G. Derks et T. Ratiu, ATTRACTING CURVES ON FAMILIES OF STATIONARY SOLUTIONS IN 2-DIMENSIONAL NAVIER-STOKES AND REDUCED MAGNETOHYDRODYNAMICS, Proceedings - Royal Society. Mathematical, physical and engineering sciences, 454(1973), 1998, pp. 1407-1444
Citations number
27
Categorie Soggetti
Multidisciplinary Sciences
Journal title
Proceedings - Royal Society. Mathematical, physical and engineering sciences
Families of stable stationary solutions of the two-dimensional incompr
essible homogeneous Euler and ideal reduced magnetohydrodynamic equati
ons are shown to be attracting for the corresponding viscous perturbat
ions of these systems, i.e. for the Navier-Stokes and the reduced visc
ous MHD equations with magnetic diffusion. Each solution curve of the
dissipative system starting in a cone around the family of stationary
solutions of the unperturbed conservative system defines a shadowing c
urve which attracts the dissipative solution in an exponential manner.
As a consequence, the specific exponential decay rates are also deter
mined. The techniques to analyse these two equations can be applied to
other dissipative perturbations of Hamiltonian systems. The method in
its general setting is also presented.