Discrete maximal regularity of time-stepping schemes for fractional evolution equations

Abstract

In this work, we establish the maximal ℓp-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order α∈ (0 , 2) , 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.