An introduction to KAM and Normal Form methods for PDEs

DATES: 16 - 20 January 2023 (5 sessions)
TIME: 9:00-11:00 (a total of 10 hours)
LOCATION: Maryam Mirzakhani Seminar Room at BCAM


ABSTRACT:
In this course I will give an introduction to the perturbative methods used in order to analyze the Dynamics of evolution PDEs, with a particular focus on dispersive PDEs and PDEs arising from fluid-dynamics (like Euler, Water Waves equations and some of their approximate models). More precisely I shall treat the following topics:
1) The problem of energy transfer to high frequency modes for linear Schrödinger type equations with time dependent potentials
2) The problem of constructing quasi-periodic solutions for nonlinear equations (Kolmogorov-Arnold-Moser theory for PDEs).

REFERENCES:

1) M. Berti, R. Montalto. Quasi-periodic standing wave solutions for gravity-capillary water waves. Memoirs of The American Mathematical Society, 263(1273), pp. 1-184 (2020).
2) P. Baldi, M. Berti, E. Haus, R. Montalto. Time quasi-periodic gravity water waves in nite depth. Inventiones Mathematicae, 214 (2), 739-911 (2018) DOI:10.1007/s00222-018-0812-2.
3) P. Baldi, R. Montalto, Quasi-periodic incompressible Euler flows in 3D, Advances in Math. 384 (2021), 107730.
4) J.M. Delort. Growth of Sobolev norms of solutions of linear Schr odinger equations on some compact manifolds. Int. Math. Res. Notices, 12: 2305{2328, 2010.
5) J. Bourgain. Green's function estimates for lattice Schr odinger operators and applications. Annals of Mathematics Studies 158, Princeton University Press, Princeton (2005).
6) S. Kuksin. Hamiltonian perturbations of innite-dimensional linear systems with imaginary spectrum. Funktsional. Anal. i Prilozhen. 21 no. 3, 22-37, 95 (1987).
7) R. Montalto. On the growth of Sobolev norms for a class of linear Schr odinger equations on the torus with superlinear dispersion. Asymptotic Analysis, 108(1-2), pp. 85-114 (2018).


*Registration is free, but mandatory before 11 January 2023. To sign-up go to https://forms.gle/AgH2mPSq4t4VSqc18 and fill the registration form.

• BCAM-Course_20230116