{"title":"带阻尼的可压缩欧拉方程的时间-渐近展开","authors":"Feimin Huang, Xiaochun Wu","doi":"10.1112/jlms.12908","DOIUrl":null,"url":null,"abstract":"<p>In 1992, Hsiao and Liu first showed that the solution to the compressible Euler equations with damping time-asymptotically converges to the diffusion wave <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mover>\n <mi>v</mi>\n <mo>¯</mo>\n </mover>\n <mo>,</mo>\n <mover>\n <mi>u</mi>\n <mo>¯</mo>\n </mover>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\bar{v}, \\bar{u})$</annotation>\n </semantics></math> of the porous media equation. Geng et al. proposed a time-asymptotic expansion around the diffusion wave <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mover>\n <mi>v</mi>\n <mo>¯</mo>\n </mover>\n <mo>,</mo>\n <mover>\n <mi>u</mi>\n <mo>¯</mo>\n </mover>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\bar{v}, \\bar{u})$</annotation>\n </semantics></math>, which is a better asymptotic profile than <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mover>\n <mi>v</mi>\n <mo>¯</mo>\n </mover>\n <mo>,</mo>\n <mover>\n <mi>u</mi>\n <mo>¯</mo>\n </mover>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\bar{v}, \\bar{u})$</annotation>\n </semantics></math>. In this paper, we rigorously justify the time-asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time-asymptotic expansion.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The time-asymptotic expansion for the compressible Euler equations with damping\",\"authors\":\"Feimin Huang, Xiaochun Wu\",\"doi\":\"10.1112/jlms.12908\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In 1992, Hsiao and Liu first showed that the solution to the compressible Euler equations with damping time-asymptotically converges to the diffusion wave <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mover>\\n <mi>v</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>,</mo>\\n <mover>\\n <mi>u</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\bar{v}, \\\\bar{u})$</annotation>\\n </semantics></math> of the porous media equation. Geng et al. proposed a time-asymptotic expansion around the diffusion wave <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mover>\\n <mi>v</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>,</mo>\\n <mover>\\n <mi>u</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\bar{v}, \\\\bar{u})$</annotation>\\n </semantics></math>, which is a better asymptotic profile than <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mover>\\n <mi>v</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>,</mo>\\n <mover>\\n <mi>u</mi>\\n <mo>¯</mo>\\n </mover>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\bar{v}, \\\\bar{u})$</annotation>\\n </semantics></math>. In this paper, we rigorously justify the time-asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time-asymptotic expansion.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12908\",\"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":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12908","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
1992 年,Hsiao 和 Liu 首次证明带阻尼的可压缩欧拉方程的解在时间上近似收敛于多孔介质方程的扩散波 ( v ¯ , u ¯ ) $(\bar{v}, \bar{u})$ 。Geng 等人提出了围绕扩散波 ( v ¯ , u ¯ ) $(\bar{v}, \bar{u})$ 的时间渐近展开,这是一个比 ( v ¯ , u ¯ ) $(\bar{v}, \bar{u})$ 更好的渐近剖面。在本文中,我们通过近似格林函数方法和能量估计严格论证了时间渐近展开。此外,时间-渐近展开精确地描述了带阻尼的可压缩欧拉方程解的大时间行为。
The time-asymptotic expansion for the compressible Euler equations with damping
In 1992, Hsiao and Liu first showed that the solution to the compressible Euler equations with damping time-asymptotically converges to the diffusion wave of the porous media equation. Geng et al. proposed a time-asymptotic expansion around the diffusion wave , which is a better asymptotic profile than . In this paper, we rigorously justify the time-asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time-asymptotic expansion.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.