A counterexample to a universal error bound for small-BV conservation laws

We give a negative answer to a proposed universal error estimate for approximate small-BV hyperbolic flows. For a smooth strictly hyperbolic two-component system with both fields genuinely nonlinear and a uniformly strictly convex entropy, we construct one fixed initial datum and approximations satisfying the prescribed conservation and entropy residual bounds on every time strip. Their $L^1$ error divided by $\sqrt\varepsilon\,|\log\varepsilon|$ is unbounded. The construction pairs opposite entropy defects in translating Burgers pulses and places countably many disjoint pulse families in a fixed interval.