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Well-posedness for equations of Benjamin-Ono type. (English) Zbl 1215.35136
Summary: The Cauchy problem $$u_t- |D|^\alpha u_x + uu_x=0$$ in $$(-T,T) \times\mathbb R$$, $$u(0)=u_0$$, is studied for $$1<\alpha<2$$. Using suitable spaces of Bourgain type, local well-posedness for initial data $$u_0\in H^s(\mathbb R)\cap \dot{H}^{-\omega}(\mathbb R)$$ for any $$s>-\frac{3}{4}(\alpha-1)$$, $$\omega:=1/\alpha-1/2$$ is shown. This includes existence, uniqueness, persistence, and analytic dependence on the initial data. These results are sharp with respect to the low frequency condition in the sense that if $$\omega<1/\alpha-1/2$$, then the flow map is not $$C^2$$ due to the counterexamples previously known. By using a conservation law, these results are extended to global well-posedness in $$H^s(\mathbb R) \cap \dot{H}^{-\omega}(\mathbb R)$$ for $$s \geq 0$$, $$\omega=1/\alpha-1/2$$, and real valued initial data.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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