Generalized flows and singular ODEs on differentiable manifolds (Q1431629)

A theory of singular ordinary differential equations on differentiable manifolds is constructed. It takes as a basis and extends Colombeau's theory of nonlinear generalized functions, its versions in a geometric context and special constructions to a nonlinear distributional geometry. The paper is organized as follows. In the introduction, a review of previous results and detailed references are given. In Section 2, some basic definitions from linear and nonlinear distributional geometry are introduced; among them the spaces of moderate and negligible nets in \({\mathcal E}(X)\), \({\mathcal G}(X):={\mathcal E}_{\mathcal M}(X)/{\mathcal N}(X)\), and \(\Gamma_{\mathcal G}(X,E):=\Gamma_{{\mathcal E}_{\mathcal M}}(X, E)/ \Gamma_{\mathcal N}(X, E)\). In Section 3, basic existence and uniqueness theorems are proved, in particular, the flow theorems in local and global context. Section 4 is devoted to the distributional setting: the relationship between the theory introduced in the paper and results obtained on the basis of the distributional approach is clarified. In Section 5, definitions of \(u\) zero-associated, pointwise-associated, model-associated, and associated with \(v\) are given for \(u\in {\mathcal G}[X, Y], v: X\to Y\) and \(X\) with a Riemannian metric, and implications for them are determined. Due to the associations introduced, the following question is analyzed in Section 6. Let \(\xi \in{\mathcal G}_0^1(X)\) be \({\mathcal G}\)-complete with \(\Phi\) its unique flow in \({\mathcal G}[{\mathbb R}\times X, X]\). If \(\Phi\) admits a shadow \(\Psi\), does it imply that \(\Psi\) has the flow property?