Editing Well-posed problem (section)

==Existence of local solutions==
The existence of local solutions is often an important part of the well-posedness problem, and it is the foundation of many estimate methods, for example the energy method below.

There are many results on this topic. For example, the [[Cauchy–Kowalevski theorem]] for Cauchy initial value problems essentially states that if the terms in a partial [[differential equation]] are all made up of [[analytic function]]s and a certain transversality condition is satisfied (the [[hyperplane]] or more generally hypersurface where the initial data are posed must be non-characteristic with respect to the partial differential operator), then on certain regions, there necessarily exist solutions which are as well analytic functions. This is a fundamental result in the study of analytic partial differential equations. Surprisingly, the theorem does not hold in the setting of smooth functions; an [[Lewy's example|example]] discovered by [[Hans Lewy]] in 1957 consists of a linear partial differential equation whose coefficients are smooth (i.e., have derivatives of all orders) but not analytic for which no solution exists. So the Cauchy-Kowalevski theorem is necessarily limited in its scope to analytic functions.