University of Birmingham > Talks@bham > Analysis Seminar > On the deterministic and probabilistic well-posedness of the cubic fourth order NLS on the circle

On the deterministic and probabilistic well-posedness of the cubic fourth order NLS on the circle

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If you have a question about this talk, please contact Dr. Maria Carmen Reguera.

We will discuss the Cauchy problem for the cubic fourth order nonlinear Schr√∂dinger equation (4NLS) on the circle from both deterministic and probabilistic points of view. In the deterministic setting, by using the short-time Fourier restriction norm method, we first prove the global existence of solutions to the renormalized 4NLS in some negative spaces. Then, we consider the uniqueness of solutions. In fact, this is the main difficulty of this problem. In this part, we establish an energy estimate on the difference of two solutions with the same initial data by implementing an infinite iteration of the normal form reductions to the energy functional. As for the probabilistic part, we will consider the renormalized 4NLS with white noise as initial data. Since the white noise is very rough, the deterministic local theory is out of reach. By introducing a (random resonant) nonlinear decomposition, we prove almost sure local well-posedness. Then, we apply Bourgain’s invariant measure argument to extend the local solutions globally in time.

This is a joint work with Tadahiro Oh (University of Edinburgh) and Nikolay Tzvetkov (Université de Cergy-Pontoise).

This talk is part of the Analysis Seminar series.

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