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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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