The aim of this lectures is to provide the main analytical tools, developed in the last 30 years, in order to handle the main questions concerning the evolution of dispersive waves. These include harmonic analytical tools (e.g. Strichartz estimates) as well as spectral theoretical results (e.g. the Virial Theorem) and some nowadays standard arguments which are ubiquitous in Nonlinear Analysis. The nonlinear Schr ̈odinger equation (NLS) will be our toy model along the lectures. At the end of the course, the students will be able to approach the research about this and related topics.
Description
The duration of the lectures will be 30 hours. The list of topics is as follows, and it will depend on the students’ feedback.
(i) Dispersive waves: group and phase velocity, dispersion relation. Linear Schr ̈odinger equation: evolution of wave packets, fundamental solution. Dispersive estimates: time decay, Strichartz estimates. An overview about other dispersive PDE (wave, Klein-Gordon, Dirac, KdV).
(ii) Nonlinear Schrödinger equation (NLS). Local Cauchy Theory by compactness arguments. Conservation laws and variational inequalities. Global Cauchy Theory in H1 for subcritical NLS with large data and for critical NLS with small data. Glassey’s blow-up.
(iii) Scattering Theory. Existence of Wave Operators, modified Wave Operators. Asymptotic completeness: Morawetz estimates, Interaction Morawetz Estimates, Pseudoconformal Transformation, Scattering in H1 and in Σ.
(iv) Schrödinger equation with potentials. Virial Theorem. Uniform Resolvent estimates and Kato-Yajima estimate. Applications to Spectral Theory. Strichartz estimates for the Schrödinger Equation with critical 0-order potentials. Scattering in Σ for NLS via Lens transform.
(v) Critical NLS with large data in H1: the Kenig-Merle strategy. Concentration compactness, construction of the critical element. Rigidity arguments.
References
[1] Cazenave T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, American Mathematical Society, Providence, 2003.
[2] Ginibre, J., and Velo, G., The global Cauchy problem for the nonlinear Schr ̈odinger equation revisited, Ann. I.H.P.,Analyse Non Lynéare 2 (1985), 309–327.
[3] Keel, M., and Tao, T., Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) no. 5, 955–980.
[4] Kenig, C., and Merle, F., Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), 645-675.
[5] Morawetz, C., Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A 306 (1968), 291–296.