Holdings: An introduction to semiflows

An introduction to semiflows:

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Bibliographic Details
Main Authors:	Milani, Albert (Author), Koksch, Norbert 1962- (Author)
Format:	Book
Language:	English
Published:	Boca Raton u.a. Chapman & Hall/CRC 2005
Series:	Chapman & Hall/CRC monographs and surveys in pure and applied mathematics 134
Subjects:	Flots (Dynamique différentiable) Flows (Differentiable dynamical systems) Fluss > Mathematik Dynamisches System
Online Access:	Inhaltsverzeichnis
Item Description:	Includes bibliographical references (p. 373-380) and index
Physical Description:	xvi, 386 p. graph. Darst. 25 cm
ISBN:	1584884584

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adam_text	K CHAPMAN & HALL/CRC MONOGRAPHS AND SURVEYS IN PURE AND APPLIED MATHEMATICS 134 AN INTRODUCTION TO SEMIFLOWS ALBERT J. MILANI NORBERT J. KOKSCH CHAPMAN & HALL/CRC A CRC PRESS COMPANY BOCA RATON LONDON NEW YORK WASHINGTON, D.C. CONTENTS 1 DYNAMICAL PROCESSES 1 1.1 INTRODUCTION 1 1.2 ORDINARY DIFFERENTIAL EQUATIONS 4 1.2.1 WELL-POSEDNESS 6 1.2.2 REGULAR AND CHAOTIC SYSTEMS 8 1.2.3 DEPENDENCE ON PARAMETERS 10 1.2.4 AUTONOMOUS EQUATIONS 11 1.3 ATTRACTING SETS 16 1.4 ITERATED SEQUENCES 20 1.4.1 POINCARE MAPS 22 1.4.2 BERNOULLI S SEQUENCES 24 1.4.3 TENT MAPS 27 1.4.4 LOGISTIC MAPS 30 1.5 LORENZ EQUATIONS 32 1.5.1 THE DIFFERENTIAL SYSTEM 33 1.5.2 EQUILIBRIUM POINTS 33 1.6 DUFFING S EQUATION 36 1.6.1 THE GENERAL MODEL 36 1.6.2 A LINEARIZED MODEL : 38 1.7 SUMMARY 40 2 ATTRACTORS OF SEMIFLOWS 41 2.1 DISTANCE AND SEMIDISTANCE 41 2.2 DISCRETE AND CONTINUOUS SEMIFLOWS 42 2.2.1 TYPES OF SEMIFLOWS . . : ; 42 2.2.2 EXAMPLE: LORENZ EQUATIONS 46 2.3 INVARIANT SETS 49 2.3.1 ORBITS 50 2.3.2 LIMIT SETS 52 2.3.3 STABILITY OF STATIONARY POINTS 55 2.3.4 INVARIANCE OF ORBITS AND O)-LIMIT SETS 59 2.4 ATTRACTORS 64 2.4.1 ATTRACTING SETS 64 2.4.2 GLOBAL ATTRACTORS 65 2.4.3 COMPACTNESS 67 2.5 DISSIPATIVITY 68 2.6 ABSORBING SETS AND ATTRACTORS 70 VN VIA 2.6.1 ATTRACTORS OF COMPACT SEMIFLOWS 70 2.6.2 A GENERALIZATION 72 2.7 ATTRACTORS VIA A-CONTRACTIONS 73 2.7.1 MEASURING NONCOMPACTNESS 74 2.7.2 A ROUTE TO A-CONTRACTIONS 79 2.8 FRACTAL DIMENSION 81 2.9 A PRIORI ESTIMATES 83 2.9.1 INTEGRAL AND DIFFERENTIAL INEQUALITIES * . 