Evolution Equations And Approximations

By: Kazufumi Ito, Franz Kappel

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Hardcover | 28 May 2002

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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.

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Preface	p. v
Dissipative and Maximal Monotone Operators	p. 1
Duality mapping and directional derivatives of norms	p. 1
Dissipative operators	p. 9
Properties of m-dissipative operators	p. 17
Perturbation results for m-dissipative operators	p. 25
Maximal monotone operators	p. 32
Convex functionals and subdifferentials	p. 40
Linear Semigroups	p. 57
Examples and basic definitions	p. 57
Cauchy problems and mild solutions	p. 66
The Hille-Yosida theorem	p. 74
The Lumer-Phillips theorem	p. 79
A second order equation	p. 83
Analytic Semigroups	p. 89
Dissipative operators and sesquilinear forms	p. 89
Analytic semigroups	p. 97
Approximation of C[subscript 0]-Semigroups	p. 115
The Trotter-Kato theorem	p. 115
Approximation of nonhomogeneous problems	p. 131
Variational formulations of the Trotter-Kato theorem	p. 135
An approximation result for analytic semigroups	p. 141
Nonlinear Semigroups of Contractions	p. 147
Generation of nonlinear semigroups	p. 147
Cauchy problems with dissipative operators	p. 153
The infinitesimal generator	p. 163
Nonlinear diffusion	p. 175
Locally Quasi-Dissipative Evolution Equations	p. 181
Locally quasi-dissipative operators	p. 183
Assumptions on the operators A(t)	p. 186
DS-approximations and fundamental estimates	p. 188
Existence of DS-approximations	p. 202
Existence and uniqueness of mild solutions	p. 208
Autonomous problems	p. 229
"Nonhomogeneous" problems	p. 231
Strong solutions	p. 238
Quasi-linear equations	p. 246
A "parabolic" problem	p. 253
The Crandall-Pazy Class	p. 263
The conditions	p. 263
Existence of an evolution operator	p. 267
Variational Formulations and Gelfand Triples	p. 285
Cauchy problems and Gelfand triples	p. 285
An approximation result	p. 301
Applications to Concrete Systems	p. 305
Delay-differential equations	p. 305
Equations of retarded type in the state space C	p. 306
State dependent delays	p. 313
Equations of neutral type	p. 319
Scalar conservation laws	p. 326
Basic assumptions and preliminaries	p. 326
Globally bounded functions	p. 331
Vanishing viscosity, quasi-dissipativity	p. 333
L[superscript 2]-considerations	p. 338
L[superscript 1]- and L[superscript [infinity]-estimates	p. 342
W[superscript 1,1]-estimates	p. 347
Uniformity with respect to v	p. 354
The limit v [characters not reproducible] 0	p. 361
The limiting operator	p. 378
The general case	p. 385
The Navier-Stokes equations	p. 386
The two-dimensional case	p. 398
The three-dimensional case	p. 403
Approximation of Solutions for Evolution Equations	p. 409
Approximation by approximating evolution problems	p. 409
Chernoff's theorem	p. 427
Operator splitting	p. 440
Semilinear Evolution Equations	p. 447
Well-posedness	p. 447
Delay equations with time and state dependent delays	p. 458
Approximation theory	p. 463
A concrete approximation scheme for delay systems	p. 469
Appendix	p. 479
Some inequalities	p. 479
Convergence of Steklov means	p. 483
Some technical results needed in Section 9.2	p. 484
Bibliography	p. 489
List of Symbols	p. 493
Index	p. 495
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Evolution Equations And Approximations

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Hardcover

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