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

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

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

Carlos Lizama ^1,, and Marina Murillo-Arcila ^2,

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.

Keywords: Maximal regularity, time-stepping schemes, discrete Volterra equations, nonlocal operators.

Mathematics Subject Classification: Primary: 49M25, 49N60; Secondary: 42A45, 47A58.

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

References:

[1]	L. Abadias, C. Lizama, P. 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.
[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 (2019), 2679-2708. doi: 10.3934/dcds.2019112.
[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.
[4]	G. Akrivis, B. 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.
[5]	H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics, 89, Birkhäuser-Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.
[6]	A. Ashyralyev, S. 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.
[7]	S. Blünck, Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), 157-176. doi: 10.4064/sm146-2-3.
[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.
[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.
[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.
[11]	B. Jin, B. 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.
[12]	T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math., 234 (2016), 241-263.
[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.
[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.
[15]	B. Kovács, B. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), 3600-3624. doi: 10.1137/15M1040918.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	C. Lubich, Convolution quadrature and discretized operational calculus I, Numer. Math., 52 (1988), 129-145. doi: 10.1007/BF01398686.
[23]	J. Prüss, Evolutionary Integral Equations and Applications, Springer, Basel Heidelberg, 1993. doi: 10.1007/978-3-0348-8570-6.

show all references

References:

[1]	L. Abadias, C. Lizama, P. 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.
[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 (2019), 2679-2708. doi: 10.3934/dcds.2019112.
[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.
[4]	G. Akrivis, B. 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.
[5]	H. Amann, Linear and Quasilinear Parabolic Problems, Monographs in Mathematics, 89, Birkhäuser-Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.
[6]	A. Ashyralyev, S. 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.
[7]	S. Blünck, Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), 157-176. doi: 10.4064/sm146-2-3.
[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.
[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.
[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.
[11]	B. Jin, B. 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.
[12]	T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math., 234 (2016), 241-263.
[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.
[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.
[15]	B. Kovács, B. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), 3600-3624. doi: 10.1137/15M1040918.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	C. Lubich, Convolution quadrature and discretized operational calculus I, Numer. Math., 52 (1988), 129-145. doi: 10.1007/BF01398686.
[23]	J. Prüss, Evolutionary Integral Equations and Applications, Springer, Basel Heidelberg, 1993. doi: 10.1007/978-3-0348-8570-6.

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