Solvability of the boundary value problem for stationary magnetohydrodynamic equations under mixed boundary conditions for the magnetic field

Алексеев, Г. В.; Brizitskii, R. V.

doi:10.1016/j.aml.2014.02.006

Cited by 23 publications

(18 citation statements)

References 10 publications

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“…So, the assumption excludes the boundary conditions such that H • n| Σ 𝜏 = q, J • n| Σ 𝜈 ≠ 0, which is very important in practice. By a similar argument, we can see that Alekseev and Brizitskii 15 and Alekseev 11 dealt, respectively, with the case J • n| 𝜕Ω = 0 and J • n| Σ 𝜈 = 0.…”

Section: Theorem 1 Let (3) Of Assumption 1 Hold H ∈ H DC (ω) Satisfy ...mentioning

confidence: 68%

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Existence of a solution to the steady Magnetohydrodynamics‐Boussinesq system with mixed boundary conditions

Kim

2022

Math Methods in App Sciences

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In this paper, we are concerned with the steady magnetohydrodynamics (MHD)‐Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include the stick, Tresca slip, leak, one‐sided leaks, pressure, vorticity, and stress conditions together, and the conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. For the electromagnetic field, mixed boundary conditions are given. Also, the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat of the fluid depend on the temperature. Special attention is paid to the meaning of non‐homogeneous mixed boundary conditions for magnetic field. For the problem involving the static pressure and stress boundary conditions for fluid, it is proven that if the data of problem are small enough, then there exists a solution and the solution with small norm is unique when the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat are independent of the temperature. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proven when a datum concerned with non‐homogeneous boundary conditions for magnetic field is small, but without smallness of other data.

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Section: Theorem 1 Let (3) Of Assumption 1 Hold H ∈ H DC (ω) Satisfy ...mentioning

confidence: 68%

“…In Alekseev and Brizitskii (2014), 11 the MHD system was studied under the homogeneous Dirichlet boundary condition for velocity and the homogeneous case of mixed boundary condition () for electromagnetic field.…”

Section: Introductionmentioning

confidence: 99%

Existence of a solution to the steady Magnetohydrodynamics‐Boussinesq system with mixed boundary conditions

Kim

2022

Math Methods in App Sciences

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“…As for the recovery of component r from the identity (32), this procedure is performed using the inf-sup condition (23) according to the standard scheme using de Rham theory (see details, for example, in [25]). For convenience of readers we offer here a brief outline of this procedure following [4]. To this end, we represent a functional as follows:…”

Section: Weak Formulation Of Problem 1 Weak Solutionmentioning

confidence: 99%

“…In most papers, the MHD equations are examined under the Dirichlet boundary condition u| Γ = g for the velocity vector, where g is a given function. Among these papers, we mention the articles [1][2][3][4][5][6] dedicated to the research of the solvability of the corresponding boundary value problems for Equations ( 1) and (2). Papers [7][8][9][10] are dedicated to the research of the solvability of both boundary value and control problems for (1) and (2).…”

Section: Introduction Statement Of the Boundary Value Problemmentioning

confidence: 99%

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

Алексеев

Brizitskii

2021

Symmetry

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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.

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“…For the first time a global solvability of Problem 1 under homogeneous boundary conditions for a velocity and electromagnetic field was proved in [11] without the requirement Σ τ ∩ Σ ν = ∅. It is essential to use mathematical thools from [12][13][14] A global solvability of the inhomogeneous mixed Problem 1 was proved in [15], see also [16,17].…”

Section: Introduction Statement Of the Boundary Value Problemmentioning

confidence: 99%

Study of control problems for the stationary MHD equations

Brizitskii

2017

J. Phys.: Conf. Ser.

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Abstract.The optimal control problems for the stationary magnetohydrodynamic equations under inhomogeneous mixed boundary conditions for a magnetic field are considered. The role of control in control xs under study is played by normal component of the magnetic field on the part of the boundary. In the capacity of cost fucntionals quadratic tracking-type functionals for a velocity, magnetic field or pressure are taken. Introduction. Statement of the boundary value problemGreat attention has been paid recently to the optimal control problems for the models of magnetohydrodynamics (MHD) for a viscous conducting incompressible fluid (see [1][2][3][4][5][6]).In this paper we study control problems for the new MHD model in which stationary equations of magnetohydrodynamic are considered under mixed boundary conditions for a magnetic field and under Dirichlet boundary condition for a velocity with tracking-type functionals. The role of controls in control problems under study is played by a normal component of the magnetic field on the part of the boundary.One of the features of this work consist in proving of the solvability of optimal control problem under minimum requirements on a normal component of the magnetic field given on the part of boundary and playing the role of control. This result, in contrast to [3], allows to use a simple L 2 -norm for the normal component of the magnetic field instead of the H 1/2 -norm on the part of the boundary by Tikhonov regularization of considered incorrect control problem. This regularization is needed to prove stability and uniqueness of the control problem's solution.Let Ω be a bounded domain of space R 3 with boundary ∂Ω consisting of two parts Σ ν and Σ τ . In this paper we study control problems for the stationary magnetohydrodynamic equations of viscous incompressible fluid

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Solvability of the boundary value problem for stationary magnetohydrodynamic equations under mixed boundary conditions for the magnetic field

Cited by 23 publications

References 10 publications

Existence of a solution to the steady Magnetohydrodynamics‐Boussinesq system with mixed boundary conditions

Existence of a solution to the steady Magnetohydrodynamics‐Boussinesq system with mixed boundary conditions

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

Study of control problems for the stationary MHD equations

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