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Manifold-valued generalized functions in full Colombeau spaces

Michael Kunzinger;  Eduard Nigsch — 2011

Commentationes Mathematicae Universitatis Carolinae

We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.

Global Gronwall estimates for integral curves on Riemannian manifolds.

Michael Kunzinger;  Hermann Schichl;  Roland Steinbauer;  James A. Vickers — 2006

Revista Matemática Complutense

We prove Gronwall-type estimates for the distance of integral curves of smooth vector fields on a Riemannian manifold. Such estimates are of central importance for all methods of solving ODEs in a verified way, i.e., with full control of roundoff errors. Our results may therefore be seen as a prerequisite for the generalization of such methods to the setting of Riemannian manifolds.

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