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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 |
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.
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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