TY - JOUR

T1 - Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains

AU - Mahalov, Alex

AU - Nicolaenko, B.

AU - Golse, F.

N1 - Funding Information:
Acknowledgment. We would like to thank Professors V. I. Arnold, C. Foia¸s, V. A. Solonnikov, H. K. Moffatt, G. Papanicolaou, and Ya. Sinai for very useful discussions. O. A. Ladyzhenskaya encouraged this work. We acknowledge the support of the AFOSR grant F49620-02-10026.

PY - 2006/7

Y1 - 2006/7

N2 - 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.

AB - 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.

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U2 - 10.1007/s10958-006-0198-3

DO - 10.1007/s10958-006-0198-3

M3 - Article

AN - SCOPUS:33744825254

SN - 1072-3374

VL - 136

SP - 3768

EP - 3777

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

IS - 2

ER -