Sharp well-posedness for the Benjamin-Ono equation. (English) Zbl 1543.35207
Authors’ 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 \(s > -\frac{1}{2}\). 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.
Reviewer: Ti-Jun Xiao (Fudan)
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35C08 | Soliton solutions |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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