Pub Date : 2022-09-29DOI: 10.1007/s11425-021-2043-7
Wenqing Tao
{"title":"The Heisenberg double of the quantum Euclidean group and its representations","authors":"Wenqing Tao","doi":"10.1007/s11425-021-2043-7","DOIUrl":"https://doi.org/10.1007/s11425-021-2043-7","url":null,"abstract":"","PeriodicalId":54444,"journal":{"name":"Science China-Mathematics","volume":"124 1","pages":"1713 - 1736"},"PeriodicalIF":1.4,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73465563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-08DOI: 10.48550/arXiv.2209.03647
Xiao Li, Zhonghua Qiao, Cheng Wang
In this paper, we study a second-order accurate and linear numerical scheme for the nonlocal Cahn-Hilliard equation. The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization, and by applying the Fourier spectral collocation to the spatial discretization. In addition, two stabilization terms in different forms are added for the sake of the numerical stability. We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme, combined with the rough error estimate and the refined estimate. By regarding the numerical solution as a small perturbation of the exact solution, we are able to justify the discrete ℓ ∞ bound of the numerical solution, as a result of the rough error estimate. Subsequently, the refined error estimate is derived to obtain the optimal rate of convergence, following the established ℓ ∞ bound of the numerical solution. Moreover, the energy stability is also rigorously proved with respect to a modified energy. The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work, and the energy stability estimate has greatly improved the corresponding result therein.
{"title":"Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation","authors":"Xiao Li, Zhonghua Qiao, Cheng Wang","doi":"10.48550/arXiv.2209.03647","DOIUrl":"https://doi.org/10.48550/arXiv.2209.03647","url":null,"abstract":"In this paper, we study a second-order accurate and linear numerical scheme for the nonlocal Cahn-Hilliard equation. The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization, and by applying the Fourier spectral collocation to the spatial discretization. In addition, two stabilization terms in different forms are added for the sake of the numerical stability. We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme, combined with the rough error estimate and the refined estimate. By regarding the numerical solution as a small perturbation of the exact solution, we are able to justify the discrete ℓ ∞ bound of the numerical solution, as a result of the rough error estimate. Subsequently, the refined error estimate is derived to obtain the optimal rate of convergence, following the established ℓ ∞ bound of the numerical solution. Moreover, the energy stability is also rigorously proved with respect to a modified energy. The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work, and the energy stability estimate has greatly improved the corresponding result therein.","PeriodicalId":54444,"journal":{"name":"Science China-Mathematics","volume":"5 1","pages":"1-24"},"PeriodicalIF":1.4,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80155954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-26DOI: 10.1007/s11425-021-1991-3
Qing Guo, Yuxia Guo, Shuangjie Peng
{"title":"New existence of multi-spike solutions for the fractional Schrödinger equations","authors":"Qing Guo, Yuxia Guo, Shuangjie Peng","doi":"10.1007/s11425-021-1991-3","DOIUrl":"https://doi.org/10.1007/s11425-021-1991-3","url":null,"abstract":"","PeriodicalId":54444,"journal":{"name":"Science China-Mathematics","volume":"168 1 1","pages":"977-1002"},"PeriodicalIF":1.4,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73078407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-23DOI: 10.1007/s11425-022-1993-x
C. Gui, Yeyao Hu, Weihong Xie
{"title":"Bubbling solutions of fourth order mean field equations on $mathbb{S}^{4}$","authors":"C. Gui, Yeyao Hu, Weihong Xie","doi":"10.1007/s11425-022-1993-x","DOIUrl":"https://doi.org/10.1007/s11425-022-1993-x","url":null,"abstract":"","PeriodicalId":54444,"journal":{"name":"Science China-Mathematics","volume":"31 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77557129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-18DOI: 10.1007/s11425-021-1963-3
C. Wang, Pengfu Wang
{"title":"The Morse index of minimal products of minimal submanifolds in spheres","authors":"C. Wang, Pengfu Wang","doi":"10.1007/s11425-021-1963-3","DOIUrl":"https://doi.org/10.1007/s11425-021-1963-3","url":null,"abstract":"","PeriodicalId":54444,"journal":{"name":"Science China-Mathematics","volume":"9 1","pages":"799-818"},"PeriodicalIF":1.4,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90568505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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