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

Gennadii Alekseev,Roman V Brizitskii

doi:10.3390/sym13112088

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.

Highlights

Statement of the Boundary Value ProblemLet us assume that Ω is a bounded domain from the space R3 and it has the boundaryΓ, which includes three parts: Γ1, Γ2 and Γ3
We investigate the boundary value problem for the steady-state MHD equations with mixed boundary conditions for the velocity vector, given the magnetic field’s tangential component on the entire boundary: Rahmat Ellahi
In [17], a boundary value problem for steady-state MHD Equations (1) and (2) is studied in the case that zero tangential components of the velocity and magnetic field together with the total pressure are given on the entire boundary of the flow domain

Summary

Introduction

Let us assume that Ω is a bounded domain from the space R3 and it has the boundary. Γ, which includes three parts: Γ1 , Γ2 and Γ3. In [17], a boundary value problem for steady-state MHD Equations (1) and (2) is studied in the case that zero tangential components of the velocity and magnetic field together with the total pressure are given on the entire boundary of the flow domain. The limiting case s = 0 corresponds to the physical scenario, when the given magnetic field’s H tangential component H × n = q on the boundary Γ is an element of the Symmetry 2021, 13, 2088 subspace of Hilbert space L2 (Γ)3 This scenario is the most preferable one from an applied point of view, especially while studying boundary control problems (see [10]).

Functional Spaces and Assumptions

Solvability of Problem 1

Findings

Discussion

Conclusions