84 2.9.2 EXPONENTIAL INEQUALITY 86 ATTRACTORS FOR SEMILINEAR EVOLUTION EQUATIONS 89 3.1 PDEES AS DYNAMICAL SYSTEMS 89 3.1.1 THE MODEL IBV PROBLEMS 89 3.1.2 CONSTRUCTION OF THE ATTRACTORS 92 3.2 FUNCTIONAL FRAMEWORK 95 3.2.1 FUNCTION SPACES 95 3.2.2 ORTHOGONAL BASES 97 3.2.3 FINITE DIMENSIONAL SUBSPACES 98 3.3 THE PARABOLIC PROBLEM 99 3.3.1 STEP 1: THE SOLUTION OPERATOR 100 3.3.2 STEP 2: ABSORBING SETS 105 3.3.3 STEP 3: COMPACTNESS OF THE SOLUTION OPERATOR 106 3.3.4 STEP 4: CONCLUSION 107 3.3.5 BACKWARD UNIQUENESS 108 3.4 THE HYPERBOLIC PROBLEM ILL 3.4.1 STEP 1: THE SOLUTION OPERATOR 112 3.4.2 STEP 2: ABSORBING SETS 114 3.4.3 STEP 3: COMPACTNESS OF THE SOLUTION OPERATOR 116 3.4.4 STEP 4: CONCLUSION 121 3.4.5 ATTRACTORS VIA A-CONTRACTIONS 121 3.5 REGULARITY 123 3.6 UPPER SEMICONTINUITY OF THE GLOBAL ATTRACTORS 132 EXPONENTIAL ATTRACTORS 135 4.1 INTRODUCTION 135 4.2 THE DISCRETE SQUEEZING PROPERTY 137 4.2.1 ORTHOGONAL PROJECTIONS 137 4.2.2 SQUEEZING PROPERTIES 138 4.2.3 SQUEEZING PROPERTIES AND EXPONENTIAL ATTRACTORS 139 4.2.4 PROOF OF THEOREM 4.5 141 4.3 THE PARABOLIC PROBLEM 143 4.3.1 STEP 1: ABSORBING SETS IN X 143 4.3.2 STEP 2: THE DISCRETE SQUEEZING PROPERTY 144 4.4 THE HYPERBOLIC PROBLEM 147 4.4.1 STEP 1: ABSORBING SETS IN X 147 IX 4.4.2 STEP 2: THE DISCRETE SQUEEZING PROPERTY 149 4.5 PROOF OF THEOREM 4.4 152 4.5.1 OUTLINE 152 4.5.2 THE CONE PROPERTY . . . . . . . : 153 4.5.3 THE BASIC COVERING STEP 155 4.5.4 THE FIRST AND SECOND ITERATES 159 4.5.5 THE GENERAL ITERATE 161 4.5.6 CONCLUSION 162 4.6 CONCLUDING REMARKS 175 INERTIAL MANIFOLDS 177 5.1 INTRODUCTION 177 5.2 DEFINITIONS AND COMPARISONS 179 5.2.1 LIPSCHITZ MANIFOLDS AND INERTIAL MANIFOLDS 179 5.2.2 INERTIAL MANIFOLDS AND EXPONENTIAL ATTRACTORS 183 5.2.3 METHODS OF CONSTRUCTION OF THE INERTIAL MANIFOLD 185 5.3 GEOMETRIC ASSUMPTIONS ON THE SEMIFLOW 189 5.3.1 THE CONE INVARIANCE PROPERTY 189 5.3.2 DECAY AND SQUEEZING PROPERTIES 191 5.3.3 CONSEQUENCES OF THE DECAY PROPERTY 193 5.4 STRONG SQUEEZING PROPERTY AND INERTIAL MANIFOLDS 195 5.4.1 SURJECTIVITY AND UNIFORM BOUNDEDNESS 195 5.4.2 CONSTRUCTION OF THE INERTIAL MANIFOLD , 197 5.5 A MODIFICATION 201 5.5.1 THE MODIFIED STRONG SQUEEZING PROPERTY 201 5.5.2 CONSEQUENCES OF THE MODIFIED STRONG SQUEEZING PROPERTY . 