<p>In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1037_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$H^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1037_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$s_0<s\le \frac{1}{4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo><</mo> <mi>s</mi> <mo>≤</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> on the line and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1037_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$s_0<s< \frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo><</mo> <mi>s</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> on the circle, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1037_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="183" /> </InlineMediaObject> <EquationSource Format="TEX">$s_0 = 3-\sqrt{33/4}\approx 0.1277$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>-</mo> <msqrt> <mrow> <mn>33</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msqrt> <mo>≈</mo> <mn>0.1277</mn> </mrow> </math></EquationSource> </InlineEquation>. Here, we adapt the strategy of Moşincat-Pilod (Pure Appl Anal 5:285–322, 2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.</p>

Unconditional deep-water limit of the intermediate long wave equation in low-regularity

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2025年2月18日

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Justin Forlano,
Guopeng Li,
Tengfei Zhao

摘要

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in $H^s$ $H^{s}$ when $s_0<s\le \frac{1}{4}$ $s_{0} < s \leq \frac{1}{4}$ on the line and $s_0<s< \frac{1}{2}$ $s_{0} < s < \frac{1}{2}$ on the circle, where $s_0 = 3-\sqrt{33/4}\approx 0.1277$ $s_{0} = 3 - \sqrt{33 / 4} \approx 0.1277$ . Here, we adapt the strategy of Moşincat-Pilod (Pure Appl Anal 5:285–322, 2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.