具有一般凸通量的标量守恒定律的最小熵条件

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
G. Cao, Guifang Chen
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For such scalar conservation laws, we prove that a single entropy-entropy flux pair <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis eta left-parenthesis u right-parenthesis comma q left-parenthesis u right-parenthesis right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>η<!-- η --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>q</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(\\eta (u),q(u))</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"eta left-parenthesis u right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>η<!-- η --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\eta (u)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript normal l normal o normal c Superscript normal infinity\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mi>L</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">l</mml:mi>\n <mml:mi mathvariant=\"normal\">o</mml:mi>\n <mml:mi mathvariant=\"normal\">c</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">L^\\infty _{\\mathrm { loc}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> that satisfy the inequality: <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"eta left-parenthesis u right-parenthesis Subscript t Baseline plus q left-parenthesis u right-parenthesis Subscript x Baseline less-than-or-equal-to mu\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>η<!-- η --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:mi>q</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>x</mml:mi>\n </mml:msub>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mi>μ<!-- μ --></mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\eta (u)_t+q(u)_x\\leq \\mu</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the distributional sense for some non-negative Radon measure <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\">\n <mml:semantics>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\mu</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Furthermore, we extend this result to the class of weak solutions in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript normal l normal o normal c Superscript p\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mi>L</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">l</mml:mi>\n <mml:mi mathvariant=\"normal\">o</mml:mi>\n <mml:mi mathvariant=\"normal\">c</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">L^p_{\\mathrm {loc}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, based on the asymptotic behavior of the flux function <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis u right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f(u)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and the entropy function <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"eta left-parenthesis u right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>η<!-- η --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\eta (u)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.</p>","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Minimal entropy conditions for scalar conservation laws with general convex fluxes\",\"authors\":\"G. Cao, Guifang Chen\",\"doi\":\"10.1090/qam/1669\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis eta left-parenthesis u right-parenthesis comma q left-parenthesis u right-parenthesis right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>η<!-- η --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>,</mml:mo>\\n <mml:mi>q</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(\\\\eta (u),q(u))</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"eta left-parenthesis u right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>η<!-- η --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\eta (u)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript normal l normal o normal c Superscript normal infinity\\\">\\n <mml:semantics>\\n <mml:msubsup>\\n <mml:mi>L</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">l</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">o</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">c</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:msubsup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L^\\\\infty _{\\\\mathrm { loc}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> that satisfy the inequality: <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"eta left-parenthesis u right-parenthesis Subscript t Baseline plus q left-parenthesis u right-parenthesis Subscript x Baseline less-than-or-equal-to mu\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>η<!-- η --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mi>t</mml:mi>\\n </mml:msub>\\n <mml:mo>+</mml:mo>\\n <mml:mi>q</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mi>x</mml:mi>\\n </mml:msub>\\n <mml:mo>≤<!-- ≤ --></mml:mo>\\n <mml:mi>μ<!-- μ --></mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\eta (u)_t+q(u)_x\\\\leq \\\\mu</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in the distributional sense for some non-negative Radon measure <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"mu\\\">\\n <mml:semantics>\\n <mml:mi>μ<!-- μ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mu</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Furthermore, we extend this result to the class of weak solutions in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript normal l normal o normal c Superscript p\\\">\\n <mml:semantics>\\n <mml:msubsup>\\n <mml:mi>L</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">l</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">o</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">c</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:mi>p</mml:mi>\\n </mml:msubsup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L^p_{\\\\mathrm {loc}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, based on the asymptotic behavior of the flux function <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"f left-parenthesis u right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>f</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">f(u)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and the entropy function <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"eta left-parenthesis u right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>η<!-- η --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\eta (u)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.</p>\",\"PeriodicalId\":20964,\"journal\":{\"name\":\"Quarterly of Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-12-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quarterly of Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/qam/1669\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly of Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/qam/1669","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

摘要

我们研究了具有一般凸通量函数的一维标量守恒定律的最小熵条件。对于这种标量守恒定律,我们证明了单个熵熵通量对(η(u),q(u))(\eta(u),q(u))与严格凸性的η(u)x≤μ\eta(u)_t+q(u)_x\leq\mu。此外,我们基于通量函数f(u)f(u)和熵函数η(u)\eta(u)在无穷大处的渐近行为,将这一结果推广到L L o c p L ^ p{\mathrm{loc}}中的一类弱解。这些证明基于一维标量守恒定律的熵解和相应的Hamilton-Jacobi方程的粘性解之间的等价性,以及补偿紧致性理论中类似使用的双线性形式和换向器估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Minimal entropy conditions for scalar conservation laws with general convex fluxes

We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair ( η ( u ) , q ( u ) ) (\eta (u),q(u)) with η ( u ) \eta (u) of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in L l o c L^\infty _{\mathrm { loc}} that satisfy the inequality: η ( u ) t + q ( u ) x μ \eta (u)_t+q(u)_x\leq \mu in the distributional sense for some non-negative Radon measure μ \mu . Furthermore, we extend this result to the class of weak solutions in L l o c p L^p_{\mathrm {loc}} , based on the asymptotic behavior of the flux function f ( u ) f(u) and the entropy function η ( u ) \eta (u) at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.

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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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