Convergent families of approximate inertial manifolds
This article is devoted to a new method of construction of Approximate Inertial Manifolds (AIM) which allows us to solve two open problems: -- Construction of a convergent sequence of AIMs when a spectral gap condition is satisfied and an exact inertial manifold exists. -- When the exact inertial manifold does not exist or it is not known to exist, explicit construction of AIMs that approximate the global attractor at an exponential order, (i.e.: at order \(\eta \sim c' e^{- cn^\alpha}\) when the dimension of the AIM is \(n)\). The existence of such approximate inertial manifolds was proven in \textit{C. Foias}, \textit{O. Manley} and \textit{R. Temam} [C. R. Acad. Sci., Paris, Sér. I 305, 497- 500 (1987; Zbl 0624.76072) and RAIRO, Modélisation Math. Anal. Numér. 22, No. 1, 93-118 (1988; Zbl 0663.76054)] but the proof was not constructive. We consider a fairly general class of evolution partial differential equations, including dissipative parabolic equations with selfadjoint or nonselfadjoint linear part, and complex amplitude equations that are not necessarily dissipative. Using the Lyapunov-Perron method, we first present a general theorem of existence of an exact inertial manifold; this result contains all the preceding ones concerning the existence of inertial manifolds in the selfadjoint and nonselfadjoint cases. We establish then our main results in this general setting: we show how the iteration mapping from the Lyapunov-Perron construction can be explicitly and effectively approximated. Using these approximate mappings we then construct our family of AIMs, which satisfy the results stated above.
scientific article; zbMATH DE number 811453