The aim of this paper is to give existence and uniqueness results for solutions to the Cauchy problem of Burgers type equations on stratified Lie groups <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {G}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">G</mi> </math></EquationSource> </InlineEquation> with the homogeneous dimension N. We consider a nonlinear function behaves like <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">$X(|u|^{\alpha -1}u)\,(\alpha >1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>α</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where X is a left invariant vector field on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {G}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">G</mi> </math></EquationSource> </InlineEquation>, and the initial data <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$u_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> belongs to the Sobolev spaces <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p_s({{\mathbb {G}}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>s</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq7.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$s\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_318_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$s\ne N/p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≠</mo> <mi>N</mi> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>.

Existence and uniqueness results for the Cauchy problem of generalized Burgers type equations on stratified Lie groups

期刊论文
发表日期：

2025年2月24日

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Hiroyuki Hirayama,
Yasuyuki Oka

摘要

The aim of this paper is to give existence and uniqueness results for solutions to the Cauchy problem of Burgers type equations on stratified Lie groups ${{\mathbb {G}}}$ $G$ with the homogeneous dimension N. We consider a nonlinear function behaves like $X(|u|^{\alpha -1}u)\,(\alpha >1)$ $X (| u |^{α - 1} u) (α > 1)$ , where X is a left invariant vector field on ${{\mathbb {G}}}$ $G$ , and the initial data $u_0$ $u_{0}$ belongs to the Sobolev spaces $L^p_s({{\mathbb {G}}})$ $L_{s}^{p} (G)$ for $1<p<\infty $ $1 < p < \infty$ , $s\ge 0$ $s \geq 0$ and $s\ne N/p$ $s \neq N / p$ .