In this work, we establish the maximal $$\ell ^p$$ℓp-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order $$\alpha \in (0,2)$$α∈(0,2), $$\alpha \ne 1$$α≠1, in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank–Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis (Math Ann 319:735–758, 2001. doi:10.1007/PL00004457) and its discrete analogue due to Blunck (Stud Math 146:157–176, 2001. doi:10.4064/sm146-2-3). These results generalize the corresponding results for parabolic problems.

中文翻译：

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分数阶演化方程时间步长方案的离散最大正则性

在这项工作中，我们为分数演化模型的几个时间步长方案建立了最大 $$\ell ^p$$ℓp-正则性，该模型涉及阶数为 $$\alpha \in (0,2)$$ 的分数导数α∈(0,2), $$\alpha \ne 1$$α≠1, 及时。这些方案包括由后向欧拉方法和二阶后向差分公式生成的卷积正交、L1 方案、显式欧拉方法和 Crank-Nicolson 方法的分数变体。分析的主要工具包括由 Weis (Math Ann 319:735–758, 2001. doi:10.1007/PL00004457) 及其离散类似物 (Stud Math 146:157–1716, 200 .doi:10.4064/sm146-2-3)。这些结果概括了抛物线问题的相应结果。