I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page:

**"A weak (in the PDE sense) solution satisfying the energy inequality necessarily coincides with a strong solution emanating from the same initial data, as long as the latter one exists."**

That got me wondering. Usually when we try to derive the energy inequalities in PDEs, we do that by using classical (i.e. strong) solutions. For example, for wave equation problem:

$$ \begin{cases} u_{tt}(x,t)-\bigtriangleup u(x,t)= f(x,t)\\[2ex] u(x,0)=u_{0}(x)\\[2ex] u_{t}(x,0)=u_{1}(x) \end{cases} $$

easy derivation of energy inequality could be found on page 58 in Struwe (of course no test functions here). Similar situation is in many other papers/books.

But I can't remember that I have ever seen the paper/book where authors have got some energy estimate involving test functions (whether it is in the derivation process or in the final inequality).

**So my question is how would anyone show that a weak solution satisfy some energy inequality?** I always thought that energy inequalities techniques are derived with strong solutions (Maybe I was wrong?). Or if anyone know reference in the literature that deals with this please write it down.

Of course maybe I understood wrong what the article from Arxiv.org was saying (in this case please correct me).