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

Author

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  • Gennadii Alekseev

    (Institute of Applied Mathematics FEB RAS, 7, Radio St., 690041 Vladivostok, Russia
    These authors contributed equally to this work.)

  • Yuliya Spivak

    (Institute of Applied Mathematics FEB RAS, 7, Radio St., 690041 Vladivostok, Russia
    These authors contributed equally to this work.)

Abstract

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.

Suggested Citation

  • Gennadii Alekseev & Yuliya Spivak, 2024. "Stability Estimates of Optimal Solutions for the Steady Magnetohydrodynamics-Boussinesq Equations," Mathematics, MDPI, vol. 12(12), pages 1-42, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1912-:d:1419021
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    References listed on IDEAS

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    1. Gennadii Alekseev, 2023. "Analysis of Control Problems for Stationary Magnetohydrodynamics Equations under the Mixed Boundary Conditions for a Magnetic Field," Mathematics, MDPI, vol. 11(12), pages 1-29, June.
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