Global well-posedness of the Benjamin--Ono equation in low-regularity spaces
نویسندگان
چکیده
منابع مشابه
Global Well-posedness of the Benjamin–ono Equation in Low-regularity Spaces
whereH is the Hilbert transform operator defined (on the spaces C(R : H), σ ∈ R) by the Fourier multiplier −i sgn(ξ). The Benjamin–Ono equation is a model for one-dimensional long waves in deep stratified fluids ([1] and [16]) and is completely integrable. The initial-value problem for this equation has been studied extensively for data in the Sobolev spaces H r (R), σ ≥ 0. It is known that the...
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In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation ∂tv − bH∂ xv + a∂ xv = cv∂xv − d∂x(vH∂xv + H(v∂xv)), where x, t ∈ R, v is a real-valued function, H is the Hilbert transform, a ∈ R, b, c and d are positive constants, is locally well-posed for initial data v(0) = v0 ∈ H(R), s ≥ 2 or v0 ∈ H(R) ∩ L(R; xdx), k ∈ Z+, k ≥ 2.
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We prove that the Benjamin-Ono equation is well-posed in H(T). This leads to a global well-posedeness result in H(T) thanks to the energy conservation. Résumé. Nous montrons que l’équation de Benjamin-Ono est bien posée dans H(T). Il découle alors de la conservation de l’énergie que la solution existe pour tout temps dans cette espace.
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We show that the Benjamin-Ono equation is globally well-posed in H s (R) for s ≥ 1. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in H s for any s [15]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.
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We prove that the Benjamin-Ono equation is globally well-posed in Hs(T) for s ≥ 0. Moreover we show that the associated flow-map is Lipschitz on every bounded set of Hs 0 (T), s ≥ 0, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on H∞ 0 (T)) cannot be of class C , α > 0, fr...
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ژورنال
عنوان ژورنال: Journal of the American Mathematical Society
سال: 2007
ISSN: 0894-0347
DOI: 10.1090/s0894-0347-06-00551-0