DETERMINING DEGREES OF FREEDOM FOR NONLINEAR DISSIPATIVE EQUATIONS

A finite set of linear functionals {l(i)}(N)(i=1) is said to be a set of determining degrees of freedom for a given PDE if when any two solutions of the PDE, u(1) and u(2), are such that (l(i) (u(1) (t) - u(2) (t)) converges to 0 as time t goes to infinity, 1 less than or equal to i less than or equal to N, then the solutions converge to each other as time goes to infinity. In this paper, we prove the existence of a large class of sets of determining degrees of freedom, with special attention to the standard sets of degrees of freedom used in the finite element method, for the Navier-Stokes equations; and provide an estimate on the size of the sets. We then extend and sharpen this result to general nonlinear dissipative evolution equation possessing an inertial manifold.