Linear and nonlinear aspects of vortices: the Ginzburg-Landau model
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2000
|
| Series: | Progress in nonlinear differential equations and their applications
39 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 329 - 335 |
| Physical Description: | X, 342 S. |
| ISBN: | 3764341335 0817641335 |
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MARC
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| 245 | 1 | 0 | |a Linear and nonlinear aspects of vortices |b the Ginzburg-Landau model |c Frank Pacard ; Tristan Rivière |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2000 | |
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| 490 | 1 | |a Progress in nonlinear differential equations and their applications |v 39 | |
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Record in the Search Index
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| adam_text | Contents
Preface ix
1 Qualitative Aspects of Ginzburg Landau Equations 1
1.1 The integrable case 2
1.2 The strongly repulsive case 3
1.3 The existence result 12
1.4 Uniqueness results 16
2 Elliptic Operators in Weighted Holder Spaces 21
2.1 Function spaces 22
2.2 Mapping properties of the Laplacian 24
2.2.1 Rescaled Schauder estimates 26
2.2.2 Mapping properties of the Laplacian
in the injectivity range 28
2.2.3 Mapping properties of the Laplacian
in the surjectivity range 36
2.3 Applications to nonlinear problems 43
2.3.1 Minimal surfaces with one catenoidal type end 43
2.3.2 Semilinear elliptic equations with isolated singularities . . 46
2.3.3 Singular perturbations for the Liouville equation 48
3 The Ginzburg Landau Equation in C 51
3.1 Radially symmetric solution on C 51
3.2 The linearized operator about the radially symmetric solution ... 53
vi Contents
3.2.1 Definition 53
3.2.2 Explicit solutions of the homogeneous problem 53
3.3 Asymptotic behavior of solutions of the homogeneous problem . . 54
3.3.1 Classification of all possible asymptotic behaviors at 0 .. 54
3.3.2 Classification of all possible asymptotic behaviors at oo . 56
3.4 Bounded solution of the homogeneous problem 61
3.5 More solutions to the homogeneous equation 68
3.6 Introduction of the scaling factor 70
4 Mapping Properties of Ce 73
4.1 Consequences of the maximum principle in weighted spaces ... 73
4.1.1 Higher eigenfrequencies 74
4.1.2 Lower eigenfrequencies 81
4.2 Function spaces 84
4.3 A right inverse for Ce in Bi {0} 86
4.3.1 Higher eigenfrequencies 87
4.3.2 Lower eigenfrequencies 91
5 Families of Approximate Solutions with Prescribed Zero Set 103
5.1 The approximate solution u 103
5.1.1 Notation 103
5.1.2 The approximate solution near the zeros 105
5.1.3 The approximate solution away from the zeros Ill
5.2 A 3JV dimensional family of approximate solutions 116
5.2.1 Definition of the family of approximate solutions 116
5.3 Estimates 117
5.4 Appendix 121
6 The Linearized Operator about the Approximate Solution u 125
6.1 Definition 125
6.2 The interior problem 126
6.3 The exterior problem 131
6.4 Dirichlet to Neumann mappings 134
6.4.1 The interior Dirichlet to Neumann mapping 134
6.4.2 The exterior Dirichlet to Neumann mapping 139
6.4.3 Gluing together the two Dirichlet to Neumann mappings . 140
6.5 The linearized operator in all Q 145
6.6 Appendix 148
7 Existence of Ginzburg Landau Vortices 151
7.1 Statement of the result 151
7.2 The linear mapping DM (o.o.o) 155
7.3 Estimates of the nonlinear terms 156
7.3.1 Estimates of Q 156
7.3.2 Estimates of Qt 157
7.3.3 Estimates of Qz 158
Contents vii
7.4 The fixed point argument 162
7.5 Further information about the branch of solutions 162
8 Elliptic Operators in Weighted Sobolev Spaces 167
8.1 General overview 167
8.2 Estimates for the Laplacian 169
8.3 Estimates for some elliptic operator in divergence form 178
9 Generalized Pohozaev Formula for p Conformal Fields 191
9.1 The Pohozaev formula in the classical framework 191
9.2 Comparing Ginzburg Landau solutions
using Pohozaev s argument 194
9.2.1 Notation 195
9.2.2 The comparison argument in the case of radially
symmetric data 196
9.3 p conformal vector fields 198
9.4 Conservation laws 200
9.4.1 Comparing solutions through a Pohozaev type formula:
the general case 203
9.4.2 Conservation laws for Ginzburg Landau equation 203
9.4.3 The Pohozaev formula 205
9.4.4 Integration of the Pohozaev formula 208
9.5 Uniqueness results 211
9.5.1 A few uniqueness results 211
9.5.2 Uniqueness results for semilinear elliptic problems .... 213
9.6 Dealing with general nonlinearities 218
9.6.1 A Pohozaev formula for general nonlinearities 218
9.6.2 Uniqueness results for general nonlinearities 220
9.6.3 More about the quantities involved in the
