Asymptotic degrees of freedom of fluid flows. Final report, July 1, 1986-December 31, 1989.

摘要

We have obtained rigorous estimates for the attractors of some basic dissipative differential equations which are within the physical or numerical ranges (e.g. 2D Navier-Stokes equations). We have shown that the ring laser cavity equations have a finite dimensional attractor. We have constructed inertial manifolds for a large class of dissipative differential equations (e.g. Kuramoto-Sivashinsky and Ginzberg-Landau equations). For a large class of equations including the 2D Navier-Stokes equations we have introduced several approximate intertial manifolds which yield new approximative ordinary differential equations with better error estimates then those of the usual Galerkin approximations. We have evidence that the new approximating schemes lead to computational improvements upon the Galerkin schemes. We have given a normal form for the Navier-Stokes which allows the explicit asymptotic integration of the equations. We have also proposed a new theoretical approach to decaying homogeneous turbulence. We also made some contribution to robust control theory which may be relevant to fluid dynamics.