Geodesic
Large time regular solutions to the MHD equations in cylindrical domains
Wisam Alame 1   ; Wojciech M. Zajączkowski 2
1 Faculty of Mathematics and Information Sciences Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland
2 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland and Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland
Applicationes Mathematicae, Tome 38 (2011) no. 2, pp. 117-146 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the $L_2$-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in $W_2^{2,1}$ without restriction on their magnitude. Then the existence follows from the Leray–Schauder fixed point theorem.
DOI : 10.4064/am38-2-1
Keywords: prove large time existence solutions magnetohydrodynamics equations slip boundary conditions cylindrical domain assuming smallness norms derivatives initial velocity magnetic field respect variable along axis cylinder able obtain estimate velocity magnetic field without restriction their magnitude existence follows leray schauder fixed point theorem

Wisam Alame  1   ; Wojciech M. Zajączkowski  2

1 Faculty of Mathematics and Information Sciences Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland
2 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland and Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland 
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     title = {Large time regular solutions to
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 the MHD equations in cylindrical domains
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Wisam Alame; Wojciech M. Zajączkowski. Large time regular solutions to
 the MHD equations in cylindrical domains. Applicationes Mathematicae, Tome 38 (2011) no. 2, pp. 117-146. doi: 10.4064/am38-2-1

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