INVARIANT MANIFOLDS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

By Jinqiao Duan, Kening Lu, B. Schmalfuss. 2003

310 citations recorded by the source corpus.

Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite- and infinite-dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite-dimensional random dynamical systems generated by stochastic partial differential equations. We first introduce a random graph transform and a fixed point theorem for nonautonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/613dab66589a6f9f8bbbfed61fe44661631f6773

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Citations in the corpus

Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space (2010, unknown): “See, for example,[8], [7], [18], and [32].”
Climate dynamics and fluid mechanics: Natural variability and related uncertainties (2008, background): “Thus RDS concepts and tools are not restricted to finite-dimensional systems [104, 105, 106].”
Pullback Attracting Inertial Manifolds for Nonautonomous Dynamical Systems (2002, background): “The above mentioned notion of inertial manifolds is translated and extended to more general classes of differential equations like nonautonomous differential equations, [GV97, WF97, LL99], retarded parabolic differential equations, [TY94, BdMCR98], or differential equations with random or stochastic”
Pathwise Random Periodic Solutions of Stochastic Differential Equations (2011, background): “In deterministic and random dynamical systems, to find the existence of stationary solutions and random periodic solutions, and to construct local stable and unstable manifolds near a hyperbolic stationary point is a fundamental problem [1,7,8,11,12,17,22].”
Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations (2013, background): “For the existence of invariant manifolds for stochastic PDEs, we refer the reader to [20, 21, 36, 37] for details.”
Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations (2004, methodology): “In [10], we used a coordinate transform to convert conjugately a stochastic partial differential equation into an infinite dimensional random dynamical system.”
Invariant Manifolds for Random Dynamical Systems with Slow and Fast Variables (2008, methodology): “Therefore, we can refer to the proof of a corresponding result in [7].”
Limiting dynamics for stochastic wave equations (2008, background): “For more about the dynamics between two conjugate RDSs, see [9].”
Invariant Manifolds for Random and Stochastic Partial Differential Equations (2009, background): “Stable, unstable, center, and inertial manifolds, as special random invariant structures, have been considered in the investigation of stochastic partial differential equations or stochastic evolutionary equations in infinite dimensional spaces [8, 12, 13, 19, 5, 15, 11, 27].”
Anticipating Random Periodic Solutions--I. SDEs with Multiplicative Linear Noise (2015, unknown): “Similarly as in [10], using |W (t+s)| ≤ Cδ,ω + |s|δ + |t|δ P − a.”
Existence and Upper Semicontinuity of Attractors for Stochastic Equations with Deterministic Non-autonomous Terms (2012, background): “The reader is also referred to [1, 17, 18, 29, 31] for the existence of random invariant manifolds.”