Renormalized Invariant Manifolds

Abstract

Invariant manifold theory for partial differential equations (PDEs) is technically challenging as we often lack spectral gaps. Motivated by recent progress using renormalization in various areas of mathematics, we introduce a spectral renormalization method for semilinear parabolic equations that creates an artificial spectral gap at high frequencies while leaving the low modes unchanged. For the resulting family of renormalized equations, classical inertial-manifold theory yields finite-dimensional invariant manifolds whose dimension diverges as the renormalization parameter tends to zero. We show that the reduced dynamics on these manifolds approximate the original semiflow through an approximate semi-conjugacy, with an $O(ε^2)$ error on bounded finite-time intervals. Under global incremental dissipativity, the semi-conjugacy is uniform in time and yields an $O(ε^2)$ Hausdorff estimate for the global attractors, if they exist. We also establish further dynamical properties of the spectral renormalization such as persistence of hyperbolic equilibria and their Morse indices. We illustrate the construction in settings with and without a natural spectral gap and embed it into a two-parameter fast-slow PDE framework, where the artificial gap enables us to construct slow manifolds for systems on bounded domains with arbitrary dimension. Thus our new approach provides a finite-dimensional dynamical approximation even in regimes where an exact inertial manifold for the original equation is not available. In particular, this provides a practical balance between fully invariant manifolds and direct estimates.