<p>This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>. New harmonic analysis tools, including averaging approximations to the exponential phase functions and trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new perturbation techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> (up to a logarithmic factor) in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> under the regularity condition <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u\in C([0,T];H^\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msup> <mi>H</mi> <mi>γ</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

An Unfiltered Low-Regularity Integrator for the KdV Equation with Solutions Below $\mathbf{H^1}$

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发表日期：

2025年3月4日

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Buyang Li,
Yifei Wu

摘要

This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below $H^1$ $H^{1}$ . New harmonic analysis tools, including averaging approximations to the exponential phase functions and trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new perturbation techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order $\gamma $ $γ$ (up to a logarithmic factor) in $L^2$ $L^{2}$ under the regularity condition $u\in C([0,T];H^\gamma )$ $u \in C ([0, T]; H^{γ})$ for $\gamma \in (0,1]$ $γ \in (0, 1]$ .

An Unfiltered Low-Regularity Integrator for the KdV Equation with Solutions Below \(\mathbf{H^1}\)

摘要