×

zbMATH — the first resource for mathematics

Approximation theorems for the propagators of higher order abstract Cauchy problems. (English) Zbl 1142.34037
Approximation theorems for differential equations play a crucial theoretical role: they provide the theoretical basis for many important existence theorems for various kinds of solutions, and they show the way in proving the convergence and error estimates for numerical methods.
In the theory of linear evolution equations, the celebrated Trotter-Kato approximation theorem provides the basis for almost all results in this direction, which is closely related to the Lax equivalence theorem well-known in numerical analysis. There exists various types of generalizations of this result for nonautonomous equations, cosine families, integrated semigroups, resolvent families, semigroups on locally continuous spaces, or for bi-continuous semigroups. In the flavor, they all sound like this: under a suitable stability condition on the approximations, the convergence of the stationary problems (also called consistency) implies the convergence of the solutions.
In the paper under review, the authors show a Trotter-Kato type approximation theorem for higher order abstract Cauchy problems in the framework of the first and last authors monograph.
The results are then applied to damped wave equations with dynamic boundary conditions. Understanding dynamic boundary conditions is crucial for problems in stochastic processes and boundary control theory and the topic of active current research. The methods used here are based on the operator matrix techniques developed by V. Casarino, K.-J. Engel, R. Nagel and G. Nickel [Integral Equations Oper. Theory 47, No. 3, 289–306 (2003; Zbl 1048.47054)].

