Discrete maximal regularity for Volterra equations and nonlocal time-stepping schemes

DISCRETE MAXIMAL REGULARITY FOR VOLTERRA EQUATIONS AND NONLOCAL TIME-STEPPING SCHEMES CARLOS LIZAMA AND MARINA MURILLO-ARCILA Abstract. In this paper we investigate conditions for maximal regularity of Volterra equations defined on the Lebesgue space of sequences `p(Z) by using Bl¨unck’s theorem on the equivalence between operator-valued `p-multipliers and the notion of R-boundedness. We show sufficient conditions for maximal `p−`qregularity of solutions of such problems solely in terms of the data. We also explain the significance of kernel sequences in the theory of viscoelasticity, establishing a new and surprising connection with schemes of approximation of fractional models. 1. Introduction In this paper we study a discrete time formulation for a class of integro partial differential equations with delay whose prototype equation is, (1) u(t) = Zt −∞ a(t−s)Au(s)ds +f(t, u(t)), t ∈R, where Ais a closed linear operator defined on a Banach space X. This model appears in linear viscoelasticity theory, more precisely in the study of viscoelastic fluids, heat conduction with memory and electrodynamics with memory, among others [23]. In such applications the operator Ais typically the negative Laplacian in X=L2(Ω),the elasticity operator, the Stokes operator, or the biharmonic ∆2,equipped with suitable boundary conditions. Equation (1) includes many important models in PDEs. For instance, taking a(t)≡1 we arrive at u0(t) = Au(t) + g(t, u(t)), t > 0, with prescribed initial condition u(0) = ϕ(t), t ≤0.A second important example is a(t) = tα−1 Γ(α)where α > 0.In such a case, we have the time-fractional PDE: (2) ∂β tu(t) = Au(t) + g(t, u(t)), t > 0, where βdepends on the value of α. The existence and uniqueness of solutions to (1), or to the equivalent initial value problem (3) u(t) = Zt 0 a(t−s)Au(s)ds + (Qu)(t) + f(t, u(t)) u(0) = ϕ(0), where (Qu)(t) := R0 −∞ a(t−s)Aϕ(s)ds has been studied extensively in recent years. 2010 Mathematics Subject Classification. 49M25; 49N60; 42A45; 47A58. Key words and phrases. Maximal regularity, time-stepping schemes, discrete Volterra equations, nonlocal operators. The first author is partially supported by FONDECYT grant number 1180041. The second author is supported by MEC, grant MTM2016-75963-P and GVA/2018/110. 1 2 C. LIZAMA AND M. MURILLO Numerical methods for time discretization of Volterra type equations (3) have been proposed by various authors. Recent developments on optimal `p−`qtime-space estimates for the corresponding linearized equations ask for time discretizations that preserve maximal regularity [4, 11, 12, 15, 18]. Indeed, it has been proved that this kind of regularity results in a discrete setting provide more precise error estimates for the numerical analysis of nonlinear PDEs [11, 13, 17]. This motivates the following first question: (Q1) Given a closed linear operator Aand a kernel a, for which time discretization scheme of (3) there is maximal regularity in the `p-setting?. On the other hand, it is well known in linear viscoelasticity theory that the kernel a(t) describes the behavior of the material under compression, which is a combination of two symbols called spring and dashpot in mechanical engineering. Some standards models are: Hookean solid, which is the prototype of spring; Newtonian fluid, being the prototype of dashpot; Kelvin-Voigt solid, that can be viewed as a spring and a dashpot in parallel; Maxwell fluid, which corresponds to a spring and a dashpot in series; Poynting Thompson solid, corresponding to Hookean solid and Maxell fluid in parallel; and power type materials, that represent an infinite series of springs and dashpots [8]. For instance, the kernel a(t) := gα(t) := tα−1 Γ(α)is the power type material function [23, pp.130.131] which coincides with a Newtonian fluid (dashpot) when α= 1 and a Hookean solid (spring) when