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Please join us for pre-seminar Tea

In this talk, I will explain how paradifferential calculus can be applied to construct quasiperiodic-in-time solution for PDEs (“KAM for PDEs”). Due to the loss of regularity caused by “small divisors”, these problems are traditionally resolved using Nash-Moser/KAM type iterative schemes. One step of the Nash-Moser scheme is to reduce a non-autonomous linear operator that involves “small divisors” into constant coefficient form, for which a Nash-Moser/KAM reducibility argument is necessary, yielding a complicated “Nash-Moser within Nash-Moser” 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 quasiperiodic-in-time solution of fully nonlinear hyperbolic systems with one spatial variable as illustrative example. This talk is based on joint works with Thomas Alazard.