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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).

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  • $\begingroup$ On a meta level: you can prove statements like the one quoted (that assuming a weak solution is such that it satisfies an energy inequality [and hence necessarily in $L^\infty H^1$]) without knowing how to prove that a weak solution does in fact satisfy an energy inequality. // That said, one way to go about this may be (which even works for wave equations) to prove that your weak solution can be obtained as the limit of strong solutions in an appropriate topology which enable the energy inequality to still hold. $\endgroup$ – Willie Wong Dec 12 '18 at 19:34
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    $\begingroup$ In the case of Leray's weak solutions, take a look at Chapter 12 of Lemarie-Rieusset's "The Navier-Stokes Problem in the 21st Century". For example, Proposition 12.1 states the "Strong Leray energy inequality". The existence of the weak solution goes through a weak limit of regularized solutions which satisfy the classical energy estimate, which implies that the energy estimate is satisfied for the limit "almost everywhere", and this can then be upgraded by weak continuity. $\endgroup$ – Willie Wong Dec 12 '18 at 19:41
  • $\begingroup$ @WillieWong: Thanks for recommending me the great book. I like the idea that weak solution can be obtained as the limit of strong solution but I am not sure how can I evaluate that kind of limit (I am thinking of the problem that is bugging me for six months, where I think this technique would be possibly very good). I would have something let's say continuous or smooth (from strong solution) that goes to something that is discontinuous(from weak solution). Do we usually solve that kind of limits directly or maybe using compactness lemma such as Rellich's or similar? $\endgroup$ – Mark Dec 23 '18 at 15:35
  • $\begingroup$ @WillieWong: Additionaly: Why necessarily $L^\infty H^1$? (I assume that this is $L_t ^\infty H_x ^1$ space). Also is that space the same as Bochner space $L(0,T;H^1)$ if we assume that $t\in[0,T]?$ $\endgroup$ – Mark Dec 23 '18 at 15:36
  • $\begingroup$ I don't think I understand your first comment; what exactly do you mean by "evaluate the limit"? // For your second comment: the statement of the energy inequality means that for any (fixed but arbitrary) $T >0$, for any $t\in [0,T]$ the solution $u(t,\cdot)$ is an $H^1$ function with bounded $H^1$ norm. So "necessarily" is "by definition". $\endgroup$ – Willie Wong Jan 7 at 19:12

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