Solvability Analysis of a Mixed Boundary Value Problem for Stationary Magnetohydrodynamic Equations of a Viscous Incompressible Fluid

Abstract

We investigate the boundary value problem for steady-state magnetohydrodynamic (MHD) equations with inhomogeneous mixed boundary conditions for a velocity vector, given the tangential component of a magnetic field. The problem represents the flow of electrically conducting viscous fluid in a 3D-bounded domain, which has the boundary comprising several parts with different physical properties. The global solvability of the boundary value problem is proved, a priori estimates of the solutions are obtained, and the sufficient conditions on data, which guarantee a solution’s local uniqueness, are determined.

Analysis of Control Problems for Stationary Magnetohydrodynamics Equations under the Mixed Boundary Conditions for a Magnetic Field

“…The first type is described by relations H • n = 0 and E × n = 0 on ∂Ω, corresponding to a perfectly conducting boundary (see, e.g., [2][3][4][5][17][18][19][20]). The second type is described by the condition H × n = 0 on ∂Ω (see [8,21,22]) corresponding to a perfectly insulating boundary.…”

“…In [28], the results obtained in [27] were generalized for the model of heat conducting magnetohydrodynamics. It should be noted that papers [29][30][31], magnetohydrodynamic equations are studied under mixed boundary conditions with respect to velocity and under the standard boundary conditions of the first type for an electromagnetic field. In [32], the author proves the existence of a very weak solution of the MHD boundary value problem using the Dirichlet boundary condition for a magnetic field.…”

“…where (31) takes place by (10), (11) if u = 0 and H = 0 or u 1 = u 2 and H 1 = H 2 in Ω. Then from (29), it follows that (divv, p) = 0 for all v ∈ H 1 0 (Ω) 3 . This means by inf-sup condition (17), that p = 0 or p 1 = p 2 in Ω.…”

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

“…Close questions on the study of the correctness of boundary value or control problems for stationary equations of magnetic hydrodynamics of viscous incompressible or heatconducting liquid in the Boussinesq approximation were investigated in [38][39][40][41][42][43]. In [44], the solvability of the initial-boundary problem for the non-stationary MHD-Boussinesq system is considered under mixed boundary conditions for velocity, magnetic field, and temperature, in the case when the viscosity coefficient, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat of the fluid depend on the temperature.…”

Stability Estimates of Optimal Solutions for the Steady Magnetohydrodynamics-Boussinesq Equations

“…A number of papers are devoted to the study of solvability of boundary value (and control) problems for MHD Equations (1) and ( 2) at b = 0 considered under boundary conditions different from those in (4). Let us mention among them the papers [29][30][31] considered under the so-called non-standard boundary conditions for velocity or pressure, as well as the papers by Meir and Alekseev et al,respectively,[32][33][34] considered under mixed boundary conditions for the magnetic field. The papers [35][36][37][38][39][40][41][42][43] are devoted to the study of the solvability of boundary value and control problems for stationary or nonstationary MHD-Boussinesq models.…”