Theoretical Analysis of Boundary Value Problems for Generalized Boussinesq Model of Mass Transfer with Variable Coefficients

Abstract

A boundary value problem is formulated for a stationary model of mass transfer, which generalizes the Boussinesq approximation in the case when the coefficients in the model equations can depend on the concentration of a substance or on spatial variables. The global existence of a weak solution of this boundary value problem is proved. Some fundamental properties of its solutions are established. In particular, the validity of the maximum principle for the substance’s concentration has been proved. Sufficient conditions on the input data of the boundary value problem under consideration, which ensure the local existence of the strong solution from the space H2, and conditions that ensure the conditional uniqueness of the weak solution with additional property of smoothness for the substance’s concentration are established.

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

“…Teh main results of our work were obtained by using a number of the author's findings. The first finding is related to the possibility of spreading the mathematical theory developed in [28] in the study of homogeneous boundary value problems for the MT model to the case of the inhomogeneous boundary value problem considered in the manuscript. This possibility was realized by using special extensions of inhomogeneous boundary data entered in (3) inside the domain Ω.…”

“…Unlike [25,26,28], we consider inhomogeneous boundary conditions for velocity and for concentration. This allows us to apply the theory, developed in our work, for a wide class of physically interesting problems of flowing binary fluid through a limited region.…”

“…Here, L ν , L λ and L ′ λ are some positive constants. To achieve our goal, we will use as in [25,26,28] the equivalence between the standard ∥ • ∥ 2,Ω and L 2 -norm ∥∆ • ∥ Ω of the Laplace operator in the space H 2 (Ω) ∩ H 1 0 (Ω) for the domain with the boundary Γ ∈ C 2 (see [54,55]). The mentioned equivalence is described by the following inequalities:…”

“…Recall that in [28] the proof of the local uniqueness of the solution to the boundary value problem under study was based precisely on the equivalence property of the norms ∥ • ∥ 2,Ω and ∥∆ • ∥ Ω , which is valid for the components [28], where the equations of the Boussinesq model are considered under homogeneous Dirichlet conditions, in our case, the specified equivalence property for components φ 1 and φ 2 does not work because they satisfy the inhomogeneous boundary condition φ i | Γ = ψ, i =1.2. However, this property remains valid for the difference φ = φ 1 − φ 2 , which belongs to H 1 0 (Ω).…”

“…The mentioned works can be divided into several groups. The first group includes works [25][26][27][28], devoted to the study of the solvability of boundary value problems for stationary equations of HMT. The second group includes works [29][30][31][32][33][34], devoted to the study of the solvability of boundary value and control problems for non-stationary Boussinesq equations of HT (or MT).…”