Qualitative properties of dispersive PDEs / / Vladimir Georgiev, Alessandro Michelangeli, Raffaele Scandone, editors

• Pubblicazione: Singapore : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (246 pages)
• Disciplina: 515.35
• Soggetto topico: Differential equations; Functional analysis
• Persona (resp. second.): MichelangeliAlessandro; ScandoneRaffaele; GeorgievVladimir
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Qualitative properties of dispersive PDEs
• ISBN: 981-19-6434-3
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996503552103316
• Lo trovi qui: Univ. di Salerno
• Serie: Springer INdAM series ; ; Volume 52.