Abstract
In this work, we establish the maximal ell ^p-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order alpha in (0,2), alpha ne 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.
Highlights
Maximal L p-regularity is an important mathematical tool in studying the existence, uniqueness and regularity of solutions of nonlinear partial differential equations of parabolic type
Beyond Hilbert spaces, an important and very useful characterization of the maximal L p-regularity was given by Weis [48] on UMD spaces in terms of the R-boundedness of a family of operators using the resolvent R(z; A) := (z − A)−1; see Theorem 1 in Sect. 2 for details
An important question from the perspective of numerical analysis is whether such maximal regularity estimates carry over to time-stepping schemes for discretizing the parabolic problem (1.1), which have important applications in numerical analysis of nonlinear parabolic problems [1,2,17,29,33]
Summary
Maximal L p-regularity is an important mathematical tool in studying the existence, uniqueness and regularity of solutions of nonlinear partial differential equations of parabolic type. The author established the maximal p-regularity for the problem, under the condition that the set {δ(z)(δ(z)− T )−1 : |z| = 1, z = 1} is R-bounded, with δ(z) = z1−α(1 − z)α, following the work of Blunck [10]. It can be interpreted as a time-stepping scheme: upon letting T = τ α A and f n = τ α gn, we get τ −αΔαun = Aun + gn. We address the following question: Under which conditions do the time discretizations of (1.3) preserve the maximal p-regularity, uniformly in the time step size τ ? We refer readers to the review [30] for details
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