{"title":"随机 Kuramoto-Sivashinsky 方程的后向问题:条件稳定性和数值解","authors":"Zewen Wang , Weili Zhu , Bin Wu , Bin Hu","doi":"10.1016/j.jmaa.2024.128988","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we consider a backward problem in time for a linear stochastic Kuramoto-Sivashinsky equation. Firstly, we present two Carleman estimates incorporating weight functions independent of the variable <em>x</em> for the stochastic Kuramoto-Sivashinsky equation. Subsequently, we employ these two Carleman estimates to establish conditional stability for the backward problem in two distinct scenarios: when <span><math><mn>0</mn><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>T</mi></math></span> and when <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span>. Lastly, we transform the backward problem in time into the minimization of a regularized Tikhonov functional. This functional is solved by the conjugate gradient algorithm based on the gradient formula tailored for the regularized functional. Numerical examples related to the recovery of continuous and discontinuous initial values illustrate the effectiveness of the conjugate gradient algorithm.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128988"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A backward problem for stochastic Kuramoto-Sivashinsky equation: Conditional stability and numerical solution\",\"authors\":\"Zewen Wang , Weili Zhu , Bin Wu , Bin Hu\",\"doi\":\"10.1016/j.jmaa.2024.128988\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, we consider a backward problem in time for a linear stochastic Kuramoto-Sivashinsky equation. Firstly, we present two Carleman estimates incorporating weight functions independent of the variable <em>x</em> for the stochastic Kuramoto-Sivashinsky equation. Subsequently, we employ these two Carleman estimates to establish conditional stability for the backward problem in two distinct scenarios: when <span><math><mn>0</mn><mo><</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>T</mi></math></span> and when <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span>. Lastly, we transform the backward problem in time into the minimization of a regularized Tikhonov functional. This functional is solved by the conjugate gradient algorithm based on the gradient formula tailored for the regularized functional. Numerical examples related to the recovery of continuous and discontinuous initial values illustrate the effectiveness of the conjugate gradient algorithm.</div></div>\",\"PeriodicalId\":50147,\"journal\":{\"name\":\"Journal of Mathematical Analysis and Applications\",\"volume\":\"543 2\",\"pages\":\"Article 128988\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-10-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Analysis and Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022247X24009107\",\"RegionNum\":3,\"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 Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24009107","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
A backward problem for stochastic Kuramoto-Sivashinsky equation: Conditional stability and numerical solution
In this paper, we consider a backward problem in time for a linear stochastic Kuramoto-Sivashinsky equation. Firstly, we present two Carleman estimates incorporating weight functions independent of the variable x for the stochastic Kuramoto-Sivashinsky equation. Subsequently, we employ these two Carleman estimates to establish conditional stability for the backward problem in two distinct scenarios: when and when . Lastly, we transform the backward problem in time into the minimization of a regularized Tikhonov functional. This functional is solved by the conjugate gradient algorithm based on the gradient formula tailored for the regularized functional. Numerical examples related to the recovery of continuous and discontinuous initial values illustrate the effectiveness of the conjugate gradient algorithm.
期刊介绍:
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
Papers are sought which employ one or more of the following areas of classical analysis:
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