The Institute for Mathematical Sciences at the National University of Singapore hosted a research program on "Nanoscale Material Interfaces: Experiment, Theory and Simulation'' from November 2004 to January 2005. As part of the program, tutorials for graduate students and junior researchers were given by leading experts in the field. This invaluable volume collects the expanded lecture notes of four of those self-contained tutorials. The topics covered include dynamics in different models of domain coarsening and coagulation and their mathematical analysis in material sciences; a mathematical and computational study for quantized vortices in the celebrated Ginzburg Landau models of superconductivity and the mean field Gross Pitaevskii equations of superfluidity; the nonlinear Schrödinger equation and applications in Bose Einstein condensation and plasma physics as well as their efficient and accurate computation; and finally, an introduction to constitutive modeling of macromolecular fluids within the framework of the kinetic theory. This volume serves to inspire graduate students and researchers who will embark upon original research work in these fields.
Author(s): Weizhu Bao, Jian-Guo Liu
Year: 2007
Language: English
Pages: 308
CONTENTS......Page 6
Foreword......Page 8
Preface......Page 10
Contents......Page 14
1.1.......Page 15
1.2. First models of domain formation and an open problem.......Page 16
2.1. Domain walls in the Allen-Cahn equation......Page 18
2.2. Domain wall dynamics by restricted gradient flow......Page 20
2.3. Punctuated equilibrium and 1D bubble bath......Page 24
2.4. Mean-field model of domain growth — The Gallay-Mielke transform......Page 26
2.5. Proof of universal self-similar behavior......Page 32
3. Models of domain coarsening in two and three dimensions......Page 36
3.1. Diffuse and sharp-interface models of nanoscale island coarsening......Page 37
3.2. Gradient structure for Mullins-Sekerka flow......Page 44
3.3. Monopole models by restricted gradient flow of surface energy......Page 46
3.4. Lifshitz-Slyozov-Wagner mean-field model......Page 48
4. Rigorous power-law bounds on coarsening rates — The Kohn-Otto method......Page 53
4.1. Basic inequalities......Page 54
4.2. Bounds on coarsening rates for the LSW mean-field model......Page 56
4.3. Bounds on coarsening rates for the monopole model......Page 57
5.2. A ‘new’ framework for dynamic scaling analysis......Page 60
5.3. Solution by Laplace transform......Page 62
5.4. Scaling solutions and domains of attraction......Page 64
5.5. The scaling attractor......Page 67
5.6. Linearization of dynamics on the scaling attractor......Page 70
References......Page 72
Contents......Page 76
1. Introduction......Page 77
2.1. What is superconductivity?......Page 79
2.2. Type-II superconductors and the vortex state......Page 81
2.3. Applications of superconductivity......Page 84
2.4. Superconductivity models and mathematical problems......Page 85
3. The mathematical theory of Ginzburg-Landau models......Page 86
3.1. The free energy postulated by Ginzburg and Landau......Page 87
3.2. The equilibrium Ginzburg-Landau models......Page 88
3.3. Time dependent Ginzburg-Landau equations......Page 90
3.4. Gauge invariance and some basic theory......Page 91
4. Numerical algorithms for Ginzburg-Landau models......Page 93
4.1. Finite element approximations......Page 95
4.2. Finite difference approximations......Page 96
4.3. Finite volume approximations......Page 98
4.4. Artificial boundary conditions......Page 101
4.5. More on time-discretization......Page 103
4.6. Multi-level, adaptive and parallel algorithms......Page 104
5.1. Phase diagrams and equilibrium solution branch......Page 105
5.2. Vortex solutions......Page 106
5.3. A rigorous result on vortex nucleation near HC1......Page 109
5.4. Effect due to spatial inhomogeneities......Page 110
6. Dynamics of quantized vortices......Page 114
6.2. Dynamics of individual vortices......Page 115
6.3. High-κ, high field dynamics......Page 116
6.4. Dynamics involving spatial inhomogeneities......Page 117
6.5. Dynamics driven by the applied current......Page 118
