Introduction to dispersive equations

The goal of the course is to introduce several tools to study the existence of solutions of nonlinear dispersive PDEs such as the Schrödinger and wave equations. When these equations are on euclidean spaces, dispersion is often sufficiently strong to ensure the existence of global in time solutions with dispersive behaviour. On the contrary, on compact manifolds dispersion manifests in a weaker way and global in time results are harder to obtain. In the last part of the course we will see how probabilistic methods can help in globalizing solutions, and even construct local in time solutions when general results of local well posedness are not available. The tentative syllabus is the following:





Timetable


Monday, 16:00-18:00,
Wednesday, 14:00-16:00, .



Zoom

link , ID riunione: 884 6642 6054, Passcode: 432826





Main Textbooks

  • [BCD] Bahouri, Chemin, Danchin: Fourier Analysis and Nonlinear Partial Differential Equations link
  • [Bou] Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys. 166, 1--26, 1994.
  • [BT] Burq, Tzvetkov: Random data Cauchy theory for supercritical wave equations I: local theory. Invent. math. 173, 449--475, 2008
  • [BGT] Burq, Gerard, Tzvetkov: Strichartz inequalities and the nonlinear Schrodinger equation on compact manifolds. American Journal of Mathematics, 126(3), 569--605, 2004
  • [C] Cazenave: Semilinear Schr\"odinger Equations, Courant Lecture Notes in Mathematics 10, 2003.
  • [ET] Erdogan, Tzirakis: Dispersive partial differential equations. London Mathematical society student text, 86, 2016.
  • [LP] Linares, Ponce: Introduction to Nonlinear Dispersive Equations. Springer–Verlag, 2009
  • [MS] Muscalu, Schlag: Classical and Multilinear Harmonic Analysis: Volume 1, link
  • [S] Sogge: Hangzhou lectures on eigenfunctions of laplacian.
  • [Tao] Tao: Nonlinear Dispersive Equations. Local and Global Analysis. CBMS Regional Conference Series in Mathematics, 106, AMS (2006).
  • [Tz] Tzvetkov: Random data wave equations, arXiv:1704.01191


Diary

    9/11: Introduction to the course. Reminders on Fourier analysis and tempered distributions .
    14/11: Sobolev spaces in Fourier and their basic properties. [LP]
    16/11: Fourier multipliers and linear constant coefficients dispersive equations. Fundamental solutions of Schrodinger [LP]
    21/11: Oscillatory integrals in 1 and higher dimensions. Applications to time decay estimates [S, Chap. 4]. See also item 7,8 below
    23/11: Littlewood-Paley decomposition. Square function [MS, Chap 8]
    28/11: Applications of Littlewood-Paley. Tame estimates and paraproduct [BCD, Ch2]
    30/11: Fractional Leinbitz rule (see A course in Harmonic Analysis Chap. 7). Strichartz estimates for Schroedinger [LP, Chap 4.]
    05/12: Strichartz estimates for Schroedinger and the endpoint, Keel and Tao link
    11/12: Nonlinear NLS, local theory [LP, Chap 5]
    14/12: Nonlinear NLS, global theory and scattering in the defocusing case. Tsutsumi, Yajima link




Additional material

  1. Dispersive wiki link
  2. Tesfahun: Time-decay estimates for the linearized water wave type equations link
  3. Bulut: AN OPTIMAL DECAY ESTIMATE FOR THE LINEARIZED WATER WAVE EQUATION IN 2D link
  4. Bourgain: On the Schrodinger maximal function in higher dimension link
  5. Carleson: Some analytic problems related to statistical mechanics link
  6. Ginibre, Velo: Generalized Strichartz Inequalities for the wave equation link
  7. Tao: Oscillatory integrals link
  8. FOURIER TRANSFORMS OF SURFACE MEASURE ON THE SPHERE link




© Tetiana Savitska 2017