Sharp well-posedness results of the Benjamin-Ono equation in $H^s(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution

Abstract

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^s(\mathbb{T},\mathbb{R})$ for any $s>-1/2$ and ill-posed for $s\le -1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^s(\mathbb{T},\mathbb{R})$ for any $s>-1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^s(\mathbb{T},\mathbb{R})$ with $s>-1/2$ are key ingredients into the proofs of these results.

arXiv commentThe paper is a greatly extended version of arXiv:1909.07314. In particular, we have included the result on ill-posedness of the Benjamin-Ono equation in $H^{-1/2}$ and added an appendix on the restriction of the Birkhoff map to the scale of Sobolev space $H^s$, $s>0$, and another one on a sharper form of ill-posedness of the Benjamin-Ono equation in $H_r^s$ with $s<-1/2$

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• On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$