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Beyer H.R. — Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations
Beyer H.R. — Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

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Название: Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

Автор: Beyer H.R.


The present volume is self-contained and introduces to the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis. Only some examples require more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Particular stress is on equations of the hyperbolic type since considerably less often treated in the literature. Also, evolution equations from fundamental physics need to be compatible with the theory of special relativity and therefore are of hyperbolic type. Throughout, detailed applications are given to hyperbolic partial differential equations occurring in problems of current theoretical physics, in particular to Hermitian hyperbolic systems. This volume is thus also of interest to readers from theoretical physics.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 288

Добавлена в каталог: 22.05.2008

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Предметный указатель
Boundary condition, Engquist — Majda      78 127
Boundary condition, Sommerfeld      78 124
Chains      146
Chains, finitely generated      148
Chains, juxtapositions      147
Commutator estimates      186 199 246
Commutator function      9
Conventions      1
Couette flow      74
Difference method      78
Equation, abstract evolution, linear      171
Equation, abstract evolution, quasilinear      221
Equation, Burgers      235
Equation, Dirac      103
Equation, Klein — Gordon in Minkowski space      11
Equation, Klein — Gordon on the real line      10
Equation, Maxwell’s equations      132
Equation, wave      8 84
Evolution operator      149
Evolution operator, associated differential equations      150
Evolution operator, finitely generated      151
Evolution operator, for the linear evolution equation      165
Evolution operator, uniqueness      150
Exponential function      35
Group, strongly continuous      63
Group, strongly continuous, associated differential equation      63
Group, strongly continuous, infinitesimal generator      63 65
Inequalities, elementary      235
Inequalities, Peetre’s      265
Inequalities, Sobolev      236
Inequalities, Sobolev’s      265
Infinitesimal generators      42
Infinitesimal generators, bounded      42
Infinitesimal generators, bounded perturbations      61
Infinitesimal generators, characterization      51
Infinitesimal generators, exponential formula      155
Infinitesimal generators, fractional powers      180
Infinitesimal generators, homogeneous IVP      42
Infinitesimal generators, inhomogeneous IVP      66 67
Infinitesimal generators, logarithm      180
Infinitesimal generators, maximality      42
Infinitesimal generators, resolvent estimates      42
Infinitesimal generators, resolvent representation      42
Infinitesimal generators, spectrum property      42
Infinitesimal generators, uniqueness      42
Integration, Bochner      30
Integration, Pettis      26
Integration, weak      26
Integration, weak, existence      26 27
Integration, weak, Fubini’s theorem      30
Integration, weak, properties      27
Integration, weak, substitution rule      34
Intertwining relation      112
Intertwining relation, equivalent conditions      114
Intertwining relation, sufficient conditions      115
kernels      137
Kernels, Convolution      137
Kernels, Volterra      137
Kernels, Volterra equation      137
Kernels, Volterra, approximation      145
Kernels, Volterra, distance      143
Kreiss condition      78
Linear operators      14
Linear operators, 1st resolvent formula      24
Linear operators, 2nd resolvent formula      24
Linear operators, accretive      54
Linear operators, accretive, characterization      55
Linear operators, accretive, closability      56
Linear operators, adjoint      17
Linear operators, closable      14
Linear operators, closed      14
Linear operators, core      14
Linear operators, densely-defined      14
Linear operators, essentially self-adjoint      17
Linear operators, extension      14
Linear operators, extension, weak vs strong      100
Linear operators, graph      14
Linear operators, mollifiers      98
Linear operators, non-closable      16
Linear operators, quasi-accretive      54
Linear operators, quasi-accretive, closability      56
Linear operators, resolvent      24
Linear operators, resolvent set      24
Linear operators, self-adjoint      17
Linear operators, self-adjoint, maximality      17
Linear operators, spectrum      24
Linear operators, symmetric      17
Linear operators, unbounded      16
Nonexpansive homomorphisms properties      117
Numerics of PDEs      78
Quantum system, conservation of probability      7
Quantum system, example      7
Quantum system, Hamilton operator      6
Quantum system, measurement      5
Quantum system, measurement, probabilities      5
Quantum system, observables      5
Quantum system, observables, expectation value      6
Quantum system, observables, spectral measure      6
Quantum system, observables, spectrum      5
Quantum system, quantization of a classical system      5
Quantum system, representation space      5
Quantum system, Schrodinger equation      7
Quantum system, states      5
Quantum system, superposition principle      6
Quantum system, time evolution, unitary      6
Regularity      105
Scalar field in black hole background      178
Scalar field in Minkowski space      11
Scalar field on the real line      10 78 123 127
Semigroup, strongly continuous      42
Semigroup, strongly continuous, approximation      54
Semigroup, strongly continuous, associated differential equation      42
Semigroup, strongly continuous, contraction      56
Semigroup, strongly continuous, infinitesimal generator      42
Semigroup, strongly continuous, integral representation      59
Semigroup, strongly continuous, quasi-contraction      56
Semigroup, uniformly continuous      42
Spaces, admissible      109
Spaces, admissible, sufficient condition      112
Spaces, Banach direct sum      13
Spaces, Hilbert direct sum      13
Spaces, Sobolev      98
Stable families of generators      155
Stable families of generators, bounded perturbations      162
Stable families of generators, criterion      160
Stable families of generators, equivalent conditions      158
Systems, Hermitian hyperbolic, linear autonomous      103
Systems, Hermitian hyperbolic, linear non-autonomous      199
Systems, Hermitian hyperbolic, quasi-linear      246
Tangent functionals, normalized      54
Theorem, Banach fixed point      215
Theorem, Banach fixed point, variation      216
Theorem, closed graph      14
Theorem, Closed graph, application      17
Theorem, E. Noether      113
Theorem, Hellinger — Toeplitz      17
Theorem, Hille — Yosida — Phillips      51
Theorem, Inverse mapping      14
Theorem, Lumer — Phillips      56
Theorem, Mazur      108
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