3 Global Well - Posedness of the Benjamin - Ono Equation in H 1 ( R )
نویسنده
چکیده
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.
منابع مشابه
Well-posedness in H for the (generalized) Benjamin-Ono equation on the circle
We prove the local well posedness of the Benjamin-Ono equation and the generalized Benjamin-Ono equation in H(T). This leads to a global wellposedness result in H(T) for the Benjamin-Ono equation.
متن کاملGlobal 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...
متن کاملWell - Posedness of the Benjamin - Ono Equation in H 1 ( R )
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.
متن کاملSharp Well-posedness Results for the Generalized Benjamin-ono Equation with High Nonlinearity
We establish the local well-posedness of the generalized BenjaminOno equation ∂tu+H∂ xu±u ∂xu = 0 in Hs(R), s > 1/2−1/k for k ≥ 12 and without smallness assumption on the initial data. The condition s > 1/2−1/k is known to be sharp since the solution map u0 7→ u is not of class Ck+1 on Hs(R) for s < 1/2 − 1/k. On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prov...
متن کاملGlobal well-posedness in the Energy space for the Benjamin-Ono equation on the circle
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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