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.