Scholarlink

<p>We study the three dimensional quantum many-body dynamics with a delta-type potential <i>N</i><sup>3<i>β</i></sup><i>V</i>(<i>N</i><sup><i>β</i></sup><i>x</i>) and a Coulomb potential. As the particle number <i>N</i> tends to infinity and the Planck’s constant <i>ħ</i> tends to zero independently, we prove the weak convergence of the quantum mass and momentum densities to the Euler–Poisson equation with the pressure before its blow-up time. The proof is based on the modulated energy method in the setting of the quantum many-body dynamics, for which the key is a quantum functional inequality according to the interaction potentials. In Golse–Paul [Comm. Pure Appl. Math., 2022, 75(6): 1332–1376], the functional inequality for the Coulomb potential is achieved based on Serfaty’s inequality [Duke Math. J., 2020, 169(15): 2887–2935]. In the mean-field regime <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11464_2024_43_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta \in \left({0,{1 \over 3}} \right)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>β</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </math></EquationSource> </InlineEquation>, we prove the quantum functional inequality for the delta-type potential under technical profile assumptions on the potential <i>V</i>(<i>x</i>).</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Quantum Mean-field Limit to the Compressible Fluids

  • 期刊论文

  • 发表日期：

    2025年1月5日

查看全文
  • Shunlin Shen,
  • Jiahao Wu

摘要

We study the three dimensional quantum many-body dynamics with a delta-type potential N3βV(Nβx) and a Coulomb potential. As the particle number N tends to infinity and the Planck’s constant ħ tends to zero independently, we prove the weak convergence of the quantum mass and momentum densities to the Euler–Poisson equation with the pressure before its blow-up time. The proof is based on the modulated energy method in the setting of the quantum many-body dynamics, for which the key is a quantum functional inequality according to the interaction potentials. In Golse–Paul [Comm. Pure Appl. Math., 2022, 75(6): 1332–1376], the functional inequality for the Coulomb potential is achieved based on Serfaty’s inequality [Duke Math. J., 2020, 169(15): 2887–2935]. In the mean-field regime \(\beta \in \left({0,{1 \over 3}} \right)\) β ∈ ( 0 , 1 3 ) , we prove the quantum functional inequality for the delta-type potential under technical profile assumptions on the potential V(x).