Sharp well-posedness for the Benjamin--Ono equation

Rowan Killip; Thierry Laurens; Monica Visan

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arXiv · 2304.00124

Sharp well-posedness for the Benjamin--Ono equation

Rowan Killip·Thierry Laurens·Monica Visan

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Abstract

The Benjamin--Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces $H^s$ for $s>-\tfrac12$. The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of G\'erard's explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.

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Rowan Killip, Thierry Laurens, Monica Visan. 2023-03-31. Sharp well-posedness for the Benjamin--Ono equation. https://arxiv.org/abs/2304.00124

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