Invariant manifolds of partially normally hyperbolic invariant manifolds in Banach spaces

Abstract: We investigate the existence and regularity of (locally) invariant manifolds nearby an approximately invariant set satisfying certain (geometric) hyperbolicity with respect to an abstract generalized' dynamical system in a Banach space; such hyperbolicity is between normal hyperbolicity and partial hyperbolicity which has being studied in the finite-dimension and in some concrete PDEs. Thegeneralized' dynamical system is allowed to be non-smooth, non-Lipschitz, or even `non-mapping', making it applicable to both well-posed and ill-posed differential equations. As an illustration, we apply our results to study the dynamics of the whiskered tori.