软势费米-狄拉克玻尔兹曼方程解的存在唯一性

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Zongguang Li
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In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript upper T Superscript normal infinity Baseline upper L Subscript v comma x Superscript normal infinity intersection upper L Subscript upper T Superscript normal infinity Baseline upper L Subscript x Superscript normal infinity Baseline upper L Subscript v Superscript 1\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^{\\infty }_{T}L^{\\infty }_{v,x}\\cap L^{\\infty }_{T}L^{\\infty }_{x}L^1_v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper bound <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F left-parenthesis t comma x comma v right-parenthesis less-than-or-equal-to StartFraction 1 Over delta EndFraction\"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>δ<!-- δ --></mml:mi> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">F(t,x,v)\\leq \\frac {1}{\\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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引用次数: 0

摘要

本文考虑了一个修正的量子玻尔兹曼方程,其量子效应由一个连续参数δ \delta测量,该参数可以从费米-狄拉克粒子的δ =1 \delta =1减小到经典粒子的δ =0 \delta =0。在软势能情况下,对于整个空间或环应的柯西问题,我们建立了函数空间L T∞L v,x∞∩L T∞L x∞L v 1 L^ {\infty _TL^ }{}{\infty _v},x {}\cap L^{\infty _TL^ }{}{\infty _xL}^{1_v}中缺陷质量小的非负温和解的整体存在唯一性;能量和熵,但允许有较大的振幅,直到可能的最大上界F(t,x,v)≤1 δ F(t,x,v) \leq\frac 1 {}{\delta。关键}是得到的估计在量子参数0 &gt;δ≤10 &gt;\delta\leq
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Existence and uniqueness of solutions to the Fermi-Dirac Boltzmann equation for soft potentials
In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter δ \delta that can decrease from δ = 1 \delta =1 for the Fermi-Dirac particles to δ = 0 \delta =0 for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space L T L v , x L T L x L v 1 L^{\infty }_{T}L^{\infty }_{v,x}\cap L^{\infty }_{T}L^{\infty }_{x}L^1_v with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper bound F ( t , x , v ) 1 δ F(t,x,v)\leq \frac {1}{\delta } . The key point is that the obtained estimates are uniform in the quantum parameter 0 > δ 1 0> \delta \leq 1 .
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来源期刊
Quarterly of Applied Mathematics
Quarterly of Applied Mathematics 数学-应用数学
CiteScore
1.90
自引率
12.50%
发文量
31
审稿时长
>12 weeks
期刊介绍: The Quarterly of Applied Mathematics contains original papers in applied mathematics which have a close connection with applications. An author index appears in the last issue of each volume. This journal, published quarterly by Brown University with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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