203 5.5.3 CONSTRUCTION OF THE INERTIAL MANIFOLD, 2 204 5.5.4 COMPARISON OF THE SQUEEZING PROPERTIES 206 5.6 INERTIAL MANIFOLDS FOR EVOLUTION;EQUATIONS 208 5.6.1 THE EVOLUTION PROBLEM : 208 5.6.2 THE SPECTRAL GAP CONDITION 209 5.6.3 THE STRONG SQUEEZING PROPERTIES 212 5.6.4 UNIFORM BOUNDEDNESS AND SURJECTIVITY 215 5.7 APPLICATIONS , 218 5.7.1 SEMILINEAR HEAT EQUATIONS 219 5.7.2 SEMILINEAR WAVE EQUATIONS 220 5.8 SEMILINEAR EVOLUTION EQUATIONS IN ONE SPACE DIMENSION 229 5.8.1 THE PARABOLIC PROBLEM 229 5.8.2 ABSORBING SETS 230 5.8.3 ADJUSTING THE NONLINEARITY 234 5.8.4 THE INERTIAL MANIFOLD 235 5.8.5 THE HYPERBOLIC PERTURBATION * 238 5.8.6 CONCLUDING REMARKS 239 6 EXAMPLES 241 6.1 CAHN-HILLIARD EQUATIONS 241 6.1.1 INTRODUCTION 242 6.1.2 THE CAHN-HILLIARD SEMIFLOWS 244 6.1.3 ABSORBING SETS 247 6.1.4 THE GLOBAL ATTRACTOR 250 6.1.5 THE EXPONENTIAL ATTRACTOR 251 6.1.6 THE INERTIAL MANIFOLD 254 6.2 BEAM AND VON KARMAN EQUATION 261 6.2.1 FUNCTIONAL FRAMEWORK AND NOTATIONS 261 6.2.2 THE BEAM EQUATION SEMIFLOW 262 6.2.3 ABSORBING SETS 263 6.2.4 THE GLOBAL ATTRACTOR 267 6.2.5 THE EXPONENTIAL ATTRACTOR 269 6.2.6 INERTIAL MANIFOLD 271 6.2.7 VON KARMAN EQUATIONS 272 6.3 NAVIER-STOKES EQUATIONS 272 6.3.1 THE EQUATIONS AND THEIR FUNCTIONAL FRAMEWORK 272 6.3.2 THE 2-DIMENSIONAL NAVIER-STOKES SEMIFLOW 275 6.3.3 ABSORBING SETS AND ATTRACTOR 276 6.3.4 THE EXPONENTIAL ATTRACTOR 278 6.4 MAXWELL S EQUATIONS 280 6.4.1 THE EQUATIONS AND THEIR FUNCTIONAL FRAMEWORK 281 6.4.2 THE QUASI-STATIONARY MAXWELL SEMIFLOW 285 6.4.3 ABSORBING SETS AND ATTRACTORS 288 7 A NONEXISTENCE RESULT FOR INERTIAL MANIFOLDS 291 7.1 THE INITIAL-BOUNDARY VALUE PROBLEM 291 7.2 OVERVIEW OF THE ARGUMENT 293 7.3 THE LINEARIZED PROBLEM 295 7.4 INERTIAL MANIFOLDS FOR THE LINEARIZED PROBLEM 298 7.5 C 1 LINEARIZATION EQUIVALENCE 303 7.6 PERTURBATIONS OF THE NONLINEAR FLOW 304 7.7 ASYMPTOTIC PROPERTIES OF THE PERTURBED FLOW 307 7.8 THE NONEXISTENCE RESULT 309 7.9 PROOF OF PROPOSITION 7.17 310 7.10 THE C 1 LINEARIZATION EQUIVALENCE THEOREMS 316 7.10.1 EQUIVALENCE FOR A SINGLE OPERATOR 316 7.10.2 EQUIVALENCE FOR GROUPS OF OPERATORS 320 APPENDIX: SELECTED RESULTS FROM ANALYSIS 323 A.I ORDINARY DIFFERENTIAL EQUATIONS 323 A.1.1 CLASSICAL SOLUTIONS 323 A.1.2 GENERALIZED SOLUTIONS 324 A. 1.3 STABILITY FOR AUTONOMOUS SYSTEMS 325 XI A.2 LINEAR SPACES AND THEIR DUALS 328 A.2.1 ORTHONORMAL BASES IN HILBERT SPACES 328 A.2.2 DUAL SPACES AND THE HAHN-BANACH THEOREM 329 A.2.3 LINEAR OPERATORS IN BANACH SPACES 330 A.2.4 ADJOINT OF A BOUNDED OPERATOR 333 A.2.5 ADJOINT OF AN UNBOUNDED OPERATOR 335 A.2.6 GELFAND TRIPLES OF HILBERT SPACES 335 A.2.7 LINEAR OPERATORS IN GELFAND TRIPLES 336 A.2.8 EIGENVALUES OF COMPACT OPERATORS 338 A.2.9 FRACTIONAL POWERS OF POSITIVE OPERATORS 340 A.2.10 INTERPOLATION SPACES . 