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Wave kinetic equations: Global solutions and long-term behavior

Lecturer: 
Ricardo Grande Izquierdo
Course Type: 
PhD Course
Academic Year: 
2024-2025
Period: 
1st April - 16th May
Duration: 
20 h
Description: 
Course description:
 
Recent progress in the theory of non-equilibrium statistical physics for nonlinear waves has brought much attention to the study of solutions to wave kinetic equations. These solutions, which capture the average evolution of large wave systems undergoing weakly nonlinear interactions, present a variety of asymptotic behaviors connected to interesting physical phenomena, such as energy cascades and Bose Einstein condensation. 
 
The aim of this course is to present recent progress in the study of such wave kinetic equations, with a particular emphasis on those coming from the cubic NLS equation (although analogies with the Nordheim equation will be discussed).
 
Course contents:
 
  1. Isotropic solutions and isotropic wave kinetic equation in 3D.
  2. Wellposedness: definition of mild and weak solutions, existence of mild solutions and blowup alternative, construction of global weak solutions.
  3. Stationary solutions and equilibria.
  4. Long-term behavior of weak solutions and condensation. Asymptotic behavior of mass and energy.
 
Time permitting, we will discuss other classes of solutions, such as self-similar solutions or solutions which display pulsating behavior, as well as several open problems in this field.
 
 
Dates and rooms for the lectures:
  • April 11, 14:00-16:00, Room 005
  • April 14, 14:00-16:00, Room 005
  • April 16, 14:00-16:00, Room 138
  • April 23, 14:00-16:00, Room 005
  • April 24, 11:00-13:00, Room 134
  • April 28, 14:00-16:00, Room 005
  • May 7, 14:00-16:00, Room 138
  • May 9, 14:00-16:00, Room 004
  • May 12, 14:00-16:00, Room 138
  • May 14, 14:00-16:00, Room 138
 
 
References:
 
[1] M. Escobedo and J. J. L. Velázquez, On the Theory of Weak Turbulence for the Nonlinear Schrödinger Equation, Memoirs of the AMS, Volume 238 (2015).
 
[2] M. Escobedo and J. J. L. Velázquez, Finite time blow-up for the bosonic Nordheim equation, Inventiones Mathematicae 200, 761–847 (2015).
 
[3] 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).
 
[4] S. Nazarenko, Wave Turbulence, Lecture Notes in Physics, Springer-Verlag Berlin (2011).
 
[5] Majda, Mclaughlin, Tabac, A One-Dimensional Model for Dispersive Wave Turbulence, J. Nonlinear Sci. Vol. 6: pp. 9–44 (1997).
 
 
Research Group: 
Mathematical Analysis, Modelling, and Applications
Next Lectures: 

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