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 $$\mathbb {T}{:}{=}\mathbb {R}/2\pi \mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> with bounded initial data $$u_0\in L^\infty (\mathbb {T},\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 $$u_0\in L^2\cap L^\infty (\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>∩</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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.