This book provides a valuable collection of contributions by distinguished scholars presenting the state of the art and some of the most significant latest developments and future challenges in the field of dispersive partial differential equations. The material covers four major lines: (1) Long time behaviour of NLS-type equations, (2) probabilistic and nonstandard methods in the study of NLS equation, (3) dispersive properties for heat-, Schrödinger-, and Dirac-type flows, (4) wave and KdV-type equations. Across a variety of applications an amount of crucial mathematical tools are discussed, whose applicability and versatility goes beyond the specific models presented here. Furthermore, all contributions include updated and comparative literature.
Author(s): Vladimir Georgiev, Alessandro Michelangeli, Raffaele Scandone
Series: Springer INdAM Series, 52
Publisher: Springer
Year: 2023
Language: English
Pages: 245
City: Singapore
Preface
Organization
Program Chairs
Contents
About the Editors
Part I Long-Time Behavior of NLS-Type Equations
A Note on Small Data Soliton Selection for Nonlinear Schrödinger Equations with Potential
1 Introduction
2 The Proof
2.1 Estimate of the Continuous Variable η
2.2 Estimate of Discrete Variables
References
Dynamics of Solutions to the Gross–Pitaevskii Equation Describing Dipolar Bose–Einstein Condensates
1 Introduction
1.1 Background
1.2 Main Results
1.2.1 Dynamics Below the Threshold
1.2.2 Dynamics Above the Threshold
1.2.3 Dynamics at the Threshold
2 Decay for Powers of Riesz Transforms and Virial Arguments
2.1 Integral Estimates for R4j
2.2 Point-Wise Estimates for R2j
2.3 Virial Identities
3 Sketch of the Proofs Below the Threshold
3.1 Scattering
3.2 Blow-up
3.3 Grow-up
4 Sketch of the Proofs Above the Threshold
5 Sketch of the Proofs at the Threshold
References
Part II Probabilistic and Nonstandard Methods in the Study of NLS Equations
Almost Sure Pointwise Convergence of the Cubic Nonlinear Schrödinger Equation on T2
1 Introduction
1.1 Notations and Terminology
2 Proof of Theorem 1
2.1 Proof of Theorem 1
3 Proof of Proposition 1
3.1 Proof of Proposition 1
4 Proof of Theorem 2
4.1 Proof of Theorem 2
4.2 Proof of Proposition 3
References
Nonlinear Schrödinger Equation with Singularities
1 Introduction
2 The Colombeau Algebra
2.1 Notion of Colombeau Solution
2.2 Compatibility
3 Existence and Uniqueness of a Singular Solution
4 Convergence Properties
References
Part III Dispersive Properties
Schrödinger Flow's Dispersive Estimates in a Regime of Re-scaled Potentials
1 Introduction and Background
2 A Preliminary Overview of Relevant Spectral Properties
3 Dispersive Estimates with -Uniform Bound
4 Outlook on Further Scaling Regimes
References
Dispersive Estimates for the Dirac–Coulomb Equation
1 Introduction
1.1 The Setup: Partial Wave Decomposition, Spectral Theory, and the Hankel Transform Method
2 Dispersive Estimates
2.1 Local Smoothing
2.2 Strichartz Estimates with Loss of Angular Derivatives
2.3 Open Problems and Related Models
References
Heat Equation with Inverse-Square Potential of Bridging Type Across Two Half-Lines
1 Introduction: The Bridging-Heat Equation in 1D
2 A Concise Review of Geometric Confinement and Transmission Protocols in a Grushin Cylinder
3 Related Settings: Grushin Planes and Almost-Riemannian Manifolds
4 A Numerical Glance at the Bridging-Heat Evolution
References
Part IV Wave- and KdV-Type Equations
On the Cauchy Problem for Quasi-Linear Hamiltonian KdV-Type Equations
1 Introduction
2 Paradifferential Calculus
3 Paralinearization
4 Linear Local Well-Posedness
5 Nonlinear Local Well-Posedness
References
Quasilinear Wave Equations with Decaying Time-Potential
1 Introduction
2 Quasilinear Wave Equations
2.1 Statement of the Main Results
2.2 Proof of Theorem 1
2.3 Proof of Theorem 2
3 Applications
4 An Existence Result
References
Hamiltonian Field Theory Close to the Wave Equation: From Fermi-Pasta-Ulam to Water Waves
1 Introduction
2 Outline of the Method and Results
2.1 Hamiltonian Field Theory
2.2 Results: Informal Presentation
3 Abstract Setting: Perturbation Theory in Poisson Systems
3.1 Poisson Formalism
3.2 Perturbation Theory
4 Hamiltonian Field Theory Close to qtt=qxx
4.1 Traveling Waves
4.2 The Generic Case
4.3 The Mechanical Case
5 Applications
5.1 The Fermi-Pasta-Ulam Problem
5.2 Water Waves
6 Conclusions and Open Problems
References
Author Index
