On smoothing properties and Tao's gauge transform of the Benjamin-Ono equation on the torus

Abstract

We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space $H^s(\mathbb{T},\mathbb{R})$ for any $s\ge 0$. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourier transform $\Phi$ for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.

Cited by in corpus (2)

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