I have a general question about the use of energy estimates to prove existence theorems for certain PDEs:
Let some energy be defined, for example as
$$E(t)= \dfrac{1}{2} \int\limits_{\mathbb{R}^n} u^T A u \ dx $$
for a function $u:[t_0, \infty ) \times \mathbb{R}^n \rightarrow \mathbb{R}$ and $A$ some matrix.
Then often, the idea of proof is to show existence for $E$ and local existence of $u$ around some $t \in [t_0, \infty )$ and that's it. For me, it seems like it should be obvious how to get the existence of an $u$ from this.
Is it obvious or do they just often omit this part of proof in the books I'm working through?