Invariant subspaces of the periodic Navier-Stokes and magnetohydrodynamicsequations: Symmetries and inverse cascades

It is shown that when the initial condition and the forcing term of the per iodic Navier-Stokes or magnetohydrodynamics equations have Fourier coeffici ents which vanish outside a certain semigroup of frequencies, the same happ ens to the solutions for all time. Subgroups of frequencies correspond to s olutions possessing certain symmetries. By taking as a semigroup the freque ncies whose Fourier components are non-negative integers, we get a class of solutions for which the higher modes do not influence the evolution of the lower ones; therefore, the phenomenon of inverse cascading cannot occur fo r them. (C) 2000 American Institute of Physics. [S0022-2488(00)03609-4].