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On well posedness for the Benjamin Ono equation

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37 pages
On well-posedness for the Benjamin-Ono equation Nicolas Burq ? and Fabrice Planchon † Abstract We prove existence and uniqueness of solutions for the Benjamin-Ono equation with data in Hs(R), s > 1/4. Moreover, the flow is hölder continuous in weaker topologies. 1 Introduction Let us consider (1.1) ∂tu+H∂ 2 xu+ u∂xu = 0, u(x, t = 0) = u0(x), (t, x) ? R 2. Here and hereafter, H is the Hilbert transform, defined by (1.2) Hf(x) = 1 pi ∫ f(y) x? y dy = 1 pi vp 1 x ? u = F?1(?isgn(?)f(?)). We will restrict ourselves to real-valued u0, for reasons which will appear later. Equation (1.1) deals with wave propagation at the interface of layers of fluids with di?erent densities (see Benjamin [2] and Ono [22]), and it belongs to a larger class of equation modeling this type of phenomena, some of which are certainly more physically relevant. Mathematically, however, (1.1) presents several interesting and challenging properties; the exact balance between the degree of the nonlinearity and the smoothing properties of the linear part precludes any hope to achieve results through a direct fixed point procedure, be it in Kato smoothing type of spaces or more elaborate conormal (Bourgain)

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