Existence and Uniqueness of Solutions

Abstract

A fundamental issue in the theory of differential equations consists of proving the existence and uniqueness of solutions. Before we discuss partial differential equations, we analyse in Section 9.1 the situation for general systems of ordinary differential equations, often called differential algebraic equations. For involutive equations it is here straightforward to extend the classical existence and uniqueness theorem. The formal theory also provides us with a natural geometric approach to the treatment of certain types of singularities.

Traditionally, (first-order) ordinary differential equations are studied via vector fields on the manifold $\mathcal{E}$ (actually, one usually restricts to the autonomous case assuming that $\mathcal{E} = \mathcal{X} \times \mathcal{U}$ and considers vector fields on $\mathcal{U}$). However, for a unified treatment of many singular phenomena it turns out to be much more useful to associate with the equation a vector field (or more precisely a distribution) in the first jet bundle $J_1 \pi$ arising very naturally from the contact structure. We will not develop a general theory of singularities but study a number of situations that have attracted much interest in the literature.

In local coordinates, one may say that the study of power series solutions underlies much of the formal theory. Hence, it is not surprising that results on analytic solutions of partial differential equations are fairly straightforward to obtain. In Section 9.2 we recall the famous Cauchy—Kovalevskaya Theorem for normal systems. The main point of the proof consists of showing that the easily obtained formal power series solution of the usual initial value problem actually converges.