α= 2. Our first key observation in this paper is that standard numerical time-steeping schemes of approximation for fractional evolution equations have the form [11]: ∂α τu(n) := τ−α(b?u)(n), where τis the step size, bis a scalar-valued sequence and the star denotes finite convolution. Examples of schemes that fit into this framework are the Backward Euler scheme, the second order backward difference formula and the L1-scheme, among others Lubich’s quadrature methods. See [11] for more details. For instance, the fractional Backward Euler scheme is obtained by convolution with the sequence b(n) = k−α(n) where kα(n) := α(α+ 1) ···(α+n−1) n!for n∈N;kα(0) := 1, α ∈R. In such a case, the discrete model (4) ∂α τu(n) = Au(n) + f(n), where fis given (but in concrete examples it corresponds to a time discretization of the nonlinear term g(·, u) in (2)) is equivalent to the abstract Volterra equation (5) u(n) = n X j=0 bτ(n−j)Au(j) + g(n), for the explicit sequence kernel bτ(n) = ταkα(n) and where, in relation to (4), g(n) := τα(kα? f)(n).See Example 4.1 below. A novel characteristic of our model (5) is that we will assume zero-padding in the past time, i.e. u(n) = 0 for all n=−1,−2, ... This is a concept used in digital signal processing and refers to adding zeros to a time domain signal to increase its length. In physical terms, it refers to the notion of causality. It is remarkable that this hypothesis for (5) coincides with the fact that Lubich’s convolution quadrature method implicitly considers zero-padding in DISCRETE MAXIMAL REGULARITY 3 the negative real axis, see [22, p.131, lines 1-2]. A second important observation is that by means of the recently defined Poisson transformation [2, 19], which represents a way of sampling time continuous systems into discrete time models, we have the remarkable relation [19, Example 3.3] kα(n) = Z∞ 0 pn(t)gα(t)dt, pn(t) := tn n!e−t. It reveals a new and surprising association between (5) and the Volterra equation u(t) = Zt 0 gα(t−s)Au(s)ds +g(t), t > 0. This relationship shows that time-stepping schemes and material functions are correlated, which in terms of the above analysis can be stated as follows: Power type material function ⇐⇒ Fractional backward Euler scheme. To be more precise, the above expression means that whereas in continuous time the kernel gα(t) represents the power material function of Volterra equations in viscoelasticity theory [23, p.131 (vi)], we have that in discrete time the sequence kernel k−α(n) is in correspondence to the characteristic function of the fractional backward Euler scheme by means of its generating function, i.e. we have, ∞ X j=0 k−α(j)ξj=δ(ξ) := (1 −ξ)α, see [11, identities (3.2)]. In particular, when α= 1 we have that δ(ξ) = (1 −ξ) is the characteristic function of the Backward Euler scheme method [11, Section 3.1] and, on the other hand, the function g1(t)≡1 is the kernel that represents a Newtonian fluid for Volterra equations in viscoelasticity theory [23, p.130 (ii)]. This correspondence is expressed as: Newtonian fluid ⇐⇒ Backward Euler scheme. In this way, we reveal new insights about a surprising relation between time-stepping schemes and linear viscoelasticity theory that, as far as we know, has not been previously observed in the existing literature. Our findings are also supported and coincident with very recent research in mechanical engineering, where methodologies to identify the desired viscoelastic behavior before choosing a real material has been developed [8]. In terms of time schemes, the connection stated in this paper can be interpreted as a methodology to identify the desired (and probably best) time stepping scheme in terms not only of the mathematical model but also of the characteristics of the real material that it models. In view of the above considerations, we address the following second question: (Q2) Which class of material type functions are time-stepping schemes