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

A. S. Makhalov   ; B. Nicolaenko   ; F. Golse

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 203-219

Résumé

Non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proven for a class of three-dimensional initial data characterized by both uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on the 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 restriction on the size of 3D initial data. After establishing strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D MHD Equations with weakly aligned uniformly large vorticity and magnetic field at $t=0$.

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