Regularization and Approximation of Second Order Evolution Equations

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Abstract
This is the publisher’s final pdf. The published article is copyrighted by the Society for Industrial and Applied Mathematics and can be found at: http://epubs.siam.org/loi/sjmaah.We give a nonstandard method of integrating the equation Bu" + Cu’ + Au = f in\ud Hilbert space by reducing it to a first order system in which the differentiated term corresponds to\ud energy. Semigroup theory gives existence for hyperbolic and for parabolic cases. When C = εA, ε ≧ 0,\ud this method permits the use of Faedo-Galerkin projection techniques analogous to the simple case of a\ud single first order equation; the appropriate error estimates in the energy norm are obtained. We also\ud indicate certain singular perturbations which can be used to approximate the equation by one which is\ud dissipative or by one to which the above projection techniques are applicable. Examples include\ud initial-boundary value problems for vibrations (possibly) with inertia, dynamics of rotating fluids, and\ud viscoelasticity