<p>Consensus behavior is a notable emergence phenomenon in nature. It is only recently that consensus behavior has been demonstrated in the Vlasov alignment and Euler alignment models with low-order power-law potentials, i.e. <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3522_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="152" /> </InlineMediaObject> <EquationSource Format="TEX">$U(r)=r^{\alpha }, \ \alpha \in [1,4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>r</mi> <mi>α</mi> </msup> <mo>,</mo> <mspace width="4pt" /> <mi>α</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Note that the attraction between particles weakens as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3522_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> grows, so it is interesting to consider the high-order power-law potential case. By some macroscopic and microscopic Lyapunov functionals, for any <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3522_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in (2,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and any long range communication weight, we establish both the weak and strong consensus and their precise convergence rates for the Vlasov alignment and Euler alignment models.</p>

The Vlasov alignment model with high order power-law potentials

期刊论文
发表日期：

2025年10月3日

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Yuepeng Li,
Zili Chen

摘要

Consensus behavior is a notable emergence phenomenon in nature. It is only recently that consensus behavior has been demonstrated in the Vlasov alignment and Euler alignment models with low-order power-law potentials, i.e. $U(r)=r^{\alpha }, \ \alpha \in [1,4)$ $U (r) = r^{α}, α \in [1, 4)$ . Note that the attraction between particles weakens as $\alpha $ $α$ grows, so it is interesting to consider the high-order power-law potential case. By some macroscopic and microscopic Lyapunov functionals, for any $\alpha \in (2,\infty )$ $α \in (2, \infty)$ and any long range communication weight, we establish both the weak and strong consensus and their precise convergence rates for the Vlasov alignment and Euler alignment models.