Date: Thursday, November 21, 2024

 

Link:  https://wayne-edu.zoom.us/j/99210289207?pwd=NFMvazN4WnZxWEpOQXQ5MXFlYlhLUT09

Title: On the wave kinetic equation and its associated hierarchy

Abstract: The wave kinetic equation is one of the fundamental models in the theory of wave turbulence, and provides a statistical description of weakly nonlinear interacting waves. This talk will address the global in time well-posedness of the (spatially inhomogeneous) wave kinetic equation and the associated wave kinetic hierarchy, which is an infinite system of coupled equations closely related to the equation itself. 

To prove well-posedness of the wave kinetic equation, we apply techniques inspired by the analysis of the Boltzmann equation – another model in statistical mechanics that describes evolution of rarefied gases.

To prove well-posedness of the wave kinetic hierarchy, we use several tools. In particular, a result from the probability theory known as the Hewitt-Savage theorem helps to lift the existence result from the equation to the hierarchy. On the other hand, uniqueness of solutions to the wave kinetic hierarchy is proved with the help of a combinatorial technique, known as the Klainerman-Machedon board game argument, which allows us to control the factorial growth of the Dyson series. 

This is a joint work with Ioakeim Ampatzoglou, Joseph K. Miller and Natasa Pavlovic.