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Convex Analysisの二,三の進展について

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

Random Dynamical Systems

General Topology-I

Annals of physics

Evolution of pattern complexity in the Cahn–Hilliard theory of phase separation

At the end of the last century the French mathematician Henri Poincare laid the foundation for what we call nowadays the qualitative theory of ordinary differential equations. Roughly speaking, this theory is devoted to studying how the qualitative behavior (e.g. the asymptotic behavior) of solutions to certain initial value problems changes as the initial condition is varied. In order to make this more explicit, consider the autonomous differential equation 

$$ x = f(x), $$

(1.1)

where f : ℝd → ℝd is a C1-mapping with f (x 0) = 0 for some x 0 ∈ ℝd. Obviously, the point x 0 is a constant solution of (1.1). But what can be said about the behavior of solutions of (1.1) starting in some small neighborhood of this point? Of course one is tempted to first study the linearized equation 

$$ x = Df(x_0 )x $$

(1.2)

near the origin, since it may be hoped that the nonlinear behavior of (1.1) near x 0 will be “basically” the same.

Linearization of Random Dynamical Systems

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation
 
where Ω⊂ℝ
 n
 , n∈{1,2,3 }, is a bounded domain with sufficiently smooth boundary, and f is cubic-like, for example f(u) =u−u

 3. Based on the results of [26] the nonlinear Cahn-Hilliard equation will be discussed. This equation generates a nonlinear semiflow in certain affine subspaces of H

 2(Ω). In a neighborhood U

 e with size proportional to e
 n
 around the constant solution , where μ lies in the spinodal region, we observe the following behavior. Within a local inertial manifold containing there exists a finite-dimensional invariant manifold
 which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1. The dimension of is proportional to e−

 n
 and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to e.

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

While the structure of the set of stationary solutions of the Cahn-Hilliard equation on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this paper, we demonstrate how rigorous computational techniques can be employed to establish computerassisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Mischaikow and Zgliczy´nski [22], and combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of specific equilibrium solutions. Sample computations for several branches are presented, which illustrate the resulting patterns.

/pdf/rigorous-numerics-for-the-cahn-hilliard-equation-on-the-unit-14gehhuaqx.pdf

Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square

We describe the fine structure of the global attractor of the Cahn–Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equi...

Thomas Wanner

Papers

Evolution of pattern complexity in the Cahn–Hilliard theory of phase separation

Linearization of Random Dynamical Systems

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square

Structure of the Attractor of the CAHN-HILLIARD Equation on a Square