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

Carlos Lizama,Marina Murillo-Arcila

doi:10.3934/dcds.2020020

Abstract

In this paper we investigate conditions for maximal regularity of Volterra equations defined on the Lebesgue space of sequences $ \ell_p(\mathbb{Z}) $ by using Blünck's theorem on the equivalence between operator-valued $ \ell_p $-multipliers and the notion of $ R $-boundedness. We show sufficient conditions for maximal $ \ell_p-\ell_q $ regularity 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.

Highlights

In this paper we study a discrete time formulation for a class of integro partial differential equations with delay whose prototype equation is, t (1)u(t) = a(t − s)Au(s)ds + f (t, u(t)), t ∈ R, −∞where A is a closed linear operator defined on a Banach space X
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]
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

Summary

Introduction

In this paper we study a discrete time formulation for a class of integro partial differential equations with delay whose prototype equation is, t (1). 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] It reveals a new and surprising association between (5) and the Volterra equation t u(t) = gα(t − s)Au(s)ds + g(t), t > 0. 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, k−α(j)ξj = δ(ξ) := (1 − ξ)α, j=0 see [11, identities (3.2)].

Analytical framework and notation

Abstract setting: A characterization of maximal p-regularity

Applications to schemes of approximation

Further Applications