Non-uniform dependence of the data-to-solution map for the Hunter–Saxton equation in Besov spaces

The Cauchy problem for the Hunter–Saxton equation is known to be locally well posed in Besov spaces \(B^s_{2,r} \) on the circle. We prove that the data-to-solution map is not uniformly continuous from any bounded subset of \(B^s_{2,r} \) to \(C([0, T]; B^s_{2,r} )\). We also show that the solution map is Holder continuous with respect to a weaker topology.