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Global well-posedness of the Benjamin–Ono equation in low-regularity spaces. (English) Zbl 1123.35055

The Benjamin-Ono initial value problem \[ \partial_t u+\mathcal{H}\partial_x^2u+\partial_x(u^2/2)=0\;\text{ on}\; \mathbb{R}_x\times\mathbb{R}_t,\quad u(0)=\varphi \] is considered, where \(\mathcal{H}\) is the Hilbert transform operator defined on the spaces \(C(\mathbb{R}:\mathcal{H}^\sigma)\), \(\sigma\in\mathbb{R}\) by the Fourier multiplier \(-i \operatorname{sgn}(\xi)\). The authors prove that this problem is globally well-posed in the Sobolev space \(H_r^\sigma(\mathbb{R})\), \(\sigma>0\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q05 Euler-Poisson-Darboux equations
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