= Maximum representation for a concave conservation law
{title2=$U(t,x)=\max_y\{U_0(y)+tg((x-y)/t)\}$}
Let $f$ be smooth and strictly concave, with $f(0)=0$ and $f'$ bijective, and let $u_0$ be smooth with compact support. For $0<t<t_*$ in the smooth <concave-flux characteristic lifespan>, the primitive $U_x=u$ solves the <Hamilton-Jacobi equation> $U_t+f(U_x)=0$ and obeys
$$
U(t,x)=\max_y\left\{U(0,y)+t g\left(\frac{x-y}{t}\right)\right\},
$$
where $g$ is the <concave Legendre dual>. The tangent inequality for a <concave function> bounds every candidate by $U(t,x)$, and the backward <characteristic curve> attains equality. Its foot $x_0$ is the unique maximizer, and $u(t,x)=h((x-x_0)/t)$ for $h=(f')^{-1}$. This statement concerns the smooth solution, without asserting a post-crossing continuation.
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