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Siberian Journal of Pure and Applied Mathematics, 2016, Volume 16, Issue 4, Pages 72–96
DOI: https://doi.org/10.17377/PAM.2016.16.408
(Mi vngu423)
 

This article is cited in 3 scientific papers (total in 3 papers)

A priori tame estimates for free boundary plasma–vacuum problem

N. V. Mandrik

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Full-text PDF (287 kB) Citations (3)
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Abstract: We study the free boundary problem for the plasma–vacuum interface in ideal compressible magnetohydrodynamics. Unlike the classical statement, when the vacuum magnetic field obeys the ${\rm div}$-${\rm rot}$ system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. This work is a continuation of the previous analysis by Mandrik and Trakhinin in 2014, where a sufficient condition on the vacuum electric field that precludes violent instabilities was found and analyzed, the well-posedness of the linearized problem in anisotropic weighted Sobolev spaces was proved under the assumption that this condition is satisfied at each point of the unperturbed nonplanar plasma-vacuum interface. Since the free boundary is characteristic, the functional setting is provide by weighted anisotropic Sobolev spaces $H^s_*$. The fact that the Kreiss–Lopatinski condition is satisfied only in a weak sense yields losses of derivaties in a priori estimates. Assuming that the mentioned above condition is satisfied at each point of the unperturbed nonplanar plasma-vacuum interface, we prove that tame estimates in $H^s_*$ holds for the linearized problem. In future we are going to use those estimates to prove the existence of solutions of the nonlinear problem.
Keywords: tame estimates, ideal compressible magnetohydrodynamics, vacuum Maxwell equations, plasma-vacuum interface.
Received: 24.12.2015
English version:
Journal of Mathematical Sciences, 2018, Volume 230, Issue 1, Pages 118–140
DOI: https://doi.org/10.1007/s10958-018-3732-1
Document Type: Article
UDC: 517.956.3
Language: Russian
Citation: N. V. Mandrik, “A priori tame estimates for free boundary plasma–vacuum problem”, Sib. J. Pure and Appl. Math., 16:4 (2016), 72–96; J. Math. Sci., 230:1 (2018), 118–140
Citation in format AMSBIB
\Bibitem{Man16}
\by N.~V.~Mandrik
\paper A priori tame estimates for free boundary plasma–vacuum problem
\jour Sib. J. Pure and Appl. Math.
\yr 2016
\vol 16
\issue 4
\pages 72--96
\mathnet{http://mi.mathnet.ru/vngu423}
\crossref{https://doi.org/10.17377/PAM.2016.16.408}
\transl
\jour J. Math. Sci.
\yr 2018
\vol 230
\issue 1
\pages 118--140
\crossref{https://doi.org/10.1007/s10958-018-3732-1}
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