CONTENTS -- Foreword -- Preface -- Lectures on Dynamics in Models of Coarsening and Coagulation Robert L. Pego -- Contents -- 1. Introduction -- 1.1. -- 1.2. First models of domain formation and an open problem. -- 2. A hierarchy of domain coarsening models in one space dimension -- 2.1. Domain walls in the Allen-Cahn equation -- 2.2. Domain wall dynamics by restricted gradient flow -- 2.3. Punctuated equilibrium and 1D bubble bath -- 2.4. Mean-field model of domain growth - The Gallay-Mielke transform -- 2.5. Proof of universal self-similar behavior -- 3. Models of domain coarsening in two and three dimensions -- 3.1. Diffuse and sharp-interface models of nanoscale island coarsening -- 3.2. Gradient structure for Mullins-Sekerka flow -- 3.3. Monopole models by restricted gradient flow of surface energy -- 3.4. Lifshitz-Slyozov-Wagner mean-field model -- 4. Rigorous power-law bounds on coarsening rates - The Kohn-Otto method -- 4.1. Basic inequalities -- 4.2. Bounds on coarsening rates for the LSW mean-field model -- 4.3. Bounds on coarsening rates for the monopole model -- 5. Smoluchowski's coagulation equations -- 5.1. Introduction -- 5.2. A 'new' framework for dynamic scaling analysis -- 5.3. Solution by Laplace transform -- 5.4. Scaling solutions and domains of attraction -- 5.5. The scaling attractor -- 5.6. Linearization of dynamics on the scaling attractor -- Acknowledgements -- References -- Quantized Vortices in Superfluids - A Mathematical and Computational Study Qiang Du -- Contents -- 1. Introduction -- 2. Superconductivity and mathematical models -- 2.1. What is superconductivity? -- 2.2. Type-II superconductors and the vortex state -- 2.3. Applications of superconductivity -- 2.4. Superconductivity models and mathematical problems -- 3. The mathematical theory of Ginzburg-Landau models.

3.1. The free energy postulated by Ginzburg and Landau -- 3.2. The equilibrium Ginzburg-Landau models -- 3.3. Time dependent Ginzburg-Landau equations -- 3.4. Gauge invariance and some basic theory -- 4. Numerical algorithms for Ginzburg-Landau models -- 4.1. Finite element approximations -- 4.2. Finite difference approximations -- 4.3. Finite volume approximations -- 4.4. Artificial boundary conditions -- 4.5. More on time-discretization -- 4.6. Multi-level, adaptive and parallel algorithms -- 4.7. Other methods -- 5. Vortex con.gurations - Analysis and simulation -- 5.1. Phase diagrams and equilibrium solution branch -- 5.2. Vortex solutions -- 5.3. A rigorous result on vortex nucleation near HC1 -- 5.4. Effect due to spatial inhomogeneities -- 6. Dynamics of quantized vortices -- 6.1. Dynamics of vortex nucleation -- 6.2. Dynamics of individual vortices -- 6.3. High-κ, high field dynamics -- 6.4. Dynamics involving spatial inhomogeneities -- 6.5. Dynamics driven by the applied current -- 6.6. Vortex state in a thin superconducting spherical shell -- 6.7. Stochastic dynamics driven by noises -- 6.8. Variants of G-L models: Lawrence-Doniach and d-wave models -- 6.9. Vortex density models -- 7. The vortex state in the Bose-Einstein condensation -- 7.1. Vortices in BEC confined in a rotating magnetic trap -- 7.2. Vortex shedding behind a stirring laser beam -- 8. Future challenges -- 9. Conclusion -- Acknowledgment -- References -- The Nonlinear Schrödinger Equation and Applications in Bose-Einstein Condensation and Plasma Physics Weizhu Bao -- Contents -- 1. Introduction -- 2. Derivation of NLSE from wave propagation -- 3. Derivation of NLSE from BEC -- 3.1. Dimensionless GPE -- 3.2. Reduction to lower dimension -- 4. The NLSE and variational formulation -- 4.1. Conservation laws -- 4.2. Lagrangian structure -- 4.3. Hamiltonian structure.

4.4. Variance identity -- 5. Plane and solitary wave solutions of NLSE -- 6. Existence/blowup results of NLSE -- 6.1. Integral form -- 6.2. Existence results -- 6.3. Finite time blowup results -- 7. WKB expansion and quantum hydrodynamics -- 8. Wigner transform and semiclassical limit -- 9. Ground, excited and central vortex states of GPE -- 9.1. Stationary states -- 9.2. Ground state -- 9.3. Central vortex states -- 9.4. Variation of stationary states over the unit sphere -- 9.5. Conservation of angular momentum expectation -- 10. Numerical methods for computing ground states of GPE -- 10.1. Gradient flow with discrete normalization (GFDN) -- 10.2. Energy diminishing of GFDN -- 10.3. Continuous normalized gradient flow (CNGF) -- 10.4. Semi-implicit time discretization -- 10.5. Discretized normalized gradient flow (DNGF) -- 10.6. Numerical methods -- 10.7. Energy diminishing of DNGF -- 10.8. Numerical results -- 11. Numerical methods for dynamics of NLSE -- 11.1. General high-order split-step method -- 11.2. Fourth-order TSSP for GPE without external driving field -- 11.3. Second-order TSSP for GPE with external driving field -- 11.4. Stability -- 11.5. Crank-Nicolson finite difference method (CNFD) -- 11.6. Numerical results -- 12. Derivation of the vector Zakharov system -- 13. Generalization and simplification of ZS -- 13.1. Reduction from VZSM to GVZS -- 13.2. Reduction from GVZS to GZS -- 13.3. Reduction from GVZS to VNLS -- 13.4. Reduction from GZS to NLSE -- 13.5. Add a linear damping term to arrest blowup -- 14. Well-posedness of ZS -- 15. Plane wave and soliton wave solutions of ZS -- 16. Time-splitting spectral method for GZS -- 16.1. Crank-Nicolson leap-frog time-splitting spectral discretizations (CN-LF-TSSP) for GZS -- 16.2. Phase space analytical solver + time-splitting spectral discretizations (PSAS-TSSP).

16.3. Properties of the numerical methods -- 16.4. Extension TSSP to GVZS -- 17. Crank-Nicolson finite difference (CNFD) method for GZS -- 18. Numerical results of GZS -- References -- Introduction to Constitutive Modeling of Macromolecular Fluids Qi Wang -- Contents -- 1. Introduction -- 2. Introduction to equilibrium thermodynamics -- 3. Introduction to statistical mechanics -- 4. Introduction to continuum mechanics -- 4.1. Material, referential, and spatial description of motion, and deformation tensors -- 4.2. Transformation under the motion x(X, t) -- 4.2.1. Line element -- 4.2.2. Surface element -- 4.2.3. Volume element -- 4.2.4. Material derivative -- 4.2.5. Transport theorems -- 4.3. Conservation laws -- 4.3.1. Eulerian description -- 4.3.2. Lagrangian description -- 4.4. Superimposed rigid body motion (SRBM) and invariant principles -- 4.5. Invariant time derivatives -- 4.6. Material symmetry -- 4.7. Clausius-Duhem inequality -- 5. Some constitutive models for .exible polymers -- 6. Introduction to polymer physics -- 6.1. Equilibrium distribution of the end-to-end vector in simple polymer models -- 6.2. Flory-Huggins Theory -- 7. Kinetic theory and the Rouse model for flexible polymers -- 7.1. Langevin equation -- 7.2. System of constraint -- 7.3. Rouse model -- References.