342 A.2.11 DIFFERENTIAL CALCULUS IN BANACH SPACES 344 A.3 SEMIGROUPS OF LINEAR OPERATORS 345 A.3.1 GENERAL RESULTS 345 A.3.2 APPLICATIONS TO PDES 347 A.4 LEBESGUE SPACES 349 A.4.1 THE SPACES I/(FT) 349 A.4.2 INEQUALITIES 350 A.4.3 OTHER PROPERTIES OF THE SPACES LP(Q) 351 A.5 SOBOLEV SPACES OF SCALAR VALUED FUNCTIONS 352 A.5.1 DISTRIBUTIONS IN FT 353 A.5.2 THESPACESH M (FT),MN 353 A.5.3 THE SPACES H S (I2),^EM O 355 A.5.4 THE SPACES H !O (^), J 6 R 0 , AND H J (FT),.? K 0 357 A.5.5 THE LAPLACE OPERATOR 358 A.6 SOBOLEV SPACES OF VECTOR VALUED FUNCTIONS 361 A.6.1 LEBESGUE AND SOBOLEV SPACES 361 A.6.2 THE INTERMEDIATE DERIVATIVES THEOREM 362 A.7 THE SPACES H(DIV,FT) AND H(CURL,FT) 363 A.7.1 NOTATIONS 364 A.7.2 THE SPACE H(DIV,FT) . : 364 A.7.3 THE SPACE H(CURL, FT) . . : 365 A.7.4 RELATIONS BETWEEN H(DIV,FT) AND H(CURL,FT) 367 A.8 ALMOST PERIODIC FUNCTIONS 371 BIBLIOGRAPHY 373 INDEX 381 NOMENCLATURE 385
adam_txt	K CHAPMAN & HALL/CRC MONOGRAPHS AND SURVEYS IN PURE AND APPLIED MATHEMATICS 134 AN INTRODUCTION TO SEMIFLOWS ALBERT J. MILANI NORBERT J. KOKSCH CHAPMAN & HALL/CRC A CRC PRESS COMPANY BOCA RATON LONDON NEW YORK WASHINGTON, D.C. CONTENTS 1 DYNAMICAL PROCESSES 1 1.1 INTRODUCTION 1 1.2 ORDINARY DIFFERENTIAL EQUATIONS 4 1.2.1 WELL-POSEDNESS 6 1.2.2 REGULAR AND CHAOTIC SYSTEMS 8 1.2.3 DEPENDENCE ON PARAMETERS 10 1.2.4 AUTONOMOUS EQUATIONS 11 1.3 ATTRACTING SETS 16 1.4 ITERATED SEQUENCES 20 1.4.1 POINCARE MAPS 22 1.4.2 BERNOULLI'S SEQUENCES 24 1.4.3 TENT MAPS 27 1.4.4 LOGISTIC MAPS 30 1.5 LORENZ' EQUATIONS 32 1.5.1 THE DIFFERENTIAL SYSTEM 33 1.5.2 EQUILIBRIUM POINTS 33 1.6 DUFFING'S EQUATION 36 1.6.1 THE GENERAL MODEL 36 1.6.2 A LINEARIZED MODEL : 38 1.7 SUMMARY 40 2 ATTRACTORS OF SEMIFLOWS 41 2.1 DISTANCE AND SEMIDISTANCE 41 2.2 DISCRETE AND CONTINUOUS SEMIFLOWS 42 2.2.1 TYPES OF SEMIFLOWS . . : ; 42 2.2.2 EXAMPLE: LORENZ' EQUATIONS 46 2.3 INVARIANT SETS 49 2.3.1 ORBITS 50 2.3.2 LIMIT SETS 52 2.3.3 STABILITY OF STATIONARY POINTS 55 2.3.4 INVARIANCE OF ORBITS AND O)-LIMIT SETS 59 2.4 ATTRACTORS 64 2.4.1 ATTRACTING SETS 