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arXiv · 2001.06896

Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle

Bradley Isom·Dionyssios Mantzavinos·Seungly Oh·Atanas Stefanov
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Abstract

For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) + \epsilon}$, $0<\epsilon \ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.

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Bradley Isom, Dionyssios Mantzavinos, Seungly Oh, Atanas Stefanov. 2020-01-19. Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle. https://arxiv.org/abs/2001.06896

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