Existence and Uniqueness: The Flow Map

In this chapter we describe in detail several general results concerning the initial value problem (IVP): $$\begin{array}{rl}x^\prime & = X(t, x)\\ x(t_0) & = c.\end{array}$$ The main result is the Existence and Uniqueness Theorem, from which many additional results can be derived. Throughout, $$X : B \to \mathbb{R}^n$$ is a time-dependent vector field on an open set $$B \subseteq \mathbb{R}^{n+1}$$ . Various continuity and differentiability conditions will be imposed on X in order to get the results, but at the start we assume, at the bare minimum, that X is continuous on B. For the sake of reference we include the proofs of most of the results, although understanding some of the details requires being comfortable with several basic ideas from functional analysis. The most fundamental and important construct to arise from the results presented here is the flow map, or simply the flow, ø generated by the vector field X. It is an indispensable notion and tool in many fields of study ranging from differential geometry to continuum mechanics. A key ingredient in the proof of existence and uniqueness theorems is the fact that any initial value problem (IVP) can be reformulated as an integral equation. To see how to do this, we need a definition.