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Holly Type Stability of Set of Solutions of Three Dimensional Stationary Incompressible Magnetohydrodynamic Equations

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Abstract

In this paper, following Holly (Ann Polon Math LIV 2:93–109, 1991; Univ Iagell Acta Math XXXI:154–174, 1994), Holly and Motyl (Inversion of the \(divdiv^{\star }\)-operator and three numerical methods in hydrodynamics, selected Problems in Mathematics. Cracow University of Technology, pp 35–94, 1995) and Motyl (Univ Iagell Acta Math XXXVIII:227–277, 2000; Ann Fac Sci Toulouse XXI(4):651–743, 2012), we study stability of solutions from the perspective of Hausdorff metric. To be precise, we construct a sequence of sets of approximate solutions for stationary MHD equations by using Galerkin approximation method and prove that this sequence of sets converges to the set of actual solutions of stationary MHD equations. The convergence is with respect to Hausdorff metric which is a distance defined on a family of sets.

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Acknowledgements

The authors are very grateful to the referees and the Editor for their valuable remarks and helpful suggestions.

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  1. Department of Mathematics, RTM Nagpur University, University Campus, Nagpur, 440033, India

    Swapna V. Uddhao & R. V. Saraykar

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  1. Swapna V. Uddhao
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  2. R. V. Saraykar
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Correspondence to Swapna V. Uddhao.

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Uddhao, S.V., Saraykar, R.V. Holly Type Stability of Set of Solutions of Three Dimensional Stationary Incompressible Magnetohydrodynamic Equations. Differ Equ Dyn Syst 29, 35–58 (2021). https://doi.org/10.1007/s12591-019-00503-w

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