January  2020, 40(1): 509-528. doi: 10.3934/dcds.2020020

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

1. 

Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Las Sophoras 173, Santiago, Estación Central, Santiago, Chile

2. 

Universitat Jaume I, Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Campus del Riu Sec s/n, 12071 Castelló, Spain

* Corresponding author: Carlos Lizama

Received  March 2019 Revised  July 2019 Published  October 2019

Fund Project: The first author is supported by FONDECYT grant 1180041.

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.

Citation: Carlos Lizama, Marina Murillo-Arcila. Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 509-528. doi: 10.3934/dcds.2020020
References:
[1]

L. AbadiasC. LizamaP. J. Miana and M. P. Velasco, Cesáro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.  doi: 10.1007/s11856-016-1417-3.  Google Scholar

[2]

L. AbadiasC. LizamaP. J. Miana and M. P. Velasco, On well-posedness of vector-valued fractional differential-difference equations, Discr. Cont. Dyn. Systems, Series A, 39 (2019), 2679-2708.  doi: 10.3934/dcds.2019112.  Google Scholar

[3]

R. P. Agarwal, C. Cuevas and C. Lizama, Regularity of Difference Equations on Banach Spaces, Springer-Verlag, Cham, 2014. doi: 10.1007/978-3-319-06447-5.  Google Scholar

[4]

G. AkrivisB. Li and C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations, Math. Comp., 86 (2017), 1527-1552.  doi: 10.1090/mcom/3228.  Google Scholar

[5]

H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics, 89, Birkhäuser-Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[6]

A. AshyralyevS. Piskarev and L. Weis, On well-posedness of difference schemes for abstract parabolic equations in $L_p([0, T ]; E)$spaces, Numer. Funct. Anal. Optim., 23 (2002), 669-693.  doi: 10.1081/NFA-120016264.  Google Scholar

[7]

S. Blünck, Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), 157-176.  doi: 10.4064/sm146-2-3.  Google Scholar

[8]

R. E. Corman, L. Rao, N. Ashwin-Bharadwaj, J. T. Allison and R. H. Ewoldt, Setting material function design targets for linear viscoelastic materials and structures, J. Mech. Des., 138 (2016), 051402, 12pp. doi: 10.1115/1.4032698.  Google Scholar

[9]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), ⅷ+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[10]

S. Elaydi, Stability and asymptoticity of Volterra difference equations: A progress report, J. Comp. Appl. Math., 228 (2009), 504-513.  doi: 10.1016/j.cam.2008.03.023.  Google Scholar

[11]

B. JinB. Li and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.  doi: 10.1007/s00211-017-0904-8.  Google Scholar

[12]

T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math., 234 (2016), 241-263.   Google Scholar

[13]

T. Kemmochi and N. Saito, Discrete maximal regularity and the finite element method for parabolic equations, Numer. Math., 138 (2018), 905-937.  doi: 10.1007/s00211-017-0929-z.  Google Scholar

[14]

V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Differential Equations, 230 (2006), 634-660.  doi: 10.1016/j.jde.2006.07.018.  Google Scholar

[15]

B. KovácsB. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), 3600-3624.  doi: 10.1137/15M1040918.  Google Scholar

[16]

D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods, Numer. Math., 135 (2017), 923-952.  doi: 10.1007/s00211-016-0821-2.  Google Scholar

[17]

B. Li and W. Sun, Maximal regularity of fully discrete finite element solutions of parabolic equations, SIAM J. Numer. Anal., 55 (2017), 521-542.  doi: 10.1137/16M1071912.  Google Scholar

[18]

B. Li and W. Sun, Maximal $L_p$ analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra, Math. Comp., 86 (2017), 1071-1102.  doi: 10.1090/mcom/3133.  Google Scholar

[19]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar

[20]

C. Lizama., $\ell_p$-maximal regularity for fractional difference equations on $UMD$ spaces., Math. Nach., 288 (2015), 2079-2092.  doi: 10.1002/mana.201400326.  Google Scholar

[21]

C. Lizama and M. Murillo-Arcila, Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations, 263 (2017), 3175-3196.  doi: 10.1016/j.jde.2017.04.035.  Google Scholar

