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Abstract

Using the explicit formula of P. G\'erard, 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 G\'erard 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 characteristics appearing in 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.