Approximation theorems for the propagators of higher order abstract Cauchy problems. (English) Zbl 1142.34037

Liang, Jin; Nagel, Rainer; Xiao, Ti-Jun

Trans. Am. Math. Soc. 360, No. 4, 1723-1739 (2008).

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)].

Reviewer: Andras Batkai (Roma)

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

Keywords:

differential equations in Banach spaces; higher order Cauchy problem; dynamic boundary condition; approximation

Citations:

Zbl 1048.47054

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