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Sharp well-posedness for the Benjamin–Ono equation

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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 \(s>-\tfrac{1}{2}\). 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 Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.

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Funding

R.K. was supported by NSF grants DMS-1856755 and DMS-2154022; M.V. was supported by NSF grant DMS-2054194. The work of T.L. was also supported by these grants.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Rowan Killip & Monica Vişan

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Thierry Laurens

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  1. Rowan Killip
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  2. Thierry Laurens
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Killip, R., Laurens, T. & Vişan, M. Sharp well-posedness for the Benjamin–Ono equation. Invent. math. 236, 999–1054 (2024). https://doi.org/10.1007/s00222-024-01250-8

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