correlated with?. Whereas question (Q2) is completely new, question (Q1) has been answered in many papers from different perspectives. For example, for the model (4) when α= 1, Ashyralyev, Piskarev and Weis [6] showed the discrete maximal regularity using the implicit Euler method; Kov´acs, Li and Lubich [15] proved the discrete maximal regularity for the Crank-Nicholson, BDF and A-stable RungeKutta methods; Kemmochi and Saito [12, 13] proved the maximal `pregularity for the θ-method, and Leykekhman and Vexler [16] proved the maximal regularity for Galerkin methods. In the fractional case, the first work is due to Lizama [20]. Recently, a further important step was given by Jin, Li and Zhou [11]. They have provided an analysis for several time-stepping schemes, including the convolution quadratures generated by the 4 C. LIZAMA AND M. MURILLO implicit Euler method and second-order backward difference formula, the L1 scheme, the explicit Euler method and a fractional variant of the Crank–Nicolson method. The paper is organized as follows: In Section 2, we introduce some basic concepts and results in the existing literature related to discrete time Fourier transforms and operatorvalued multiplier theorems that will be later needed. In Section 3, we answer question (Q1). We introduce the new concept of 1regular sequences that together with the R-boundedness of the symbol of the equation ensures maximal regularity of the (linearized) discretization scheme (3) with zero-padding in the past time, in the `p-setting. Using this new concept of 1-regularity, our first main contribution in this paper is to unify earlier approaches to (Q1) in a simpler way. More concretely, we prove the following theorem: Theorem Let Xbe a UMD space, 1 < p < ∞, and b:Z→Csuch that b(n) = 0 for all n∈Z−.Suppose that bis 1-regular and satisfies that the z-transform of bconverges on the complex unit disk and their radial limit exists for all t∈T0:= (−π, π)\ {0}.Moreover, bb(t)6= 0 for all t∈T0.Assume that (1 bb(t))t∈T0 ⊂ρ(A) and the set {(I−bb(t)A)−1:t∈T0}is R-bounded. Then the solution of the equation (6) u(n) = n X j=0 b(n−j)Au(j) + f(n), n ∈N0, defined as 0 by negative values of Z,satisfies the following maximal regularity estimate kuk`p(Z;X)+kb ? Auk`p(Z;X)≤Ckfk`p(Z;X). We also obtain an easier computable condition for the Hilbert space case, where Rboundedness can be replaced by norm boundedness. Our strategy of proof follows closely the recent works [11, 15, 19, 21] and employs the (discrete) operator-valued Fourier multiplier technique developed by Blunck [3, 7]. In Section 3, we show that our model (6) includes an important class of schemes of approximation given by the convolution quadrature method introduced by C. Lubich in 1988 [22], by identifying kernels b(n) with characteristic functions of each scheme by means of the discrete time Fourier transform. For example, we obtain the Explicit and Implicit Euler scheme, the Backward Euler and the Fractional Crank-Nicholson scheme among others and, as an application of our results, we provide maximal regularity estimates for the aforementioned time-stepping schemes. Moreover, we address question (Q2) and we obtain a direct correlation between time-stepping schemes and material type functions in viscoelasticity theory. 2. Analytical framework and notation In this section, we present some notations and preliminary results that will be needed throughout this work. Except for Lemma 2.4 below, all the quoted results are contained in the existing literature on the subject. Let Xbe a Banach space. We denote by S(Z;X) the space of vector-valued sequences f:Z→Xsuch that for each k∈N0there exists a constant Ck>0 satisfying pk(f) := supn∈Z|n|kkf(n)k< Ckand if X=Rwe denote the space as S(Z).We write as Cn per(R;X), n ∈ N0,the space of all 2π-periodic X-valued and n-times continuously differentiable functions defined in R.In what follows, we will denote T:= (−π, π) and