Pohozaev identity 221
10 The Role of Zeros in the Uniqueness Question 225
10.1 The zero set of solutions of Ginzburg Landau equations 225
10.2 A uniqueness result 234
10.2.1 Preliminary results 235
10.2.2 The proof of Theorem 10.1 249
11 Solving Uniqueness Questions 253
11.1 Statement of the uniqueness result 253
11.2 Proof of the uniqueness result 254
11.2.1 Geometric modification of the family vE 254
11.2.2 Estimating the Z,2 norm of |«e| |5«| 257
11.2.3 Pointwise estimates for m£ — vs and ue — vs 268
11.2.4 Final arguments to prove that ue = vs 273
11.3 A conjecture of F. Bethuel, H. Brezis and F. Helein 277
viii Contents
12 Towards Jaffe and Taubes Conjectures 279
12.1 Statement of the result 279
12.1.1 Preliminary remarks 280
12.1.2 The uniqueness result 283
12.2 Gauge invariant Ginzburg Landau critical points with one zero . . 284
12.3 Proof of Theorem 12.2 297
12.3.1 The Coulomb gauge 297
12.3.2 Preliminary results 300
12.3.3 The Pohozaev formula 308
12.3.4 The end of the proof 310
References 329
Index of Notation 337
Index 341
|
| any_adam_object | 1 |
| author | Pacard, Frank Rivière, Tristan |
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| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.355 |
| dewey-sort | 3515 3355 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV013304354 |
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| indexdate | 2024-07-09T18:43:27Z |
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| language | English |
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| physical | X, 342 S. |
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| series | Progress in nonlinear differential equations and their applications |
| series2 | Progress in nonlinear differential equations and their applications |
| spelling | Pacard, Frank Verfasser aut Linear and nonlinear aspects of vortices the Ginzburg-Landau model Frank Pacard ; Tristan Rivière Boston [u.a.] Birkhäuser 2000 X, 342 S. txt rdacontent n rdamedia nc rdacarrier Progress in nonlinear differential equations and their applications 39 Literaturverz. S. 329 - 335 Tourbillons (mécanique des fluides) ram Équations différentielles ram Differential equations, Nonlinear Vortex-motion Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd rswk-swf Wirbel Physik (DE-588)4128386-7 gnd rswk-swf Ginzburg-Landau-Gleichung (DE-588)4157356-0 gnd rswk-swf Ginzburg-Landau-Gleichung (DE-588)4157356-0 s DE-604 Wirbel Physik (DE-588)4128386-7 s Nichtlineare Differentialgleichung (DE-588)4205536-2 s Rivière, Tristan Verfasser aut Progress in nonlinear differential equations and their applications 39 (DE-604)BV007934389 39 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009070814&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pacard, Frank Rivière, Tristan Linear and nonlinear aspects of vortices the Ginzburg-Landau model Progress in nonlinear differential equations and their applications Tourbillons (mécanique des fluides) ram Équations différentielles ram Differential equations, Nonlinear Vortex-motion Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Wirbel Physik (DE-588)4128386-7 gnd Ginzburg-Landau-Gleichung (DE-588)4157356-0 gnd |
| subject_GND | (DE-588)4205536-2 (DE-588)4128386-7 (DE-588)4157356-0 |
| title | Linear and nonlinear aspects of vortices the Ginzburg-Landau model |
| title_auth | Linear and nonlinear aspects of vortices the Ginzburg-Landau model |
| title_exact_search | Linear and nonlinear aspects of vortices the Ginzburg-Landau model |
| title_full | Linear and nonlinear aspects of vortices the Ginzburg-Landau model Frank Pacard ; Tristan Rivière |
| title_fullStr | Linear and nonlinear aspects of vortices the Ginzburg-Landau model Frank Pacard ; Tristan Rivière |
| title_full_unstemmed | Linear and nonlinear aspects of vortices the Ginzburg-Landau model Frank Pacard ; Tristan Rivière |
| title_short | Linear and nonlinear aspects of vortices |
| title_sort | linear and nonlinear aspects of vortices the ginzburg landau model |
| title_sub | the Ginzburg-Landau model |
| topic | Tourbillons (mécanique des fluides) ram Équations différentielles ram Differential equations, Nonlinear Vortex-motion Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Wirbel Physik (DE-588)4128386-7 gnd Ginzburg-Landau-Gleichung (DE-588)4157356-0 gnd |
| topic_facet | Tourbillons (mécanique des fluides) Équations différentielles Differential equations, Nonlinear Vortex-motion Nichtlineare Differentialgleichung Wirbel Physik Ginzburg-Landau-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009070814&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV007934389 |
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