Inertial manifolds and finite-dimensional reduction for dissipative PDEs

Abstract

Abstract. These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mané projection theorems. The recent counterexamples which show that the underlying dynamics may be in a sense infinite-dimensional if the spectral gap condition is violated as well as the discussion on the most important open problems are also included.

Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions

“…Throughout the work we will use the notation u + := P N u and u − := Q N u for a given element u ∈ H. The next proposition collects the standard dissipativity and smoothing properties of the solution semigroup associated with equation (2.1), see [10,23,22] for more details.…”

“…Then the desired inertial form can be constructed just by restricting the considered PDE to the invariant manifold, see [9,15,19]. However, the existence of an inertial manifold requires rather strong spectral gap assumptions which are usually satisfied only for parabolic equations in space dimension one and, despite a big permanent interest and many results obtained in this direction, the finite-dimensional reduction for the case where the IM does not exist remains unclear and there are even some evidence that the dissipative dynamics may be infinite-dimensional in this case, see [6,14,23] and reference therein.…”

“…In Section 3, for the reader convenience, we recall the invariant cone and squeezing properties as well as give the proof of the IM existence theorem for our class of equations under the assumption that the cone and squeezing properties are satisfied (following mainly [23]). …”

Параболическое Уравнение С Нелокальной Диффузией Без Гладкого Инерциального Многообразия

“…На основе модификации леммы 1 в недавних рабо-тах [9], [13] получена общая конструкция абстрактного уравнения (1.1) с нелинейной компонентой ∈ ∞ и показателем нелинейности = 0 без гладкого инерциального многообразия. Там же с помощью других (более деликатных) соображений постро-ено уравнение вида (1.1) с ∈ ∞ , не допускающее даже липшицева инерциального многообразия.…”

“…Эти результаты, по-видимому, могут быть обобщены на общий слу-чай показателя нелинейности ∈ [0, 1). Контрпримеры [9], [12], [13] не слишком естественны, хотелось бы предъявить какое-либо физически значимое полулиней-ное параболическое уравнение, не обладающее свойством асимптотической конеч-номерности. Эта задача решается ниже.…”