this is a comment to Deane Yang, but apparently it was too long so here is a separate answer. My background is in numerical solution of PDEs
1) while I know about this, it is not at all well-known by people who numerically solve PDEs.
2) this is not true. Most computations are systems of PDEs. I think most computations are done with systems where there are no actual theory, i.e. existence and uniqueness results. Think about Navier-Stokes. Many systems are NS coupled with for example convection diffusion type systems (small amounts of material in the flow etc). Then there are liquid crystals, Maxwell, elasticity, flow coupled with elasticity etc. Of course when the computers were slower one had to simplify to get a scalar equation and then hope that it gives something reasonable.
Of course Cauchy-Kovalevskaia as such is irrelevant because one wants the solutions in Sobolev spaces. But the whole formal theory started as a generalization of CK.
3) this is not true. For example there are systems which are not elliptic initially but whose involutive forms are elliptic. This gives a priori regularity results and existence results.
Also one could argue that the word "determined" (and over/underdetermined) can't be defined in general without formal theory.