{"title":"软势费米-狄拉克玻尔兹曼方程解的存在唯一性","authors":"Zongguang Li","doi":"10.1090/qam/1681","DOIUrl":null,"url":null,"abstract":"In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta\"> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that can decrease from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta equals 1\"> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\delta =1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the Fermi-Dirac particles to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\delta =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <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>. The key point is that the obtained estimates are uniform in the quantum parameter <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 greater-than delta less-than-or-equal-to 1\"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">0> \\delta \\leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence and uniqueness of solutions to the Fermi-Dirac Boltzmann equation for soft potentials\",\"authors\":\"Zongguang Li\",\"doi\":\"10.1090/qam/1681\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"delta\\\"> <mml:semantics> <mml:mi>δ<!-- δ --></mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that can decrease from <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"delta equals 1\\\"> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\delta =1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the Fermi-Dirac particles to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"delta equals 0\\\"> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\delta =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <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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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 LT∞Lv,x∞∩LT∞Lx∞Lv1L^{\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)≤1δF(t,x,v)\leq \frac {1}{\delta }. The key point is that the obtained estimates are uniform in the quantum parameter 0>δ≤10> \delta \leq 1.
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