Nonuniform dependence and well posedness for the periodic Hunter-Saxton equation

Abstract

It is proved that the flow map for the Hunter-Saxton (HS) equation from the homogeneous Sobolev space $\dot{H}^s({\mathbb{T}})$ into the space $C([0,T], \dot{H}^s({\mathbb{T}}))$ is continuous but not uniformly continuous on bounded subsets. To demonstrate this sharpness of continuity, two sequences of bounded solutions to the HS equation are constructed whose distance at the initial time converges to zero and whose distance at any later time is bounded from below by a positive constant. To achieve this result, appropriate approximate solutions are chosen and then the actual solutions are found by solving the Cauchy problem with initial data taken to be the value of approximate solutions at time zero. Then, using well-posedness estimates, it is shown that the difference between solutions and approximate solutions is negligible.