The Cauchy problem for the Benjamin-Ono equation in L^2 revisited (Joint work with Luc Molinet)

The Cauchy problem for the Benjamin-Ono equation in L^2 revisited (Joint work with Luc Molinet)

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Didier Pilod, Universidade Federal do Rio de Janeiro / University of Chicago
Fine Hall 314

The Benjamin-Ono equation models the unidirectional evolution of weakly nonlinear dispersive internal long waves at the interface of a two-layer system, one being infinitely deep. The Cauchy problem associated to this equation presents interesting mathematical difficulties and has been extensively studied in the recent years. In a recent work (2007), Ionescu and Kenig proved well-posedness for real-valued initial data in L^2(R). In this talk, we will give another proof of Ionescu and Kenig's result, which moreover provides stronger uniqueness results. In particular, we prove unconditional well-posedness in H^s(R), for s > 1/4 . Note that our approach also permits to simplify the proof of the global well-posedness in L^2(T) by Molinet (2008) and yields unconditional well-posedness in H^{1/2}(T). Finally, it is worthwhile to mention that our technique of proof also apply for a higher-order Benjamin-Ono equation. We prove that the associated Cauchy problem is globally well-posed in the energy space H^1(R).