Year: 2026
Type: article
Abstract: Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin–Ono equation on the circle $$\mathbb {T}{:}{=}\mathbb {R}/2\pi \mathbb {Z}$$ T : = R / 2 π Z with bounded initial data $$u_0\in L^\infty (\mathbb {T},\mathbb {R})$$ u 0 ∈ L ∞ ( T , 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 $$u_0\in L^2\cap L^\infty (\mathbb {R})$$ u 0 ∈ L 2 ∩ L ∞ ( 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.
Author: Ola Mæhlen
Institution: Université Paris-Saclay
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4416255727