Compensated compactness for system of conservation laws?

Question

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly there are some works which use it for $N\times N$ systems; for instance

1- Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems, Pierangelo Marcati and Bruno Rubino.

2- Convergence of a relaxation scheme for hyperbolic systems of conservation laws, Corrado Lattanzio, Denis Serre.

3- Existence and Uniqueness of Solutions for some Hyperbolic Systems of Conservation Laws, Arnaud Heibig.

Does Div-Curl lemma work in such cases? Could anyone give me any explanation about it?

Answer 1

So far, only the div-curl Lemma could be used when applying compensated compactness to systems of conservation laws. But it has two flaws.

1. Because we only have divergences but no curls in general, we cannot handle multi-dimensional systems. Only one-dimensional systems can be considered by this method.
2. The idea behind Tartar's and DiPerna's use of compensated compactness is that, because of nonlinearity, a system displays some ellipticity. This feature could in principle be used to establish compactness of (approximate) solutions. But since the systems under consideration are not elliptic (they are hyperbolic), the amount of ellipticity is marginal. To carry out the method, we therefore need much more than just one information. Actually, we need infinitely many. This is why the CC has been successful only for systems having infinitely many independent entropy-entropy flux pairs. This covers the scalar case as well as the $2\times2$ case. Generic systems of large size usually have only finitely many independent entropies and CC cannot be fruitful. A notable exception is the class of rich systems, to which CC can be efficiently applied. This is the class of systems of conservation laws which admits a diagonalisation in terms of Riemann invariants. The rich class contains the class of Temple systems, which is the object of Heibig's article.