<p>We consider a periodic higher-order nonlinear Schrödinger equation with the nonlinearity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_315_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$u^k \partial _xu$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mi>k</mi> </msup> <msub> <mi>∂</mi> <mi>x</mi> </msub> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>, where <i>k</i> is a natural number. We prove the norm inflation in a subspace of the Sobolev space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_315_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$H^s(\mathbb {T})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_315_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$s \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. In particular, the Cauchy problem is ill-posed in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_315_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$H^s(\mathbb {T})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_315_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$s \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle

期刊论文
发表日期：

2025年3月12日

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Toshiki Kondo,
Mamoru Okamoto

摘要

We consider a periodic higher-order nonlinear Schrödinger equation with the nonlinearity $u^k \partial _xu$ $u^{k} \partial_{x} u$ , where k is a natural number. We prove the norm inflation in a subspace of the Sobolev space $H^s(\mathbb {T})$ $H^{s} (T)$ for any $s \in \mathbb {R}$ $s \in R$ . In particular, the Cauchy problem is ill-posed in $H^s(\mathbb {T})$ $H^{s} (T)$ for any $s \in \mathbb {R}$ $s \in R$ .