T0:= (−π, π)\{0}.The space DISCRETE MAXIMAL REGULARITY 5 of test functions is the space C∞ per(T;X) := Tn∈N0Cn per(R;X).When X=Rwe simply write C∞ per(T). For each f∈`p(Z;X) we define the map (7) Tf(ψ) := hTf, ψi:= X n∈Z f(n)ψ(n), ψ ∈ S(Z), and it follows that Tf∈ S0(Z, X) = {T:S(Z)→X:Tis linear and continuous}. Remark 2.1.This mapping identifies `p(Z;X) with a subspace of S0(Z;X).Thus, a function f∈`p(Z;X) will be identified with Tf∈ S0(Z, X). There exists another natural mapping that identifies C∞ per(T;X) with a subspace of D0(T;X) = {T:C∞ per(T)→X:Tis linear and continuous}which assigns to each S∈C∞ per(T;X) the linear map LS(ϕ) := hLS, ϕi:= 1 2πZπ −π ϕ(t)S(t)dt, ϕ ∈C∞ per(T), and we have LS∈ D0(T;X). Definition 2.2. The discrete time Fourier transform F:S(Z;X)→C∞ per(T;X) is defined by Fϕ(t)≡bϕ(t) := ∞ X j=−∞ e−ijtϕ(j), t ∈(−π, π] and the inverse transform is given by (8) F−1ϕ(n)≡ˇϕ(n) := 1 2πZπ −π ϕ(t)eintdt, n ∈Z, where ϕ∈C∞ per(T;X). The discrete time Fourier transform (DTFT) between the spaces of distributions S0(Z;X) and D0(T;X) then follows as: (9) hFT, ψi≡F(T)(ψ) := b T(ψ)≡ hT, ˇ ψi, T ∈ S0(Z;X), ψ ∈C∞ per(T), whose inverse F−1:D0(T;X)→ S0(Z;X) is given by hF−1L, ψi≡F−1(L)(ψ) := ˇ L(ψ)≡ hL, b ψi, L ∈ D0(T;X), ψ ∈ S(Z). We finally present a technical lemma which will be necessary througout the paper. We first need the following definition. Definition 2.3. Given u:Z→Xand v:Z→Cthe convolution product between uand v is defined as (u∗v)(n) := n X j=−∞ u(n−j)v(j) = ∞ X j=0 u(j)v(n−j), n ∈Z, whenever the series converges. Moreover, the convolution of a distribution T∈ S0(Z, X) with a sequence a:N0→Cis defined by (10) hT∗a, ϕi:= hT, a ◦ϕi, ϕ ∈ S(Z), where (a◦ϕ)(n) := ∞ X j=0 a(j)ϕ(j+n), n ∈Z. 6 C. LIZAMA AND M. MURILLO Observe that a◦ϕ∈ S(Z).We denote (11) H(n) = 1 if n∈N0; 0 otherwise, the Heaviside sequence. For a:N0→Cand v:N0→Xwe denote by (12) (v ? a)(n) := n X j=0 v(n−j)a(j), n ∈N0, the finite convolution. We will need the following Lemma that corresponds to a weaker version of [21, Lemma 2.2]. Lemma 2.4. Let u, v :Z→Xbe given and a:N0→Cwhich is defined by 0for negative values of n. We assume that the series ˜a(z) := ∞ X n=0 a(n)zn,|z|<1, converges on the complex unit disk, and that the radial limit ˆa(t) = lim r%1˜a(re−it)exists for all t∈T0. Suppose that hu, ˇϕi=hv, (ϕ·ba−ˇ )ifor all ϕ∈C∞ per(T), where (ϕ·ba−ˇ )(n) := 1 2πZπ −πba(−t)ϕ(t)eintdt, n ∈Z. Then u(n) = H(n)(v ? a)(n)for all n∈Z. Proof. Given ϕ∈C∞ per(T) and by hypothesis, we obtain (ϕ·ba−ˇ )(n) = lim r%1 1 2πZπ −πh∞ X j=0 a(j)reijtieintϕ(t)dt = ∞ X j=0 a(j)1 2πZπ −π ei(j+n)tϕ(t)dt = ∞ X j=0 a(j) ˇϕ(j+n)=(a◦ˇϕ)(n), for all n∈Z. Hence hu, ˇϕi=hv, (ϕ·ba−ˇ )i=hv, a ◦ˇϕi=hv∗a, ˇϕi, for all ϕ∈C∞ per(T),that is X n∈Z (v∗a)(n) ˇϕ(n) = X n∈Z u(n) ˇϕ(n). Choosing ϕk(t) := e−ikt, k ∈Zwe find that (13) u(n)=(v∗a)(n) = n X j=−∞ v(n−j)a(j) = Pn j=0 v(n−j)a(j) if n∈N0; 0 otherwise, proving the Lemma.  We recall the notion of R-bounded sets and `p-multipliers in the space B(X, Y ) of bounded linear operators from Xinto Yendowed with the uniform operator topology. DISCRETE MAXIMAL REGULARITY 7 Definition 2.5. Let Xand Ybe Banach spaces. A subset Tof B(X, Y ) is called R-bounded if there is a constant c > 0 such that (14) k(T1x1, ..., Tnxn)kR≤ck(x1, ..., xn)kR, for all T1, ..., Tn∈ T , x1, ..., xn∈X, n ∈N,where k(x1, ..., xn)kR:= 1 2nX j∈{−1,1}n n X j=1 jxj, for x1, ..., xn∈X. See [3, Section 2.2] and [9] for more information about R-boundedness. We next recall the concept of `p-multiplier. Definition 2.6. [21] Let X,Ybe Banach spaces, 1 <p<∞.A function M∈C∞ per(T,B(X, Y )) is an `p-multiplier (from Xto Y) if there exists a bounded operator T:`p(Z;X)→`p(Z;Y) such that (15) X n∈Z (Tf)(n) ˇϕ(n) = X n∈Z (ϕ·M−ˇ )(n)f(n) for all f∈`p(Z;X) and all ϕ∈C∞ per(T).Here (ϕ·M−ˇ )(n) := 1 2πZπ −π eintϕ(t)M(−t)dt, n ∈Z. We now recall the following Fourier multiplier theorem for operator-valued symbols given by S. Blunck [7, 