Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains |
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Authors: | A Mahalov B Nicolaenko F Golse |
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Institution: | (1) Department of Mathematics and Statistics, Arizona State University, USA;(2) U.M.R. of Mathematics, University of Paris-7, France;(3) Laboratoire J. L. Lions, University of Paris-6, France |
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Abstract: | The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional
initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional
assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach
of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the
context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any
restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations,
we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned
uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.
Dedicated to the memory of O. A. Ladyzhenskaya
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 203–219. |
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