Some closure results for inertial manifolds

Suppose that the family of evolution equationsdu/dt+Au+f _N (u)=0 possesses inertial manifolds of the same dimension for a sequence of nonlinear termsf _N withf _N f in the C⁰ norm. Conditions are found to ensure that the limiting equationdu/dt+Au+f(u)=0 also possesses an inertial manifold. There are two cases. The first, where the manifolds for the family have a bounded Lipschitz constant, is straightforward and leads to an interesting result on inertial manifolds for Bubnov-Galerkin approximations. When the Lipschitz constant is unbounded, it is still possible to prove the existence of an exponential attractor of finite Hausdorff dimension for the limiting equation. This more general result is applied to a problem in approximate inertial manifold theory discussed by Sell (1993).For Paul Glendinning, with thanks.