Kato inequality for scalar conservation laws (${\partial }_t|u-v|+{\partial }_{x}Q(u,v)\le 0$)

For bounded entropy solutions of the same scalar law, $Q(u,v)=\mathrm{sgn}(u-v)(f(u)-f(v))$ satisfies the displayed distributional inequality, with initial value $|u_0-v_0|$ in its test-function form. The doubling of variables for scalar conservation laws, local translation continuity and the averaged initial trace of an entropy solution prove it. Its significance is comparison between two solutions rather than an inequality against a constant state.

If $|f^{\prime }|\le M$ on the solutions' state range, the Kato inequality for scalar conservation laws implies the displayed estimate for almost every $s\in [0,t]$. Approximate the shrinking interval by smooth cutoffs $W$ satisfying $W_s+M|W_x|\le 0$ and multiply by a temporal cutoff ending at $s$. Since $|Q|\le M|u-v|$, the lateral flux cannot increase the integral. The estimate gives uniqueness and a finite propagation speed without global $L^1$ assumptions on the data.

Subtract the weak equation for $u-v$ from its Kato inequality for scalar conservation laws and divide by two. The result is an inequality for $(v-u{)}_{+}$ with a flux bounded in magnitude by $M(v-u{)}_{+}$. The same shrinking-interval argument proves $u_0\ge v_0\Rightarrow u\ge v$ almost everywhere. Choosing the constant solution $v=0$ proves preservation of nonnegativity even when $f(0)\ne 0$. Two-sided absolute-value contraction alone should not be mistaken for this one-sided comparison proof.