Long-time validity of the Korteweg-de Vries approximation for the Boussinesq equation

We consider the Korteweg-de Vries (KdV) approximation for the good Boussinesq equation. Building upon the low-regularity local-in-time justification established by Hong and Yang \cite{HY2024}, we prove that this approximation remains valid on logarithmically long time intervals. The key ingredient is a persistence of regularity argument, which allows the local approximation to be iterated using the rescaled conservation law for the Boussinesq equation.