3]. This theorem provides sufficient conditions to ensure when an operatorvalued symbol is a multiplier, and allows to establish an equivalence between `p-multipliers and the notion of Rboundedness for the UMD class of Banach spaces. For more information about these spaces see [5, Section III.4.3-III.4.5]. Theorem 2.7. [7, Theorem 1.3] and [21] Let p∈(1,∞)and let X, Y be UMD spaces. Let M∈C∞ per(T0,B(X;Y)) such that the sets {M(t) : t∈T0}and (1 −eit)(1 + eit)M0(t) : t∈T0, are both R-bounded. Then Mis an `p-multiplier (from Xto Y) for 1<p<∞. The converse of Blunck’s theorem also holds for any Banach spaces X, Y as follows: Theorem 2.8. [7, Proposition 1.4] Let p∈(1,∞)and let X, Y be Banach spaces. Let M:T→ B(X;Y)be an operator valued function. Suppose that there is a bounded operator TM:lp(Z;X)→lp(Z;Y)such that (15) holds. Then the set {M(t) : t∈T} is R-bounded. 3. Abstract setting: A characterization of maximal `p-regularity Let b:N0→Cbe given and Xbe a Banach space. We denote: `p 0(N0;X) := {f∈`p(Z;X) : f(n) = 0 for all n=−1,−2, ...}. For a given vector-valued sequence f:N0→Xwe consider the abstract discrete equation (16) u(n) = n X j=0 b(n−j)Au(j) + f(n), n ∈N0, 8 C. LIZAMA AND M. MURILLO where Ais a closed linear operator with domain D(A) defined in a Banach space X. Observe that using the notation of the finite convolution we have u=b ? Au +f=Au ? b +f. We also recall that by [D(A)] we denote the domain of Aendowed with the graph norm. Definition 3.1. Let 1 <p<∞be given. We say that (16) has maximal `p 0-regularity if for each f∈`p 0(N0;X) there exists a unique solution u∈`p 0(N0; [D(A)]) of (16). For a sequence b:N0→Cextended to negative subscripts nby 0, we assume the following hypothesis: (H)b: The z-transform ˜ bof bconverges on the complex unit disk: ˜ b(z) := ∞ X n=0 b(n)zn,|z|<1, and the radial limit ˆ b(t) = lim r%1 ˜ b(re−it) exists for all t∈T0. We introduce the following definition. Observe that, in some sense, it corresponds to the discrete counterpart of the notion of k-regularity introduced in the paper [14]. See also [23]. Definition 3.2. Let k∈N0be given. A sequence b:Z→Cis called k-regular if there exists a constant c > 0 such that |((1 + eit)(1 −eit))n[ˆ b(t)](n)| ≤ c|ˆ b(t)|for all 1 ≤n≤kand all t∈T0. Remark 3.3.It follows immediately from the definition that if ˆ b(t) is 1-regular and there exists a sequence asuch that ˆa(t)ˆ b(t) = 1 then ais 1-regular, too. Example 3.4. An example of a k-regular sequence is given by the sequence defined as (17) b(n) = 1 2nif n∈N0; 0 otherwise. The 1-regularity follows easily since ˆ b(t) = 2(2 −e−it)−1and (1 + eit)(1 −eit)[ˆ b(t)]0 ˆ b(t)= −(e−it−1) (2−e−it)≤2. The case k > 1follows analogously. In the following result, we show the equivalence between the R-boundedness of the operator valued symbol of the equation (16) defined by (1 −ˆ b(t)A)−1and the fact that it is an `pmultiplier. Theorem 3.5. Let Xbe a UMD space, 1<p<∞,b:N0→Csatisfying hypothesis (H)b. Suppose that bis 1-regular, ˆ b(t)6= 0 for all t∈T0and (1 ˆ b(t))t∈T0 ⊂ρ(A), and denote M(t) := (1 −ˆ b(t)A)−1,then the following assertions are equivalent: (i) M(t)is an `p-multiplier from Xto [D(A)]. (ii) {M(t)}t∈T0is R-bounded. Proof. (ii) =⇒(i) We first prove that the set {(eit −1)(eit + 1)M0(t)}t∈T0is R-bounded. Given t∈T0and observing that M(t) = 1 ˆ b(t)[1 ˆ b(t)−A]−1we obtain that M0(t) = −M(t)[ˆ b(t)]0 ˆ b(t)+M(t)2[ˆ b(t)]0 ˆ b(t)t∈T0. (18) DISCRETE MAXIMAL REGULARITY 9 Therefore, (1 −eit)(1 + eit)M0(t) = −(1 −eit)(1 + eit)M(t)[ˆ b(t)]0 ˆ b(t) + (1 −eit)(1 + eit)M(t)2[ˆ b(t)]0 ˆ b(t), t ∈T0. From [3, Proposition 2.2.5], the hypothesis and the 1-regularity of bwe conclude that the set {(1 −eit)(1 + eit)M0(t) : t∈T0}is Rbounded and the claim is proven. Finally, using Theorem 2.7 we obtain (i). (i) =⇒(ii) By hypothesis we have that there exists a bounded operator Tsuch that (15) holds. Now, (ii) holds as a consequence of Theorem 2.8.  