Continuity Properties of the Data-To-Solution Map for the Hunter-Saxton Equation

It is proved that the data-to-solution map for the Hunter-Saxton (HS) 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, approximate solutions that satisfy this property are chosen, after which actual solutions are found by solving the HS 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.