Through January 12, 2024 Eckhart Hall, Room 202
Thomas Duyckaerts (Sorbonne Paris Nord University) will give three lectures as part of our annual Zygmund-Calderon Lecture Series.
Lecture 1: Generalized Soliton Resolution for Nonlinear Dispersive Equations
According to the soliton resolution conjecture, solutions of nonlinear dispersive equations should behave asymptotically, for infinitely large times, as the sum of decoupled solitary waves and a radiation term. In this lecture, we will first give the history of this conjecture. We will next report on progresses made on focusing wave equations in the last 10 years, including the proof of the soliton resolution for radially symmetric solutions of the energy-critical equation. We will also give examples of solutions that do not fit into a strict soliton resolution scenario, leading to a weaker conjecture on the expected asymptotic behaviors of solutions of dispersive equations.
This lecture can be followed by non-specialist. It is based on joint works with Charles Collot, Carlos E. Kenig, Yvan Martel, Frank Merle, Giuseppe Negro.
Lecture 2: On Classification of Non-Radiative Solutions for Various Energy-Critical Wave Equations
In the modern theory of dispersive nonlinear equations, the study of the asymptotic dynamics of solutions is often reduced to the proof of a rigidity theorem, classifying solutions with a specific nondispersive behavior. In this lecture, we will focus on the classification of non-radiative solutions of wave equations, that are solutions such that their energy in the exterior of a wave cone vanishes asymptotically in both time directions. We will see that this classification depends crucially on the dimension, and that it is a powerful tool to solve the soliton resolution conjecture for the energy-critical nonlinear equations in the radial case.
This lecture is based on joint works with Charles Collot, Carlos E. Kenig, Yvan Martel, Frank Merle.
Lecture 3: Classification of Radial Nonlinear Waves Outside a Ball
This lecture concerns a simple model of nonlinear dispersive equation where one can give a complete description of the dynamics, in the spirit of the soliton resolution conjecture presented in Lecture I. We will consider radial solutions of the focusing non-linear wave equation outside a ball, in dimension 3 of space, with Dirichlet conditions at the boundary. We will show that any global radial solution of the equation is written asymptotically as the sum of a stationary solution and a solution of the linear wave equation, and that the set of initial data leading to a given stationary solution is a submanifold of finite explicit codimension.
This lecture is based on joint work with Jianwei Yang.
