What is an example of vector field $c: \mathbb R_+ \times \mathbb R^N \to \mathbb R^N$ with $c \in L^1(\mathbb{R}_+, BV(\mathbb R^N))$ without bounded divergence $div_x c$ but such that there exists a distributional solution $u$ of $$u_t + div(cu) = 0$$ with $$\alpha \le u \le \beta, $$ $\alpha, \beta >0$?
2
-
$\begingroup$ Would a local statement make you happy? In other words are you interested in large $t$, large $|x|$, or is it really important for you that the bounds on $u$ hold globally in $R_+\times R^N$? Would you be happy e.g. with $c\in L^1_{loc}(R_+,BV)$? $\endgroup$– leo monsaingeonCommented Aug 6, 2020 at 8:29
-
$\begingroup$ @leomonsaingeon Yes, I'd be interested in a local example too. $\endgroup$– RikuCommented Aug 6, 2020 at 8:48
Add a comment
|