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.

  • $\begingroup$ Mmh, is this not a consequence of the compactness of Sobolev embeddings, 'dualised'? $\endgroup$
    – Leo Moos
    Apr 8, 2021 at 20:06
  • $\begingroup$ @LeoMoos So what theorem are you referring to precisely? $\endgroup$
    – Ma Joad
    Apr 8, 2021 at 20:12
  • 3
    $\begingroup$ 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? $\endgroup$
    – Leo Moos
    Apr 8, 2021 at 20:18
  • 1
    $\begingroup$ Whoops, I meant to write $W^{-1,2}(U)$, sorry. $\endgroup$
    – Leo Moos
    Apr 8, 2021 at 20:28
  • $\begingroup$ Yes, I second Leo Moos' statement. @Leo Moos: please make this an answer? $\endgroup$ Apr 9, 2021 at 10:55

1 Answer 1


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 $\{ \partial_j w_k = \phi \partial_j u_k + u_k \partial_j \phi \mid k \in \mathbf{N} \}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.