6.6. Vortex state in a thin superconducting spherical shell......Page 120
6.7. Stochastic dynamics driven by noises......Page 123
6.8. Variants of G-L models: Lawrence-Doniach and d-wave models......Page 126
6.9. Vortex density models......Page 129
7. The vortex state in the Bose-Einstein condensation......Page 133
7.1. Vortices in BEC confined in a rotating magnetic trap......Page 134
7.2. Vortex shedding behind a stirring laser beam......Page 138
8. Future challenges......Page 139
9. Conclusion......Page 143
References......Page 144
Contents......Page 154
1. Introduction......Page 156
2. Derivation of NLSE from wave propagation......Page 157
3. Derivation of NLSE from BEC......Page 159
3.2. Reduction to lower dimension......Page 161
4.1. Conservation laws......Page 163
4.2. Lagrangian structure......Page 164
4.3. Hamiltonian structure......Page 165
4.4. Variance identity......Page 166
5. Plane and solitary wave solutions of NLSE......Page 170
6. Existence/blowup results of NLSE......Page 171
6.2. Existence results......Page 172
6.3. Finite time blowup results......Page 173
7. WKB expansion and quantum hydrodynamics......Page 174
8. Wigner transform and semiclassical limit......Page 175
9. Ground, excited and central vortex states of GPE......Page 177
9.2. Ground state......Page 178
9.3. Central vortex states......Page 180
9.4. Variation of stationary states over the unit sphere......Page 181
9.5. Conservation of angular momentum expectation......Page 182
10.2. Energy diminishing of GFDN......Page 184
10.3. Continuous normalized gradient flow (CNGF)......Page 186
10.4. Semi-implicit time discretization......Page 187
10.5. Discretized normalized gradient flow (DNGF)......Page 190
10.6. Numerical methods......Page 191
10.7. Energy diminishing of DNGF......Page 194
10.8. Numerical results......Page 196
11.1. General high-order split-step method......Page 202
11.2. Fourth-order TSSP for GPE without external driving field......Page 203
11.4. Stability......Page 205
11.6. Numerical results......Page 208
12. Derivation of the vector Zakharov system......Page 214
13. Generalization and simplification of ZS......Page 218
13.1. Reduction from VZSM to GVZS......Page 219
13.2. Reduction from GVZS to GZS......Page 221
13.3. Reduction from GVZS to VNLS......Page 223
13.4. Reduction from GZS to NLSE......Page 224
14. Well-posedness of ZS......Page 225
15. Plane wave and soliton wave solutions of ZS......Page 226
16. Time-splitting spectral method for GZS......Page 227
16.1. Crank-Nicolson leap-frog time-splitting spectral discretizations (CN-LF-TSSP) for GZS......Page 229
16.2. Phase space analytical solver + time-splitting spectral discretizations (PSAS-TSSP)......Page 231
16.3. Properties of the numerical methods......Page 234
16.4. Extension TSSP to GVZS......Page 237
17. Crank-Nicolson finite difference (CNFD) method for GZS......Page 239
18. Numerical results of GZS......Page 240
References......Page 246
Contents......Page 254
1. Introduction......Page 255
2. Introduction to equilibrium thermodynamics......Page 256
3. Introduction to statistical mechanics......Page 263
4. Introduction to continuum mechanics......Page 269
4.1. Material, referential, and spatial description of motion, and deformation tensors......Page 270
4.2.1. Line element......Page 275
4.2.3. Volume element......Page 276
4.2.4. Material derivative......Page 277
4.2.5. Transport theorems......Page 278
4.3.1. Eulerian description......Page 279
4.3.2. Lagrangian description......Page 280
4.4. Superimposed rigid body motion (SRBM) and invariant principles......Page 281
4.5. Invariant time derivatives......Page 282
4.6. Material symmetry......Page 283
4.7. Clausius-Duhem inequality......Page 284
5. Some constitutive models for .exible polymers......Page 288
6.1. Equilibrium distribution of the end-to-end vector in simple polymer models......Page 292
6.2. Flory-Huggins Theory......Page 294
7. Kinetic theory and the Rouse model for flexible polymers......Page 300
7.2. System of constraint......Page 303
7.3. Rouse model......Page 305
References......Page 307