64 2.4.2 GLOBAL ATTRACTORS 65 2.4.3 COMPACTNESS 67 2.5 DISSIPATIVITY 68 2.6 ABSORBING SETS AND ATTRACTORS 70 VN VIA 2.6.1 ATTRACTORS OF COMPACT SEMIFLOWS 70 2.6.2 A GENERALIZATION 72 2.7 ATTRACTORS VIA A-CONTRACTIONS 73 2.7.1 MEASURING NONCOMPACTNESS 74 2.7.2 A ROUTE TO A-CONTRACTIONS 79 2.8 FRACTAL DIMENSION 81 2.9 A PRIORI ESTIMATES 83 2.9.1 INTEGRAL AND DIFFERENTIAL INEQUALITIES * . 84 2.9.2 EXPONENTIAL INEQUALITY 86 ATTRACTORS FOR SEMILINEAR EVOLUTION EQUATIONS 89 3.1 PDEES AS DYNAMICAL SYSTEMS 89 3.1.1 THE MODEL IBV PROBLEMS 89 3.1.2 CONSTRUCTION OF THE ATTRACTORS 92 3.2 FUNCTIONAL FRAMEWORK 95 3.2.1 FUNCTION SPACES 95 3.2.2 ORTHOGONAL BASES 97 3.2.3 FINITE DIMENSIONAL SUBSPACES 98 3.3 THE PARABOLIC PROBLEM 99 3.3.1 STEP 1: THE SOLUTION OPERATOR 100 3.3.2 STEP 2: ABSORBING SETS 105 3.3.3 STEP 3: COMPACTNESS OF THE SOLUTION OPERATOR 106 3.3.4 STEP 4: CONCLUSION 107 3.3.5 BACKWARD UNIQUENESS 108 3.4 THE HYPERBOLIC PROBLEM ILL 3.4.1 STEP 1: THE SOLUTION OPERATOR 112 3.4.2 STEP 2: ABSORBING SETS 114 3.4.3 STEP 3: COMPACTNESS OF THE SOLUTION OPERATOR 116 3.4.4 STEP 4: CONCLUSION 121 3.4.5 ATTRACTORS VIA A-CONTRACTIONS 121 3.5 REGULARITY 123 3.6 UPPER SEMICONTINUITY OF THE GLOBAL ATTRACTORS 132 EXPONENTIAL ATTRACTORS 135 4.1 INTRODUCTION 135 4.2 THE DISCRETE SQUEEZING PROPERTY 137 4.2.1 ORTHOGONAL PROJECTIONS 137 4.2.2 SQUEEZING PROPERTIES 138 4.2.3 SQUEEZING PROPERTIES AND EXPONENTIAL ATTRACTORS 139 4.2.4 PROOF OF THEOREM 4.5 141 4.3 THE PARABOLIC PROBLEM 143 4.3.1 STEP 1: ABSORBING SETS IN X\ 143 4.3.2 STEP 2: THE DISCRETE SQUEEZING PROPERTY 144 4.4 THE HYPERBOLIC PROBLEM 147 4.4.1 STEP 1: ABSORBING SETS IN X\ 147 IX 4.4.2 STEP 2: THE DISCRETE SQUEEZING PROPERTY 149 4.5 PROOF OF THEOREM 4.4 152 4.5.1 OUTLINE 152 4.5.2 THE CONE PROPERTY . . . . . . .': 153 4.5.3 THE BASIC COVERING STEP 155 4.5.4 THE FIRST AND SECOND ITERATES 159 4.5.5 THE GENERAL ITERATE 161 4.5.6 CONCLUSION 162 4.6 CONCLUDING REMARKS 175 INERTIAL MANIFOLDS 177 5.1 INTRODUCTION 177 5.2 DEFINITIONS AND COMPARISONS 179 5.2.1 LIPSCHITZ MANIFOLDS AND INERTIAL MANIFOLDS 179 5.2.2 INERTIAL MANIFOLDS AND EXPONENTIAL ATTRACTORS 183 5.2.3 METHODS OF CONSTRUCTION OF THE INERTIAL MANIFOLD 185 5.3 GEOMETRIC ASSUMPTIONS ON THE SEMIFLOW 189 5.3.1 THE CONE INVARIANCE PROPERTY 189 5.3.2 DECAY AND SQUEEZING PROPERTIES 191 5.3.3 CONSEQUENCES OF THE DECAY PROPERTY 193 5.4 STRONG SQUEEZING PROPERTY AND INERTIAL MANIFOLDS 195 5.4.1 SURJECTIVITY AND UNIFORM BOUNDEDNESS 195 5.4.2 CONSTRUCTION OF THE INERTIAL MANIFOLD , 197 5.5 A MODIFICATION 201 5.5.1 THE MODIFIED STRONG SQUEEZING PROPERTY 201 5.5.2 CONSEQUENCES OF THE MODIFIED STRONG SQUEEZING PROPERTY . 