On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations
https://doi.org/10.17586/2220-8054-2017-8-2-202-215
Abstract
We improve some recent estimates of the rate of convergence for product approximations of solution operators for linear non-autonomous Cauchy problem. The Trotter product formula approximation is proved to converge to the solution operator in the operator-norm. We estimate the rate of convergence of this approximation. The result is applied to diffusion equation perturbed by a time-dependent potential.
Keywords
Evolution equations,
non-autonomous Cauchy problem,
solution operators (propagators),
Trotter product approximation,
operatornorm convergence,
convergence rate,
operator splitting
Supporting agencies: The preparation of the paper was supported by the European Research Council via ERC-2010-AdG No. 267802 (“Analysis of Multiscale Systems Driven by Functionals”).
About the Authors
H. Neidhardt
WIAS Berlin
Germany
Germany
Mohrenstr. 39, D-10117 Berlin
A. Stephan
Humboldt Universitat zu Berlin Institut fur Mathematik
Germany
Germany
Unter den Linden 6, D-10099 Berlin
V. A. Zagrebnov
Universite d’Aix-Marseille and Institut de Mathematiques de Marseille (I2M)
France
France
UMR 7373, CMI – Technopole Chateau-Gombert, 13453 Marseille
References
1. Kato T. Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan, 1953, 5, P. 208–234.
2. Phillips R.S. Perturbation theory for semi-groups of linear operators. Trans. Amer. Math. Soc., 1953, 74, P. 199–221.
3. Tanabe H. Equations of evolution. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.
4. Trotter H.F. On the product of semi-groups of operators. Proc. Amer. Math. Soc., 1959, 10, P. 545–551.
5. Engel K.-J., Nagel R. One-parameter semigroups for linear evolution equations. Springer-Verlag, New York, 2000.
6. Ichinose T., Tamura H. Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations. Osaka J. Math., 1998, 35 (4), P. 751–770.
7. Neidhardt H., Stephan A., Zagrebnov V.A. Convergence rate estimates for the Trotter product approximations of solution operators for non-autonomous Cauchy problems, 2016, URL: https://arxiv.org/abs/1612.06147.
8. Cachia V., Zagrebnov V.A. Operator-norm convergence of the Trotter product formula for holomorphic semigroups. J. Operator Theory, 2001, 46, P. 199–213.
9. Kato T. Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995.
10. Neidhardt H., Zagrebnov V.A. Linear non-autonomous Cauchy problems and evolution semigroups. Adv. Differential Equations, 2009, 14 (3–4), P. 289–340.
11. Pazy A. Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, 1983.
12. Pr¨uss J., Schnaubelt R. Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time. J. Math. Anal. Appl., 2001, 256 (2), P. 405–430.
Review
For citations:
Neidhardt H., Stephan A., Zagrebnov V.A. On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations. Nanosystems: Physics, Chemistry, Mathematics. 2017;8(2):202-215. https://doi.org/10.17586/2220-8054-2017-8-2-202-215
Views:
8
Email this article 