Characterizations of the Crandall--Pazy Class of $C_0$-semigroups on Hilbert Spaces and Their Application to Decay Estimates
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| Format: | Preprint |
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2024
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| _version_ | 1866916647618478080 |
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| author | Wakaiki, Masashi |
| author_facet | Wakaiki, Masashi |
| contents | We investigate immediately differentiable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$ satisfying $\sup_{0 < t <1} t^{1/β}\|Ae^{-tA}\| < \infty$ for some $0 < β\leq 1$. Such $C_0$-semigroups are referred to as the Crandall--Pazy class of $C_0$-semigroups. In the Hilbert space setting, we present two characterizations of the Crandall--Pazy class. We then apply these characterizations to estimate decay rates for Crank--Nicolson schemes with smooth initial data when the associated abstract Cauchy problem is governed by an exponentially stable $C_0$-semigroup in the Crandall--Pazy class. The first approach is based on a functional calculus called the $\mathcal{B}$-calculus. The second approach builds upon estimates derived from Lyapunov equations and improves the decay estimate obtained in the first approach, under the additional assumption that $-A^{-1}$ generates a bounded $C_0$-semigroup. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizations of the Crandall--Pazy Class of $C_0$-semigroups on Hilbert Spaces and Their Application to Decay Estimates Wakaiki, Masashi Functional Analysis Numerical Analysis We investigate immediately differentiable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$ satisfying $\sup_{0 < t <1} t^{1/β}\|Ae^{-tA}\| < \infty$ for some $0 < β\leq 1$. Such $C_0$-semigroups are referred to as the Crandall--Pazy class of $C_0$-semigroups. In the Hilbert space setting, we present two characterizations of the Crandall--Pazy class. We then apply these characterizations to estimate decay rates for Crank--Nicolson schemes with smooth initial data when the associated abstract Cauchy problem is governed by an exponentially stable $C_0$-semigroup in the Crandall--Pazy class. The first approach is based on a functional calculus called the $\mathcal{B}$-calculus. The second approach builds upon estimates derived from Lyapunov equations and improves the decay estimate obtained in the first approach, under the additional assumption that $-A^{-1}$ generates a bounded $C_0$-semigroup. |
| title | Characterizations of the Crandall--Pazy Class of $C_0$-semigroups on Hilbert Spaces and Their Application to Decay Estimates |
| topic | Functional Analysis Numerical Analysis |
| url | https://arxiv.org/abs/2410.19387 |