what is the essence of the blowup technique and bubbling analysis in PDE or geometric analysis? when we use the blowup technique and bubbling analysis? what are the essence and pespective? could you give some examples to explain them or some reference？ thank you
 A: In geometric analysis, I guess the first well-known blow-up argument was given in R. Schoen's solution to Yamabe problem. He used a Green function to blow up the metric near a point $x$ so that the punctured ball $B(x,r)\setminus \{x\}$ becomes an asymptotically flat end of a complete non-compact manifold. (The idea of Schoen comes from his study of "mass".)
In this case, because the blow-up factor is a positive function, the new metric differs from the original one only by a conformal change. In other cases, we usually use constants to blow-up the metric because we would like to understand the metric itself, rather than its conformal class. 
For example, when studying geometric flows such as the Ricci flow or the mean curvature flow, "blow-up" means to dilate the metric near singularities. The motivation is to "look into the detail" and "kill the lower order terms". In Perelman's work, he can prove that after a suitable dilation, every finite-time singularity of 3-dimensional Ricci flow on a compact manifold must be a cylinder, ball or capped cylinder.(It's called Canonical Neighborhood Theorem.) On the other hand, there is also a blow-down argument which could be used to study the asymptotic behavior. 
This is an (partial) answer from geometric viewpoint.
