# Weakly convergent sequence and compensated compactness

This question is about a claim made in the proof of theorem 2.1.1 in the book Hyperbolic Conservation Laws and the Compensated Compactness Method by Yunguang Lu. (For simplicity I will only write done a special case.) The general idea is that weak convergence + compactness of derivatives gives some form of (weak) continuity of a quadratic form.

Assume that $$u^k\in L^2(U)$$ converges weakly to $$0$$ and the sequence $$\partial_{x_j} u^k$$ is contained in a compact subset of $$H^{-1}(U).$$ Then for a cutoff function $$\phi\in C_c^\infty(U), w^k = \phi u^k,$$ we have $$\partial_{x_j} w^k =\phi \partial_{x_j}u^k + u^k\partial_{x_j}\phi.$$ The book (page 11) says that: the first term $$\phi \partial_{x_j}u^k$$ is contained in a compact subset of $$H^{-1} (U);$$ therefore $$\partial_{x_j} w^k$$ is contained in a compact subset of $$H^{-1} (U).$$ So we can extract a strongly convergent subsequence, and so on.

It's obvious that $$\phi \partial_{x_j}u^k$$ is contained in a compact set by assumption. But how does this imply $$\partial_{x_j} w^k$$ is also contained in a compact set? After all, we do not really know at this point if $$u^k\partial_{x_j}\phi$$ is contained in a compact set.

• Mmh, is this not a consequence of the compactness of Sobolev embeddings, 'dualised'? Apr 8, 2021 at 20:06
• @LeoMoos So what theorem are you referring to precisely? Apr 8, 2021 at 20:12
• I was referring to the map $L^2(U) \to W^{-1}(U)$, which is compact essentially because of the Rellich-Kondrachov theorem. Via this embedding, weak convergence of $(u_k \mid k \in \mathbf{N})$ in $L^2(U)$ ought to give strong convergence in $W^{-1}(U)$, no? Apr 8, 2021 at 20:18
• Whoops, I meant to write $W^{-1,2}(U)$, sorry. Apr 8, 2021 at 20:28
• Yes, I second Leo Moos' statement. @Leo Moos: please make this an answer? Apr 9, 2021 at 10:55

Let $$U \subset \mathbf{R}^n$$ be a bounded open domain. Then by the Rellich--Kondrachov theorem the embedding $$W_0^{1,2}(U) \to L^2(U)$$ is a compact map. Therefore the map $$L^2(U) \to W^{-1,2}(U) = (W_0^{1,2})^*(U)$$ that sends $$u \in L^2(U)$$ to the functional $$v \in W_0^{1,2}(U) \mapsto \int_U uv$$ is compact also.

Let $$(u_k \mid k \in \mathbf{N})$$ be a sequence of functions in $$L^2(U)$$ as in the question, converging weakly in $$L^2(U)$$ to some $$u \in L^2(U)$$. Given $$\phi \in C_c^\infty(U)$$, the sequence $$u_k \partial_j \phi$$ is weakly convergent in the same sense. Therefore $$\{ u_k \partial_j \phi \mid k \in \mathbf{N} \}$$ is contained in a compact subset of $$W^{-1,2}(U)$$, and so is $$\{ \partial_j w_k = \phi \partial_j u_k + u_k \partial_j \phi \mid k \in \mathbf{N} \}$$.