We often get a priori estimates on a given PDE and use its idea to construct and regularize weak solution.

For example,

Let $u$ be a smooth solution of heat equation $u_t-\Delta u=0$. $$0=\int u\cdot u_t-u\cdot\Delta u=\int \frac{1}{2}|u|_t^2+|\nabla u|^2\Rightarrow\frac{1}{2}\|u\|_{C^0L^2}^2+\|u\|_{L^2H^1}^2=\frac{1}{2}\|u(,0)\|_{L^2}^2$$

This is a naive equality on heat equation that does not require $u_{xx}$ and $u_t$ of $u$ in result, but require them in the process.

For weak formulation we would actually multiply the integrand by some $\phi$ and do something similar, with somewhat more care.

Sometimes we would also lose equality and get inequality instead(it does not matter on showing regularity, though)

Here I have two questions.

Question 1) Does that kind of naive calculation always give us the right result?

Saying in converse, do there exist PDEs which have some a priori regularity inequality for smooth solution, but that actually fails for weak formulation?

Question 2) Is there a way to construct weak solution directly from weak a priori estimates of a given PDE, not using something like weak formulation?

(we might use different notion of weak solution this case)

-You may answer either of these.


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