On the singular limit of the Camassa-Holm equation

We study the singular limit of the Camassa–Holm equation as the length scale $\mathrm{\ell }$ tends to zero. Although the formal limit is the Burgers equation with flux $3u^2\mathrm{/}2$, we show that, for a class of smooth initial data, convergence to entropy solutions fails after the shock formation, even along subsequences in local space-time $L^1$.