Calendar
Thursday, October 31, 2024
Time  Items 

All day 

3pm 

4pm 
10/31/2024  4:00pm In this talk, I will explain how paradifferential calculus can be applied to construct quasiperiodicintime solution for PDEs (“KAM for PDEs”). Due to the loss of regularity caused by “small divisors”, these problems are traditionally resolved using NashMoser/KAM type iterative schemes. One step of the NashMoser scheme is to reduce a nonautonomous linear operator that involves “small divisors” into constant coefficient form, for which a NashMoser/KAM reducibility argument is necessary, yielding a complicated “NashMoser within NashMoser” formalism. However, it is discovered that paradifferential calculus can be used to completely avoid such formalism, yielding “fixed point” style proof. In particular, it is possible to reduce the nonlinear equation itself into constant coefficient form (modulo smoothing remainder), not just its linearization. This is because paradifferential operators share all the algebraic structures of (pseudo)differential operators while gain back regularities due to J.M. Bony’s paralinearization process. I will use the existence problem for quasiperiodicintime solution of fully nonlinear hyperbolic systems with one spatial variable as illustrative example. This talk is based on joint works with Thomas Alazard. Location:
KT 207
