A few thoughts, maybe you find one of them convincing:
Blow-up is analogous to taking the derivative.
Say you have a function $u \colon B_1 \to \mathbb{R}$ and define the blow-up sequence $u_r(x) := u(rx)$. This sequence (or rather family) encodes a lot about the infinitesimal behavior of $u$ at $0$. For example:
- $u$ has a limit $a \in \mathbb{R}$ at $0$ iff $u_r \rightrightarrows a$ on $B_1$ as $r \to 0$;
- $u$ is differentiable at $0$ (with differential $L$) iff $\frac{u_r-u(0)}{r} \rightrightarrows L$ on $B_1$ as $r \to 0$.
So one would be expect that the "limit" of $u_r$ (whatever kind of limit we consider) can have more structure (be more rigid) than $u$, and thus easier to analyze, but at the same time still say something interesting about $u$. And if we're lucky, the knowledge of blow-ups at different points implies something about $u$ in general (similar to how e.g. bounding the derivative produces a Lipschitz constant).
Problems of geometric origins are scale-invariant.
Not all PDEs are scale-invariant, but those in geometric analysis often are. For example, it is obvious that a surface of minimal area will still have this property if we zoom in. Once we have scale-invariance, why shouldn't we exploit it? There's probably a ton of literature on how it can help (also from a more functional analytic viewpoint), but one natural idea is this. Say we want to prove some claim (e.g. a regularity result) by contradiction. If the claim is false, it is useful to locate the point $x$ and scale $r$ at which things go wrong, and looking at $u_{x,r}(y) := u(x+ry)$ instead of $u$ often helps in reaching a contradiction in the limit. This was (as far as I know) introduced by Leon Simon in Schauder estimates by scaling (link). I know $\varepsilon$-regularity for the harmonic map flow was proved in this manner, and there are probably other similar examples.
Why does it work?
There is a number of ingredients needed if you want to use the approach you describe:
- scale-invariance of the problem (see above)
- monotonicity formula (this will typically imply that $u_r$ is a bounded sequence, and that its limit should have additional symmetry)
- compactness theorem (bounded norm will give you weak convergence, but most often you need stronger convergence to say something useful about the limit)
An $\varepsilon$-regularity theorem also helps in making this work, as it implies that any singular behavior is tied to concentration of energy. And if energy is concentrated on some ball $B_r(x)$, it's only natural to rescale it to $B_1(0)$ and proceed with the rescaled object. That's just an argument I find intuitive - do look up the MO discussion linked by Deane (link) for deeper insights.
