The zero-dispersion limit for the Benjamin--Ono equation on the circle

Abstract

Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin--Ono equation on the circle $\mathbb{T}=\mathbb{R}/2\pi \mathbb{Z}$ with bounded initial data $u_0\in L^{\infty }(\mathbb{T},\mathbb{R})$. The result generalizes the work of L. Gassot, who focused on periodic bell-shaped data, and complements the work of Gérard and X. Chen who identified the zero-dispersion limit on the line with $u_0\in L^2\cap L^{\infty }(\mathbb{R})$. Here, as well as in the mentioned cases, the characterization agrees with the one first obtained by Miller--Xu for bell-shaped data on the line: The zero-dispersion limit is given as an alternating sum of the branches of the multivalued solution of Burgers' equation. From this characterization, we compute regularity properties of the zero-dispersion limit, including maximum principles and an Oleinik estimate.

19 pages, 1 figure