Consider the transport equation with smooth coefficient $a \in C^1(\mathbb{R}\times \mathbb{R}^+)$ given by
\begin{align} u_t+(a(x,t)u)_x=0. \end{align}
A weak solusolution $u \in C(\mathbb{R}^+;L^{1}(\mathbb{R}))$ satifies the PDE in the sense of distribution. Since it is transport equation, one would expect any weak solution should automatically satisfy the Kruzkov entropy condition.
How to prove this rigorously for any $u \in C(\mathbb{R}^+;L^{1}(\mathbb{R}))$?.
P.S. If $u \in C^1(\mathbb{R}\times \mathbb{R}^+)$ is a weak solution, then a standard computaion shows that $u$ satisfies enropy condition for any $\eta \in C^1(\mathbb{R})$ convex and then by a density argument we can choose $\eta(u)=|u-k|$ by a limiting argument.