Large time regular solutions to the MHD equations in cylindrical domains

Wisam Alame¹ ; Wojciech M. Zajączkowski²

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.

Résumé

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.

Détail

DOI : 10.4064/am38-2-1

Mots-clés : 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