Regularity, determining modes and Galerkin methods

AbstractFor abstract evolutionary equations in a Banach space X, suppose that A is an invariant set, which is compact, for example. We present conditions which ensure that, if u(t)∈A, t∈R, is a solution, then the map t↦u(t) is as smooth in t as the vector field. We obtain also in the abstract setting a generalization of known results on determining modes. It is shown how regularity in t can be used to obtain regularity in space if the equation is generated by a partial differential equation. Applications are given to the linearly damped wave equation and the weakly damped Schrödinger equation. The method of proof relies heavily upon a generalized Galerkin method