A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form d nu/dt = F(nu), in the Banach space, X, of all bounded continuous functions of the variable s is an element of R with values in certain finite-dimensional linear space. This new evolution ODE, named determining form, induces an infinite-dimensional dynamical system in the space X which is noteworthy for two reasons. One is that F is globally Lipschitz from X into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form nu(t, s) = nu(0)(t + s), correspond exactly to initial data v0 that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter). (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4766459]