MSC:
34G10 Linear differential equations in abstract spaces
47D09 Operator sine and cosine functions and higher-order Cauchy problems
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
47D06 One-parameter semigroups and linear evolution equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kevin T. Andrews, K. L. Kuttler, and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl. 197 (1996), no. 3, 781 – 795. · Zbl 0854.34059 · doi:10.1006/jmaa.1996.0053 · doi.org
[2] Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, Basel, 2001. · Zbl 0978.34001
[3] H. T. Banks and D. J. Inman, On damping mechanisms in beams, ASME Trans. 58 (1991), 716-723. · Zbl 0741.73030
[4] András Bátkai and Klaus-Jochen Engel, Abstract wave equations with generalized Wentzell boundary conditions, J. Differential Equations 207 (2004), no. 1, 1 – 20. · Zbl 1063.35104 · doi:10.1016/j.jde.2003.12.005 · doi.org
[5] Robert Wayne Carroll and Ralph E. Showalter, Singular and degenerate Cauchy problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Mathematics in Science and Engineering, Vol. 127.
[6] Valentina Casarino, Klaus-Jochen Engel, Rainer Nagel, and Gregor Nickel, A semigroup approach to boundary feedback systems, Integral Equations Operator Theory 47 (2003), no. 3, 289 – 306. · Zbl 1048.47054 · doi:10.1007/s00020-002-1163-2 · doi.org
[7] Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. · Zbl 0952.47036
[8] Joachim Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations 18 (1993), no. 7-8, 1309 – 1364. · Zbl 0816.35059 · doi:10.1080/03605309308820976 · doi.org
[9] H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. · Zbl 0564.34063
[10] Angelo Favini, Giséle Ruiz Goldstein, Jerome A. Goldstein, and Silvia Romanelli, \?\(_{0}\)-semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc. 128 (2000), no. 7, 1981 – 1989. · Zbl 0947.47034
[11] Angelo Favini, Gisèle R. Goldstein, Jerome A. Goldstein, and Silvia Romanelli, Generalized Wentzell boundary conditions and analytic semigroups in \?[0,1], Semigroups of operators: theory and applications (Newport Beach, CA, 1998) Progr. Nonlinear Differential Equations Appl., vol. 42, Birkhäuser, Basel, 2000, pp. 125 – 130. · Zbl 0971.47023
[12] Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, and Silvia Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ. 2 (2002), no. 1, 1 – 19. · Zbl 1043.35062 · doi:10.1007/s00028-002-8077-y · doi.org
[13] Angelo Favini and Enrico Obrecht, Conditions for parabolicity of second order abstract differential equations, Differential Integral Equations 4 (1991), no. 5, 1005 – 1022. · Zbl 0735.34043
[14] Cícero Lopes Frota and Jerome A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, J. Differential Equations 164 (2000), no. 1, 92 – 109. · Zbl 0979.35105 · doi:10.1006/jdeq.1999.3743 · doi.org
[15] Jerome A. Goldstein, On the convergence and approximation of cosine functions, Aequationes Math. 11 (1974), 201 – 205. · Zbl 0282.47012 · doi:10.1007/BF01832857 · doi.org
[16] Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. · Zbl 0592.47034
[17] Ciprian G. Gal, Gisèle Ruiz Goldstein, and Jerome A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evol. Equ. 3 (2003), no. 4, 623 – 635. Dedicated to Philippe Bénilan. · Zbl 1058.35139 · doi:10.1007/s00028-003-0113-z · doi.org
[18] Kazufumi Ito and Franz Kappel, The Trotter-Kato theorem and approximation of PDEs, Math. Comp. 67 (1998), no. 221, 21 – 44. · Zbl 0893.47025
[19] Tosio Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad. 35 (1959), 467 – 468. · Zbl 0095.10502
[20] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0148.12601
[21] Marjeta Kramar, Delio Mugnolo, and Rainer Nagel, Theory and applications of one-sided coupled operator matrices, Conf. Semin. Mat. Univ. Bari 283 (2002), 1 – 29.
[22] Thomas G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Functional Analysis 3 (1969), 354 – 375. · Zbl 0174.18401
[23] Thomas G. Kurtz, A general theorem on the convergence of operator semigroups, Trans. Amer. Math. Soc. 148 (1970), 23 – 32. · Zbl 0194.44103
[24] John Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983), no. 2, 163 – 182. · Zbl 0536.35043 · doi:10.1016/0022-0396(83)90073-6 · doi.org
[25] Jin Liang, Rainer Nagel, and Ti-Jun Xiao, Nonautonomous heat equations with generalized Wentzell boundary conditions, J. Evol. Equ. 3 (2003), no. 2, 321 – 331. · Zbl 1026.35043 · doi:10.1007/978-3-0348-7924-8_17 · doi.org
[26] J.-L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961 (French). · Zbl 0098.31101
[27] C. Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators, J. Integral Equations Appl. 2 (1990), no. 2, 269 – 280. · Zbl 0739.47016 · doi:10.1216/JIE-1990-2-2-263 · doi.org
[28] Carlos Lizama, On approximation and representation of \?-regularized resolvent families, Integral Equations Operator Theory 41 (2001), no. 2, 223 – 229. · Zbl 1011.45006 · doi:10.1007/BF01295306 · doi.org
[29] Serge Nicaise, The Hille-Yosida and Trotter-Kato theorems for integrated semigroups, J. Math. Anal. Appl. 180 (1993), no. 2, 303 – 316. · Zbl 0791.47035 · doi:10.1006/jmaa.1993.1402 · doi.org
[30] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[31] V. V. Vasil\(^{\prime}\)ev and S. I. Piskarev, Differential equations in Banach spaces. II. Theory of cosine operator functions, J. Math. Sci. (N. Y.) 122 (2004), no. 2, 3055 – 3174. Functional analysis. · Zbl 1115.34058 · doi:10.1023/B:JOTH.0000029697.92324.47 · doi.org
[32] Thomas I. Seidman, Approximation of operator semi-groups, J. Functional Analysis 5 (1970), 160 – 166. · Zbl 0186.45803
[33] H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887 – 919. · Zbl 0099.10302
[34] Ti-Jun Xiao and Jin Liang, The Cauchy problem for higher-order abstract differential equations, Lecture Notes in Mathematics, vol. 1701, Springer-Verlag, Berlin, 1998. · Zbl 0915.34002
[35] Ti-Jun Xiao and Jin Liang, Approximations of Laplace transforms and integrated semigroups, J. Funct. Anal. 172 (2000), no. 1, 202 – 220. · Zbl 0977.47034 · doi:10.1006/jfan.1999.3545 · doi.org
[36] Ti-Jun Xiao and Jin Liang, A solution to an open problem for wave equations with generalized Wentzell boundary conditions, Math. Ann. 327 (2003), no. 2, 351 – 363. · Zbl 1035.35061 · doi:10.1007/s00208-003-0457-2 · doi.org
[37] Ti-Jun Xiao and Jin Liang, Complete second order differential equations in Banach spaces with dynamic boundary conditions, J. Differential Equations 200 (2004), no. 1, 105 – 136. · Zbl 1059.34034 · doi:10.1016/j.jde.2004.01.011 · doi.org
[38] Ti-Jun Xiao and Jin Liang, Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc. 356 (2004), no. 12, 4787 – 4809. · Zbl 1086.34051
[39] Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.