Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors

By James C. Robinson. 2001

351 citations recorded by the source corpus.

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Citations in the corpus

Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity (2013, unknown): “Some useful properties concerning b that we will use in the next sections are the following (see [22] or [23]): b(u, v,w) = −b(u,w, v) for all u, v, w ∈ V, which also implies that b(u, v, v) = 0 for all u, v ∈ V.”
Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems (2011, background): “The study of the long-time behavior of infinite dimensional dynamical systems or semigroups generated by autonomous partial differential equations can be usually reduced to the description of the compact invariant set attracting all bounded subsets of the phase space called the global attractor (see”
Random attractors for stochastic reaction–diffusion equations on unbounded domains (2009, background): “It is worth mentioning that in deterministic case differential equations of this type were extensively studied in both autonomous and nonautonomous cases and in both bounded domains and unbounded domains [15–29].”
Pullback attractors for a semilinear heat equation in a non-cylindrical domain (2008, background): “[18]) Let X ⊂ Y be Banach spaces such that X is reflexive and the injection of X in Y is compact.”
Kernel sections and uniform attractors of multi-valued semiprocesses☆ (2007, background): “4) Thus, by basic results in the theory of dynamical systems (see [12,17,26,29], etc.”
EVENTUAL COMPACTNESS FOR SEMIFLOWS GENERATED BY NONLINEAR AGE-STRUCTURED MODELS (2004, methodology): “We refer to Hale [20], Temam [36], Babin and Vishik [11], Robinson [34], Ladyzhenskaya [25], Sell and You [35], Zhao [46] for nice surveys on global attractor theory.”
Bayesian Solution Uncertainty Quantification for Differential Equations (2013, background): “This system has a unique solution whose trajectory lies on a bounded region of the phase space (e.g., Robinson, 2001, pp. 271–272), and is unknown in closed form.”
Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations (2006, unknown): “Moreover, from the properties of b (see [9] or [10]), and the definition of FN , one easily obtains the existence of a constant C1 > 0 only dependent on Ω such that |bN (u, v, w)| ≤ C1 |u|1/4‖u‖3/4|v|1/4‖v‖3/4‖w‖, ∀u, v, w ∈ V, |bN (u, v, w)| ≤ NC1 ‖u‖‖w‖, ∀u, v, w ∈ V.”
A result on the existence of global attractors for semigroups of closed operators (2007, background): “[1, 2, 3, 4, 6, 7, 8]), the existence of the global attractor A is usually established for asymptotically compact semigroups within the assumption that S(t) ∈ C(X ,X ) for every fixed t ∈ K.”