203 5.5.3 CONSTRUCTION OF THE INERTIAL MANIFOLD, 2 204 5.5.4 COMPARISON OF THE SQUEEZING PROPERTIES 206 5.6 INERTIAL MANIFOLDS FOR EVOLUTION;EQUATIONS 208 5.6.1 THE EVOLUTION PROBLEM : 208 5.6.2 THE SPECTRAL GAP CONDITION 209 5.6.3 THE STRONG SQUEEZING PROPERTIES 212 5.6.4 UNIFORM BOUNDEDNESS AND SURJECTIVITY 215 5.7 APPLICATIONS , 218 5.7.1 SEMILINEAR HEAT EQUATIONS 219 5.7.2 SEMILINEAR WAVE EQUATIONS 220 5.8 SEMILINEAR EVOLUTION EQUATIONS IN ONE SPACE DIMENSION 229 5.8.1 THE PARABOLIC PROBLEM 229 5.8.2 ABSORBING SETS 230 5.8.3 ADJUSTING THE NONLINEARITY 234 5.8.4 THE INERTIAL MANIFOLD 235 5.8.5 THE HYPERBOLIC PERTURBATION * 238 5.8.6 CONCLUDING REMARKS 239 6 EXAMPLES 241 6.1 CAHN-HILLIARD EQUATIONS 241 6.1.1 INTRODUCTION 242 6.1.2 THE CAHN-HILLIARD SEMIFLOWS 244 6.1.3 ABSORBING SETS 247 6.1.4 THE GLOBAL ATTRACTOR 250 6.1.5 THE EXPONENTIAL ATTRACTOR 251 6.1.6 THE INERTIAL MANIFOLD 254 6.2 BEAM AND VON KARMAN EQUATION 261 6.2.1 FUNCTIONAL FRAMEWORK AND NOTATIONS 261 6.2.2 THE BEAM EQUATION SEMIFLOW 262 6.2.3 ABSORBING SETS 263 6.2.4 THE GLOBAL ATTRACTOR 267 6.2.5 THE EXPONENTIAL ATTRACTOR 269 6.2.6 INERTIAL MANIFOLD 271 6.2.7 VON KARMAN EQUATIONS 272 6.3 NAVIER-STOKES EQUATIONS 272 6.3.1 THE EQUATIONS AND THEIR FUNCTIONAL FRAMEWORK 272 6.3.2 THE 2-DIMENSIONAL NAVIER-STOKES SEMIFLOW 275 6.3.3 ABSORBING SETS AND ATTRACTOR 276 6.3.4 THE EXPONENTIAL ATTRACTOR 278 6.4 MAXWELL'S EQUATIONS 280 6.4.1 THE EQUATIONS AND THEIR FUNCTIONAL FRAMEWORK 281 6.4.2 THE QUASI-STATIONARY MAXWELL SEMIFLOW 285 6.4.3 ABSORBING SETS AND ATTRACTORS 288 7 A NONEXISTENCE RESULT FOR INERTIAL MANIFOLDS 291 7.1 THE INITIAL-BOUNDARY VALUE PROBLEM 291 7.2 OVERVIEW OF THE ARGUMENT 293 7.3 THE LINEARIZED PROBLEM '295 7.4 INERTIAL MANIFOLDS FOR THE LINEARIZED PROBLEM 298 7.5 C 1 LINEARIZATION EQUIVALENCE 303 7.6 PERTURBATIONS OF THE NONLINEAR FLOW 304 7.7 ASYMPTOTIC PROPERTIES OF THE PERTURBED FLOW 307 7.8 THE NONEXISTENCE RESULT 309 7.9 PROOF OF PROPOSITION 7.17 310 7.10 THE C 1 LINEARIZATION EQUIVALENCE THEOREMS 316 7.10.1 