MATH 565 "Introduction to Nonlinear Dispersive Equations"

Time
Location
Instructor
Office Hours
E-mail
MW 12:30-13:45 269 Bausch & Lomb Building Dan-Andrei Geba MW 11:15-12:15, 806 Hylan Building dangeba@math.rochester.edu

Description

Informally, a partial differential equation is called dispersive if different frequencies in the equation tend to propagate at different velocities, dispersing the solution over time. Prime examples of such equations are the nonlinear Schrodinger and the Korteweg-de Vries type equations. The field of nonlinear dispersive equations has experienced tremendous growth for the past 30 or so years and this course aims to introduce some of the fundamental ideas which have been developed in this period.

The main focus is on the study of wellposedness issues for this type of equations, with wellposedness meaning the existence and uniqueness of solutions and the continuity of the data-to-solution map. A central role in this investigation is played by harmonic analysis techniques, which include interpolation theory, Fourier analysis, Littlewood-Paley theory, etc. This is one of the reasons this class builds on what was studied in MATH 472.

There are three excellent sources for this material which will be followed to various degrees:

1. "Nonlinear Dispersive Equations: Local and Global Analysis" (AMS, 2006) by T. Tao.

2. "Introduction to Nonlinear Dispersive Equations" (2nd edition, Springer, 2015) by F. Linares and G. Ponce. 1st edition available online in the UR libraries system.

3. "Dispersive Partial Differential Equations: Wellposedness and Applications" (Cambridge University Press, 2016) by M. B. Erdogan and N. Tzirakis.

Weekly schedule:


Week of Topic
1/15 Formal definition of dispersive equations, physical background for the NLS and the KdV equations, analytic toolbox (Lebesgue spaces, Holder inequality, analytic interpolation, Young inequality, convolution estimate, Hardy-Littlewood-Sobolev inequality, properties of the Fourier transform, space of smooth functions, space of smooth functions with compact support, space of Schwarz functions).
1/22 Analytic toolbox (space of tempered distributions, Sobolev spaces, Littlewood-Paley theory, TT* argument, oscillatory-type integrals), informal derivation for the conservation of the L^2 norm for NLS and KdV, integral formulation using Duhamel's principle.
1/29 Linear propagators associated to NLS and KdV and their dispersive properties, Strichartz-type estimates for linear NLS and KdV.
2/5 Strichartz-type estimates for NLS and KdV, Christ-Kiselev lemma, scaling analysis, local well-posedness for NLS by energy methods.
2/12 Local well-posedness for NLS by energy methods and Strichartz-type estimates.
2/19 Sharp local well-posedness for the L^2-critical NLS, extensions of local solutions to global ones.
2/26 Energy methods for KdV: a priori Sobolev bounds, existence by parabolic regularization, uniqueness, extensions of local solutions to global ones.
3/4 Oscillatory integral-type estimates for KdV: Kato smoothing and maximal function estimates.
3/18 Local well-posedness for KdV by oscillatory integral-type methods, initial considerations for the restricted norm method.
3/25 Linear estimates for KdV which are key pieces for the restricted norm method.
4/1
4/8
4/15
4/22
4/29