On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says

Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to a conservation law with viscosity term converges boundedly almost everywhere on $\mathbb R^m \times [0,T)$ to some function $U,$ and $\mu_k \to 0.$ Then $U$ is a weak solution to the conservation law without viscosity term. Moreover, $U$ is entropy admissible.

Question: What does "converges boundedly almost everywhere" mean?

Note: $U_\mu$ should be smooth according to theory of heat equations, but no other assumptions are made about $U_\mu.$

  • 1
    $\begingroup$ The natural guess is that there is some norm and a costant $M$ such that $U_{\mu_k} \to U$ pointwise a.e. and $||U_{\mu_k}|| <M$ for all $\mu_k$. $\endgroup$ Jul 14 at 14:19
  • $\begingroup$ I would guess similar things. But then it is not clear which norm to use. This term is used several times, but without definition. $\endgroup$
    – Ma Joad
    Jul 14 at 14:21

This should mean that $U_{\mu_k} \to U$ almost everywhere on $\mathbb{R}^m \times [0,T)$, and moreover the sequence of functions $U_{\mu_k}$ is uniformly bounded: $$\sup_k \sup_{(x,t) \in \mathbb{R}^m \times [0,T)} |U_{\mu_k}(x,t)| < \infty.$$ I suppose that the functions $U_{\mu_k}$ take their values in $\mathbb{R}$ or $\mathbb{C}$ or some other obvious normed space.

This is standard terminology; see for instance Section 10.5 of:

Ghorpade, Sudhir R.; Limaye, Balmohan V., A course in calculus and real analysis, Undergraduate Texts in Mathematics. Cham: Springer (ISBN 978-3-030-01399-8/hbk; 978-3-030-01400-1/ebook). ix, 538 p. (2018). ZBL1403.26001.

In context, it is probably fine to replace the sup over $(x,t)$ with the essential supremum.


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