Theorem 3.6. Let Xbe a UMD space, 1<p<∞,b:N0→Csatisfying hypothesis (H)b. Suppose that bis 1-regular, ˆ b(t)6= 0 for all t∈T0and (1 ˆ b(t))t∈T0 ⊂ρ(A), and the set {M(t) := (1 −ˆ b(t)A)−1}t∈T0is R-bounded, then the set {AN(t) := ˆ b(t)A(1 − ˆ b(t)A)−1}t∈T0is an `p-multiplier from Xto X. Proof. We observe that N(t) = ˆ b(t)M(t) and AN(t) = M(t)−I. Therefore {AN(t)}t∈T0is an R-bounded set. Note that (1 −eit)(1 + eit)N0(t) = (1 −eit)(1 + eit)[ˆ b(t)]0 ˆ b(t) ˆ b(t)M(t) + (1 −eit)(1 + eit)M0(t)ˆ b(t) = (1 −eit)(1 + eit)[ˆ b(t)]0 ˆ b(t)N(t) + (1 −eit)(1 + eit)M0(t)ˆ b(t), and using (18) we get (1 −eit)(1 + eit)N0(t) = (1 −eit)(1 + eit)[ˆ b(t)]0 ˆ b(t)N(t) −(1 −eit)(1 + eit)[M(t)[ˆ b(t)]0+M(t)2[ˆ b(t)]0] = (1 −eit)(1 + eit)[ˆ b(t)]0 ˆ b(t)N(t) −(1 −eit)(1 + eit)[ˆ b(t)]0 ˆ b(t)N(t) + (1 −eit)(1 + eit)[ˆ b(t)]0 ˆ b(t)N(t)M(t) = (1 −eit)(1 + eit)[ˆ b(t)]0 ˆ b(t)N(t)M(t), and since bis 1-regular, we conclude from Theorem 2.7 the assertion.  The following theorem shows sufficient conditions for maximal `p 0-regularity. It corresponds to the main abstract result of this paper. 16 C. LIZAMA AND M. MURILLO Now, taking into account the identity Z∞ 0 e−λttβdt =Γ(β+ 1) λβ, β > 0, λ > 0,we obtain Z∞ 0 pn(t)gα(t)e−atdt =kα(n) (a+ 1)n+α, a > −1. In particular, with a= 2 we get R∞ 0pn(t)gα(t)e−2tdt =kα(n) 3n+αand with a=−1/2we have R∞ 0pn(t)gα(t)e−1 2tdt = 2αkα(n).Therefore, the convolution property of the Poisson transform [19] produces Z∞ 0 pn(t)[(gαe−1 2·)∗(gαe−2·)](t)dt = n X j=0 2αkα(n−j)kα(j) 3j+α=aα(n). This identity shows a correspondence between aα(n)and the kernel aα(t) := [(gαe−1 2·)∗(gαe−2·)](t) = Zt 0 gα(t−s)e−1 2(t−s)gα(s)e−2sds, which, to our knowledge, cannot be clearly identified with any material function in viscoelasticity theory. However, this is in some sense similar to a material function of the form Power type ×Maxell fluid. Example 4.3. (Explicit Euler scheme) Let α > 0be given. The symbol δ(t) = (1 −ξ)α ξ defines the explicit Euler difference scheme as it has been studied in [11, Section 5] for the discretization of the fractional equation (33). Define the sequence (38) a(n) = kα(n−1) if n∈N 0 otherwise. Using the generation formula (31) we obtain ba(t) = e−it (1 −e−it)α, t ∈T. Note that (1 + eit)(1 −eit)[ba(t)]0 ba(t)=−i(1 + eit)(1 −eit) + iα(1 + eit) which shows 1-regularity of the sequence a(n)for all α > 0. Also, we have δ(t) = 1 ba(t)which means that condition (H)ais verified. Compared with Example 4.1, the Volterra model has in this example the slightly different form u(n) = n−1 X j=0 ταkα(n−j)Au(j) + n−1 X j=0 ταkα(n−j)f(j), but concerning the comparison with viscoelasticity theory, the same correlation than in Example 4.1 applies. This example shows that same material functions can define somewhat different approximation schemes. Example 4.4. (Fractional Crank-Nicholson scheme) Let 0< α < 2be given. The symbol δ(t) = τ−α(1 −ξ)α 1−α 2+α 2ξ DISCRETE MAXIMAL REGULARITY 17 defines the fractional Crank-Nicholson scheme [11, Section 6]. We define the sequence (39) a(n) =    τα(1 −α 2) if n= 0; τα(1 −α 2)kα(n) + ταα 2kα(n−1) if n= 1,2, ... 0 otherwise. For α= 1 it corresponds to the Crank-Nicholson scheme with step τ, that is δ(t) = 2 τ 1−ξ 1+ξ. See [15, Section 3]. The identity 1 2 1+ξ 1−ξ=−1 2+1 1−ξshows that a(n) = −τ 2δ0,n +τ, n ∈N0, and 0otherwise. Therefore, this scheme behaves essentially as the backward Euler scheme except by the gap in n= 0.Hence, it should correspond to a Newtonian fluid. In case α > 0the description of the sequence kernel in (39) shows the same behavior as a power type material function. Compare it with Example 4.1. Using the generation formula (31) we obtain after a simple computation ba(t) = τα1−α 2+α 2e−it (1 −e−it)α, t ∈T. Note that (1 + eit)(1 −eit)[ba(t)]0 ba(t)=−i(α/2)e−it(1 −eit)(1 + eit) 1−α 2+α 2e−it +iα(1 + eit) which shows 1-regularity of the sequence a(n)for all 0< α < 2.Moreover, it is clear that the condition (H) is satisfied. In fact, using the Cauchy product, we find b(n) = τ−α2 2−α n X j=0 k−α(n−j)(−α)j (2 −α)j, for each n∈N0and defined by 0otherwise. We summarize this example with the following correspondences Crank-Nicholson scheme ⇐⇒ Newtonian fluid and Fractional Crank-Nicholson scheme ⇐⇒ Power type material function. We remark that the above correspondence of the Crank-Nicholson scheme, that shows coincidence with the Euler scheme, is not at all surprising as it was also observed by Jin, Li and Zhen [11, Section 1] but in the context of `p-maximal regularity. 