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 Lattanzioany, 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?

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?