an a priori estimate
/ a priori -> ah pree-OR-ee /
Here is a clever and slightly paradoxical move at the heart of PDE theory. Before you know that a solution even exists, you can often prove that IF one exists, it cannot be too big — its size is controlled by the size of the data. That advance bound, derived from the equation itself without solving it, is an a priori estimate (Latin for 'from before', meaning beforehand, in advance of knowing the solution explicitly). It is the workhorse for proving uniqueness, stability, and ultimately existence, all without ever writing the solution down.
The most important kind is the energy estimate, and its recipe is short enough to walk through. Take the heat equation u_t = k u_xx on 0 < x < L with u zero at both ends. Multiply the whole equation by u and integrate over the bar. The left side becomes the time-derivative of the integral of (1/2) u^2 — a natural 'energy'. On the right, integrating u times k u_xx by parts and using the zero boundary values turns it into minus k times the integral of u_x^2, which is never positive. So the energy of the solution can only decrease: the integral of u^2 at any later time is bounded by its initial value. That single inequality does triple duty. Uniqueness: if two solutions share the same data, their difference has zero initial energy, so zero energy forever, so the solutions coincide. Stability: the difference of two solutions is bounded by the difference of their data — continuous dependence. And such bounds are the first step toward existence by approximation.
Why this is the standard route, rather than a trick: a priori estimates work even when you have no formula for the solution, which is the usual situation for hard or nonlinear PDEs. You commit to the structure of the equation — its sign, its conservation or dissipation of energy — and let that structure bound the answer. The catch to be honest about is the word 'if': an a priori estimate assumes a (sufficiently smooth) solution exists in order to do the integration by parts, so on its own it delivers uniqueness and stability, while turning those bounds into an actual existence proof needs an extra construction (approximation, fixed-point, or compactness). The estimate is the engine; existence is assembled around it.
Energy method for u_t = k u_xx, u(0,t)=u(L,t)=0: define E(t) = integral from 0 to L of (1/2) u^2 dx. Then E'(t) = integral of u u_t = k integral of u u_xx = -k integral of u_x^2 <= 0. So E(t) <= E(0). Two solutions with the same initial data have difference w with E_w(0)=0, hence E_w(t)=0, hence w=0 — uniqueness, straight from the estimate.
Multiply by u, integrate, integrate by parts: a positive energy bounded by the data yields uniqueness and stability.
An a priori estimate presupposes a solution in order to bound it, so by itself it proves uniqueness and stability but not existence; existence has to be built separately (by approximation or fixed-point methods) and only then combined with the estimate.