{"title":"具有时间和空间相关扰动的一维阻尼可压缩欧拉方程的全局存在性和炸毁问题","authors":"","doi":"10.1016/j.na.2024.113658","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider the 1D Euler equation with time and space dependent damping term <span><math><mrow><mo>−</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mi>v</mi></mrow></math></span>. It has long been known that when <span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient <span><math><mi>a</mi></math></span> satisfies the following condition <span><span><span><math><mrow><mrow><mo>|</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>|</mo></mrow><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are integrable functions with <span><math><mi>t</mi></math></span> and <span><math><mi>x</mi></math></span>. Under this condition, we show the global existence and the blow-up with small initial data, when <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span> respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001779/pdfft?md5=9f5946837a904defdc71f1e5354348c9&pid=1-s2.0-S0362546X24001779-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation\",\"authors\":\"\",\"doi\":\"10.1016/j.na.2024.113658\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we consider the 1D Euler equation with time and space dependent damping term <span><math><mrow><mo>−</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mi>v</mi></mrow></math></span>. It has long been known that when <span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient <span><math><mi>a</mi></math></span> satisfies the following condition <span><span><span><math><mrow><mrow><mo>|</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>|</mo></mrow><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are integrable functions with <span><math><mi>t</mi></math></span> and <span><math><mi>x</mi></math></span>. Under this condition, we show the global existence and the blow-up with small initial data, when <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span> respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.</p></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001779/pdfft?md5=9f5946837a904defdc71f1e5354348c9&pid=1-s2.0-S0362546X24001779-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001779\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001779","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文考虑的是一维欧拉方程,其阻尼项-a(t,x)v 与时间和空间有关。众所周知,当 a(t,x) 为正常数或 0 时,解分别在时间上全局存在或在有限时间内炸毁。在本文中,我们将证明这些结果在与时间和空间相关的扰动方面是不变的。我们假设系数 a 满足以下条件 |a(t,x)-μ0|≤a1(t)+a2(x),其中 μ0≥0,a1 和 a2 是与 t 和 x 有关的可积分函数。在此条件下,我们分别证明了当 μ0>0 和 μ0=0 时的全局存在性和小初始数据下的炸毁。证明的关键在于利用特征曲线将空间划分为与时间相关的区域。
Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation
In this paper, we consider the 1D Euler equation with time and space dependent damping term . It has long been known that when is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient satisfies the following condition where and and are integrable functions with and . Under this condition, we show the global existence and the blow-up with small initial data, when and respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.
期刊介绍:
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