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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 $L^{\infty}$ solutions to the scalar conservation law $u_{t}+(f(u))_{x}=0$ with initial data in $L^{\infty}$. Here we do not need to have a strictly hyperbolic conservation law. Perhaps just assume $f\in C^{2}$. 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 $2\times 2$ 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.

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2 Answers 2

Compensated compactness helps when one needs to find the limit of $u_n \cdot v_n$, where the sequences of vector fields $u_n$ and $v_n$ converge weakly in $L^2$: $u_n\rightharpoonup u$, $v_n\rightharpoonup v$. If none of the sequences converges strongly in $L^2$, the vector fields can still possess some additional properties which would compensate for the lack of strong convergence. For instance, the following result and its versions are often used in the homogenization theory.

Lemma. Let $u_n$, $v_n\in L^2(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb R^d$ and let $u_n\rightharpoonup 0$, $v_n\rightharpoonup 0$ in $L^2(\Omega)$. Assume that $$\mbox{curl }v=0\qquad \mbox{and}\qquad \mbox{div } u_n\to w\quad \mbox{in }\ H^{-1}(\Omega).$$ Then $u_nv_n\to uv$ in the weak-* topology of $L^1(\Omega)$.

The lemma can be used to justify the convergence of solutions of the Dirichlet problem with strongly oscillating coefficients $$\mbox{div }\left(a\left(\frac{x}{\varepsilon}\right)\nabla u_\epsilon\right)=f,\qquad \left.u\right|_{\partial\Omega}=0,$$ to the solutions of some averaged problem (see, e.g., "Homogenization of differential operators and integral functionals" by V. V. Jikov, S. M. Kozlov, O. A. Oleinik).

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In addition to proving the existence of solutions to a large class of nonlinear partial differential equations, among them conservation laws, compensated compactness is also used to show the convergence of numerical methods to approximate those solutions. Given an equation $P(u)=0$, one construcsts by whatever method (finite differences, finite elements, spectral,...) a sequence of approximations $u_n$. Then one tries to prove:

  1. the sequence $\{u_n\}$ converges in a some sense to some function $u$;
  2. this function $u$ is a solution of the original problem;
  3. an eror estimate for $u-u_n$.

Then, if sufficiently good a priori estimates are known for $u_n$, the method of compensated compactness can give 1. and 2.This plan is carried out for instance here and here,

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