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May  2004, 4(2): 465-478. doi: 10.3934/dcdsb.2004.4.465

The long-time evolution of mean field magnetohydrodynamics

Manuel Núñez 1,

1. 

Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain

Received  October 2002 Revised  September 2003 Published  February 2004

The equations of mean field magnetohydrodynamics with constant mean velocity are proved to posses solutions bounded in the $H^{1}$-norm for all time, and a compact attractor whose dimension is estimated. It is shown that depending on the functional form of the so-called alpha term the attractor may reduce to zero or be a larger set. If, as usual in physical situations, there exists a set of solutions with a minimum size $N$, the dimension of this set decreases rapidly with increasing $N$. Finally, the dependence of the dimension on the magnetic diffusivity is analyzed, suggesting that the evolution of a magnetic field under the mean field equation is much more restricted than the one deduced from the full magnetohydrodynamic system.
Keywords: attractor., fractal dimension, Mean field magnetohydrodynamics.
Mathematics Subject Classification: 35K55, 37L30, 76F20, 76W0.
Citation: Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465
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