Maria Ntekoume: Critical well-posedness for the derivative nonlinear Schrodinger equation: disper...

(15 novembre 2024/November 15, 2024) Colloque des sciences mathématiques du Québec/CSMQ. www.crmath.ca/....
Maria Ntekoume (Concordia University): Critical well-posedness for the derivative nonlinear Schrodinger equation: dispersion and integrability
Abstract: The focus of this talk is the derivative nonlinear Schr\"odinger equation (DNLS), a PDE arising as a model in magnetohydrodynamics. It is a nonlinear dispersive equation, meaning that, in the absence of a boundary, solutions to the underlying linear flow tend to spread out in space as they evolve in time. It is also known to be completely integrable: in addition to a conserved mass and energy, it has an infinite hierarchy of conserved quantities. These intriguing features have captured the interest of mathematicians and have played an important role in the investigation of the well-posedness of DNLS, that is, the question of whether solutions exist, are unique, and depend continuously on the initial data. However, until recently not much was known regarding the evolution of rough and slowly decaying initial data. We will discuss why previous methods failed to solve this problem and recent progress towards closing this gap, culminating in our proof of sharp well-posendess in the critical space in joint work with Benjamin Harrop-Griffiths, Rowan Killip, and Monica Visan.