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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Bibliographic Details
Published in:Mathematics (Basel) 2024-01, Vol.12 (3), p.391
Main Authors: Alekseev, Gennadii, Soboleva, Olga
Format: Article
Language:English
Subjects:
Analysis
Approximation
binary fluid
Boundary conditions
Boundary value problems
Boussinesq approximation
Dirichlet problem
Existence theorems
Fluid mechanics
generalized Boussinesq model of mass transfer
global solvability
Heat conductivity
Inhomogeneity
inhomogeneous boundary conditions
local uniqueness
Mass transfer
Maximum principle
Nonlinear theories
Velocity
Viscosity
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12030391