Constant flux for a one-dimensional divergence-form equation ($a(x)u^{\prime }(x)=J$)

On an interval, the weak equation $(a{u}^{\prime }{)}^{\prime }=0$ makes $a{u}^{\prime }$ constant almost everywhere. If $\lambda \le a\le \Lambda$, the derivative energy on concentric subintervals satisfies $E(r)\le (\Lambda /\lambda {)}^2(r/R)E(R)$. This supplies dyadic energy decay even though a one-dimensional annulus is disconnected.