<p>Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin–Ono equation on the circle <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {T}{:}{=}\mathbb {R}/2\pi \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>:</mo> <mo>=</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mi>π</mi> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> with bounded initial data <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u_0\in L^\infty (\mathbb {T},\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u_0\in L^2\cap L^\infty (\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo>∩</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

The Zero-Dispersion Limit for the Benjamin–Ono Equation on the Circle

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2026年5月2日

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Ola Mæhlen

摘要

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 $u_0\in L^\infty (\mathbb {T},\mathbb {R})$ $u_{0} \in L^{\infty} (T, R)$ . 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 $u_0\in L^2\cap L^\infty (\mathbb {R})$ $u_{0} \in L^{2} \cap L^{\infty} (R)$ . 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.