 Solution

Set $H(a,b)=|a-b|$ and $Q(a,b)=\mathrm{sgn}(a-b)(f(a)-f(b))$. Both are symmetric in $a,b$. For fixed $(s,y)$ apply the entropy inequality for $u$ at the constant level $k=v(s,y)$ with test $\Phi (t,x,s,y)$, then integrate over $(s,y)$. This producesHere the first integral is over both time-space pairs and the second over the three remaining variables. Repeat with $v$, level $u(t,x)$, integrating over $(t,x)$, to obtain the analogous terms with ${\Phi }_s,{\Phi }_y$ and initial value $v_0$.

Adding and using symmetry giveswhere $B_u=\int H(u_0(x),v(s,y))\Phi (0,x,s,y)dsdxdy$ and $B_v=\int H(u(t,x),v_0(y))\Phi (t,x,0,y)dtdxdy$. All integrals are finite on the test's compact support because the states are bounded. The levels being inserted are constants relative to the variables in each entropy inequality; there is no differentiation of the merely measurable other solution. Fubini's theorem and continuity of $H,Q$ justify the integration. This is doubling of variables for scalar conservation laws.