Comparing weak and strong solutions of a PDE problems

Question

A few days ago I was reading the paper:

"Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system" - Feireisl, Jin, Novotny, 2012 [Arxiv].

It deals with the relative energy inequalities for the Navier-Stokes system. Basicaly using them, we could compare weak and the strong solution of the Navier-Stokes problem.

So I started to think about the topic of comparing. Let's say we have Cauchy problem:

$$ (1) \hspace{0.5cm} u_t+f(u)_x=0 $$ $$ (2) \hspace{0.5cm} u(x,0)=u_0 (x) $$

where $x \in A \subseteq \mathbb{R}$, $t \in [0,T]$ and $u \in \mathbb{R}^n, n > 1$. Initial condition $(2)$ could be smoooth (e.g. in $C_0^{\infty}$) or discontinuous (such as in Riemann or Generalized Riemann problem), in $L^{\infty}, H^m,...$ take your pick. Also, I've written the system of equations in the conservation law form because I work on them currently. In the place of $(1)$ it could stand any other system that is not in the conservation law form.

I have two questions.

1. Is it possible to use relative energy inequalities to compare weak and strong solutions of other systems rather then Navier-Stokes? By weak solution I mean in the distributional sense but any other notion of weak solution would be fine. I am pretty sure that the answer is yes - I just would like to see it in the some other paper that doesn't deal with the Navier-Stokes problem.

2. Besides relative energy inequalities, are there any other ways of comparing weak and strong solutions of some system of PDEs?

If anyone knows some paper that deals with this topic or if one has some ideas/examples about this, share it with the rest of us.

Answer 1

Yes, this is a classical approach in nonlinear PDEs. Here is one example: Sometimes the starting and suitable notion of weak solution is so weak that one can prove existence but not uniqueness. The typical statement then becomes: if, among these possibly many weak solutions, there exists a strong one, then it is the unique only (weak) one. The real problem then becomes: can one prove existence of at least one strong solution? (and as we ll know this is worth one million dollars for Navier-Stokes, roughly speaking!)

It is often the case in practice that one tries writing down a Grönwall lemma for the difference of two solutions and try to prove that if they agree at time $t=0$ then their difference must be zero for all times. The computation may usually require at least one integration by parts. If the solutions are not smooth enough (too weak) then one cannot justify this integration by parts legitimately, but if at least one of the two solutions is smooth enought then everything suddently becomes rigorous.

As a starting point for a bibliographical search I suggest googling "weak-strong uniqueness" and "modulated energy".