EQUIVALENCE FOR A SINGLE OPERATOR 316 7.10.2 EQUIVALENCE FOR GROUPS OF OPERATORS 320 APPENDIX: SELECTED RESULTS FROM ANALYSIS 323 A.I ORDINARY DIFFERENTIAL EQUATIONS 323 A.1.1 CLASSICAL SOLUTIONS 323 A.1.2 GENERALIZED SOLUTIONS 324 A. 1.3 STABILITY FOR AUTONOMOUS SYSTEMS 325 XI A.2 LINEAR SPACES AND THEIR DUALS 328 A.2.1 ORTHONORMAL BASES IN HILBERT SPACES 328 A.2.2 DUAL SPACES AND THE HAHN-BANACH THEOREM 329 A.2.3 LINEAR OPERATORS IN BANACH SPACES 330 A.2.4 ADJOINT OF A BOUNDED OPERATOR 333 A.2.5 ADJOINT OF AN UNBOUNDED OPERATOR 335 A.2.6 GELFAND TRIPLES OF HILBERT SPACES 335 A.2.7 LINEAR OPERATORS IN GELFAND TRIPLES 336 A.2.8 EIGENVALUES OF COMPACT OPERATORS 338 A.2.9 FRACTIONAL POWERS OF POSITIVE OPERATORS 340 A.2.10 INTERPOLATION SPACES . 342 A.2.11 DIFFERENTIAL CALCULUS IN BANACH SPACES 344 A.3 SEMIGROUPS OF LINEAR OPERATORS 345 A.3.1 GENERAL RESULTS 345 A.3.2 APPLICATIONS TO PDES 347 A.4 LEBESGUE SPACES 349 A.4.1 THE SPACES I/(FT) 349 A.4.2 INEQUALITIES 350 A.4.3 OTHER PROPERTIES OF THE SPACES LP(Q) 351 A.5 SOBOLEV SPACES OF SCALAR VALUED FUNCTIONS 352 A.5.1 DISTRIBUTIONS IN FT 353 A.5.2 THESPACESH M (FT),MN 353 A.5.3 THE SPACES H S (I2),^EM O 355 A.5.4 THE SPACES H !O (^), J 6 R 0 , AND H J (FT),.? K 0 357 A.5.5 THE LAPLACE OPERATOR 358 A.6 SOBOLEV SPACES OF VECTOR VALUED FUNCTIONS 361 A.6.1 LEBESGUE AND SOBOLEV SPACES 361 A.6.2 THE INTERMEDIATE DERIVATIVES THEOREM 362 A.7 THE SPACES H(DIV,FT) AND H(CURL,FT) 363 A.7.1 NOTATIONS \ 364 A.7.2 THE SPACE H(DIV,FT) . : 364 A.7.3 THE SPACE H(CURL, FT) . . : 365 A.7.4 RELATIONS BETWEEN H(DIV,FT) AND H(CURL,FT) 367 A.8 ALMOST PERIODIC FUNCTIONS 371 BIBLIOGRAPHY 373 INDEX 381 NOMENCLATURE 385
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id	DE-604.BV020849541
illustrated	Illustrated
index_date	2024-07-02T13:19:24Z
indexdate	2024-07-09T20:26:35Z
institution	BVB
isbn	1584884584
language	English
lccn	2004055145
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-014171271
oclc_num	55765524
open_access_boolean
owner	DE-384 DE-11 DE-188
owner_facet	DE-384 DE-11 DE-188
physical	xvi, 386 p. graph. Darst. 25 cm
publishDate	2005
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publisher	Chapman & Hall/CRC
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series	Chapman & Hall/CRC monographs and surveys in pure and applied mathematics
