{"title":"Hilbert空间上$$\\varvec{\\exp (A^{-1}t)A^{-1}}$$的衰减率及初始数据光滑的Crank-Nicolson格式","authors":"Masashi Wakaiki","doi":"10.1007/s00020-023-02748-1","DOIUrl":null,"url":null,"abstract":"Abstract This paper is concerned with the decay rate of $$e^{A^{-1}t}A^{-1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for the generator A of an exponentially stable $$C_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup on a Hilbert space. To estimate the decay rate of $$e^{A^{-1}t}A^{-1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable $$C_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup whose generator is normal.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Decay Rate of $$\\\\varvec{\\\\exp (A^{-1}t)A^{-1}}$$ on a Hilbert Space and the Crank–Nicolson Scheme with Smooth Initial Data\",\"authors\":\"Masashi Wakaiki\",\"doi\":\"10.1007/s00020-023-02748-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract This paper is concerned with the decay rate of $$e^{A^{-1}t}A^{-1}$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for the generator A of an exponentially stable $$C_0$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup on a Hilbert space. To estimate the decay rate of $$e^{A^{-1}t}A^{-1}$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable $$C_0$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup whose generator is normal.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-11-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00020-023-02748-1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00020-023-02748-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
研究Hilbert空间上指数稳定的$$C_0$$ c0 -半群的生成子A的衰减率($$e^{A^{-1}t}A^{-1}$$ e A - 1)和(A - 1)。为了估计$$e^{A^{-1}t}A^{-1}$$ e A - 1 t A - 1的衰减率,我们应用了有界泛函演算。利用这个估计和Lyapunov方程,我们还研究了具有光滑初始数据的Crank-Nicolson格式的量化渐近行为。一个类似的论证被应用于多项式稳定的$$C_0$$ C 0 -半群,它的生成器是正常的。
Decay Rate of $$\varvec{\exp (A^{-1}t)A^{-1}}$$ on a Hilbert Space and the Crank–Nicolson Scheme with Smooth Initial Data
Abstract This paper is concerned with the decay rate of $$e^{A^{-1}t}A^{-1}$$ eA-1tA-1 for the generator A of an exponentially stable $$C_0$$ C0 -semigroup on a Hilbert space. To estimate the decay rate of $$e^{A^{-1}t}A^{-1}$$ eA-1tA-1 , we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable $$C_0$$ C0 -semigroup whose generator is normal.