[22]

C. Lubich, Convolution quadrature and discretized operational calculus I, Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686.  Google Scholar

[23]

J. Prüss, Evolutionary Integral Equations and Applications, Springer, Basel Heidelberg, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

show all references

References:
[1]

L. AbadiasC. LizamaP. J. Miana and M. P. Velasco, Cesáro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.  doi: 10.1007/s11856-016-1417-3.  Google Scholar

[2]

L. AbadiasC. LizamaP. J. Miana and M. P. Velasco, On well-posedness of vector-valued fractional differential-difference equations, Discr. Cont. Dyn. Systems, Series A, 39 (2019), 2679-2708.  doi: 10.3934/dcds.2019112.  Google Scholar

[3]

R. P. Agarwal, C. Cuevas and C. Lizama, Regularity of Difference Equations on Banach Spaces, Springer-Verlag, Cham, 2014. doi: 10.1007/978-3-319-06447-5.  Google Scholar

[4]

G. AkrivisB. Li and C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations, Math. Comp., 86 (2017), 1527-1552.  doi: 10.1090/mcom/3228.  Google Scholar

[5]

H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics, 89, Birkhäuser-Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[6]

A. AshyralyevS. Piskarev and L. Weis, On well-posedness of difference schemes for abstract parabolic equations in $L_p([0, T ]; E)$spaces, Numer. Funct. Anal. Optim., 23 (2002), 669-693.  doi: 10.1081/NFA-120016264.  Google Scholar

[7]

S. Blünck, Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), 157-176.  doi: 10.4064/sm146-2-3.  Google Scholar

[8]

R. E. Corman, L. Rao, N. Ashwin-Bharadwaj, J. T. Allison and R. H. Ewoldt, Setting material function design targets for linear viscoelastic materials and structures, J. Mech. Des., 138 (2016), 051402, 12pp. doi: 10.1115/1.4032698.  Google Scholar

[9]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), ⅷ+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[10]

S. Elaydi, Stability and asymptoticity of Volterra difference equations: A progress report, J. Comp. Appl. Math., 228 (2009), 504-513.  doi: 10.1016/j.cam.2008.03.023.  Google Scholar

[11]

B. JinB. Li and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), 101-131.  doi: 10.1007/s00211-017-0904-8.  Google Scholar

[12]

T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math., 234 (2016), 241-263.   Google Scholar

[13]

T. Kemmochi and N. Saito, Discrete maximal regularity and the finite element method for parabolic equations, Numer. Math., 138 (2018), 905-937.  doi: 10.1007/s00211-017-0929-z.  Google Scholar

[14]

V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Differential Equations, 230 (2006), 634-660.  doi: 10.1016/j.jde.2006.07.018.  Google Scholar

[15]

B. KovácsB. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), 3600-3624.  doi: 10.1137/15M1040918.  Google Scholar

[16]

D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods, Numer. Math., 135 (2017), 923-952.  doi: 10.1007/s00211-016-0821-2.  Google Scholar

[17]

B. Li and W. Sun, Maximal regularity of fully discrete finite element solutions of parabolic equations, SIAM J. Numer. Anal., 55 (2017), 521-542.  doi: 10.1137/16M1071912.  Google Scholar

[18]

B. Li and W. Sun, Maximal $L_p$ analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra, Math. Comp., 86 (2017), 1071-1102.  doi: 10.1090/mcom/3133.  Google Scholar

[19]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar

[20]

C. Lizama., $\ell_p$-maximal regularity for fractional difference equations on $UMD$ spaces., Math. Nach., 288 (2015), 2079-2092.  doi: 10.1002/mana.201400326.  Google Scholar

[21]

C. Lizama and M. Murillo-Arcila, Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations, 263 (2017), 3175-3196.  doi: 10.1016/j.jde.2017.04.035.  Google Scholar

[22]

C. Lubich, Convolution quadrature and discretized operational calculus I, Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686.  Google Scholar

[23]

J. Prüss, Evolutionary Integral Equations and Applications, Springer, Basel Heidelberg, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

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