Mathematics

A large system of weakly-interacting waves is generally governed by a large number of differential equations which describe the dynamics of each wave. Due to their complexity and the large number of waves,

the precise description of an individual wave may be difficult, and it might not be representative of the behavior of the system as a whole. What is sometimes possible is a statistical description of such a system, which provides information about its typical behavior. We will present such a description in the case where the underlying "microscopic" system of nonlinear waves satisfies the cubic Schrödinger equation.

Course contents:

I. Resonances and quasi-resonances. Formal derivation of Wave Kinetic Equation from the cubic Schrödinger equation.

II. Introduction to Gaussian Hilbert Spaces. Isserlis' theorem. Gaussian hypercontractivity estimates.

III. Picard iteration, remainder, and reduction to a combinatorics problem.

IV. Feynman diagrams, pairings and counting estimates.

V. Well-posedness of WKE in the 3D isotropic setting. Fluxes and special solutions. Long term behavior.

 

1. C. Collot and P. Germain, Derivation of the homogeneous kinetic wave equation: longer time scales,  arXiv: 2007.03508 (2020). 
2. Y. Deng and Z. Hani, On the Derivation of the Wave Kinetic equation for NLS, Forum of Math. Pi 9 (2021), e6, 1-37. 
3. Y. Deng and Z. Hani, Full Derivation of the Wave Kinetic equation, to appear in Invent. Math., arXiv: 2104.11204 (2021). 
4. M. Escobedo and J.J.L. Velázquez, On the Theory of Weak Turbulence for the Nonlinear Schrödinger Equation, Memoirs of the American Mathematical Society, Volume 238 (2015). 
5. P. Germain, A. Ionescu, M.-B.Tran, Optimal local well-posedness theory for the kinetic wave equation, Journal of Functional Analysis, 279 (4), 108570 (2020). 
6. Svante Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics Vol. 129, Cambridge University Press, 1997.
7. Sergey Nazarenko, Wave Turbulence, Lecture Notes in Physics, Springer-Verlag Berlin.

Rooms:

• 05/02/2024 - Room A-134
• 07/02/2024 - Room A-128, 129
• 12/02/2024 - Room A-134
• 14/02/2024 - Room A-134
• 19/02/2024 - Room A-134
• 21/02/2024 - Room A-132
• 26/02/2024 - Room A-134
• 28/02/2024 - Room A-133
• 04/03/2024 - Room A-134
• 07/03/2024 - Room A-134