5. Further Applications In this section we give some applications of Theorem 3.7 that are connected with the different schemes of approximation considered so far. They also improve analogous results of earlier papers on this subject; cf. [11]. Example 5.1. We verify the conditions provided in Theorem 3.7 in order to show the existence and uniqueness of `p 0(N0, L2(R)) solutions for the following equation (40) u(n, x) = n X j=0 1 2n−j ∂u(j, x) ∂x +f(n, x)x∈R, n ∈N0 where we set in equation (16) b(n) = 1 2n,A=∂ ∂x with domain D(A) = W1,2(R), and f∈`p 0(N0, L2(R)). Observe by Remark 3.4 that bis k-regular for all k∈Nand bb(t) = 18 C. LIZAMA AND M. MURILLO 2(2−e−it)−16= 0 for all t∈T0.See also [10] where this kernel bhas been previously considered. Moreover, since the operator Au =u0with domain D(A) = W1,2(R)generates a contraction C0-group on L2(R)there exists M > 0such that the following estimate holds: (41) ||(z−A)−1|| ≤ M <(z),for all zsuch that <(z)>0. It is easy to see that for equation (40) we have 1 ˆ b(t)=1 2(2 −e−it)and a simple computation shows that <(1 2(2 −e−it)) = 1 −1 2cos(t)>0. Using inequality (41) it follows that: sup t∈T0 1 |ˆ b(t)| 1 ˆ b(t)−A!−1= sup t∈T0 1 2|(2 −e−it)|1 2(2 −e−it)−1−A−1<∞. Then Theorem 3.7 asserts the existence of a unique solution u∈`p 0(N0;W1,2(R)) for each f∈`p 0(N0;L2(R)) satisfying a maximal regularity estimate. Using Example 4.1 we obtain the following result which complements [11, Theorem 5] as it reflects the backward Euler scheme for the fractional evolution equation ∂α tu(t) = Au(t) + ∂α tf(t), t > 0, where the backward Euler approximation is given by [19, 20] ∆αu(n) = τ−α∆m(km−α∗u)(n), m −1< α < m. Compared with [11, Theorem 5] (see also [15, Theorem 3.1] in case α= 1) our hypothesis are weaker. Moreover, we do not make any restriction on the values of α > 0. Corollary 5.2. Let Xbe a UMD space, τ, α > 0and let Abe a closed linear operator defined in Xsuch that {τ−α(1−e−it)α}t∈T0⊂ρ(A)and the set {(τ−α(1−e−it)α−A)−1}t∈T0 is R-bounded. Then, for any f∈`p 0(N0, X)the backward Euler scheme u(n) = n X j=0 ταkα(n−j)Au(j) + f(n), n ∈N0, has a unique solution u∈`p 0(N0;X)that satisfies the following maximal `p 0-regularity estimate ∞ X n=0 ku(n)kp X!1/p +τα ∞ X n=0 kkα? u(n)kp X!1/p ≤C ∞ X n=0 kf(n)kp X!1/p , where the constant Cis independent of f, τ, α and A. Remark 5.3.If Ais an R-sectorial operator on Xof angle απ/2 (where 0 < α < 2) then the hypothesis on Aof the above corollary are automatically satisfied. Remark 5.4.From Example (4.3) we have that the explicit Euler method represented by the sequence kernel a(n) = ταkα(n−1) satisfies all the hypothesis of Theorem 3.7. Consequently, an analogous corollary about maximal `p 0-regularity for the explicit Euler scheme still holds. The following theorem is correllated with [11, Theorem 6]. It makes use of Example 4.2. Corollary 5.5. Let Xbe a UMD space, τ, α > 0and let Abe a closed linear operator defined in Xsuch that {(2τ)−α(3 −e−it)α(1 −e−it)α}t∈T0⊂ρ(A)and the set {(2τ)−α(3 −e−it)α(1 − DISCRETE MAXIMAL REGULARITY 19 e−it)α−A)−1}t∈T0is R-bounded. Then, for any f∈`p 0(N0, X)the second order backward difference scheme u(n) = 2τ 3αn X j=0 n−j X l=0 kα(n−j−l)3−lkα(l)Au(j) + f(n) =2τ 3αn X l=0 3n−lkα(n−l)A(kα? u)(l) + f(n), n ∈N0, has a unique solution u∈`p 0(N0;X)that satisfies the following maximal `p 0regularity estimate ∞ X n=0 ku(n)kp X!1/p + (2τ 3)α ∞ X n=0 k n X l=0 3n−lkα(n−l)A(kα? u)(l)kp X!1/p ≤C ∞ X n=0 kf(n)kp X!1/p , where the constant Cis independent of f, τ, α and A. Acknowledgments The first author was partially supported by FONDECYT, Grant No 1180041. The second author was supported by was supported by MEC, grant MTM2016-75963-P and GVA/2018/110. References [1] L.Abadias, C. Lizama, P.J. Miana and M.P. Velasco. Ces´aro sums and algebra homomorphisms of bounded operators. Israel J. Math., 216 (1) (2016), 471–505. [2] L.Abadias, C. Lizama, P.J. Miana and M.P. Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discr. Cont. Dyn. Systems, Series A, 39 (5) (2019), 2679–2708. [3] R. P. Agarwal, C. Cuevas and C. Lizama. Regularity of Difference Equations on Banach Spaces, SpringerVerlag, Cham, 2014. [4] G. Akrivis, B. Li and C. Lubich. Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comp. 86 (306) (2017), 1527–1552. [5] H. Amann. Linear and Quasilinear Parabolic Problems, Monographs in Mathematics, 89, Birkh¨auserVerlag, Basel, 1995. [6] A. Ashyralyev, S. Piskarev, and L. Weis. On well-posedness of difference schemes for abstract parabolic equations in Lp([0, T ]; E)spaces. Numer. Funct. Anal. Optim., 23 (7-8) (2002), 669–693. [7] S. Bl¨unck. Maximal regularity of discrete and continuous time evolution equations. Studia Math. 146 (2) (2001), 157–176. [8] R.E. Corman, L. Rao, N. Ashwin-Bharadwaj, J.T. Allison, R.H. Ewoldt. Setting material function design targets for linear viscoelastic materials and structures. J. Mech. Des. 138 (5) (2016), 051402 [9] R. Denk, M. Hieber and J. Pr¨uss. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166 (788), 2003. [10] S. Elaydi. Stability and asymptoticity of Volterra difference equations: A progress report. J. Comp. Appl. Math 228 (2009) 504–513. [11] B. Jin, B. Li and Z. Zhou. Discrete maximal regularity of time-stepping schemes for fractional evolution equations. Numer. Math. 138 (1) (2018), 101–131. [12] T. Kemmochi. Discrete maximal regularity for abstract Cauchy problems. Studia Math. 234 (3) (2016), 241–263. [13] T. Kemmochi and N. Saito. Discrete maximal regularity and the finite element method for parabolic equations. Numer. Math. 138 (4) (2018), 905–937. [14] V. Keyantuo and C. Lizama. H¨older continuous solutions for integro-differential equations and maximal regularity. J. Differential Equations, 230 (2006), 634–660. [15] B. Kov´acs, B. Li and C. Lubich. A-stable time discretizations preserve maximal parabolic regularity. SIAM J. Numer. Anal. 54 (6) (2016), 3600–3624. 20 C. LIZAMA AND M. MURILLO [16] D. Leykekhman and B. Vexler. Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math., 135(3) (2017), 923–952. [17] B. Li and W. Sun. Maximal regularity of fully discrete finite element solutions of parabolic equations. SIAM J. Numer. Anal. 55 (2) (2017), 521–542. [18] B. Li and W. Sun. Maximal Lpanalysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra. Math. Comp. 86 (305) (2017), 1071–1102. [19] C. Lizama. The Poisson distribution, abstract fractional difference equations, and stability. Proc. Amer. Math. Soc. 145 (9) (2017), 3809-3827. [20] C. Lizama. `p-maximal regularity for fractional difference equations on UMD spaces. Math. Nach., 288 (17/18) (2015), 2079–2092. [21] C. Lizama and M. Murillo-Arcila. Maximal regularity in `pspaces for discrete time fractional shifted equations J. Differential Equations. 263 (6) (2017), 3175–3196. [22] C. Lubich. Convolution quadrature and discretized operational calculus I. Numer. Math. 52 (1988), 129– 145. [23] J. Pr¨uss. Evolutionary Integral Equations and Applications. Springer, Basel Heidelberg, 1993. (C. Lizama) Departamento de Matem´ atica y Ciencia de la Computaci´ on, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile E-mail address:[email protected] (M. Murillo) Institut de Matem` atiques i Aplicacions de Castell´ o (IMAC), Universitat Jaume I, Campus del Riu Sec s/n, 12071 Castell´ o, Spain E-mail address:[email protected]