Evolution Equations and Approximations

This text presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille-Yosida), nonlinear (Crandall-Liggett) and time-dependent (Crandall-Pazy) theorems. The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter-Kato theorem and the Lie-Trotter type product formula give a mathematical framework for the convergence analysis of numerical approximations of solutions to a general class of partial differential equations. This work contains examples demonstrating the applicability of the generation as well as the approximation theory. In addition, the Kobayashi-Oharu approach of locally quasi-dissipative operators is discussed for homogeneous as well as non-homogeneous equations. Applications to the delay differential equations, Navier-Stokes equation and scalar conservation equation are given. This book presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille-Yosida), nonlinear (Crandall-Liggett) and time-dependent (Crandall-Pazy) theorems.The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter-Kato theorem and the Lie-Trotter type product formula give a mathematical framework for the convergence analysis of numerical approximations of solutions to a general class of partial differential equations. This book contains examples demonstrating the applicability of the generation as well as the approximation theory.In addition, the Kobayashi-Oharu approach of locally quasi-dissipative operators is discussed for homogeneous as well as nonhomogeneous equations. Applications to the delay differential equations, Navier-Stokes equation and scalar conservation equation are given. This book presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille-Yosida), nonlinear (Crandall-Liggett) and time-dependent (Crandall-Pazy) theorems. The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter-Kato theorem and th 3.1 Dissipative operators and sesquilinear forms3.2 Analytic semigroups; Chapter 4. Approximation of Co-Semigroups; 4.1 The Trotter-Kato theorem; 4.2 Approximation of nonhomogeneous problems; 4.3 Variational formulations of the Trotter-Kato theorem; 4.4 An approximation result for analytic semigroups; Chapter 5. Nonlinear Semigroups of Contractions; 5.1 Generation of nonlinear semigroups; 5.2 Cauchy problems with dissipative operators; 5.3 The infinitesimal generator; 5.4 Nonlinear diffusion; Chapter 6. Locally Quasi-Dissipative Evolution Equations; 6.1 Locally quasi-dissipative operators 9.1 Delay-differential equations9.1.1 Equations of retarded type in the state space C; 9.1.2 State dependent delays; 9.1.3 Equations of neutral type; 9.2 Scalar conservation laws; 9.2.1 Basic assumptions and preliminaries; 9.2.2 Globally bounded functions; 9.2.3 Vanishing viscosity quasi-dissipativity; 9.2.4 L2-considerations; 9.2.5 L1- and Loo-estimates; 9.2.6 W 1-1 -estimates; 9.2.7 Uniformity with respect to v; 9.2.8 The limit v 0; 9.2.9 The limiting operator; 9.2.10 The general case; 9.3 The Navier-Stokes equations; 9.3.1 The two-dimensional case; 9.3.2 The three-dimensional case 6.2 Assumptions on the operators A(t)6.3 DS-approximations and fundamental estimates; 6.4 Existence of DS-approximations; 6.5 Existence and uniqueness of mild solutions; 6.6 Autonomous problems; 6.7 ""Nonhomogeneous"" problems; 6.8 Strong solutions; 6.9 Quasi-linear equations; 6.10 A ""parabolic"" problem; Chapter 7. The Crandall - Pazy Class; 7.1 The conditions; 7.2 Existence of an evolution operator; Chapter 8. Variational Formulations and Gelfand Triples; 8.1 Cauchy problems and Gelfand triples; 8.2 An approximation result; Chapter 9. Applications to Concrete Systems Preface; Contents; Chapter 1. Dissipative and Maximal Monotone Operators; 1.1 Duality mapping and directional derivatives of norms; 1.2 Dissipative operators; 1.3 Properties of m-dissipative operators; 1.4 Perturbation results for m-dissipative operators; 1.5 Maximal monotone operators; 1.6 Convex functionals and subdifferentials; Chapter 2. Linear Semigroups; 2.1 Examples and basic definitions; 2.2 Cauchy problems and mild solutions; 2.3 The Hille-Yosida theorem; 2.4 The Lumer-Phillips theorem; 2.5 A second order equation; Chapter 3. Analytic Semigroups Annotation Ito (North Carolina State U.) and Kappel (U. of Graz, Austria) offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K. and Y. Kobayashi and S. haru. They also explore abstract approximation results for evolution equations. Their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses. Annotation c. Book News, Inc., Portland, OR (booknews.com) Annotation Ito (North Carolina State U.) and Kappel (U. of Graz, Austria) offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K. and Y. Kobayashi and S. Oharu. They also explore abstract approximation results for evolution equations. Their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses. Annotation copyrighted by Book News, Inc., Portland, OR Chapter 10. Approximation of Solutions for Evolution Equations10.1 Approximation by approximating evolution problems; 10.2 Chernoff's theorem; 10.3 Operator splitting; Chapter 11. Semilinear Evolution Equations; 11.1 Well-posedness; 11.2 Delay equations with time and state dependent delays; 11.3 Approximation theory; 11.4 A concrete approximation scheme for delay systems; Appendix; A.l Some inequalities; A.2 Convergence of Steklov means; A.3 Some technical results needed in Section 9.2; Bibliography; List of Symbols; Index In this chapter we present the basic facts on dissipative resp. monotone operators in Hilbert or Banach spaces which will be used in the following chapters.