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arXiv · 2004.04857

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

P. Gérard·T. Kappeler·P. Topalov
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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.

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P. Gérard, T. Kappeler, P. Topalov. 2020-04-09. Sharp well-posedness results of the Benjamin-Ono equation in $H^{s}(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution. https://arxiv.org/abs/2004.04857

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