Regularity results for a model in magnetohydrodynamics with imposed pressure

Poirier, J.; Seloula, Nour El Houda

doi:10.5802/crmath.113

Cited by 4 publications

(5 citation statements)

References 12 publications

(29 reference statements)

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“…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. In [33], the MHD boundary value problem is studied in the case where a pressure and zero tangential velocity component are specified on an entire boundary. A local solvability of boundary value and boundary control problems for the model of magnetomicropolar flow is proved in paper [6], where the optimality systems are also derived and analyzed for the control problem.…”

Section: Statement Of the Boundary Value Problem And Notationmentioning

confidence: 99%

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

Алексеев

2023

Mathematics

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The optimal control problems for stationary magnetohydrodynamic equations under the inhomogeneous mixed boundary conditions for a magnetic field and the Dirichlet condition for velocity are considered. The role of controls in the control problems under study is played by normal and tangential components of the magnetic field given on different parts of the boundary and by the exterior current density. Quadratic tracking-type functionals for velocity, magnetic field or pressure are taken as cost functionals. The global solvability of the control problems under consideration is proved, an optimality system is derived and, based on its analysis, a mathematical apparatus for studying the local uniqueness and stability of the optimal solutions is developed. On the basis of the developed apparatus, the local uniqueness of solutions of control problems for specific cost functionals is proved, and stability estimates of optimal solutions are established.

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Section: Statement Of the Boundary Value Problem And Notationmentioning

confidence: 99%

“…Let assumptions (i), (ii) be satisfied. Then the operator Φ : X → Y, which is defined by ( 32), (33), is the isomorphism of the space X into Y.…”

Section: Lifting Of the Velocity And Magnetic Field Solvability Of Pr...mentioning

confidence: 99%

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

Алексеев

2023

Mathematics

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“…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.…”

Section: Introduction and Formulation Of The Boundary Value Problemmentioning

confidence: 99%

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

Alekseev,

Spivak

2024

Mathematics

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This paper develops the mathematical apparatus of studying control problems for the stationary model of magnetic hydrodynamics of viscous heat-conducting fluid in the Boussinesq approximation. These problems are formulated as problems of conditional minimization of special cost functionals by weak solutions of the original boundary value problem. The model under consideration consists of the Navier–Stokes equations, the Maxwell equations without displacement currents, the generalized Ohm’s law for a moving medium and the convection-diffusion equation for temperature. These relations are nonlinearly connected via the Lorentz force, buoyancy force in the Boussinesq approximation and convective heat transfer. Results concerning the existence and uniqueness of the solution of the original boundary value problem and of its generalized linear analog are presented. The global solvability of the control problem under study is proved and the optimality system is derived. Sufficient conditions on the data are established which ensure local uniqueness and stability of solutions of the control problems under study with respect to small perturbations of the cost functional to be minimized and one of the given functions. We stress that the unique stability estimates obtained in the paper have a clear mathematical structure and intrinsic beauty.

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“…The existence of a weak solution in W 1,p (Ω) with 3 2 < p < 2 is proved by applying Banach's fixed-point theorem over the linearized problem. Some results of this work are announced in [21] 2 Main results…”

Section: Introductionmentioning

confidence: 99%

Regularity results for a model in magnetohydrodynamics with imposed pressure

Poirier¹,

Seloula²

2023

Preprint

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The magnetohydrodynamics (MHD) problem is most often studied in a framework where Dirichlet type boundary conditions on the velocity field is imposed. In this Note, we study the (MHD) system with pressure boundary condition, together with zero tangential trace for the velocity and the magnetic field. In a threedimensional bounded possibly multiply connected domain, we first prove the existence of weak solutions in the Hilbert case, and later, the regularity in W 1,p (Ω) for p ≥ 2 and in W 2,p (Ω) for p ≥ 6/5 using the regularity results for some Stokes and elliptic problems with this type of boundary conditions. Furthermore, under the condition of small data, we obtain the existence and uniqueness of solutions in W 1,p (Ω) for 3/2 < p < 2 by using a fixed-point technique over a linearized (MHD) problem.

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Regularity results for a model in magnetohydrodynamics with imposed pressure

Cited by 4 publications

References 12 publications

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

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

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

Regularity results for a model in magnetohydrodynamics with imposed pressure

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