series2	Chapman & Hall/CRC monographs and surveys in pure and applied mathematics
spelling	Milani, Albert Verfasser (DE-588)1027129056 aut An introduction to semiflows Albert J. Milani ; Norbert J. Koksch Boca Raton u.a. Chapman & Hall/CRC 2005 xvi, 386 p. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC monographs and surveys in pure and applied mathematics 134 Includes bibliographical references (p. 373-380) and index Flots (Dynamique différentiable) Flows (Differentiable dynamical systems) Fluss Mathematik (DE-588)4489499-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Fluss Mathematik (DE-588)4489499-5 s Dynamisches System (DE-588)4013396-5 s DE-604 Koksch, Norbert 1962- Verfasser (DE-588)121901548 aut Chapman & Hall/CRC monographs and surveys in pure and applied mathematics 134 (DE-604)BV013350872 134 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014171271&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Milani, Albert Koksch, Norbert 1962- An introduction to semiflows Chapman & Hall/CRC monographs and surveys in pure and applied mathematics Flots (Dynamique différentiable) Flows (Differentiable dynamical systems) Fluss Mathematik (DE-588)4489499-5 gnd Dynamisches System (DE-588)4013396-5 gnd
subject_GND	(DE-588)4489499-5 (DE-588)4013396-5
title	An introduction to semiflows
title_auth	An introduction to semiflows
title_exact_search	An introduction to semiflows
title_exact_search_txtP	An introduction to semiflows
title_full	An introduction to semiflows Albert J. Milani ; Norbert J. Koksch
title_fullStr	An introduction to semiflows Albert J. Milani ; Norbert J. Koksch
title_full_unstemmed	An introduction to semiflows Albert J. Milani ; Norbert J. Koksch
title_short	An introduction to semiflows
title_sort	an introduction to semiflows
topic	Flots (Dynamique différentiable) Flows (Differentiable dynamical systems) Fluss Mathematik (DE-588)4489499-5 gnd Dynamisches System (DE-588)4013396-5 gnd
topic_facet	Flots (Dynamique différentiable) Flows (Differentiable dynamical systems) Fluss Mathematik Dynamisches System
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014171271&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV013350872
work_keys_str_mv	AT milanialbert anintroductiontosemiflows AT kokschnorbert anintroductiontosemiflows

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