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2002 | Buch

Dynamics of Evolutionary Equations

verfasst von: George R. Sell, Yuncheng You

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

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Über dieses Buch

The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. Dynamical issues arise in equations that attempt to model phenomena that change with time. The infi­ nite dimensional aspects occur when forces that describe the motion depend on spatial variables, or on the history of the motion. In the case of spatially depen­ dent problems, the model equations are generally partial differential equations, and problems that depend on the past give rise to differential-delay equations. Because the nonlinearities occurring in thse equations need not be small, one needs good dynamical theories to understand the longtime behavior of solutions. Our basic objective in writing this book is to prepare an entree for scholars who are beginning their journey into the world of dynamical systems, especially in infinite dimensional spaces. In order to accomplish this, we start with the key concepts of a semiflow and a flow. As is well known, the basic elements of dynamical systems, such as the theory of attractors and other invariant sets, have their origins here.

Inhaltsverzeichnis

Frontmatter
1. The Evolution of Evolutionary Equations
Abstract
“May you live in exciting times!” This traditional Chinese saying aptly describes the environment surrounding the basic developments in mathematics, physics, and chemistry over the past four centuries. From the founding of the European Academies of Sciences during the era of Peter the Great and Napoleon, to the founding of the National Science Foundation in the United States of America during the presidency of Harry Truman, governments have realized the importance of scientific research.1 To trace the implications of this academic research, as it affects the evolution of evolutionary equations, we begin in 1601 in Prague with the appointment of Johannes Kepler (1571–1630) to the position of Imperial Mathematician of the Holy Roman Empire, after the death of his predecessor, Tycho Brahe (1546–1601).
George R. Sell, Yuncheng You
2. Dynamical Systems: Basic Theory
Abstract
The basic concept underlying the study of dynamics in infinite dimensional spaces is that of a semiflow, or as it is sometimes called, a semigroup. This semiflow is a time-dependent action on the ambient space, which we assume to be a complete metric space W, for example, a Banach space or a Fréchet space. One should think of the semiflow as a mechanism for describing the solutions of an underlying evolutionary equation. This evolutionary equation is oftentimes the abstract formulation of a given partial differential equation or, sometimes, an ordinary differential equation with time delays. In this chapter we will examine some basic properties of semiflows. Our principal objective is to describe the longtime dynamics in terms of the invariant sets, the limit sets, and the attractors of the semiflow. A comprehensive theory of global attractors is included here. Later in this volume, we will develop the connections between the semiflow and the underlying evolutionary equation.
George R. Sell, Yuncheng You
3. Linear Semigroups
Abstract
In this chapter we describe the basic notions of a linear C 0-semigroup and the related concepts of an infinitesimal generator. These concepts form infinite dimensional versions of solutions of the finite dimensional linear ordinary differential equation ə t x = Ax.In particular, the C 0-semigroup corresponds to the solution operator or the fundamental solution matrix, and the infinitesimal generator corresponds to the linear coefficient matrix A. We will see that the C 0-semigroups are linear prototypes of the semiflows described in Chapter 2.
George R. Sell, Yuncheng You
4. Basic Theory of Evolutionary Equations
Abstract
In the last chapter, we presented a theory describing solutions of a linear evolutionary equation ə t u + Au = 0, where \( {\partial _t}u = \frac{d}{{dt}}u \) on a Banach space W, in terms of C o-semigroups. As we have seen, this theory allows one to construct mild solutions of many linear partial differential equations with constant coefficients. Our objective in this chapter is to generalize this theory so that it applies first to the linear inhomogeneous equation
$$ {\partial _t}u + Au = f(t) $$
(40.1)
and then the nonlinear evolutionary equation
$$ {\partial _t}u + Au = F(u) $$
(40.2)
.
George R. Sell, Yuncheng You
5. Nonlinear Partial Differential Equations
Abstract
In this chapter we turn our attention to the study of the dynamical properties of solutions of nonlinear partial differential equations. We are especially interested here in those nonlinear evolutionary equations which arise in the analysis of two broad classes of partial differential equations: parabolic evolutionary equations and hyperbolic evolutionary equations. While our usage of the terms “parabolic” and “hyperbolic” in this context is motivated by related concepts arising in the basic classification of partial differential equations, we will attribute these terms, instead, to certain dynamical features of the linear ancestry of the underlying nonlinear problems. More precisely, the linear prototypes of the partial differential equations of interest here include the heat equation and the wave equation:
$${\partial _t}u - \nu \Delta u = 0\;and\;\partial _t^2u - \nu \Delta u = 0,$$
on a suitable domain Ω in ℝ d , with various boundary conditions and initial conditions.
George R. Sell, Yuncheng You
6. Navier-Stokes Dynamics
Abstract
Longtime dynamical issues arise in many areas in the world about us. For example, one finds them in various fluid flows as illustrated by 1) heat transfer and its effects on global climate modeling and weather prediction; 2) flows of multiphased fluids and oil recovery; 3) behavior of chemical solutes in lakes, harbors and river basins; 4) geothermal consequences of magma flows; and 5) fluid flows in thin films. In order to understand the modeling of such phenomena, one needs good mathematical tools coupled with sound insight into the physics and the chemistry of the problems.
George R. Sell, Yuncheng You
7. Major Features of Dynamical Systems
Abstract
In this chapter we will examine a number of the classical issues arising in the study of the dynamics of differential equations on Banach spaces. The corresponding theories for the finite dimensional problems appear in a number of sources, as noted in the Commentary Section. Our emphasis here will be on the infinite dimensional theory, in the context of general nonlinear evolutionary equations. Most of the applications will be to the theory of solutions of partial differential equations.
George R. Sell, Yuncheng You
8. Inertial Manifolds: The Reduction Principle
Abstract
In the previous chapters, we have seen several illustrations of finite dimensional structures within the infinite dimensional dynamical systems. For example, many dissipative systems have global attractors, and oftentimes, the attractor A has finite Hausdorff and fractal dimensions. During the last few years it has been shown that some infinite dimensional nonlinear dissipative evolutionary equations have inertial manifolds. We will give the definition shortly.
George R. Sell, Yuncheng You
Backmatter
Metadaten
Titel
Dynamics of Evolutionary Equations
verfasst von
George R. Sell
Yuncheng You
Copyright-Jahr
2002
Verlag
Springer New York
Electronic ISBN
978-1-4757-5037-9
Print ISBN
978-1-4419-3118-4
DOI
https://doi.org/10.1007/978-1-4757-5037-9