 7, Radio St., 690041 Vladivostok, RussiaWe 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.https://www.mdpi.com/2227-7390/12/3/391generalized Boussinesq model of mass transferbinary fluidinhomogeneous boundary conditionsglobal solvabilitymaximum principlelocal uniqueness |
| spellingShingle | Gennadii Alekseev Olga Soboleva Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer Mathematics generalized Boussinesq model of mass transfer binary fluid inhomogeneous boundary conditions global solvability maximum principle local uniqueness |
| title | Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer |
| title_full | Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer |
| title_fullStr | Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer |
| title_full_unstemmed | Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer |
| title_short | Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer |
| title_sort | inhomogeneous boundary value problems for the generalized boussinesq model of mass transfer |
| topic | generalized Boussinesq model of mass transfer binary fluid inhomogeneous boundary conditions global solvability maximum principle local uniqueness |
| url | https://www.mdpi.com/2227-7390/12/3/391 |
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