I am trying to understand Compensated compactness. I am new to this area. I have the following doubts to start with
1) I have been reading many books where its been written in differnet ways. So What is compensated compactness? is it a name of a particular theorem or its a general term used for a kind of compactness result?
2) How does it help in understanding conservation laws.?
3) Is there any book/notes which briefly explains compensated compactness which is good enough to understand conservation laws.
4)I read that
"If $(u^n_1, u^n_2,....u^n_k) \rightarrow(u_1, u_2,....u_k)$ and $(v^n_1, v^n_2,....v^n_k) \rightarrow(v_1, v_2,....v_k)$ weakly in $L^2$. Such that $\operatorname{div}\textbf{u^n}$ and $\operatorname{curl} \textbf{v^n}$ are bounded in $L^2$ then $\sum_{i=1}^k u^n_iv^n_i \rightarrow \sum_{i=1}^k u_iv_i$ in the sense of distribution"
Is this result called compesated compactness? If so why the name compensated compactness? what did we compensate here?
Thanks
I used this method a long time ago so I maybe make some mistake here. As I recall the compensated compactness is the functional analytic method that is mainly used for the solving scalar and 2x2 systems of conservation laws (more detail are in the books given bellow).
These days I use the vanishing viscosity method a lot, so I would try to explain the role of compensated compactness in one of the problems I was working on: Let's say you have two systems of conservation laws:
The original system:
$$u_t + f(u)_x =0 \hspace{1cm} (1)$$
And the approximation system:
$$u_t^{\epsilon} + f(u^{\epsilon})_x =\epsilon u^{\epsilon}_{xx} \hspace{1cm} (2)$$
We would like to show that the solution of $(2)$ converges to the solution of $(1)$ i.e. $u^{\epsilon} \rightarrow u$ weakly, when $\epsilon \rightarrow 0$ (this would be the vanishing viscosity method). Establishing this convergence is not trivial.
This weak limit $u$ actually provides a distributional solution to the system $(1)$. The proof relies on a compensated compactness argument - based on the representation of the weak limit in terms of Young measures...
References in the literature that helped me long time ago with the method of compensated compactness are:
D. Serre, Systems of conservation laws 2, 2000 - Chapter 9
C. Dafermos, Hyperbolic conservation laws in continuum physics, $3^{rd}$ edition, 2010 - Chapter 16
R.J. DiPerna, Convergence of approximate solutions to conservation laws, 1983 - Section 3
L.C. Evans, Weak convergence methods for nonlinear partial differential equations, 1990 - Section 5