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Sharp well-posedness results of the Benjamin-Ono equation in $H^{s}(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution by Gérard, P.; Kappeler, T.; Topalov, P. is a scholarly article available to read on EtoBox.

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

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.

Author

Gérard, P.; Kappeler, T.; Topalov, P.

Published

2020

Language

EN