Geodesic
On the eigenvalues and eigenfunctions for a free boundary problem for incompressible viscous magnetohydrodynamics
Piotr Kacprzyk 1   ; Wojciech M. Zajączkowski 2
1 Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology S. Kaliskiego 2 00-908 Warszawa, Poland <a href="https://orcid.org/0000-0003-1504-5394">ORCID: 0000-0003-1504-5394</a>
2 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland <a href="https://orcid.org/0000-0003-1229-2162">ORCID: 0000-0003-1229-2162</a>
Applicationes Mathematicae, Tome 47 (2020) no. 1, pp. 99-131 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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The motion of incompressible magnetohydrodynamics (mhd) in a domain bounded by a free surface and coupled through it with an external electromagnetic field is considered. Transmission conditions for electric currents and magnetic fields are prescribed on the free surface. In this paper we show the idea of the proof of local existence by the method of successive approximations. For this we need linearized problems: the Stokes system for the velocity and pressure and the linear transmission problem for the electromagnetic field. We do not prove the local existence of solutions to the original problem but we show existence of a fundamental basis of functions for the linearized problems. Once we have such a basis, the existence of solutions to the linear problems can be shown by the Faedo–Galerkin method, as in other papers of Kacprzyk. The existence of solutions of the linear systems can also be shown by the method of regularizer.
DOI : 10.4064/am2358-10-2018
Keywords: motion incompressible magnetohydrodynamics mhd domain bounded surface coupled through external electromagnetic field considered transmission conditions electric currents magnetic fields prescribed surface paper idea proof local existence method successive approximations linearized problems stokes system velocity pressure linear transmission problem electromagnetic field prove local existence solutions original problem existence fundamental basis functions linearized problems once have basis existence solutions linear problems shown faedo galerkin method other papers kacprzyk existence solutions linear systems shown method regularizer

Piotr Kacprzyk  1   ; Wojciech M. Zajączkowski  2

1 Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology S. Kaliskiego 2 00-908 Warszawa, Poland <a href="https://orcid.org/0000-0003-1504-5394">ORCID: 0000-0003-1504-5394</a>
2 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland <a href="https://orcid.org/0000-0003-1229-2162">ORCID: 0000-0003-1229-2162</a> 
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     title = {On the eigenvalues and eigenfunctions for a free boundary problem for incompressible viscous magnetohydrodynamics},
     journal = {Applicationes Mathematicae},
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Piotr Kacprzyk; Wojciech M. Zajączkowski. On the eigenvalues and eigenfunctions for a free boundary problem for incompressible viscous magnetohydrodynamics. Applicationes Mathematicae, Tome 47 (2020) no. 1, pp. 99-131. doi: 10.4064/am2358-10-2018

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