= Solution
If $u_0=v_0$, the right-hand side of the <local L1 contraction for scalar conservation laws> estimate is zero. Applying it on a countable collection of rational intervals and final times proves $u=v$ almost everywhere in space-time. Thus
$$
\boxed{\text{bounded entropy solutions with the same data are unique}.}
$$
To deduce nonnegativity one needs a one-sided comparison, rather than merely putting $v=0$ in the absolute-value estimate. The difference $w=u-v$ satisfies the weak equation with flux $f(u)-f(v)$. Subtract its weak integral identity from the <Kato inequality for scalar conservation laws> and divide by two. The result is the same inequality for
$$
w_-=(|w|-w)/2,\qquad q_-=(Q(u,v)-f(u)+f(v))/2.
$$
When $w\ge0$ both expressions vanish; when $w<0$, $|q_-|\le Mw_-$. The shrinking-interval proof therefore bounds the negative part by its initial negative part. This is <order preservation for scalar entropy solutions>.
The constant $v=0$ is an <entropy solution> for every $f$, even if $f(0)\ne0$, because the constant flux has zero spatial <derivative>. If $u_0\ge0$, its initial negative part is zero, and comparison gives
$$
\boxed{u(t,x)\ge0\quad\text{almost everywhere}.}
$$
This explicitly establishes the additional sign conclusion without incorrectly identifying uniqueness alone with order preservation.
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