Modelling of the interaction of lower and higher modes in two-dimensional MHD-equations

A new method of approximating the solutions of the magnetohydrodynamic equations for a long time by means of approximate inertial manifolds has been proposed. This approximation scheme has been derived directly from the MHD-equations without any phenomenological considerations. The last two inequalities of Section 5 show that the distance between any solution to the MHD-equations and the approximate inertial manifold is smaller than the distance to the flat space q = 0 by a factor λm+1-1/2 for the velocity and μm+1-1/2 for the magnetic field. Our arguments have been yielded an improvement of the distance to the manifold ℳ0 in the L2-norm. The estimates in the W1,2-norm will be one of our subjects for further investigation.