This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of solutions to the scalar conservation law with initial data in . Here we do not need to have a strictly hyperbolic conservation law. Perhaps just assume . The sources I used for my presentation were from Yunguang Lu's book Hyperbolic Conservation Laws and the Compensated Compactness Method and Dafermos's book Hypberbolic Conservation Laws in Continuum Physics.
I am aware that Compensated Compactness can be used to prove existence of solutions to systems of conservation laws. I haven't looked over this proof, but I believe Lu covers this.
My question is as follows: Where else is the method of Compensated Compactness used? If you can say, if possible, could you give a rough, brief sketch of how the method was used? Any input/thoughts would be appreciated.